# Asset Allocation
Mark Kritzman, September 21, 2016
Asset allocation is one of the most important and difficult challenges we face as investors. Thanks to Harry Markowitz, we have an elegant and widely accepted theory to guide us, though implementation in the face of real world complexities is less straightforward than theory might suggest. In this white paper, we describe how to determine allocation to broad asset classes given the complexities of the real world.
There are four steps to asset allocation (Figure 1). We must first identify eligible asset classes. Then we need to estimate their expected returns, volatilities, and correlations. Next we must isolate the subset of efficient portfolios that offer the highest expected returns for different levels of risk. Finally we need to select the specific portfolio that matches our tolerance for risk.
## 1. Eligible Asset Classes
What constitutes an asset class? First, we should expect an asset class to improve our portfolio’s efficiency either by raising its expected return or by lowering its risk. Consider commodities, for example. We might believe that their expected return is insufficient to raise our portfolio’s expected return because advances in technology tend to outpace depletion of resources, thereby lowering commodity prices. However, because commodities offer diversification against financial assets, especially during periods of high unanticipated inflation, their inclusion in our portfolio might lower risk thus offsetting their expected reduction of return.
We should also require homogeneity among the components of an asset class so that we do not forego opportunities for diversification. If an asset class comprises dissimilar components, then by investing in it we implicitly impose the unnecessary and potentially harmful constraint that the components must be held in the same relative proportions as their weights in the asset class. We should be able to achieve a more efficiently diversified portfolio if we partition the dissimilar components into multiple asset classes.
Finally, an asset class should be sufficiently large to absorb a meaningful fraction of our portfolio. If we were to invest in an asset class with inadequate capacity, we would likely drive up the cost of investment and reduce our portfolio’s liquidity. The consequence might be to lower our portfolio’s expected return and increase its risk to the point at which the proposed asset class would no longer improve our portfolio’s efficiency.
For illustrative purposes we will consider the following asset classes in our asset allocation analysis.
* U.S. stocks
* Foreign stocks
* U.S. bonds
* Real estate
* Commodities
* Cash equivalents
Keep in mind, though, that these are but a few of the many other asset classes available to us.
## 2. Estimating Expected Returns, Standard Deviations, and Correlations
The standard approach to asset allocation is based on portfolio theory, which requires us to estimate expected returns, standard deviations, and correlations.\[1] To estimate expected returns, we start by assuming markets are fairly priced; therefore, expected returns represent fair compensation for the degree of risk each asset class contributes to a broadly diversified market portfolio. These returns are called equilibrium returns, and we estimate them by first calculating the beta of each asset class with respect to a broad market portfolio based on historical standard deviations and correlations. Then we estimate the expected return for the market portfolio and the risk-free return. We calculate the equilibrium return of each asset class as the risk-free return plus its beta times the excess return of the market portfolio. Admittedly, the markets are seldom, if ever, in equilibrium, but the pull in this direction is powerful and persistent. Moreover, we can easily adjust the expected return of each asset class to accord with our views about departures from fair value.
Suppose we estimate the market’s expected return to equal 7.5% and the risk-free return to equal 4.0%.\[2] Given these estimates, together with estimates of beta based on monthly returns from January 1997 through December 2006, we derive the equilibrium returns shown in Table 1 below
> Table 1: Expected Return
| Asset Class | $$\beta$$ | Equilibrium | Views | Confidence | Blend |
| ---------------- | :-------: | :---------: | :----: | :--------: | :----: |
| U.S. stocks | 1.65 | 9.85% | 9.85% | - | 9.85% |
| Foreign stocks | 1.70 | 10.02% | 10.02% | - | 10.02% |
| U.S. bonds | 0.30 | 5.04% | 6.00% | 100% | 6.00% |
| Real estate | 0.86 | 7.01% | 7.50% | 50% | 7.25% |
| Commodities | 0.30 | 5.05% | 6.00% | 50% | 5.52% |
| Cash equivalents | 0.00 | 4.00% | 4.00% | - | 4.00% |
We may expect some asset classes to produce returns that differ from those that would occur if markets were in equilibrium and perfectly integrated, especially if they are not typically arbitraged against other asset classes. Suppose we expect real estate to return 7.5% and commodities to return 6.0% and that we assign as much confidence to these views as we do their equilibrium returns. We can blend these views to derive our expected returns. The final column of Table 1 shows the expected returns for each of the asset classes in our analysis.
We also need to estimate the standard deviations of the asset classes as well as the correlations between each pair of asset classes. These values, shown in Tables 2 and 3, are estimated from monthly returns for the period beginning in January 1977 and ending in December 2006.
It is important to note that standard deviations and correlations are not always stable through time. It is therefore useful to separate historical returns into those returns associated with normal times and those associated with periods of market turbulence.\[3] This separation allows us to estimate these values for each regime and to stress test portfolios by measuring exposure to loss based on risk characteristics that prevail during turbulent periods.
Table 2 shows estimates for standard deviations for both normal and turbulent regimes, while Table 3 shows the correlations for both regimes. As you might suspect, volatility rises during times of turbulence.
> Table 2: Expected Risk
| Asset Class | Normal | Turbulent |
| ---------------- | :----: | :-------: |
| U.S. stocks | 16.69% | 22.99% |
| Foreign stocks | 18.75% | 24.46% |
| U.S. bonds | 5.83% | 8.99% |
| Real estate | 15.22% | 22.26% |
| Commodities | 17.40% | 24.18% |
| Cash equivalents | 0.95% | 1.41% |
> Table 3a: Normal Regime Correlation Coefficients
| | U.S. stocks | Foreign stocks | U.S. bonds | Real estate | Commodities |
| ---------------- | :---------: | :------------: | :--------: | :---------: | :---------: |
| Foreign stocks | 0.5704 | | | | |
| U.S. bonds | 0.2468 | 0.1286 | | | |
| Real estate | 0.4594 | 0.3080 | 0.2041 | | |
| Commodities | -0.0327 | 0.1493 | -0.0046 | -0.0633 | |
| Cash equivalents | 0.0038 | -0.0445 | 0.1968 | -0.0477 | -0.1682 |
> Table 3b: Turbulent Regime Correlation Coefficients
| | U.S. stocks | Foreign stocks | U.S. bonds | Real estate | Commodities |
| ---------------- | :---------: | :------------: | :--------: | :---------: | :---------: |
| Foreign stocks | 0.5797 | | | | |
| U.S. bonds | 0.1762 | 0.1220 | | | |
| Real estate | 0.5491 | 0.3379 | 0.2104 | | |
| Commodities | -0.0590 | 0.0229 | 0.0665 | -0.1007 | |
| Cash equivalents | 0.0580 | -0.0190 | 0.1170 | 0.0627 | -0.1307 |
## 3. Efficient Portfolios
With this information, we use optimization to combine asset classes efficiently, so that for a particular level of expected return the efficiently combined asset classes offer the lowest level of risk, measured as standard deviation. A continuum of these portfolios plotted in dimensions of expected return and standard deviation is called the efficient frontier. Figure 2 shows a hypothetical efficient frontier.
Based on our earlier assumptions for expected returns, standard deviations, and correlations, we have derived three specific efficient portfolios: one for a conservative investor, one for an investor with a moderate appetite for risk, and one for an aggressive investor. We used the standard deviations and correlations from the normal periods to derive these portfolios. We use the risk values from both the normal and turbulent periods to measure their exposure to loss. Table 4 shows the three optimal portfolios.
> Table 4: Optimal Portfolios
| Asset class | Conservative | Moderate | Aggressive |
| ---------------- | :----------: | :------: | :--------: |
| U.S. stocks | 22.86% | 35.23% | 48.15% |
| Foreign stocks | 16.59% | 24.22% | 32.19% |
| U.S. bonds | 49.95% | 32.81% | 14.89% |
| Real estate | 3.85% | 2.59% | 1.28% |
| Commodities | 6.75% | 5.16% | 3.49% |
| Cash equivalents | 0.00% | 0.00% | 0.00% |
| | Conservative | Moderate | Aggressive |
| ------ | :----------: | :------: | :--------: |
| Return | 7.60% | 8.37% | 9.17% |
| Risk | 7.77% | 10.12% | 12.86% |
The composition of these portfolios should not be surprising. The conservative portfolio has nearly a 50% allocation to U.S. bonds. The moderate portfolio is well diversified and not unlike many institutionally managed portfolios. The aggressive portfolio, by contrast, has more than an 80% allocation to U.S. and foreign stocks. It should be comforting to note that we did not impose any constraints on the optimizer to arrive at these portfolios. The process of employing an equilibrium perspective for estimating expected returns yielded nicely behaved results.
## 4. The Optimal Portfolio
The final step is to select the portfolio that best suits our tolerance for risk, which we call the optimal portfolio. The theoretical approach for identifying the optimal portfolio is to specify how many units of expected return we are willing to give up to reduce our portfolio’s risk by one unit. If, for example, we are willing to give up $$\frac{1}{2}$$unit of expected return to lower portfolio variance (the squared value of standard deviation) by one unit, our risk aversion would equal $$\frac{1}{2}$$. Risk aversion is the reciprocal of risk tolerance. We would then draw a line with a slope of $$\frac{1}{2}$$ and find the point of tangency between this line and the efficient frontier (with risk defined as variance rather than standard deviation). The portfolio located at this point of tangency is theoretically optimal because its risk/return trade-off matches our preference for balancing risk and return. In practice, however, we do not know intuitively how many units of return we are willing to sacrifice in order to lower variance by one unit. Therefore, we need to translate combinations of expected return and risk into metrics that are more intuitive.
Because returns are approximately normally distributed, we can easily estimate the probability that a portfolio with a particular expected return and standard deviation will experience a certain loss over a particular horizon.4 Alternatively, we can estimate the largest loss a portfolio might experience given a certain level of confidence. We call this measure value at risk. We can also rely on the assumption of normality to estimate the likelihood that a portfolio will grow to a particular value at some future date.
Investors typically measure exposure to loss at the end of their investment horizon. This view of risk ignores what may happen along the way. Investors should think about risk differently. They should care about exposure to loss throughout their investment horizon and not just at its conclusion. We therefore focus on two additional risk measures to evaluate these portfolios: within-horizon probability of loss and continuous value at risk.
Within-horizon probability of loss measures the likelihood that an investment will depreciate to a particular level from inception to any point during a specified horizon and not just at the conclusion. Value at risk measured conventionally gives the worst outcome at a chosen probability at the end of an investment horizon. By contrast, continuous value at risk gives the worst outcome at a chosen probability from inception to any time throughout an investment horizon. As we shall soon see, these two risk measures reveal that within-horizon exposure to loss is substantially greater than investors typically assume.
Figure 3 shows the likelihood of a 10% loss over a five-year investment horizon for the three efficient portfolios. These probability measures are presented for a normal regime, in which the risk parameters are based on the entire sample of returns, and for a turbulent regime, which uses the turbulent sub-sample of returns.
End-of-horizon estimates of exposure to loss drastically understate a portfolio’s vulnerability to losses along the way. The moderate investor has only about a 1% chance of losing 10% or more at the end of five years, but there is a 15% chance that the portfolio will depreciate by at least that amount at some point along the way, and it increases to 36% if we expect a turbulent period to prevail. These are huge differences.
Figure 4 shows conventional value at risk (end-of-horizon) and continuous value at risk (within-horizon) for both a normal regime and a turbulent regime, measured at a 1% confidence level. Again, we see drastic difference in exposure to loss depending on whether we focus on the end of the horizon or the interim period as well. For example, the worst outcome at the end of the horizon for a moderate investor given a 1% probability and assuming a normal regime is a 10% loss. In comparison, the worst outcome at any point throughout the horizon is that the portfolio will decline by as much as 22%. If a turbulent regime prevails the worst outcome is a 35% loss.
If exposure to loss were our only consideration, we would choose the conservative portfolio, but by doing so we might forego upside opportunity. One way to assess the upside potential of these portfolios is to simulate the distribution of future wealth associated with investment in each of them.
Suppose, for example, our current portfolio’s value equals $1,000,000 and we plan to contribute $25,000 this year and grow our contributions annually by 5%. We can simulate how our portfolio’s value will change through time with a procedure known as Monte-Carlo simulation. This procedure generates possible future outcomes by drawing random numbers from a theoretical distribution such as a normal or lognormal distribution with a pre-specified expected return and standard deviation. If we wish to simulate outcomes that are relatively far into the future, it is important that we recognize the effect of compounding, which leads to a lognormal distribution rather than a normal distribution. Compared to a normal distribution, which is symmetrical, a lognormal distribution has a longer right tail than left tail and an average value that exceeds the median value.
To perform a Monte-Carlo simulation of our wealth 15 years forward, assuming we invest in the moderate portfolio and contribute annually beginning with $25,000 and growing this amount by 5% per year, we proceed as follows
1. We draw a random return from a log-normal distribution with an expected return of 8.37% and a standard deviation of 10.12%, the expected return and risk of the moderate portfolio.
2. We then add 1 to the randomly selected return and multiply this value by $1,000,000, the fund’s initial value.
3. Next we add the $25,000 contribution.
4. We then draw a second return from the theoretical distribution, add 1, and multiply this value by the new value of the fund including the contribution.
5. Next we add our second contribution, which is now $26,250.
6. We repeat steps one through five until 15 random returns are selected and we record the ending value of the portfolio.
7. We repeat steps one through six 5,000 times to generate 5,000 15-year sequences.
8. Finally, we rank the 5,000 final values from highest to lowest and note the values at the percentiles in which we are interested
Figure 5 shows the probable terminal wealth at different confidence levels in both nominal and real terms, assuming an inflation rate of 3.0%. For example, there is a 5% chance that our nominal wealth could grow to as much as $7 million in fifteen years, assuming we invest in the moderate portfolio and make the annual contributions specified above. After inflation, this amount falls to about $4.5 million. We should be 95% confident of ending up with about $2.3 million which would be worth $1.6 million after inflation. And we should expect to end up with $4.1 million dollars nominally and $2.7 after inflation.
We can also use Monte Carlo simulation to estimate the distribution of our future income. We follow the same procedures as described above for estimating our future wealth, but we assume that the wealth at the end of each of the 5,000 paths is invested in an annuity at a particular interest rate for a specified period of time. In addition, we can specify an amount to be left at the conclusion of the annuity period.
Figure 6 shows the distribution of future income assuming an annuity rate of 3.5% and a remainder to be left of $250,000. Our expected annual income in nominal terms is about $340,000, and adjusted for inflation equals about $175,000. There is a 5% chance it could be as high as $590,000 and $300,000 respectively, and we should be 95% confident that it will at least be $195,000 nominally or $100,000 adjusted for inflation.
By mapping the expected returns and standard deviations onto estimates of exposure to loss and the distribution of future wealth and income, we should have a clear idea of the merits and limitations of these portfolios. It is important to keep in mind, though, that there is no universally optimal portfolio; it is specific to each investor. If our focus is to avoid losses, the conservative portfolio might be optimal. If, instead, we believe that we can endure significant losses along the way in exchange for greater opportunity to grow wealth and future income, then we might choose the aggressive portfolio. If our goal is to limit exposure to loss, yet still maintain a reasonable opportunity to grow wealth and income, then perhaps the moderate portfolio would suit us best.
## Conclusion
Asset allocation is a complex process, yet one we cannot ignore. We strongly urge investors to approach asset allocation with discipline, structure, and internal consistency. Moreover, we recommend that investors evaluate portfolio choices based not on statistical abstractions, but rather on intuitive interpretations of their risk and return attributes.
## Endmatter
1. See, for example, H. Markowitz, “Portfolio Selection,” *Journal of Finance*, March 1952. In 1990 Markowitz was awarded the Nobel Prize in Economics for his development of portfolio theory.
2. The current risk-free return is readily observable. There are a variety of methods for estimating the expected return of a diversified market portfolio. For example, we might adjust the historical risk premium to accord with current risk levels and add this adjusted risk premium to the current risk-free return.
3. See, for example, Chow, G.,E. Jacquier, M. Kritzman, and K. Lowrey, “Optimal Portfolios in Good Times and Bad,” *Financial Analyst Journal*, May/June 1999.
4. In theory, returns are lognormally distributed owing to the effect of compounding.
5. For more detail about these risk measures, see Kritzman, M. and D. Rich, “The Mismeasurement of Risk”, *Financial Analysts Journal*, May/June 2002
#### Disclaimer
> This material is not intended to provide professional or investment advice and you are advised to seek independent professional advice prior to investing in any products or strategies described herein or recommended by Windham Capital Management, LLC. In addition, this constitutes neither an offer to buy or sell any securities, nor a solicitation of an offer to buy or sell interests or shares in any fund or strategy. Past performance, including any projection or forecast, are not necessarily indicative of future or likely performance of any investment products. No assurance may be given that the strategies’ investment objectives will be achieved. Investments are subject to investment risks including possible loss of principal amount invested.
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# Asset Allocation and Factor Investing
An Integrated Approach to Portfolio Construction
When it comes to portfolio construction, the debate of asset allocation versus factor investing can grow quite heated. Those who choose to build portfolios from asset classes argue that they are easy to observe and are easily investible — unlike factors. These investors also believe that portfolios from asset classes are more stable out of sample compared to portfolios composed from factors. Investors who prefer to allocate to factors argue that asset classes are defined arbitrarily and do not capture the fundamental determinants of performance as effectively as factors do. In addition, some investors prefer to invest in factors because they believe that factors carry risk premiums that are not directly available from asset classes.
We propose a compromise that allows investors to get the best of both approaches. By integrating asset allocation and factor investing, we can preserve the benefit of investing in observable and directly accessible units, while capturing our preferred factor exposures.
## Integrating Asset Allocation and Factor Investing
We start with the traditional approach to portfolio construction, known as mean-variance optimization, which was introduced by Harry Markowitz in 1952 through his [modern portfolio theory](/portfolio-construction/modern-portfolio-theory). The equation for mean-variance optimization is as follows
You’ll notice that this equation does not consider factors. In order to do so, we need to expand the objective function to include an additional term that reflects **aversion to deviating from a factor profile.**
Of course, in order to do so, we need to build a factor profile. One of the challenges in building a factor profile is that factors are not always measured in the same unit, so instead we must record their changes as logarithms (shown in the equation below).
Another challenge is that many factors are relatively stable because, unlike assets, they are not traded and thus not subject to investor uncertainty. The problem with trying to build a portfolio that is sensitive to a particular factor profile is that the covariances between the factor returns and asset returns are going to be too low to capture any significant sensitivities. Therefore, to build a portfolio that is reasonability sensitive to a comparatively stable factor profile, we need to rescale the factor returns by multiplying them by a constant.
## Summary
1. We estimate the expected returns and covariances of the asset classes in which we wish to invest
2. We identify factors to which we seek exposure (positive or negative)
3. We create factor time series by recording changes in the factor values (measured in log logarithms) and rescaling them to create a portfolio that is reasonably sensitive to changes in factor values
4. We define a factor profile by calculating a weighted average of the rescaled factor returns in accordance with our factor preferences
5. We estimate the covariances between the assets and the factor profile
6. We solve for a factor-sensitive optimal portfolio by maximizing an expanded objective function that incorporates both aversion to absolute risk and aversion to deviations from the factor profile
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To learn more about this approach, we recommend you read the full article on which this summary is based on, [**Asset Allocation and Factor Investing: An Integrated Approach**](http://jpm.iijournals.com/content/44/4/32), published in the 2018 Quantitative Special Issue of The Journal of Portfolio Management.
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# Asset Allocation in Taxable Portfolios
Modeling Taxable Portfolios
Taxes can consume a substantial portion of returns in an individual’s portfolio, and it is important to consider assets on an after-tax basis. It allows us to find an individual’s optimal portfolio — which may vary significantly from person to person. Considering assets on an after-tax basis also allows us to estimate the future value of a portfolio.
In this post, we will outline the steps to convert pre-tax return and risk into after-tax values. Next, we will identify optimal portfolios on a pre- and after-tax basis. Finally, we will simulate future wealth on a pre- and after-tax basis.
## Calculating After-Tax Return and Risk
For U.S. based investors, not all investments are created equal when it comes to taxes. Realized gains and income are classified and taxed at different rates. After-tax expected return and risk should be used when making asset allocation decisions.
The effective tax rate is the average annual percentage of total return that will be paid in taxes. Total returns include both income and capital appreciation. In a taxable account, income can be taxed at a qualified dividend rate or an individual’s marginal tax rate. Capital appreciation can be taxed at short-term capital gains rates or long-term capital gains rates depending on the holding period and turnover rate.
In a tax-deferred 401k or traditional IRA account, all distributions are taxed as income during retirement. Both the principal and the gains are taxed. When making asset allocation decisions, investors should consider assets in all investment accounts. We will show in a later example how to discount the value of the 401k in order to consider it simultaneously with taxable accounts.
For this example, we will us the tax assumptions summarized in Table 1 to calculate the effective tax rates in a taxable account.
> Table 1: Tax Assumptions
| | Assumptions |
| ------------------------------------- | :---------: |
| Qualified dividends tax rate | 15% |
| Marginal tax rate | 35% |
| Long-term capital gains tax rate | 15% |
| Short-term capital gains tax rate | 35% |
| Average holding period of asset class | 10 years |
We will use five asset classes and assume an average holding period of 10 years. The asset classes and their characteristics are shown in Table 2, below. We have assumed that commodities are taxed annually at a blended rate of 60% long-term capital gains and 40% short-term capital gains. We have also assumed the annual turnover of passive strategies to be negligible.
> Table 2: Investment Assumptions
| Asset class | Annual Turnover | Pre-tax Expected Total Return | Dividends and Realized Long-term Gains | Unqualified Dividends and Short-Term Gains |
| ------------------------- | :-------------: | :---------------------------: | :------------------------------------: | :----------------------------------------: |
| Passive U.S. Stock Index | 0% | 9.6% | 2.0% | 0.0% |
| Active Foreign Stock Fund | 100% | 10.3% | 5.3% | 4.5% |
| Commodities | 100% | 6.7% | 4.0% | 2.7% |
| Passive REIT Index | 0% | 7.5% | 3.0% | 2.0% |
| Passive U.S. Bond Index | 0% | 5.1% | 0.0% | 5.1% |
Using the assumptions in Table 1 and Table 2, we can calculate an effective tax rate. These values are summarized in Table 3 below.
> Table 3: Effective Tax Rates
| Asset class | Taxable Account | Tax-deferred |
| ------------------------- | :-------------: | :----------: |
| Passive U.S. Stock Index | 12% | 0% |
| Active Foreign Stock Fund | 25% | 0% |
| Commodities | 23% | 0% |
| Passive REIT Index | 19% | 0% |
| Passive U.S. Bond Index | 35% | 0% |
With the effective tax rate, we can estimate after-tax return and risk for each investment in the taxable account.
> Table 4: After-Tax Return and Risk
| Asset class | After-tax Return | After-tax Risk |
| ------------------------- | :--------------: | :------------: |
| Passive U.S. Stock Index | 8.5% | 13.5% |
| Active Foreign Stock Fund | 7.7% | 12.2% |
| Commodities | 5.2% | 10.8% |
| Passive REIT Index | 6.0% | 12.9% |
| Passive U.S. Bond Index | 3.3% | 2.6% |
With these tax-adjusted capital market forecasts, we can now perform an optimization or calculate the after-tax return of a given portfolio.
## Asset Allocation
Let’s assume we have the same asset classes and tax characteristics as used above. Let’s also assume we have $400,000 in a taxable account and $500,000 in a 401k account, and we plan to retire in 10 years. Our objective is to maximize our after-tax wealth for a given level of risk.
As mentioned before, we must make adjustments to the value of the 401k account to consider it simultaneously with the taxable portion of our portfolio. In a 401k, the principal and the gains are taxed at the time of distribution. We can calculate the after-tax present value of the 401k account by deducting the present value of the tax paid in retirement. The net worth of the 401k becomes (1- effective tax rate in retirement) × (401k account balance). We will assume a 20% effective tax rate during retirement. The value of the 401k on an after-tax basis is now $400,000. Our after-tax wealth is $800,000.
We will first find an optimal portfolio on a pre-tax basis, and then allocate equally in both accounts.
> Table 5: Pre-tax Optimal Portfolio
| | Taxable | 401k | Total |
| ----------------------------------------------------- | :-----: | :---: | :---: |
| Passive U.S. Stock Index | 17.4% | 17.4% | 17.4% |
| Active Foreign Stock Fund | 24.6% | 24.6% | 24.6% |
| Commodities | 10.6% | 10.6% | 10.6% |
| Passive REIT Index | 7.2% | 7.2% | 7.2% |
| Passive U.S. Bond Index | 40.2% | 40.2% | 40.2% |
| | | | |
| Pre-tax expected return ($900,000 pre-tax wealth) | | | 7.5% |
| After-tax expected return ($800,000 after-tax wealth) | | | 6.6% |
| After-tax expected risk | | | 6.5% |
> Table 6: After-tax Optimal Portfolio
| | Taxable | 401k | Total |
| ----------------------------------------------------- | :-----: | :---: | :---: |
| Passive U.S. Stock Index | 55.4% | 0.0% | 27.7% |
| Active Foreign Stock Fund | 0.0% | 26.2% | 13.1% |
| Commodities | 28.5% | 0.0% | 14.2% |
| Passive REIT Index | 16.1% | 0.0% | 8.0% |
| Passive U.S. Bond Index | 0.0% | 73.8% | 36.9% |
| | | | |
| Pre-tax expected return ($900,000 pre-tax wealth) | | | 7.4% |
| After-tax expected return ($800,000 after-tax wealth) | | | 6.8% |
| After-tax expected risk | | | 6.5% |
The expected return for the after-tax optimal portfolio is 0.2% higher than the pre-tax optimal. This excess return is gained primarily from a reduction in taxes.
Since we typically can’t access our 401k until we turn 59½, we may be tempted to create a portfolio that has an aggressive allocation in the 401k account and a conservative allocation in the taxable account. Table 7 shows a portfolio that was optimized on a pre-tax basis to be conservative in the taxable account and aggressive in the 401k account.
> Table 7: Pre-tax Optimal with an Aggressive 401k
| | Taxable | 401k | Total |
| ----------------------------------------------------- | :-----: | :--: | :---: |
| Passive U.S. Stock Index | 8% | 27% | 17.5% |
| Active Foreign Stock Fund | 11% | 38% | 24.5% |
| Commodities | 8% | 13% | 10.5% |
| Passive REIT Index | 4% | 10% | 7.0% |
| Passive U.S. Bond Index | 70% | 10% | 40.0% |
| | | | |
| Pre-tax expected return ($900,000 pre-tax wealth) | | | 7.5% |
| After-tax expected return ($800,000 after-tax wealth) | | | 6.5% |
| After-tax expected risk | | | 6.5% |
## Wealth Planning
We adjust returns for taxes to generate more accurate estimates of future expected wealth. Using the portfolio from Table 6, we perform a Monte-Carlo simulation of wealth 10 years from today. If we ignore taxes and perform a simulation, we find that we are 70% certain that the portfolio will be worth at least $1,199,000 adjusted for inflation. This value overstates wealth because it is pre-tax and ignores any tax payments that may occur during the 10 year period.
Using the after-tax present value of the $800,000 and the after-tax expected return and risk we are 70% certain the portfolio will be worth at least $1,017,000 after taxes and adjusted for inflation. In this example, ignoring taxes overestimates future wealth by 18%.
## Conclusion
We have shown that taxes impact expected return, expected risk, and the optimal asset allocation for a taxable investor. We have also shown that, by ignoring taxes, we may be misled to overestimate the investor’s future wealth. Further, we should consider all investment accounts simultaneously and resist the temptation of taking excessive risk in our 401k accounts.
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# Asset Allocation versus Security Selection: Evidence from Global Markets
Which activity is more important: asset allocation or security selection? The answer could surprise you.
A common debate in the investment management field regards asset allocation and security selection. Which is more important? Which causes the greatest dispersion in wealth? The overwhelming consensus within the industry is that asset allocation is more important when assembling your portfolio. When taking a closer look, however, it becomes clear that those who subscribe to that belief fail to distinguish between the consequences of investor behavior, and the opportunity set offered by the capital markets.
Before we get into the methodology that brought us to a result which defies popular opinion, let’s take a look at how that popular opinion was formed, as well as a few people who have gone against it.
## How did we get here?
**1986** – Gary Brinson and his colleagues release a study titled “Determinants of Portfolio Performance,” which attributes the performance of 91 large corporate pension plans to three investment activities: policy, timing, and security selection. They performed regression analyses which revealed that asset allocation policy, on average, accounted for over 93.6% of total return variation through time, and in no case less than 75.5%.
**1991** – Brinson *et al*. updates the study and found that asset allocation policy still accounted for more than 90% of return variation.
**2000** – Ibbotson and Kaplan publish an article that supported the Brinson *et al*. result, but demonstrates that only 40% of return differences across funds is attributable to asset mix policy.
**2000** – Ankrim and Hensel object to the Brinson *et al*. methodology, because it attributes the returns from 0% up to the policy portfolio return to asset allocation. This attribution assumes implicitly that the default exposure is 100% cash.
**2000**– Jahnke argues that it is difficult to measure the importance of asset allocation because the answer depends on many factors, such as the extent to which investors engage in active asset allocation and security selection, investment expenses and skill.
In order to assess the relative importance of asset allocation and security selection, it is necessary to move *beyond* historical performance, because these results depend on two separate influences: the investment opportunities available from variation in asset class and security returns, and the extent to which investors chose to exercise discretion in exploiting these opportunities.
Which brings us to question number two.
## Which is more important, asset allocation or security selection?
**For the purpose of this investigation, importance is defined as** **the extent to which a particular investment activity causes dispersion in wealth**
In order to solve for this question, we present a simple mathematical model in which the potential for dispersion is measured as the tracking error between two investments (investments that differ by security composition, or investments that differ by asset class composition). Since individual securities are more volatile than asset classes, one may assume that security selection creates more dispersion unless the securities are perfectly correlated. On the contrary, those who subscribe to the belief that asset allocation creates more dispersion, must also believe that high correlations between securities outweigh their high individual volatilities.
In order to put these theories to the test, we start with two asset classes each made up of two separate securities. If we first assume the four securities are uncorrelated with each other, then security selection would be more important than asset allocation because the securities would be riskier than the asset classes containing them resulting in less dispersion among the asset classes. What we are able to deduce from this exercise is that it’s only when the correlation between the asset classes is substantially less than the correlation between the individual securities within the asset classes that dispersion among the asset classes is greater than that among the securities.
When only a few asset classes are considered, the associations between standard deviation, correlation, and tracking error seem clear. However when considering numerous asset classes and weighing them among hundreds of securities with a wide range of volatilities and correlations, the associations become less obvious. When you contemplate the real-world conditions, it is necessary to resolve the question of *importance* with a simulation procedure called bootstrapping.
Bootstrapping is a procedure by which new samples are generated from an original dataset by randomly selecting observations from the original data set. It differs from Monte Carlo simulation in that it draws randomly from an empirical sample, whereas Monte Carlo simulation draws randomly from a theoretical distribution.
The importance of security selection is measured by holding a constant asset mix at a 60/30/10 allocation among stocks, bonds and cash, and calculating variation in return due purely to variation among randomly diversified stock portfolios. The importance of asset allocation is measured by holding constant security weights and calculating variation in return owing purely to expected allocation of 60/30/10 to stocks, bonds, and cash respectively.
The following table demonstrates the extent to which a talented investor (top 25th or 5th percentile) would improve upon average performance by engaging in asset allocation and security selection. It also shows the extent to which an unskilled investor (bottom 75th or 95th percentile) would perform, depending on the choice of investment discretion.
> Percentile Performance Annualized Difference From Average (1988 - 2001), U.S. Investor
| | Asset Allocation | Security Selection |
| ---- | :--------------: | :----------------: |
| 5th | 0.6% | 1.6% |
| 25th | 0.3% | 0.7% |
| 75th | 0.3% | -0.7% |
| 95th | 0.5% | -1.6% |
**So, when questioning the relative importance of asset allocation versus security selection, the answer is clear:**
{% hint style="info" %}
Random variation among individual securities within a portfolio causes considerably more return variation than asset allocation.
{% endhint %}
# Defining an Asset Class
Asset allocation is one of the most important decisions investors face, however there are no universally accepted criteria that define exactly what an asset class is. Some investments take on the status of an asset class because managers feel that investors are more inclined to allocate funds to products if they are defined as an asset class, rather than merely as an investment strategy. Alternatively, the investment industry tends to overlook investment categories that legitimately qualify as an asset class because investors are reluctant to defy tradition.
## What are the real consequences of NOT defining an asset class?
The undefined nature of an asset class reduces the efficiency of the asset allocation process in at least two ways.
1. If dissimilar investments are wrongly grouped together into an asset class, the portfolio will not be diversified efficiently.
2. If an asset class is inappropriately partitioned into redundant components, the investor will be required to deploy resources unproductively to analyze irrelevant expected returns, standard deviations, and correlations.
Furthermore, the investor may waste additional resources in search of relevant investment managers. For these reasons, it is important to establish criteria for the purpose of identifying legitimate asset classes.
## Proposed Asset Classes
The list of proposed asset classes is long and diverse. The traditional candidates are:
* Domestic Stocks
* Foreign Stocks
* Real Estate
* Domestic Bonds
* Foreign Bonds
* Cash Equivalents
The stock and bonds are often subdivided into more specific groups:
* Large Cap Stocks
* Mid Cap Stocks
* Small Cap Stocks
* Growth Stocks
* Value Stocks
* Financial Stocks
* Developed Market Foreign Stocks
* Emerging Market Foreign Stocks
* Long Term Government Bonds
* Long Term Corporate Bonds
* Intermediate Term Government Bonds
* Intermediate Term Corporate Bonds
* High Yield Bonds
* Municipal Bonds
* Developed Market Foreign Bonds
* Emerging Market Foreign Bonds
Finally, there are the so-called alternative investments:
* Commodities
* Currencies
* Hedge Funds
* Managed Futures
* Market Neutral Funds
* Private Equity
* Timberland
* Venture Capital
**We propose four criteria for determining asset class status:**
1. An asset class should be relatively independent of other classes in the investor’s portfolio
2. An asset class should be expected to raise the utility of the investor’s portfolio without requiring selection skill
3. An asset class should be comprised of homogeneous investments
4. An asset class should have the capitalization capacity to absorb a meaningful fraction of the investor’s portfolio
###
### **(1) Relative Independence**
Relative independence address whether or not a new asset class will help diversify more efficiently. An asset class will not improve diversification if combinations of asset classes already in the portfolio could duplicate the new asset class’s risk characteristics. The redundancy doesn’t have to be with a single asset class, but with any linear combination of asset classes.
#### How can we test for independence?
We can test for independence of a proposed asset class by identifying the combination of asset classes that minimizes tracking error with the proposed new asset class. We call this portfolio a **mimicking portfolio**. Then, we judge whether the tracking error of the mimicking portfolio is sufficiently large to suggest independence. Tracking error is computer as the square root of the average of the squared differences between the mimicking portfolio’s returns and the returns of the proposed asset class. In other words, it is the standard deviation of the return differences.
For example, suppose we allocate our portfolio among U.S. stocks, foreign stocks, U.S. long-term bonds, and U.S. cash equivalents. We want to determine whether or not we should include U.S. intermediate-term bonds in this portfolio. Based on monthly returns from the beginning of 1985 through the end of 1997, a mimicking portfolio consisting of 0.21% U.S. stocks, 0.56% foreign stocks, 48.86% long-term bonds, and 52.38% short-term instruments produces tracking error of only 1.32% with intermediate-term bonds. Compare this value to the tracking error between each of the current asset classes and its mimicking portfolio constructed from other assets in the portfolio, based on the same historical return sample.
| | Tracking Error of Mimicking Portfolios
with Current Asset Classes
|
| ---------------------- | :----------------------------------------------------------------------------: |
| U.S. Stocks | 12.15% |
| Foreign Stocks | 16.34% |
| Long-term Bonds | 6.46% |
| Short-term Instruments | 6.94% |
It is reasonable to conclude from these tracking errors that immediate-term bonds are redundant to the other asset classes and therefore should not be considered as an asset class in this situation.
### (2) Expected Utility
The second criterion for asset class status raises two critical distinctions: the distinction between expected utility and expected return, and the distinction between random selection and skillful selection on the part of the investor.
Expected utility refers to happiness or satisfaction, which comes from either the expectation of higher returns or of less risk. Consider commodities, for example. You may believe that their expected return is insufficient to raise the expected return due technological advances outpacing the depletion of scarce resources. However, because commodities offer diversification against financial assets, especially in environments of high unanticipated inflation, their inclusion in a portfolio might lower risk sufficiently enough to *more than offset* their expected reduction of return.
$$
\text{Expected Utility}=\text{Expected Return}-\text{Risk Aversion}\times \text{Variance}
$$
Suppose our portfolio consists of a single asset with an expected return of 10.00% and a standard deviation of 12.00%. Also assume that our risk aversion equals 1.5, which indicates that we are willing to give up 1.5 units of expected return in order to lower portfolio variance by one unit. With these assumptions, and remembering that variance equals standard deviation squared, we calculate expected utility to equal 7.84% $$\left( 0.10-1.5 \times 0.122 \right)$$.
Let’s assume that we estimate commodities to have an expected return of 9% and a standard deviation of 12%. At first, it doesn’t appear that an allocation to commodities would improve the risk/return profile of our portfolio because commodities have the same risk but less expected return. However, we must not ignore the correlation of commodities within our portfolio. Suppose we estimate the correlation of commodities with our portfolio to equal 5.00%. If we shifted 38% of our portfolio’s assets to commodities, its expected return would decline from 10.00% to 9.62%, but its standard deviation would fall from 12.00% to 8.92%. Based on risk aversion of 1.5, the expected utility of this new portfolio equals 8.43% $$\left( 0.0962-1.5 \times 0.08922 \right)$$. Given our willingness to exchange expected return for risk reduction, we would be happier to allocate some of our portfolio to commodities even though this shift reduces the expected return we expect to achieve. The point is that expected return, by itself, is insufficient for gauging an asset’s impact on investor satisfaction.
#### Random Selection vs. Skillful Selection
The distinction between random selection and skillful selection is subtle, yet important. An asset class should raise a portfolio’s expected utility not because the investor is skillful in identifying superior portfolio managers within that asset class. Rather the investor should expect the asset class to raise utility even if managers within an asset class are selected randomly. It does not follow, however, that an asset class is disqualified if improvement in expected utility requires skillful managers within the asset class.
Assume that passive exposure to domestic stocks is expected to raise a portfolio’s utility. Domestic stocks therefore might qualify as an asset class. Let’s also assume that the top quartile growth stock managers would also be expected to raise utility. If growth stock managers on average are note expected to raise utility, and only a skillful investor would be able to identify top quartile growth stock managers before the fact, than top quarter growth stock managers would not qualify as an asset class.
#### Are Hedge Funds an Asset Class?
Finally, let’s consider hedge funds. Let us first acknowledge the conjecture that hedge funds would raise a portfolio’s utility because their managers, *on average*, perform better than do non-hedge fund investors who invest in the same assets. In other words, random compositions of the assets that typically constitute hedge funds are not expected to raise utility, but a random selection of hedge fund managers would be. This unusual conjecture arises from the fact that hedge funds typically require a lock-up period; that is, their investors are precluded from withdrawing funds for a pre-specified period of time. Therefore, hedge fund managers are in a position to collect a liquidity premium, which may explain why their performance exceeds the average performance of the assets in which they invest. If this conjecture were indeed true, a naïve or random selection of hedge fund managers, requiring no skill selection on the part of the investor, would be expected utility because hedge fund managers as a group are skillful in extracting a liquidity premium. Hence, hedge funds might qualify as a legitimate asset class, even if their constituent investments do not.
### (3) Homogeneity
The requirement for homogeneity among the components of an asset ensures that we do not ignore opportunities for diversification. If an asset class comprises dissimilar components, then by investing in that asset class we implicitly impose the unnecessary and potentially harmful constraint that the components must be held in the same relative proportions as their weights in the asset class. We could achieve a more efficient portfolio if we partitioned the dissimilar components into multiple asset classes.
For example, foreign stocks are typically viewed as a single asset class. For this example, let’s define foreign stocks as a 40% allocation to German stocks, a 30% allocation to UK stocks, and a 30% allocation to Japanese stocks—all unhedged. This construction is not significantly different in its risk profile from the EAFE index. Based on the returns, standard deviations, and correlations from the beginning of 1980 through the end of 1997, a portfolio comprised of 25% U.S. stocks, 50% U.S. bonds, and 25% foreign stocks, has an expected return of 14.54% and a standard deviation of 9.20%.
Now suppose we partition foreign stocks into six components: hedged and unhedged German stocks, hedged and unhedged UK stocks, and hedged and unhedged Japanese stocks. If we retain the same weights in U.S. stocks, U.S. bonds, and foreign stocks as a group, but allow the allocations within foreign stocks to vary among the six components, we could achieve a more efficient portfolio. By recognizing that foreign stocks are not homogeneous, we can segment this asset class into several homogeneous components and improve our portfolio’s expected return and risk significantly.
### (4) Capacity
The final criterion for asset class status — that it has to be sufficiently large to absorb a meaningful fraction of our portfolio — is self-evident. If we invested in an asset class with inadequate capacity, we would likely drive up the cost of investment and reduce our portfolio’s liquidity. The consequence might be to lower our portfolio’s expected return and increase its risk to the point at which the proposed asset class would no longer be expected to improve our portfolio’s utility.
# An Interview with Harry Markowitz
Conversations with the father of modern portfolio theory
Windham's CEO, Mark Kritzman, recently interviewed Nobel Prize winner Harry M. Markowitz to discuss his background at the University of Chicago, the Cowles Commission, and the RAND Corporation; his many contributions not only to modern portfolio theory but also to other fields; and his views on the 2008 global financial crisis. This interview provides some insight into the life and work of one of the most influential men in the history of finance, as well as some “words of wisdom” for future generations.
Members and non-members of the CFA can read this interview in the [**Financial Analysts Journal Fourth Quarter 2017, Volume 73, Issue 4**](https://www.cfainstitute.org/en/research/financial-analysts-journal/2017/an-interview-with-nobel-laureate-harry-m-markowitz).
# MIT Masters of Finance Convocation
Windham's Mark Kritzman addresses MIT Graduates, June 7, 2018
Good morning and congratulations. As many of you know, I’m not used to speaking without a PowerPoint presentation, but surprisingly Heidi didn’t show much enthusiasm for that idea. When I mentioned my concern to my wife, she said: “Don’t worry. You’re good at giving eulogies.” So if what I say sounds a bit sad, you’ll understand why.
Of course, I’m joking. This is such a happy occasion. Happy for you students because you are about to graduate from the world’s best masters of finance program. Happy for you parents because your children are about to embark upon what almost certainly will be exceptionally successful careers. And happy for MIT because your coming success will reflect so favorably upon us.
To the students, I would like to remind you of the huge debt of gratitude you owe to your parents for the support, guidance, and comfort they have given to you over the years. I don’t mean to downplay your own skills and hard work, but I’m sure you will all agree that your parents deserve much credit for helping you arrive at this happy occasion.
To the parents, you should be so proud of your children. First, for qualifying for admission to one of the most selective graduate finance programs anywhere. And second, for successfully fulfilling all of the requirements for graduation. I can assure you that I would not have wanted to sit for the exams some of my colleagues subjected your children to. That they are sitting here today is a testament to their extraordinary abilities and effort.
You should also be proud that your children have chosen finance as their profession. In the wake of the financial crisis, we often hear politicians and people in the media criticize finance, but they are disingenuous. Finance is a powerful force for good. Like in all professions, some people in finance have misbehaved, but the overwhelming preponderance of financial professionals work honorably every day to make the world a better place. They work so that people who labor in the real economy are able to transfer their risk to others, which enables them to deliver their goods and services to many more people than would occur otherwise. And they work to ensure that prices are set properly so that capital is allocated efficiently, which raises the quality of life for everyone. The virtues of finance may not be readily apparent in how we perceive our daily lives, but look around the world and ponder the relative economic well-being of nations, and then consider the relative sophistication of their financial systems. As my colleague, Andrew Lo likes to say, finance is to the real economy what the circulatory system is to the human body. That strikes me as a fitting analogy, especially for those of you parents here who might have preferred that your children had studied medicine.
Now please indulge me for a few minutes as I describe MIT’s position in the history of finance. As some of you realize, much of modern finance was invented by professors from MIT. Think of the legendary economist, Paul Samuelson. Yes, he is better known for his contributions in other fields of economics. In fact, he referred to finance as his Sunday activity. I guess we should be thankful that he played tennis instead of golf, which would have taken up more of his time. So on his Sundays he laid out the efficient markets hypothesis. Through his work in pricing warrants, he paved the way for the eventual discovery of the options pricing formula. And by exploring the connection between time and risk, he profoundly changed the way people think about risk – and the way many people now invest. Then there was Franco Modigliani who revolutionized corporate finance with his invariance propositions – that the value of the firm does not depend on dividend policy or whether it is financed with equity or debt. I once asked Franco if he realized at the time how important his work would become, and he said, “Yes, I knew right away because my colleagues were so upset with me.” You see, he overturned much of his colleague’s prior work. And, of course, there is the ground-breaking discovery of the options pricing formula by Fischer Black and Myron Scholes and separately by Bob Merton. Stew Myers wrote the book on corporate finance – literally, along with developing the key theories of capital structure, budgeting, and valuation. And then, of course, John Cox made so many contributions to options pricing and term structure modeling. That’s a pretty impressive list of accomplishments.
Now let’s think about the major innovations that occurred away from MIT. Yes, there were a few – for example, portfolio selection by Harry Markowitz, the two-fund separation theorem by James Tobin, the capital asset pricing model by Bill Sharpe, John Linter, and Jan Mossin. (Of course, we could claim that it was invented here by Jack Treynor, who was studying under Franco Modigliani at the time). Then we have arbitrage pricing theory by Steve Ross, who later did spend half his career here, and Ken Arrow’s work on the role of securities. Yes these innovations occurred elsewhere, but what you may not realize is that they were validated here at MIT by Bob Merton’s work in continuous time finance. Bob showed how to reconcile portfolio selection and the separation theorem with expected utility theory. He resolved the issue of the instability of the CAPM and APT. And he showed how to reconcile Ken Arrow’s work, which presumes few securities and many markets with the empirical reality of many securities and few markets.
My point is that there is no institution that has a stronger claim as the birthplace of modern finance than MIT. And the work here continues in such important ways. Leonid Kogan is exploring the link between a firm’s economic activity and its stock price behavior. Andrew Lo’s adaptive markets hypothesis extends the neoclassical foundation to incorporate behavioral considerations. Debbie Lucas is doing important work to help governments embrace the teachings of modern finance. And Jiang Wang is extending the theory and practice of finance to China. Although MIT has Massachusetts as part of its name, it is a global institution that seeks to make the entire world a better place.
These are just a few examples. Everyone on the finance faculty is contributing to the ever growing legacy of MIT.
Why do I dwell on this? First because I think a little substance is good in a talk that is otherwise pretty fluffy. But mainly because I want you to understand the lineage that you are now part of. There is no stronger brand in finance than MIT, and you will now carry that brand with you for the rest of your lives. You are ambassadors of MIT finance, so make us proud.
Now let me turn to the typical purpose of a speech like this, which is to impart wisdom. I could give you practical advice about how to navigate the challenges of advancing your careers, especially given the mistakes I’ve made. But that’s not what I want to focus on this morning. Please come and visit me another time for that kind of advice.
Instead, I would like to convey just two messages to you. The first is be honest. Not only is it the right thing to do, but being trustworthy is a huge part of your human capital, alongside your intellect, your training, and your brand. If you cannot be trusted, you will bear a huge cost that will follow you throughout your careers. And trust me – no pun intended – people will learn very quickly whether or not they can trust you. So it is in your interest to be trustworthy.
The second message is about how you treat people. You are exceptional. You are highly intelligent, you have a great work ethic, and now – as evidenced by your presence here this morning, you are extremely well-trained. The odds are quite strong that you will be very successful and attain a high professional status. You will meet many important people, but you will also meet people who will not have achieved a lofty professional or social station. My message to you is to treat everyone the same – which is with dignity, respect, and kindness, whether they are your superiors or your subordinates.
I have little doubt that you will be successful, but will you be happy? Because, despite your best intentions, it is likely that you will often find yourself in conflict, both in your professional lives and in your personal lives. My advice again is to be respectful. You may disagree with some people, but it does not mean that they are not sincere in their beliefs. And, moreover, if you show respect, you will have a better chance of resolving your conflict.
This will be hard to do on a consistent basis. You are entering a very competitive profession. You will almost certainly encounter conflict, and when you do, you will be tempted to treat people who are in conflict with you as objects – objects whom you see as interfering with your plans and not worthy of respect. This attitude will not help you resolve conflict. You will be much more successful in dealing with conflict or even avoiding it by being respectful. And the easiest way to be respectful is to see others not as objects but as people who have aspirations, needs, concerns, and fears, just as you do. The more you see the humanity in people, I firmly believe that you will have a much happier and more peaceful life. And that may be the true measure of success.
Congratulations and please do stay in touch.
*MFin Covocation:*\
*MIT Sloan School of Management*\
*06/07/18*
# Future Value
When to use arithmetic and geometric returns for expected values
Suppose we want to estimate the future value of an investment based on its return history. This problem, at first glance, might seem pedestrian. Yet it involves subtleties that confound many financial analysts.
Some analysts argue that the best guide for estimating future value is the arithmetic average of past returns. Others claim that the geometric average provides a better estimate of future value. The correct answer depends on what it is about future value that we want to estimate.
Let us proceed with a quick review of the arithmetic and geometric averages.
## Averages
The arithmetic average is simply the sum of the holding-period returns divided by the number of returns in the sample.
The geometric average is calculated by adding 1 to the holding-period returns, multiplying these values together, raising the product to the power of 1 divided by the number of returns, and then subtracting 1. The geometric average is sometimes called the constant rate of return, or the annualized return.
We can also compute the geometric average by converting holding-period returns into continuous returns. A continuous return, when compounded continuously, yields the same wealth we would achieve by investing at the holding-period return without compounding. It equals the natural logarithm of the quantity 1 plus the holding-period return. If the holding-period return equals 10%, for example, the continuous return equals 9.53%. If we were to invest $1.00 at an annual rate of 9.53% compounded continuously throughout the year, it would grow to $1. 10 by the end of the year.
We calculate the geometric average from continuous returns by raising $$e$$ (2.7182), the base of the natural logarithm, to the power of the arithmetic average of the continuous returns and subtracting 1.
Table 1 shows how to compute the arithmetic and geometric averages. We compute the arithmetic average by summing the values in the first column and dividing by 4. It equals 8.00%.
We can compute the geometric average in two ways. We can multiply the values in the second column, which yields 1.3206, then take the fourth root of 1.3206 and subtract 1 to arrive at the geometric average, 7.20%. Alternatively, we can compute the arithmetic average of the third column, raise e to this value, and subtract 1 to arrive again at 7.20%. It follows, therefore, that the arithmetic average of the logarithms of the quantities 1 plus the holding-period returns equals the logarithm of the quantity 1 plus the geometric average.
Here is how to interpret the geometric average. If we invest $1.00 in this sequence of returns, our dollar will grow to $1.3206. We would achieve the same terminal value by investing $1.00 at a constant rate of 7.20% for the four periods.
## Expected Value
Now consider our earlier question. Which average should we use to estimate future value, assuming we wish to base our estimate on past returns? The question as I have posed it is too vague. We must be more precise about what we wish to know about future value.
If our goal is to estimate an investment’s expected value either one period forward or many periods forward, we should use the arithmetic average of holding-period returns. We estimate expected value from past returns by adding 1 to this average and compounding this quantity forward.
In order to see why the arithmetic average is used to estimate expected value, consider an investment that has a 50% chance of increasing by 25% and a 50% chance of decreasing by 5%. After one period, there is an even chance that a dollar will grow to $1.25 or decline to $0.95. The expected value after one period thus equals $1.10, which in turn equals 1 plus the arithmetic average of the two possible returns.
After two periods, there are four equally likely outcomes. The investment can increase to $1.25 after the first period and then increase to $1.5625 after the second period or decrease to $1.1875. It can first decrease to $0.95 after the first period and then increase to $1.1875 or decrease further to $0.9025. Figure A diagrams these four possible paths.
The expected value after two periods, which equals the probability-weighted outcome, equals 1.2100. It corresponds precisely to the quantity 1 plus the arithmetic average of 10% raised to the second power. The geometric average of a 25% increase followed by a 5% decrease or a 5% decrease followed by a 25% increase equals 8.9725%. If we add 1 to the geometric average and compound it forward for two periods, we arrive at a terminal value of 1.1875, which does not equal the expected value.
The expected value is higher than the value we would have achieved had we invested in the returns on which the arithmetic average is based.
This result might seem paradoxical. The intuition is as follows. The expected value assumes that there is an equal chance of experiencing any of the possible paths. A path of high returns raises the expected value over multiple periods more than a path of equal-magnitude low returns lowers it. This disproportionate effect is the result of compounding. Suppose the high return is 10% while the low return is -10%. Two consecutive high returns produce a 21% increase in value, while two consecutive low returns produce a decrease in value of only 19%.
Here’s how to interpret expected value. Suppose we observe 10 years of monthly returns and wish to estimate how much wealth we should expect to achieve if we were to draw randomly from these 120 monthly returns, replacing each of the returns that is drawn. If we were to invest in the 120 returns that we selected from the sample (without yet observing them), we should expect our investment to grow at a rate equal to the quantity 1 plus the arithmetic average of the sample of monthly returns, raised to the 120th power minus 1 or, equivalently, 1 plus the arithmetic average of the yearly returns from the sample, raised to the 10th power minus 1.
If we were to repeat this experiment many times, the average of the cumulative wealths generated from the sequences of randomly selected returns would indeed converge to the wealth predicted by the compounded arithmetic average. You can verify this result with a random-number generator.
## Distribution of Future Value
Suppose we ask the following questions about future value. What is the likelihood or probability that an investment will grow or fall to a particular value? Or what value should we be 50% confident of achieving or failing to achieve? The answers to these questions depend on the geometric average.
Let’s start with the assumption that the logarithms of the quantities 1 plus the holding-period returns are normally distributed. This assumption implies that the returns themselves are lognormally distributed. It follows that the normal deviate used to estimate the probability of achieving a particular future value is calculated from the mean (arithmetic average) and standard deviation of these logarithms.
A normal deviate measures distance from the mean in standard deviation units. It is the number that we look up in a normal distribution table to estimate the probability of achieving or falling short of a particular value.
Suppose we wish to estimate the likelihood that $1 million will grow to equal $1.5 million over five years, based on the past annual returns of a particular investment. If we believe that the logarithms of the quantities 1 plus these returns are normally distributed, we can proceed by computing the mean and standard deviation of these logarithms, as Table 2 shows.
We compute the normal deviate as follows:
The logarithm of the quantity 1.5 million divided by 1 million (40.5465%) is the continuous five-year return required in order for $1 million to grow to $1.5 million. It corresponds to an annualized continuous growth rate of 8.1093%. The quantity 5 times 6.4695% is the expected five-year continuous return. The normal deviate measures how far away the required five-year continuous return is from the expected five-year continuous return. It is 0.3372 standard-deviation units away. If we look up this value in a normal distribution table, we see that there is a 38.6% chance that $1 million will grow to $1.5 million, based on the past returns of this investment.
Suppose we wish to know the likelihood that our investment will generate a loss over five years. We compute the normal deviate as:
There is a 9.17% chance that this investment will lose money, on average, over five years.
What value should we expect to equal or exceed with 50% confidence? This value is called the median. Half the values are expected to exceed the median and half are expected to fall short of the median. A 50% probability of occurrence corresponds to a normal deviate of 0.00. The normal deviate equals 0.00 only when the required continuous return equals the expected continuous return. Thus we should expect with 50% confidence to equal or exceed the value that corresponds to the expected five-year continuous return. We find this value by raising $$e$$ , the base of the natural logarithm, to the power 5 times the expected annualized continuous return. Thus the median wealth equals $1,381,924.26. There is a 50% chance that the value in five years will exceed this value and a 50% chance that it will fall short of this value.
The geometric average is relevant for estimating probabilities, because the average of the logarithms equals the logarithm of the quantity 1 plus the geometric average.
The center of the probability distribution of terminal values is found by compounding the initial value at the geometric average. When we compound at the geometric average, we determine the future value for which there is an equal chance of exceeding or failing to exceed. Here is the logic behind this result:
1. The logarithms of the quantities 1 plus the holding-period returns are assumed to be normally distributed.
2. The mean (arithmetic average) of these logarithms equals the logarithm of the quantity 1 plus the geometric average of the holding-period returns.
3. The expected multi- period continuous return therefore equals the number of periods times the logarithm of the quantity 1 plus the geometric average.
4. We convert the expected multi-period continuous return into median wealth by raising *e* to the power of the multi-period continuous return.
5. The quantity $$e$$ raised to the power of the multi-period continuous return is exactly equal to the initial wealth compounded forward at the geometric average.
## Summary
The bottom line is that we should compound at the arithmetic average if we wish to estimate an investment’s expected value. We should compound at the geometric average, however, if we wish to estimate the likelihood that an investment will exceed or fall below a target value.
# The Cost of Socially Responsible Investing
As we evolve into a more political and environmentally conscious society, the concept of “socially responsible investing” is becoming an increasingly popular investment option. A socially responsible investor is one who chooses not to invest in certain companies whose behavior they judge to be at odds with the social good as they perceive it. Some proponents of socially responsible investing claim that “good” companies – those in harmony with the social good – perform as well *or better* than “bad” companies. They conclude, therefore, that socially responsible investing is without cost and may even enhance performance. However, one may argue that if the motive to invest is due to higher expected returns, it is not in fact “socially responsible” investing.
As with any method of investing, it is important to be aware of the costs. Suppose, for example, that exclusion of tobacco companies reduces return. Some may argue that this cost is worth incurring because it focuses attention on the issue, or because it may influence company behavior. Others may argue that it would be more effective to collect the return of the tobacco investment, and deploy it directly towards policies designed to curb smoking or treat lung cancer. Without knowledge of the expected cost, it would be difficult to accurately evaluate this trade-off.
### How do we measure the cost of socially responsible investing?
{% hint style="info" %}
We employ a [Monte-Carlo](/general/monte-carlo-simulation) simulation.
{% endhint %}
We use this technique to compare the performance of a skilled investor without restrictions with the performance of a skilled investor with restrictions (a socially responsible investor).
We simulate the cost of socially responsible investing under various scenarios. We vary the investment universes to model four common benchmarks: the S\&P 500, the MSCI EAFE, the MSCI World, and the MSCI All Country World Index. We select portfolios of 100 and 250 securities, assuming a variety of skill levels. Additionally, we estimate the costs associated with excluding 10%, 20%, and 30% of the securities due to not meeting *socially responsible* criteria.
As you can see in Exhibit 1, an investor with 52% skill who selects from a universe comparable to the S\&P 500, and who foregoes investment in 10% of the securities, should expect to give up 0.08% annually to an investor of the same skill, who is not socially responsible. This cost rises with the investor’s skill level, the number of excluded securities, and the cross-sectional dispersion.
Exhibit 2 reveals that the cost of socially responsible investing rises with the number of securities in the portfolio. For instance, an investor with 54% skill who selects 100 securities from a universe similar to the S\&P 500 incurs a cost of 0.37% when excluding 20% of the securities. If this investor increased the number of securities from 100 to 250, the cost would rise to 0.53%. This is because the investor must substitute progressively lower ranked securities as she expands the number of securities in the portfolio.
Of course, when expressing cost in terms of percentages, it can be challenging to truly understand the impact. Exhibits 3 and 4 convert those percentages into their cumulative dollar values. These tables assume:
* 100 or 250 securities
* 52% skill
* 20% exclusion
* An initial portfolio value of $1 billion
* An underlying market return of 8.0% for periods of five, ten, and twenty years
These tables reveal a harsh reality about socially responsible investing. A socially conscious investor can either exclude “bad” companies from their portfolio and sacrifice vast sums of wealth, or invest in an unrestricted fashion and deploy those returns directly toward reforming those social wrongs.
Based on the results of a Monte-Carlo simulation comparing socially restricted investments against unrestricted investments, it is clear that the cost of socially responsible investing is substantial — even for skilled investors.
# The Hidden Cost of Active Management
Investors are well aware of the incremental transaction costs managers incur as they seek to replace securities perceived to be overvalued with those perceived to be undervalued. Moreover, it is no secret that active funds charge much higher fees than passive funds designed to track market indexes. What investors may not be as aware of, however, is that there is a **hidden cost associated with most active funds.** The typical active fund is more than 90% correlated with the market, yet their relatively high active management fee is applied not only to the fund’s active component, but to its market component as well. Rather than pay active fees on total assets, including those that provide market exposure, an investor could achieve essentially the same result before fees by allocation most of the portfolio to an index fund with the residual allocated to a pure alpha fund that nets out market exposure. The following example illustrates this hidden cost.
**Table 1** shows the monthly returns and values of a hypothetical actively managed fund and an index fund, assuming an initial investment of $10 million. The index fund serves as the benchmark for the active fund. In this example, the active fund generates a 2.00% alpha with active risk equal to 3.23% for a respectable information ratio of 0.62. If this performance is consistent with past results, it would not be unreasonable for the fund to charge a fee of 100 basis points, or more.
Sounds pretty standard, right? We might be tempted to hire this talented manager, but let us first consider an alternative. Suppose we instruct the manager to deliver a fund that comprises only the active bets of the portfolio. The manager would do this by first putting the capital to work in a short-term fund. Let’s suppose this fund earns 4.00% annually. The manager then sells short the index fund and uses the proceeds of these short sales to purchase the stocks that are expected to outperform, and levers these exposures 12 to 1. By employing this approach, the manager delivers a pure alpha stream rather than the composite of market returns and alpha that make up the active fund. **Table 2** shows the returns of this pure alpha fund (again, assuming an initial investment of $10 million).
The $10 million investment in the short-term fund compounds at 0.33% per month for a cumulative annual return of 4.00%. The initial exposure to the active fund equals $120 million (12:1 leverage), while the initial exposure to the index fund equals negative $120 million (again, 12:1 leverage). The value of the pure alpha fund each period, therefore, equals the sum of the short term investment fund position and the active fund and index fund positions.
The pure alpha fund produces an annual return of 28.00%, which equals 12 times the active fund’s 2.00% alpha plus 4.00% from the funds invested in the short-term investment fund. The annualized standard deviation of the pure alpha fund is slightly less than 12 times the active fund’s active risk, because it is exposed to the short-term investment fund. Thus the pure alpha fund produces and information ratio of 0.79 compared to an information ratio of 0.62 for the active fund.
Now, let’s consider combining a low cost investment in an index fund with investment in the pure alpha fund, instead of investing in the active fund. **Table 3** shows the returns and values of a 90/10 mix of the index fund and the pure alpha fund.
This peculiar mix of a $9,000,000 initial investment in the index fund, together with an initial investment of $1,000,000 in the pure alpha fund, produces precisely the same return (10.00%) as the active fund, and it achieves this at less risk—17.39% versus 19.29% for the active fund. More importantly, the returns of this strategy are 99.85% correlated with the active fund returns. It is almost a perfect substitute for the active fund. Now, let’s compare the costs of these two strategies.
As stated earlier, the active fund charges 100 basis points, which is applied to the average of the beginning and ending values. Therefore, the cost of the active fund is $105,000, as shown:
$$
\frac{\left( 10,000,000+11,000,000\right)}{2}\times0.01=105,000
$$
Let’s suppose the index fund charges 20 basis points and the pure alpha fund charges 250 basis points. The premium relative to the active fund compensates for the slight increase in complexity associated with netting out the market exposure to isolate the active bets. With these assumptions, the fee of this combined strategy is substantially lower — only $47,220, as shown:
$$
\frac{\left( 9,000,000+9,720,000\right)}{2}\times0.02+\frac{\left( 1,000,000+1,280,000\right)}{2}\times0.025=47,220
$$
The chart below summaries the benefits of combining an index fund with a pure alpha fund. This blended approach delivers 100% of the active fund’s return, incurs only 90% of its risk, and is only 45% as expensive.
# Performance Fees
Institutional investors continue to allocate a significant and growing fraction of their portfolios to alternative investments such as hedge funds, private equity, and real estate, notwithstanding the challenges these relatively illiquid assets posed to them during the financial crisis. We suspect that investors now pay considerable attention to their liquidity needs and factor these needs into their decision to invest in alternative investments.
We are fairly confident, though, that most investors do not fully appreciate the consequences of a second feature of alternative investments. They typically charge performance fees. It turns out that the expected return of a group of funds that charges performance fees is less than the average of the funds’ expected returns. And it also true that an individual fund’s standard deviation or the standard deviation of a group of funds, net of performance fees, understates risk. Together, these oversights lead investors to allocate more to alternative investments than they should, if they base their allocations on expected return and risk.
## The Expected Return of the Average is Less Than the Average of the Expected Returns
Managers of alternative investments typically charge a performance fee that is a percentage of profits — but not losses — relative to a benchmark, along with a base fee that is a fixed percentage of assets under management. Often the base fee is deducted from the profits before the performance fee is applied. If, for example, the base fee equals 2% and the performance fee equals 20%, a manager who produces a 10% return in excess of the benchmark on a $100 million portfolio will collect a $2 million base fee (2% × $100,000,000) and a $1.6 million performance fee (20% × ($10,000,000 – $2,000,000) for a total fee of $3.6 million. The investor’s return net of fees, therefore, is 6.40%.
Now suppose an investor hires two managers who each charge a base fee of 2% and a performance fee of 20%. Assume as well that these managers both have expected returns of 10% in excess of the benchmark. The expected fee for each manager is 3.60% (2% + 20% × (10% – 2%)). Therefore, the investor might expect an aggregate return net of fees from these two managers equal to 6.40%. This expectation would be justified, however, only if both managers’ returns exceed the base fee. If, instead, one manager produces an excess return of 30% and the other a –10% excess return, and an equal amount of capital is allocated to each manager, the investor would pay an average fee of 4.80% rather than 3.60%, and the average return to the investor would equal 5.20% rather than 6.40%, even though the managers still have an average excess return of 10%. The difference between 4.80% and 3.60% is known as the asymmetry penalty.
This result is specific to the assumptions of this example. Nevertheless, it is easy to determine the typical reduction in expected return by applying Monte Carlo simulation. Let’s consider investment in 10 funds, each of which has an expected excess return of 7%, a standard deviation of 15%, and a correlation of 0% with the other funds. Let’s also assume that LIBOR equals 4%. In order to estimate the impact of asymmetry, we proceed as follows:
1. We first draw 1,000 random returns for each fund, assuming their returns are normally distributed and uncorrelated.
2. Next, we apply the base fee and performance fee to the 1,000 returns for each fund and compute the average of their returns net of fees.
3. We then compute the average return across the 10 funds for the 1,000 random draws, and we apply the base fee and performance fee to these average returns. The average of these net returns reveals the performance that we should expect if we were reimbursed for underperformance or if we used a single multistrategy fund instead of separate funds.
4. Finally, we compute the difference between the results of steps 2 and 3, which equals the asymmetry penalty.
With the above assumptions, the asymmetry penalty equals about 0.70%. If the funds’ correlations were higher, the penalty would be smaller, and if they were lower, the penalty would be higher. This asymmetry penalty is a hidden fee arising from the fact that investors pay for outperformance but are not reimbursed for underperformance. This effect, in principle, would be somewhat muted because most performance fee arrangements include clawback provisions that require funds to offset prior losses before collecting performance fees. In most cases, though, underperforming managers are either terminated or the performance fees are reset without reimbursement for prior losses.
## Standard Deviation Understates Risk
There is yet another subtle and unpleasant feature of performance fees. The standard deviation of funds that charge performance fees understates risk, because it is reduced by the attenuation of upside performance. When a fund outperforms its benchmark, the upside return is cut by the amount of the performance fee, but because losses are not reimbursed, downside deviations are unaffected. Based on simulation, an individual fund that charges a 2% base fee and a 20% performance fee and that has a 15% standard deviation would produce about a 13% standard deviation net of performance fees. Thus, performance fees, given these assumptions, have the effect of understating risk by about 2%. If we treat these 10 funds as a group, the risk is understated by about 0.65%, because of the effect of diversification. These results are summarized as follows:
An investor who ignores these two effects would expect an individual fund, which is a member of a group of 10 funds, to have an expected return net of fees equal to 7.70% and volatility equal to 13%.By taking into account the asymmetry penalty and the mismeasurement of risk, this fund instead has an expected return of 7% and normalized downside volatility equal to 15%. If an investor treated the 10 funds as a single asset, the return comparison would remain the same, but the correct volatility estimate would equal 4.75% instead a presumed estimate of 4.10%. Either way, the correct return-to-risk ratio is more than 20% less than an investor would presume by ignoring the effects of performance fees.
These features of performance fees do not imply that investors should avoid alternative investments or other funds that charge performance fees. Rather, investors should account for these features, along with other considerations, such as illiquidity, when determining the appropriate allocation to such funds. We suspect most do not.
# Monte-Carlo Simulation
Fortune tellers, palm readers, the Farmer’s Almanac, financial analysts. What do these things have in common? They all attempt to anticipate the future. While some use a crystal ball, the ridges of their subject’s palm, or sunspots, financial analysts use numeric models and mathematical techniques to generate simulations of their client’s financial future.
There are two kinds of forecasting models: deterministic models and stochastic models. Deterministic models assume a fixed relationship between the inputs and the output, while stochastic models depends on inputs that are influenced by chance. Deterministic models can be solved analytically via mathematical formulas, while stochastic models require numerical solutions. To solve stochastic models numerically, one must try various values for the model’s parameters and variables. When these variables come from a sequence of random numbers, the solution is called Monte-Carlo simulation.
Monte-Carlo simulation was originally introduced by financial analysts John Von Neumann and Stanislaw Ulam while working on the Manhattan Project at the Los Alamos National Laboratory. They invented a procedure of substituting a random sequence of numbers into equations to solve problems regarding the physics of nuclear explosions. The term Monte-Carlo was inspired by the gambling casinos in Monaco.
Monte-Carlo simulation is performed with a sequence of numbers that are distributed uniformly, are independently of each other, and are random. Random sequences can be generated using mathematical techniques such as the [mid-square method](http://www3.nd.edu/~mcbg/tutorials/2006/tutorial_files/randomNum/howItworks.html), but most spreadsheet software includes random-number generators. It is important to note that most financial analysis applications generate random variables that are not distributed uniformly. In order to perform a Monte-Carlo simulation, the sequence of uniformly distributed random numbers must be transformed into a sequence of normally distributed random numbers.
### Applying Monte-Carlo Simulation to the Stock Market
Imagine we invest $100,000 to an S\&P 500 index fund in which the dividends are then reinvested, and we want to predict the value of that investment 10 years from now. As previously mentioned, we start by generating a series of 10 random numbers that are uniformly distributed. However, we assume that the S\&P’s returns are normally distributed and we therefore need to transform our sequence. This transformation can be accomplished easily by applying the Central Limit Theorem.
The following table shows the relative frequencies of two variables, X and Y.
| Value | X | Y | (X+Y)/2 |
| :---: | :-: | :-: | :-----: |
| 1.0 | 1/6 | 1/6 | 1/36 |
| 1.5 | 0 | 0 | 1/18 |
| 2.0 | 1/6 | 1/6 | 1/12 |
| 2.5 | 0 | 0 | 1/9 |
| 3.0 | 1/6 | 1/6 | 5/36 |
| 3.5 | 0 | 0 | 1/6 |
| 4.0 | 1/6 | 1/6 | 5/36 |
| 4.5 | 0 | 0 | 1/9 |
| 5.0 | 1/6 | 1/6 | 1/12 |
| 5.5 | 0 | 0 | 1/18 |
| 6.0 | 1/6 | 1/6 | 1/36 |
It is clear to see that while neither X nor Y are normally distributed on their own, their average beings to approach a normal distribution. Therefore, we can create a sequence of random numbers that is normally distributed by taking the averages of many sequences of random, uniformly distributed numbers.
| Uniform Sequence | Average of 30 Uniform Squences |
| :--------------: | :----------------------------: |
| 0.6471 | 0.4965 |
| 0.4162 | 0.4336 |
| 0.5691 | 0.5747 |
| 0.2006 | 0.4477 |
| 0.4685 | 0.5014 |
| 0.7442 | 0.5126 |
| 0.9439 | 0.4930 |
| 0.5556 | 0.6040 |
| 0.2480 | 0.5267 |
| 0.3644 | 0.4721 |
Figure 1 shows the relative frequency of both sequences, and we can see from this graph that the average of the 30 uniformly distributed sequences approached a normal distribution.
The next step in Monte-Carlo simulation is to scale the normally distributed sequence so it has a standard deviation of one and a mean of zero. By dividing each observation by 0.05 (the theoretical standard deviation), and then subtracting 10 (the theoretical mean of this sequence) from each other its observations, we can achieve this transformation.
Next, we must rescale our sequence to reflect our assumptions about the mean return and standard deviation of the S\&P 500. For this exercise, let us believe that the average return of the S\&P 500 is 12% and its standard deviation is 20%. We rescale our standardized normal distribution by multiplying each observation by our assumption of 20% for the standard deviation, and adding to this value our assumption of 12% for its average return. Now we have a sequence of returns that we can then use to simulate our investments performance (column E).
To carry out the Monte-Carlo simulation, we then link the sequence of random returns and multiply the result by 100,000 (our investment) to derive an estimate of its value in 10 years. For example, the simulated returns in Table 4 yield a value of $333,810.
We repeat the entire process, beginning with the generation and averaging of 30 random sequences. We proceed until we generate a sufficiently large quantity of estimates. The distribution of these estimates is the solution to our problem. The figure below shows the frequency distribution of the terminal value of $100,000 over 10 years, resulting from 100 simulations.
While many problems can be solved using an analytical solution, Monte-Carlo simulation is immensely more simple for problems that are too complex to be described by equations. However, in order for Monte-Carlo simulation to obtain a reliable result, it must be repeated enough times.
# The Fallacy of 1/N
Equally Weighted Portfolios versus Optimized Portfolios
The battle between equally weighted portfolios and optimized portfolios is one that has produced an inaccurate assumed victor. Previous research has shown that equally weighted portfolios outperform optimized portfolios, suggesting that optimization adds no value in the absence of informed inputs and resulting in a naïve distrust of the portfolio optimization process. We, however, challenge both this suggestion and the research that lead to it. Optimized portfolios are designed to maximize expected return for a chosen level of risk by accounting for differences in expected returns, standard deviations, and correlations. The 1/N approach to investing assigns asset weights based purely on the number of asset classes, N, and ignores all other information. We understand why the concept of 1/N is so appealing:
1. It avoids concentrated positions.
2. It never underperforms the worst performing asset.
3. It always invest in the best performing asset.
4. It captures the size alpha because it overweights small-cap stocks and underweights large-cap stocks.
In 2009, a study by DeMiguel, Garlappi, and Uppal (DGU) was published that seemed to provide definitive proof that equally weighted portfolios outperformed optimized portfolios. They performed out-of-sample backtests on seven empirical data sets, and used 14 different methods to estimate inputs, including Bayesian estimation and moment restrictions designed to reduce estimation error. They found that, on average, 1/N portfolios generated Sharpe ratios that are 50% higher than those of the mean-variance optimized portfolios.
We believe that the perceived failure of optimization arises from overreliance on short term return samples to estimate expected returns. The DGU study used rolling 60- and 120- month return samples in modeling expected returns, which are prone to small-sample error and in many cases are implausible. No thoughtful investor would blindly extrapolate historical means estimated over such short samples.
{% hint style="info" %}
Think of optimization as a sophisticated navigation system. 1/N is the navigational equivalent of wandering aimlessly in search of our destination, ignoring not only the GPS but also any posted road signs. We argue that it is better to use the GPS; we just need to specify the destination. If our goal is to drive to the beach, but we instruct the GPS to take us to the office, we should not fault the GPS for directing us to the office. Those who favor 1/N think of optimization this way.
**Kritzman, Mark P. “A Practitioner’s Guide to Asset Allocation.” A Practitioner’s Guide to Asset Allocation, John Wiley & Sons, Inc., 2017, pp. 61–61.**
{% endhint %}
For our study, we discounted the notion that error maximization explains the finding that 1/N produces superior portfolios than optimization. We used simple models of expected returns that assumed no forecasting skill, and used no constraints other than the long-only constraint. With 13 datasets comprising 1,028 data series, we constructed over 50,000 optimized portfolios and evaluated their out-of-sample performance. We grouped them into three categories: [asset class](https://www.windhamlabs.com/insights/defining-asset-class/), beta, and alpha. These categories correspond to the investment process that most institutional investors follow: first, asset/liability management; then, beta allocation; and finally, the search for alpha. **Table 1** shows our datasets. We used monthly data except for the 500 stocks, for which we used daily data to accommodate the shorter period and the larger covariance matrix.
For each dataset, we compared the out-of-sample performance of the market portfolio, the 1/N portfolio, and the optimized portfolios. To measure the performance of the optimized portfolios, we forecasted risk and return only on the basis of information available at the time of portfolio construction—making every forecast out of sample. Then, we invested in the optimal allocation that maximized the trade-off between expected return and risk.
For the asset/liability simulations, we assumed five-year holding periods with annual rebalancing and measured average performance for all five-year holding periods (we reoptimized monthly for all other simulations). We used the Sharpe ratio as a measure of performance, and did not impose any constraints (other than the long-constraint). This was for two reasons: (1) We wanted our results to be directly comparable to previous studies, and (2) we wanted to evaluate the performance of optimization in its simplest form.
We used three approaches to estimate expected returns. Although incredibly simple, these expected returns have an important difference from most of the expected returns used in previous studies: **they do not rely on short rolling samples of realized returns, which often imply implausible expectations.**
1. First, we generated the minimum-variance portfolio. We designed this to determine whether we could improve risk-adjusted performance by simply optimizing to reduce risk on the basis of pure extrapolation of the covariance matrix. In this experiment, expected returns were constant for all assets.
2. Second, for each asset, we estimated a risk premium over a long data sample and assumed that it remained constant throughout the backtest. To do so, we used data available before the backtest start date. **Table 2** shows our assumptions for the asset/liability optimizations. For the betas, we simply used the first 50 years of each database.
3. Third, in the spirit of classical statistics, we used a growing sample that included all available out-of-sample data.
To estimate expected volatilities and correlations, we used the monthly rolling 5-, 10-, and 20-year covariance matrices, as well as the all-data approach (all matrices were equally weighted).
**Table 3** provides a summary of our experiments. Due to a lack of available data, we focused on the minimum-variance approach and the five-year covariance matrix for the alpha portfolios; given the large number of securities involved, we used a daily covariance matrix for the security selection experiment.
**Figure 1** shows the results for our asset/liability management optimizations. We included in-sample optimization results, which show how we would perform if we knew the true parameters of the distribution. Figure 1 shows that optimized portfolios significantly outperform the 1/N portfolio. We found that even without any ability to forecast returns, optimization of the covariance matrix alone adds value.
**Figure 2** shows the results for our beta universes. For these backtests, we averaged Sharpe ratios across beta universes. The results in DGU (2009) were based on this dataset, which is notorious for the exceptional performance of the 1/N portfolio, as evidenced by its high Sharpe ratio as compared with that of the market portfolio. Nonetheless, the optimized portfolios outperformed 1/N. Since we only report averages, we note that in some cases (e.g., when allocating among size deciles), 1/N outperformed optimization. But out-of-sample results are often noisy; hence, we are mostly interested in the average performance across backtests.
**Figure 3** shows the results for our alpha universes. We assumed that the market portfolios for hedge funds and asset managers were equally weighted, and used the S\&P 500 as the market portfolio for security selection and the S\&P GSCI as the market portfolio for commodities. Optimization of the covariance matrix, without estimates of expected return, once again significantly improved out-of-sample performance as compared with the 1/N portfolio.
The minimum-variance portfolio performed well in our asset class, beta, and alpha simulations. In some cases, it outperformed optimization with expected return. This is plausible for a few reasons. First, our expected return models did not assume any forecasting skill. Second, introducing expected returns does not necessarily increase the Sharpe ratio of high-return-high-risk portfolios when leverage is not allowed.
## Conclusion
By performing this study we were able to conclude that optimized portfolios generate superior out-of-sample performance compared with equally weighted portfolios. We showed that reliance on small historical samples for estimating returns often leads to views that would be rejected by any savvy investor. Although in this study we focused on mean-variance optimization, new technology in portfolio construction allows for increased flexibility and provides greater insight into portfolio behavior and performance.
We would also like to point out some practical problems with 1/N beyond those highlighted in our experiment.
1. Equal weighting is not sensitive to return and risk estimates, but is entirely dependent on the choice of the asset class universe. Each asset class is assigned equal importance regardless of how many there are. The 1/N approach essentially transfers the risk of input estimation error to the risk of selecting the right [asset classes](/asset-allocation/defining-an-asset-class).
2. The 1/N heuristic offers only one portfolio, no matter their attitude towards risk. The [efficient frontier](/portfolio-construction/multi-goal-optimization) allows investors the choice of a wide variety of portfolios, each catered towards different appetites for risk.
3. Equal weighting ignores the capacity of each asset class as well as a variety of other considerations, which might favor one asset class over another.
Essentially, 1/N makes sense only for investors who believe they have no insight into the expected returns and risk of asset classes. Investors who have access to historical data, and who are capable of sound judgement to apply optimization, should identify the portfolio that best suits their investment goals.
# Modern Portfolio Theory
Breaking down Modern Portfolio Theory
The year is 1952. Roughly 4.2% of the United States population invests in the stock market. Majority of the country considers investing a way for affluent individuals to gamble with their ample means. The Fed lifted the cap on interest rates one year prior, and stocks are now outperforming bonds by 25 percentage points. Index investing has yet to regain popularity since the crash of 1929, and international investing is nonexistent.
Meanwhile, 25-year old Harry Markowitz is putting together his doctoral thesis. He comes across an idea that will (over time) change the landscape of finance and earn him a Nobel Prize.
## The Article
The idea behind his seminal article, *Portfolio Selection*, was born of coincidence. While in a waiting room, a broker casually suggested Markowitz apply his studies in statistics to the stock markets. After reading *The Theory of Investment Value* by John Burr Williams, Markowitz considered two ideas that were, to his disbelief, being ignored: risk and diversification.
Modern Portfolio Theory operates under the following assumptions:
1. **Investors are risk averse**
2. **Investors are rational**
3. **All investors have access to the same information**
4. **Returns are normally distributed**
Modern portfolio theory (MPT) provides investors with a portfolio construction framework that maximizes returns for a given level of risk, through diversification. MPT reasons that investors should not concern themselves with an individual investment’s expected return, but rather the weighted average of the expected returns of a portfolio’s component securities as well as how individual securities move together. Markowitz consequently introduced the concept of covariance to quantify this co-movement.
“Markowitz reasoned that investors should not choose portfolios that maximize expected return (in isolation), because this criterion alone ignores the principle of diversification. He proposed that investors should instead consider variances of return, along with expected returns, and choose portfolios offering the highest expected return for a given level of variance.”3 These portfolios were deemed “efficient.” For given levels of risk, there are multiple combinations of asset classes (portfolios) that maximize expected return. Markowitz displayed these portfolios across a two-dimensional plane showing expected return and standard deviation, which we now call the [**efficient frontier**](https://wpahelp.windhamlabs.com/optimization/multi-goal-optimization)**.** Though all portfolios along the efficient frontier are “efficient,” the portfolio that maximizes returns for a particular investor’s level of risk is known as the **Optimal Portfolio**.
## The Impact of Modern Portfolio Theory
Those familiar with the philosophies and solutions here at Windham can likely deduce the impact of MPT on investing today. The application of modern portfolio theory is known as [**mean-variance optimization**](https://wpahelp.windhamlabs.com/optimization/multi-goal-optimization)**,** which plays a key role in investing for both individual investors to large institutions.
However, it took a long time for MPT implementation across the field. The initial article, *Portfolio Selection*, was laden with complex graphs and equations that proved difficult for practitioners to understand. For decades, investors clung to their “gut instincts” in investing, and as a result MPT remained stuck inside academic circles. In 1970, William F. Sharpe published his book titled *Portfolio Theory and Capital Markets,* which introduced the [**Capital Asset Pricing Model**](https://wpahelp.windhamlabs.com/expected-returns/equilibrium-returns) (CAPM) and built on Markowitz’s theory. Together, CAPM and MPT contributed to the rise of index investing.
## Recognizing Markowitz and Modern Portfolio Theory
In 1990, Harry Markowitz, William Sharpe, and Merton Miller, won the Nobel Prize in Economics. Their innovative work paved the way for quantitative financial analysis as well as many of the investing trends we use today. Markowitz’s belief in diversification has influenced the economy worldwide, as alternative [asset classes](https://www.windhamlabs.com/insights/defining-asset-class/) such as publically traded REITs and commodities have proven to be efficient ways to diversify portfolio risk.
The concepts introduced in Markowitz’s early work provided the foundation for the analyses and investing practices that we use today. His work has especially influenced the work we do at Windham, as we provide our users with the tools for a comprehensive understanding of risk throughout the [asset allocation and portfolio construction process](https://www.windhamlabs.com/products/windham-portfolio-advisor.html).
## References
1. Stocks Then And Now: The 1950s And 1970s. (2008, October 01). Retrieved March 23, 2018, from
2. Lauricella, T. (2012, February 19). ‘Retro’ Investing-Look Back to Get Ahead. Retrieved March 25, 2018, from
3. [Kinlaw, W., Kritzman, M. P., & Turkington, D. (2017). A Practitioners Guide to Asset Allocation. Hoboken, NJ: Wiley.](https://www.amazon.com/dp/B071YX5CD5/ref=dp-kindle-redirect?_encoding=UTF8\&btkr=1) (pg 9)
# Multi-goal Optimization
Investors are often must decide whether to consider absolute return and risk, or relative return and risk. Sophisticated investors, however, employ optimization in an attempt to be sensitive to both. Many address this dual concern of absolute and relative return and risk by *constraining* the asset weights in the optimization process, which is time-consuming and produces sub-optimal portfolios.
Imagine for a moment, a new approach. An approach that allows investors to simultaneously consider absolute and relative performance, yields a higher expected return, in a more efficient process. Here at Windham Labs, we offer an innovative optimization technique: Multi-goal Optimization.
Multi-goal Optimization combines mean-variance analysis, which characterizes assets and portfolios by their expected *absolute* return and standard deviation, with tracking error—which gives a measurement of the volatility of relative returns. This technique allows investors to identify efficient allocations that consider both absolute and relative performance. Rather than producing an efficient frontier in two dimensions, Multi-goal Optimization produces an efficient surface in three dimensions: expected return, standard deviation, and tracking error.
### Benefits of Multi-goal Optimization
Multi-goal optimization typically yields an expected result that is superior to constrained mean-variance optimization.
* For a given *expected return*, multi-goal optimization produces a portfolio with a lower standard deviation and less tracking error
* For a given *standard deviation*, multi-goal optimization produces a portfolio with a higher expected return and less tracking error
* For a given *tracking error*, multi-goal optimization produces a portfolio with a higher expected return and lower standard deviation
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Higher expected return, lower standard deviation, less tracking error
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Multi-goal Optimization not only identifies allocations that are efficient based simultaneously on expected absolute and relative performance, but also estimates return distributions in both absolute and relative dimensions. Analysis of joint-probability distributions generated by Multi-goal Optimization allow the investor to minimize the likelihood of failing to achieve an absolute target, while at the same time underperforming a benchmark.
The traditional approach of imposing allocation constraints on the mean-variance optimization process is an inefficient way to address performance goals. Multi-goal Optimization, which encompasses both absolute and relative measures of risk in an unconstrained optimization process, typically produces a **superior solution** through a **more efficient** process.
## Video Presentation
To help visualize this concept, play the video presentation below
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Multi-goal Optimization
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# Optimal Rebalancing with the Markowitz-Van-Dijk Heuristic
Institutional investors usually employ mean–variance analysis to determine optimal portfolio weights. Almost immediately upon implementation, however, the portfolio’s weights become sub-optimal as changes in asset prices cause the portfolio to drift away from the optimal targets.
In an idealized world (without transaction costs) investors would rebalance continually to the optimal weights. In the presence of transaction costs, investors must trade-off the cost of suboptimality with the cost of restoring the optimal weights. Most investors employ heuristics to manage this trade off. Some implement calendar-based rebalancing policies, in which they rebalance each month, quarter, or year. Others impose tolerance bands in which they rebalance when the exposure to any asset class drifts more than two percentage points from its target, for example. While these approached are better than not rebalancing at all, they’re arbitrary.
*Is total cost minimized by tolerance bands of one-or-two percentage points? Should equity asset classes have wider bands than fixed-income asset classes? Should investors rebalance in periods when there has been little drift?*
We propose that investors implement a rebalancing policy that minimizes explicitly the sum of transaction costs and suboptimality costs, including the expected future costs associated with each decision, at each point in time. **Rebalancing is a multi-period problem: A decision to rebalance today, or not, has implications for expected suboptimality and transaction costs in the future.**
### Dynamic Programming and Rebalancing
In 2006, Sun, Fan, Chen, Schouwenaars, and Albota showed us how to use dynamic programming to develop a rebalancing road map. This approach gives the optimal rebalancing decision for every possible combination of portfolio weights that may arise at each decision point across the investment horizon. They showed that this approach outperforms calendar and tolerance band approaches by reducing both transaction costs and suboptimality costs. Unfortunately, this solution suffers from the *curse of dimensionality* — it is intractable for portfolios with more than a few asset classes.
### Optimal Reblaancing Using Markowitz-Van-Dijk
Markowitz and van Dijk (2003) introduced a quadratic heuristic to rebalance portfolios to capture changes in expected returns of assets through time. In 2009, Kritzman, Myrgren, and Page applied this approach, which they called the Markowitz-van Dijk (MvD) heuristic, to the rebalancing problem. Through this process, they show that the MvD solution is actually quite close to the dynamic programming solution for small numbers of assets– AND that it scales manageable to several hundred assets.
The 2009 article, “Optimal Rebalancing: A Scalable Solution,” explores the process by which the MvD heuristic applied to the rebalancing problem. These tests showed that the MvD heuristic performs almost as well as dynamic programming for up to four assets and *better* than dynamic programming for five assets. In theory, of course, dynamic programming always yields the best result. Unfortunately, however, we have no way of determining how the MvD heuristic would compare to the unobservable “correct” dynamic programming solution. However, based on these tests. the MvD heuristic is a better option for rebalancing portfolios with more than a few assets.
### Applications
The scalability of MvD opens the door to several new applications of portfolio rebalancing. Passive managers could use the MvD heuristic to optimize the trade-off between tracking error and transaction costs. Quantitative asset managers could use it to minimize alpha decay between rebalancing dates. Plan sponsors could benefit from the MvD heuristic in very meaningful ways, as they are continually confronted with asset mix rebalancing decisions. Furthermore, plan sponsors could customize the optimal rebalancing process to existing tolerance bands, tracking error targets, cash inflows, and benefit payments.
# Portable Alpha
The conventional approach to building investment portfolios, which relies on the typical hierarchy of investment decisions, imposes constraints on active management. Under most circumstances, these constraints are unnecessary and produce mean-variance inefficient portfolios. By using portable alphas, managers can strike an efficient balance between alpha and beta exposures to create mean variance-efficient portfolios for their clients.
Portfolio managers can take advantage of a new approach to portfolio composition designed to eliminate inefficient constraints by using portable alphas. Portable alphas allow managers to strike an efficient balance between active and passive exposures to create mean-variance efficient portfolios.
### What are Alpha and Beta?
There are two potential sources of return and risk in an actively managed portfolio: alpha and beta. Any investment portfolio can be decomposed into an alpha portfolio and a beta portfolio.
One type of return is passive return, or **beta**, which is the compensation for bearing the systematic risks embedded in each asset class. The beta portfolio is selected by allocating assets based on passive benchmarks. The portfolio of exposures to passive benchmarks is the investor’s beta portfolio, which is sometimes called the “policy portfolio.”
The other type is active return, or **alpha**, which is expected return earned without bearing systematic risk. This source of risk and return is the away-from-benchmark positions taken by the active manager on behalf of the investor. When all the manager’s alpha-seeking positions are combines, the result is the investor’s alpha portfolio.
Therefore, a given portfolio can be viewed as two sub-portfolios:
* An assortment of benchmark portfolios, generally chosen by the investor (sometimes with the assistance of a consultant), called the **beta portfolio**
* An assortment of active portfolios, chosen by the investor’s investment managers, collectively called the **alpha portfolio**
### **How to Build a Better Portfolio Using Alpha and Beta**
Investment managers can create mean-variance efficient portfolios by taking a modular approach to portfolio construction. First, the structure of the beta sub-portfolio is optimized, and then the structure of the alpha sub-portfolio is optimized. Moreover, the alpha sub-portfolio can often be made self-financing which gives it *portability*. Due to the mutual independence of the alpha and beta sub-portfolios, they can be combined to create a mean-variance efficient portfolio.
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Want to learn how to implement portable alphas into your portfolio construction process? The [**Windham Portfolio Advisor**](https://www.windhamlabs.com/products/windham-portfolio-advisor.html) includes technology to introduce portable alphas to portfolios efficiently and cost effectively, one of the features that sets it apart from the other portfolio and risk management products.
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# Understanding Estimation Error
Mark Kritzman, June 20, 2016
When investors build portfolios, they begin with a long history of returns of the assets to be included in the portfolio. They use these historical returns to compute volatilities and correlations, which they typically extrapolate to estimate future volatilities and correlations. Although they might also use these historical returns to guide their estimates of expected returns, it is uncommon to extrapolate historical means. More often, investors rely on fundamental analysis or other information to estimate expected returns.\[1]
## Sources of Error
Investors base their risk estimates on histories that are typically decades-long; however, the investment horizon they are attempting to characterize usually ranges from one to five years. This estimation process, therefore, exposes investors to three sources of error: small-sample error, independent-sample error, and interval error.
### Small Sample Error and Independent-Sample Error
Small-sample error arises because the realization of volatilities and correlations from a small sample of returns will differ from the volatilities and correlations of the large sample from which it is selected. But investors are not concerned with a small sample within a large sample; rather, they care about the volatilities and correlations of a future small sample that is independent of the large historical sample. Therefore, investors also face independent-sample error because the volatilities and correlations of a future sample will differ from those of the prior historical sample, regardless of the size of the future sample. Finally, investors face a much-neglected error called interval error.
### Interval Error
Interval error arises because investors usually base their calculations of volatilities and correlations on monthly returns and then extend these estimates to annual or multi-year horizons. In so doing, they assume that returns are independent and identically distributed, which implies that volatility scales with the square root of time such that the volatility of annual returns, for example, is equal to the volatility of monthly returns multiplied by the square root of 12. This assumption also implies that correlations are invariant to the return interval used to estimate them. Unfortunately, there is no evidence to support this assumption. Therefore, these mapping heuristics give a distorted estimate of longer-horizon volatilities and correlations. Together, these three sources of error cause out-of-sample volatilities and correlations to differ, often markedly, from their historical values.
Investors typically address estimation error in two ways. One approach is to blend individual estimates with their cross-sectional average or some other prior belief. This approach, called Bayesian shrinkage, has the effect of making the estimates of volatilities and correlations more similar to each other, which consequently causes the portfolios along the efficient frontier to be more self-similar. Another common approach for addressing estimation error is resampling. Resampling is a process by which optimal weights are generated many times from a distribution of inputs and then averaged to determine the final portfolio. Resampling also causes portfolios along the efficient frontier to be more alike.
## Adjusting for Stability
We propose a new approach for dealing with estimation error called stability-adjusted optimization. Rather than reduce sensitivity to errors, we argue that investors should measure the relative stability of volatilities and correlations and treat this information as a distinct component of risk.
Here is how we proceed. We begin with a large sample of historical returns for the assets included in our portfolio. We then select all possible overlapping sub samples of a size equivalent to our investment horizon, and we compute covariance matrices\[2] for all of these small samples using return intervals equal to the duration of our investment horizon.
Next we subtract the covariances in all of the sub samples from the covariances of that part of the original large sample that does not include the respective sub samples. We use the heuristics described earlier to convert the covariances of each sub sample’s complementary sample to the same interval of our investment horizon. This leaves us with error matrices for all of the sub samples.
### Reflections
These error matrices reflect small-sample error because the sub samples are smaller than the complementary portion of the large sample. They reflect independent sample error because the small samples are independent of their large-sample complements. And they reflect interval error because the small-sample covariances are estimated from longer interval returns that account for lagged correlations, whereas the complementary-sample covariances are estimated from shorter-interval returns and extended to reflect longer-interval returns by using the heuristics described earlier.
We then select a base case small sample, which could be the median sub sample, for example, and we add the error matrices to the covariance matrix of the base case sub sample. Then, assuming normality, we generate return samples from all of the error-adjusted covariance matrices, and we combine them into a new large sample of returns, which by virtue of this process reflects the relative stability of the asset covariances.
This process for generating a stability-adjusted return distribution yields a distribution that has fatter tails than a normal distribution, because the variances of the error-adjusted small samples will differ from each other. These fatter tails shouldn’t present a problem to mean-variance optimization, though, as long as we can reasonably describe investor preferences with mean and variance. If that is not the case, however, we must resort to a portfolio construction process known as full-scale optimization. This process for generating a stability-adjusted return distribution is depicted in Exhibit 1.
## Full-scale Optimization
Full-scale optimization identifies the optimal portfolio by trial and error. We start by selecting a particular utility function, which do not need to be well-approximated by mean and variance. Then, we choose a set of portfolio weights and apply them every period to the asset returns in the stability-adjusted return sample to compute the utility associated with those weights for every period. We then sum utility across all periods and record this value. Next, we choose a different set of portfolio weights and apply them to the sample returns to compute their total utility across all periods. We proceed in this fashion until we arrive at the portfolio composition that yields the highest utility across all periods. This full-scale approach to optimization may be computationally expensive; nonetheless, it accounts for every feature of the data, even beyond kurtosis and skewness.
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Windham Labs' Windham Portfolio Advisor includes Full-scale Optimization features
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### Application Examples
We apply stability-adjusted optimization in two settings. We use it to allocate a portfolio across four assets classes: U.S. stocks, U.S. Treasuries, U.S. Corporates, and commodities based on monthly returns from February 1973 through December 2015.\[3] We also apply it to replicate the S\&P 500 Index using 20 randomly selected stocks, two from each GICS sector. We use weekly returns from January 2006 through December 2015. In both examples, we assume both power utility and kinked utility, which applies to investors who face thresholds.
Exhibit 2 below reveals that the stability-adjusted portfolio displayed significantly less volatility and downside risk during the global financial crisis than if we were to ignore errors or rely on Bayesian shrinkage.\[4]
The third exhibit’s results are even more striking. It reveals that explicitly accounting for the relative stability of covariances markedly reduces quarterly tracking error and downside tracking error compared to an approach that either ignore errors or one that relies on Bayesian shrinkage.\[5]
## Summary
We described three sources of errors investors face when relying on a long historical sample of returns to characterize the risk of a future smaller sample. We then introduced a process for measuring the relative stability of asset covariances. Next, we showed how to use this information to create a stability-adjusted return sample. Finally, we applied this approach to construct a portfolio of asset classes as well as an index-replicating portfolio. We then presented evidence showing that adjusting for stability yields better behaved portfolios than commonly used alternative approaches. We hope to refine this research and to test stability-adjusted optimization across a wider set of applications.
## Endmatter
1. This essay is based on the following article: Kritzman, M. and D. Turkington. “Stability-Adjusted Portfolios.” *The Journal of Portfolio Management, Special QES Issue 2016*, Vol. 42, No. 5: pp 113-122.
2. Covariances combine estimates of volatility and correlation.
3. We used the following indexes as proxies for the asset classes: S\&P 500 Index, Barclays U.S. Treasury Index, Barclays U.S. Corporate Index, and the S\&P/GSCI Commodities Index. Also we assumed we assumed the expected returns of U.S. stocks, U.S. Treasuries, U.S. corporates, and commodities were equal to 9%, 4%, 5%, and 5%, respectively.
4. This exhibit is taken from: Kritzman, M. and D. Turkington. “Stability-Adjusted Portfolios.” *The Journal of Portfolio Management, Special QES Issue 2016*, Vol. 42, No. 5: pp 113-122. Please refer to this article for greater detail about stability-adjusted optimization.
5. This exhibit is taken from: Kritzman, M. and D. Turkington. “Stability-Adjusted Portfolios.” *The Journal of Portfolio Management, Special QES Issue 2016*, Vol. 42, No. 5: pp 113-122. Please refer to this article for greater detail about stability-adjusted optimization.
#### Disclaimer
> This material is not intended to provide professional or investment advice. You are advised to seek independent professional advice prior to investing in any products or strategies described herein or recommended by Windham Capital Management, LLC. This constitutes neither an offer to buy or sell any securities, nor a solicitation of an offer to buy or sell interests or shares in any fund or strategy. Past performance, including any projection or forecast, are not necessarily indicative of future or likely performance of any investment products. No assurance may be given that the strategies’ investment objectives will be achieved. Investments are subject to investment risks including possible loss of principal amount invested.
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[Get in touch with us](https://forms.monday.com/forms/embed/c14792072797e0cf95b7add5094283e1) to see how Windham Labs can help your practice!
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# A New Index of the Business Cycle
This summary is based on the paper "A New Index of the Business Cycle" by William Kinlaw, Mark Kritzman, and David Turkington, MIT Sloan School Working Paper 5908-20.
We introduce a new index of the business cycle that uses the Mahalanobis distance to measure the statistical similarity of current economic conditions to past episodes of recession and robust growth. Our index has several important features that distinguish it from the Conference Board’s leading, coincident, and lagging indicators. It is efficient because, as a single index, it conveys reliable information about the path of the business cycle. It gives an independent assessment of the state of the economy because it is constructed from variables that are different than those used by the NBER to identify recessions. It is strictly data driven; hence, it is unaffected by human bias or persuasion. It gives an objective assessment of the business cycle because it is expressed in units of statistical likelihood. And it explicitly accounts for the interaction, along with the level, of the economic variables from which it is constructed.
## The Mahalanobis Distance
The Mahalanobis distance was introduced in 1927 and modified in 1936 to analyze resemblances in human skulls among castes in India. It was rediscovered in 1999 to measure turbulence in the financial markets, and it has since been applied to diagnose diseases and to detect anomalies in self-driving vehicles. We apply the Mahalanobis distance to measure the similarity of a set of economic variables to past episodes of recession and robust growth.
## The Mahalanobis Distance and the Business Cycle
In this application, we define the Mahalanobis distance as show in the equation below
$$
d=(x-\mu) \Sigma^{-1} (x-\mu)'
$$
In the equation, $$d$$ equals the Mahalanobis distance, $$x$$ equals the values of a set of economic variables at each point in time, $$\mu$$ equals the average values of those variables during past episodes of recession or robust growth, and $$\Sigma^{-1}$$ equals the inverse of the covariance matrix of those values during periods of recession or robust growth. Whereas Mahalanobis sought to determine if a set of dimensions for a skull was more plausibly associated with one caste versus another, we seek to determine if the values for a set of economic variables are more closely associated with the values that prevailed during past recessions or periods of robust growth. We focus on robust growth rather than growth because it is important that the regimes be symmetrically opposite each other and sufficiently separated from each other.
We construct our index using the following economic variables.
* Industrial Production (one-year percentage change, measured monthly)
* Non-farm Payrolls (one-year percentage change, measured monthly)
* Return of the Stock Market (one-year return, measured monthly)
* Slope of the Yield Curve (10-year rate minus the Federal Funds Rate)
We define periods as robust growth as months in which the year-over-year percentage change in industrial production ranked above the 75th percentile relative to the prior 10 years. We then proceed as follows.
We isolate two sub-samples from the historical observations starting in January 1926: those that qualify as recessions and those that qualify as robust growth. The data prior to 1956 includes revisions that were not available at the time. For each month starting in January 1956, we only use data (including prior revisions) that were available at that point in time.
1. We then calculate the means and covariances for each sub-sample.
2. Next, we calculate the Mahalanobis distance of each month’s observations from each sub-sample.
3. We convert these distances, $$d$$, into likelihoods, $$l$$,using the multivariate normal probability density function (PDF) as shown. In this formula, the covariance matrix $$\Sigma$$equals that of the sub-sample for which we are measuring distance and likelihood.\
\
$$l(d)= \frac{e^{-d/2}}{\sqrt{det(2\pi\Sigma)}}$$
4. We rescale the likelihood of recession by dividing it by the sum of the recession and robust growth likelihoods. We interpret this rescaled likelihood of recession as a probability.
5. We repeat this process for each month of our sample.
Exhibit 1 presents a time series of our index of the business cycle (solid black line), which we refer to as the KKT Index, beginning in January 1956 and ending in November 2019 (the period for which all observations are out of sample). This line measures how much more likely it is that the conditions at any point in time are associated with recession instead of with robust growth.
The dashed line shows the Conference Board’s Index of Coincident Indicators. This time series begins in January 1982, and its values are indicated by the right axis. A value of 0 indicates neutral economic conditions, whereas large negative values coincide with recessions. (We inverted the scale to coincide with the KKT Index.)
Exhibit 2 presents an event study of the KKT index. The shaded bar represents the events which are either recessions or periods of robust growth that occurred since 1956. The width of the bar is not relevant. These events varied by duration. The left side of the bar represents the beginning of the events while the right side represents the end of the events, irrespective of their duration.
The dark line shows the level of the KKT Index leading up to, during, and following recessions. The light line shows level of the index leading up to, during, and following periods of robust growth. Because our index is constructed as the relative likelihood of recessions, we should expect it to be low during periods of robust growth, which it is.
Exhibit 3 compares the level of the KKT Index to realizations of recessions within various time spans. We are interested in analyzing periods when the probability of recession is rising. Therefore, we require that the standardized shift of the index – defined as its current level minus its average over the past year, divided by its standard deviation over the past year – is greater than 1.
> Exhibit 3: KKT Index and Recession Realizations
| Above threshold: | 50% | 60% | 70% | 80% | 90% | Unconditional
Frequency
|
| ---------------- | :-: | :-: | :-: | :-: | :-: | :----------------------------------: |
| This month | 35% | 42% | 52% | 61% | 86% | 13% |
| Next 1m | 40% | 48% | 57% | 65% | 91% | 13% |
| Next 3m | 43% | 50% | 60% | 66% | 91% | 13% |
| Next 6m | 54% | 61% | 70% | 77% | 91% | 17% |
| Next 12m | 68% | 74% | 83% | 86% | 91% | 24% |
| Next 18m | 75% | 78% | 84% | 86% | 91% | 30% |
We highlight the row corresponding to the realization of recessions over the subsequent six months for various levels of the index. We also report the unconditional frequency of recession for the various time spans. Exhibit 6 reveals that when the index exceeded 50%, 54% of the time a recession occurred within the next six months. When it exceeded 60%, a recession occurred 61% of the time within the next six months. When the index exceeded 70% the frequency of recessions was 70%. When it exceeded 80%, recessions occurred 77% of the time. And when it exceeded 90%, recessions followed 91% of the time. The correspondence between the index level and the incidence of recessions is remarkably strong; in fact, the correlation exceeds 99%, and the slope of the relationship equals one. By comparison, the unconditional likelihood of a recession within any six-month period is only 17%.
We next present the same analysis for the yield curve. Specifically, we show the incidence of recessions that occur over varying time spans once the yield curve becomes inverted and its one-year standardized shift is below -1.
> Exhibit 4: Yield Curve and Recession Realizations
| Above Threshold: | 0% | Unconditional
Frequency
|
| ---------------- | :-: | :-------------------------------: |
| This month | 12% | 13% |
| Next 1m | 14% | 13% |
| Next 3m | 18% | 13% |
| Next 6m | 29% | 17% |
| Next 12m | 46% | 24% |
| Next 18m | 73% | 30% |
Exhibit 4 shows the yield curve to be a much less reliable indicator of subsequent recessions than the KKT Index, especially for short horizons. It is only informative for a horizon of 18 months, and even for that horizon, it is less reliable than the KKT Index.
## Summary
We apply the Mahalanobis distance to construct a new index of the business cycle. Specifically, we measure the statistical similarity of economic conditions each month to economic conditions that prevailed during prior periods of recession and robust growth. We then construct the index as the likelihood of recession relative to the likelihood of robust growth.
We argue that our index is more efficient, more objective, and more informative than the Conference Board’s indexes of leading, coincident, and lagging indicators.
## Working Paper
The complete working paper is available on SSRN:
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## Video Presentation
You can also watch the following presentation by Mark Kritzman
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## References
1. Chow, G., E. Jacquier, K. Lowry, and M. Kritzman. 1999. “Optimal Portfolios in Good Times and Bad.” *Financial Analysts Journal*, vol. 55, no. 3 (May/June): 65–73.
2. Mahalanobis, P.C. 1927. “Analysis of Race-Mixture in Bengal.” *Journal of the Asiatic Society of Bengal*, vol. 23: 301–333.
3. Mahalanobis, P. C. 1936. “On the Generalised Distance in Statistics.” *Proceedings of the National Institute of Sciences of India*, vol. 2, no. 1: 49–55.
## Disclaimer
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**Past performance is no guarantee of future results. It is not possible to invest directly in an index.**
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> This material is not intended to provide professional or investment advice, and you are advised to seek independent professional advice prior to investing in any products or strategies described herein or recommended by Windham Capital Management, LLC or its operating technology division Windham Labs. Nothing should be construed as a recommendation to buy, sell or hold any investment, nor a solicitation of an offer to buy or sell interests or shares in any fund or strategy. Past performance, including any projection or forecast, is not necessarily indicative of future or likely performance of any investment products. No assurance is given that the strategies’ investment objectives will be achieved. Investments are subject to investment risks including possible loss of principal amount invested.
# Event Studies
An event study is used to measure the relationship between an event that affects securities and the return of those securities. For example, events such as a regulatory change or market shock may affect many securities simultaneously. On the other hand, events such as a policy change or stock split may only impact certain securities.
Event studies are often used to test the “efficient market hypothesis.” The efficient market hypothesis is a theory based on research put forward by Eugene Fama, and states that asset prices fully reflect all information available. Abnormal returns that persist after an event occurs, or abnormal returns that are associated with an *anticipated* event contradict the efficient market hypothesis. Event studies are also valuable in gauging the significance of an event.
### How to Perform an Event Study
1. **Define the event and identify the time period.**\
The timing of the event is not necessarily the period during which the event occurred, but may be the investment period preceding the announcement of the event
2. **Arrange the security performance data relative to the timing of the event.**\
If information about the event is released completely on a specific day with enough time for traders to react, the day of the announcement is ZERO. Then, measurement periods preceding and following the event are selected. For example, if the 90 trading days preceding the event and the 10 days following the event are designated as the pre= and post-event periods, the pre-event trading days would be labeled *t – 90, t – 89,* and so on, and the post-event trading days would be labeled *t + 1, t +2,* and so on.
3. **Separate the security-specific component of return from the security’s total return during the pre-event measurement period.**
4. **Estimate the standard deviation of the daily security-specific returns during the pre-event measurement period from 90 days before the event announcement through the day before the announcement (*****t – 90 through t – 1*****).**
5. **Isolate the security-specific return during the event and post-event periods.**\
To estimate the security-specific return each day during these periods, subtract from each security’s total return the security’s alpha and beta times the market’s return on that day.
6. **Aggregate the security-specific returns and standard deviations across the sample of securities on the event day and the post-event days.**\
Sum the security-specific returns for each day and divide by the number of securities in the sample.
7. **Test the hypothesis that the security-specific returns on the event day and the post-event days differ significantly from zero.**\
If the event is unanticipated and the *t-*statistic is significant on the day of the event but insignificant on the days following, you can reasonably conclude that the event does affect security returns but does not contradict the efficient market hypothesis. If, on the other hand, the *t-*statistics continue to be significant on the post-event days, you may conclude that the market is inefficient in that it does not quickly absorb new information. Therefore, the event may contradict the efficient market hypothesis.
### Issues in Measuring Events
How to measure an event is not always obvious. For example, suppose that the event is an annual earnings announcement. The announcement that annual earnings are $3.00 a share is meaningless unless it goes against market expectations. Moreover, the market’s expectations are conditioned by earlier information releases pertaining to earnings. Therefore, the first issue in measuring the event is to separate the unanticipated component of the announcement from the anticipated component.
Another issue with measuring events pertains to the influence of confounding factors. Suppose the event is the announcement of a change in dividend policy. For some securities, this announcement may coincide with an announcement about earnings. This coincident information is called a *confounding event*, or an event that might distort the effect of the event on the security’s return.
# Factor Analysis
Financial analysts are concerned with common sources of risk that contribute to changes in security prices, called **factors**. By identifying these factors, analysts may be able to control a portfolio’s risk more efficiently, and perhaps even improve its return.
This post will discuss the first of two common approached used to identify factors. The first, called **factor analysis**, allows analysts to isolate factors by observing common variations in the returns of different securities. These factors are merely statistical constructs that represent some underlying source of risk (which may or may not be observable). The second approach, called **cross-sectional regression analysis**, requires that we define a set of attributes that measure exposure to an underlying factor and determine whether or not differences across security returns correspond to differences in these security attributes.
## Factor Analysis
Let us first begin with an analogy that will highlight the insight behind factor analysis. Suppose we wish to determine whether or not there are common sources of intelligence in students, based on the grades of 100 students in the following nine courses: algebra, biology, calculus, chemistry, composition, French, geometry, literature, and physics.
* First, we compute the correlation between the algebra grades of all 100 students and their grades in each of the other eight courses
* Next, we compute the correlations between the biology grades of all 100 students and their grades in each of the other seven courses
* We continue until we have computed the correlations between the grades of every pair of correlations – 36 in all (shown below).
| | Bio. | Calc. | Chem. | Comp. | Fre. | Geo. | Lit. | Phy. |
| ----------- | :--: | :---: | :---: | :---: | :--: | :--: | :--: | ---- |
| Algebra | 0.41 | .93 | .52 | .31 | .35 | .88 | .29 | .59 |
| Biology | | .39 | .94 | .49 | .44 | .50 | .31 | .90 |
| Calculus | | | .42 | .29 | .33 | .95 | .38 | .60 |
| Chemistry | | | | .37 | .41 | .47 | .40 | .91 |
| Composition | | | | | .87 | .28 | .94 | .35 |
| French | | | | | | .32 | .89 | .46 |
| Geometry | | | | | | | .38 | .55 |
| Literature | | | | | | | | .43 |
That all of these correlations are positive suggests the presence of a pervasive factor (probably related to study habits). In addition to this factor, there appear to be three other factors or commonalities in performance.
First, the variation in algebra grads is highly correlated with the variation on calculus and geometry grades. Moreover, performance in calculus is highly correlated with performance in geometry. The three grades, however, are not nearly as highly correlated with the grades in ANY of the other six courses. Therefore, we conclude that there is a common aptitude that underlies performance in these three courses.
Second, performance in biology is highly correlated with performance in chemistry and physics, and performance in chemistry is highly correlated with performance in physics. Again, performance in these courses does not correspond as closely with performance in any other course. We may therefore again conclude that there is a common source of aptitude associated with biology, chemistry, and physics.
Finally, the grades in composition, French, and literature are all highly correlated with each other, but not with the grades of any other courses. This leads us to deduce the presence of a third factor.
Our next task is to identify these factors, which is where our intuition comes into play. We may reasonable conclude that one of the common sources of scholastic aptitude is skill in mathematics or quantitative methods, because we observe high correlations between the three math courses. Aptitude in science appears to be another common factor, given the high correlations in the three science courses. Finally, verbal aptitude seems to be another common factor, due to the high correlations in French, composition, and literature.
We do not actually observe the underlying factors; we merely observe that a student who performs well in algebra is more likely to perform well in geometry or calculus than in French. From this observation, we infer that there is a particular aptitude that helps to explain performance in algebra, calculus, and geometry—but not in French. **The aptitude is the factor.**
We should note that these results do not imply that performance in a certain course is explained by a single factor. If such were the case, we would only observe correlations of 1 and 0. This point is underscored by the fact that the variation in physics grades (science) is more highly correlated with performance in math courses than it is with French, literature, or composition. This result is intuitively pleasing in that physics depends more on mathematics than French, literature, or composition. We may therefore conclude that performance in physics is primarily explained by aptitude in science, but that it is also somewhat dependent on aptitude in math as well.
### Factors in Stock Returns
Ok, so how do we apply this thought process to the stock market? Let’s assume instead that we wish to determine the factors that underlie performance in the stock market. We begin by calculating the daily returns of a representative sample of stocks during some period. *(In this study, the stocks are analogous to courses, the days in the period are analogous to students, and the returns are analogous to grades!).*
To isolate the factors that underlie stock market performance, we begin by computing the correlations between the daily returns of each stock and the returns on every other stock. Then, we seek out groups consisting of stocks that are highly correlated with each other, but not with stocks outside the group.
For example, we might observe that stock 1’s returns are highly correlated with the stocks of 12, 21, 39, 47, 55, 70, and 92, and that the returns of all the other stocks in this group are all highly correlated with each other. From this observation, we may conclude that the returns of these stocks are explained, at least in part, by a common factor. We proceed to isolate groups of stocks whose returns are highly correlated with each other, until we isolate all the groups that seem to respond to a common source of risk.
Our next task is to identify the underlying source of risk for each group. Suppose that a particular group consists of utility companies, financial companies, and a few other companies that come from miscellaneous industries but that all have especially high debt-to-equity ratios. We might reasonably conclude that interest rate risk is a common source of variation in the returns of this group of stocks. Another group might be dominated by stocks whose earnings depend on the level of energy prices; we may thus deduce that the price of energy is another source of risk. Yet another group might include companies across many different industries that all derive a large fraction of their earnings from foreign operations; we might conclude that exchange risk is another factor.
We must first rely on our intuition to identify the factor that underlies the common variation in returns among the member stocks. Then, we can test our intuition as follows:
1. We define a variable that serves as a proxy for the *unanticipated* change in the factor value
2. We regress the returns of stocks that seem to depend on our hypothesized factor with the unanticipated component of the factor value
It is important that we isolate the unanticipated component of the factor value, because stock prices should not respond to an anticipated change in a factor. It is new information that causes investors to reappraise the prospects of a company.
Suppose we identify inflation as a factor. If the Consumer Price Index is expected to rise 0.5% in a given month, and it rises precisely by that amount, the prices of inflation-sensitive stocks should not change in response. If, however, the CPI rises 1.5%, then the prices of these stocks should change in response. In order to test whether or not a particular time series represents a factor, we must model the unanticipated component of its changes.
A reasonable approach for modeling the unanticipated component of inflation is to regress inflation on its prior values under the assumption that the mark’s outlook is conditioned by past experience. The errors, or residuals, from this regression represent the unanticipated component of inflation. We thus regress these residuals on the returns of the stocks we believe to be dependent on an inflation factor to determine is inflation is indeed a factor.
This approach is heuristic, designed to expose factors by identifying groups of stocks with common price variations. Its intuitive appeal is offset by the fact that it produces factors that explain only part of the variation in returns. Moreover, these factors are not necessarily independent of each other.
With a more advanced mathematical technique (*called maximum likelihood factor analysis*), we can identify several linear combinations of securities, comprised of both long and short positions, that explain virtually all the covariation in the returns of a sample of securities. These linear functions are called **eigenvectors**, and the sensitivity of a particular security to an eigenvector is called an **eigenvalue**.
Instead of groups of highly correlated stocks, this approach yields precise linear combinations of stocks that represent independent sources of common variation in returns. In effect, the eigenvectors are the factors. Not only are the factors derived in this fashion independent of each other, but we can derive as many factors as necessary to explain as much of the covariation in a portfolio as we would like.
In order to label these factors, we proceed as described earlier. We determine whether or not the returns of these linear combinations of stocks correlate with the unanticipated changes in the variables that proxy for the factors. Within this context, we represent a security’s return as follows:
$$
R\_i=\alpha\_i+\beta\_{i1}F\_1+\beta\_{i2}F\_2+...+\beta\_{in}F\_n+\xi\_i
$$
where $$R\_i$$ is the return of security $$i$$ , $$\alpha\_i$$ is a constant, $$\beta\_{in}$$ is the sensitivity of security $$i$$ to the $$n^{th}$$ factor, $$F$$ , and $$\xi\_i$$ is the unexplained component of security $$i$$ 's return.
### Issues of Interpretation
Factors derived through factor analysis, whether we employ the heuristic approach or the more formal approach, are not always amenable to interpretation. It may be that a particular factor cannot be proxied by a measurable economic or financial variable. Instead, the factor may reflect a combination of several influences, some perhaps offsetting, that came together in a particular way unique to the selected measurement period and the chosen sample of securities. In fact, factors may not be definable.
We thus face the following trade-off with factor analysis. Although we can account for nearly all of a sample’s common variation in return with independent factors, we may not be able to assign meaning to these factors, or even know if they represent the same sources of risk from period to period or sample to sample. Next, we’ll consider an alternative procedure called cross-sectional regression analysis.
## Cross-Sectional Regression Analysis
As we described earlier, factor analysis reveals covariation in returns, and challenges us to identify the sources of covariation. Cross-sectional regression analysis, on the other hand, requires us to specify the sources of return and *challenges us to affirm that these sources correspond to differences in return.*
We proceed as follows. Based on our intuition and prior research, we hypothesize attributes that we believe correspond to differences in stock returns. For example, we might believe that highly leveraged companies perform differently from companies with low debt, or that performance varies according to industry affiliation. In either case, we are defining an *attribute*—not a factor. The factor that causes low-debt companies to perform differently from high-debt companies most likely has something to do with interest rates. Industry affiliation, of course, measures sensitivity to factors that affect industry performance (such as military spending or competition).
Once we specify a set of attributes that we feel measure sensitivity to the common sources of risk, we perform the following regression. We regress the returns across a large sample of stocks during a given period—say a month—on the attribute values of the stocks as of the beginning of that month. Then, we repeat this regression over many different periods. If the coefficients of the attribute values are not zero and are significant in a sufficiently high number of the regressions, we conclude that differences in return across the stocks relate to the differences in their attribute values.
According to this approach, a security’s return in a particular period equals:
$$
R\_i=\alpha+\theta\_{i1}\gamma\_1+\theta\_{i2}\gamma\_2+...+\theta\_{in}\gamma\_n+\xi\_i
$$
where $$R\_i$$ is the return of security $$i$$ , $$\alpha\_i$$ is a constant, $$\gamma\_n$$ is the marginal return to attribute $$n$$ , $$\theta\_{in}$$ is the attribute $$n$$of security $$i$$, and $$\xi\_i$$ is the unexplained component of security $$i$$ 's return.
It is not necessary for the coefficient $$\gamma$$ in the above formula to be significantly positive or negative on average over all the regressions. The attribute $$\theta$$ is a measure of sensitivity to some underlying factor. Suppose the attribute it affiliation with industries that benefit from military spending. If there is an unexpected increase in military spending in a particular period, the coefficient of this attribute will be positive. If military spending declines, the coefficient will be negative. The average value for the coefficient over many regressions may be zero, but the attribute would still be important if the coefficient were not zero in a large number of the regressions.
We can measure the extent to which a coefficient is significant in a particular regression by its t-statistic. The t-statistic equals the value of the coefficient divided by its standard error. A t-statistic of 1.96 implies that the likelihood of observing a significant coefficient by chance is only 5%. In order to be confident that a particular attribute helps to explain differences across security returns, we should observe a t-statistic for its coefficient of 1.96 or greater in more than 5% of regressions. Otherwise, it is possible that the attribute occasionally appears significant merely by chance.
### Which is better?
There are pros and cons to both factor methods. Through factor analysis, we can isolate independent sources of common variation in returns that explain nearly all of a portfolio’s risk. It is not always possible, however, to attach meaning to these sources of risk. They may represent accidental and temporary confluences of myriad factors. Because we cannot precisely define these factors, it is difficult to know whether they are stable or simply an artifact of the chosen measurement period or sample.
As an alternative to factor analysis, we can define a set of security attributes we know are observable and readily measurable and, through cross-sectional regression analysis, test them to determine if they help explain differences in returns across securities. With this approach we know the identity of the attributes, but we are limited in the amount of return variation we are able to explain. Moreover, because the attributes are typically codependent, it is difficult to understand the true relationship between each attribute and the return. Which approach is more appropriate depends on the importance we attach to the identity of the factors versus the amount of return variation we hope to explain with independent factors.
## Why Bother With Factors?
At this point, you may be questioning why we bother to search for factors or attributes in the first place. Why not address risk by considering the entire covariance matrix instead?
There are two reasons why we might prefer to address risk through a limited number of factors. A security’s sensitivity to a common source of risk may be more stable than its sensitivity to the returns of all the other securities in the portfolio. If this is true, then we can control a portfolio’s risk more reliably by managing its exposure to these common sources.
The second reason has to do with parsimony. If we can limit the number of sources of risk, we might find that it is easier to control risk and to improve return simply because we are faced with fewer parameters to estimate.
# Financial Turbulence Risk Management
The Mahalanobis Distance in Risk Management
The majority of investors look to their domestic equity markets as the main engine of growth for their portfolios, and then search for other assets to diversify this exposure. Typically, investors consider only average correlations when measuring an assets diversification benefits, though average correlations tend to be misleading. For instance, when both U.S. and non-U.S. equities produce returns greater than one standard deviation above their means, their correlation is much lower than when both markets produce returns more than one standard deviation below their means. The difference between these correlations can explain why so many investors who were confident in the diversification of their portfolios suffered immense losses during the 2007 financial crises. This kind of surprise can be avoided by using conditional measures that take into account the behavior of assets during turbulent sub periods, rather than relying on average measures of risk.
Mahalanobis (1927, 1936) introduced a methodology to analyze human skulls, and that methodology has since been used to measure financial turbulence. This methodology can also be used to stress-test portfolios, construct turbulence-resistant portfolios, and to scale exposure to risk to improve performance.
## Measuring Financial Turbulence
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**Financial turbulence** - a condition in which asset prices, given their historical patterns of behavior, behave in an uncharacteristic fashion.
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The method we use to measure turbulence, as mentioned earlier, was originally used to measure the human skull to determine distances and resemblances between various castes and tribes in India. A nearly identical formula was then derived to detect turbulence in financial markets by substituting asset returns for skull characteristics. By doing so, you can determine the statistical rarities of a cross section of returns on the basis of their historical multivariate distributions. The statistical measure of financial turbulence, or “turbulence index” is formally defined as
$$
d\_t=(y\_t-\mu) \Sigma^{-1} (y\_t-\mu)'
$$
$$
d\_t = \text{turbulence for a particular time period }t\\
y\_t = \text{vector of asset returns for period }t\\
\mu = \text{sample average of historical asset returns}\\
\Sigma = \text{sample covariance matrix of asset returns}
$$
Turbulence can be calculated for any group of *n* return series a user may choose. **Figure 1** shows this statistical measure of turbulence for a simple example with two return series—stocks and bonds. Each point represents the returns of stocks and bonds for a particular period, and the center of the ellipse represents the average of the joint returns of stocks and bonds. The ellipse itself represents a tolerance boundary that encloses a certain percentage of the bivariate Gaussian distribution of stock and bond returns. All points on the ellipse have equal Mahalanobis distances from the center.
This boundary also represents the threshold that separates “turbulent” from “quiet” observations. Points inside the ellipse represent return combinations associated within quiet periods, because the observations are not particularly unusual.
There are two particular implied advantages to measuring turbulence this way (over the commonly used indicator of financial stress—volatility):
1. Using this statistical measure, turbulence can be estimated for any set of assets, rather than only for assets with liquid option markets
2. This measure captures interactions among combinations of assets, in addition to the magnitude of the assets’ returns
It may be tempting to assume that the volatility of an index comprising the assets used to measure turbulence captures the same information as this measure, because such a volatility estimate incorporates both the volatility of the individual assets and their correlations with each other. However, as you can see in **Figure 2**, summarizing the data in an index would sacrifice the higher-dimensional information captured in the turbulence index.
The loosely clustered circles in the scatter plot are the returns of two assets with relatively high volatility and a negative correlation. The tightly clustered squares are the returns of two assets with relatively low volatilities and a positive correlation. An index comprising the circle assets has the same volatility as an index comprising the square assets, yet the turbulence estimates of each index’s assets are very different.
Depending on the data or particular application, the distinction between returns belonging to distinct turbulent and non-turbulent regimes may seem arbitrary. We can just as well characterize returns along a continuum ranging from calm to turbulent. If we follow this approach, we find that our mathematical measure of turbulence coincides incredibly closely with well-known turbulent financial events.
**Figure 3** shows a turbulence index for which we used monthly returns of six asset-class indices: U.S. stocks, non-U.S. stocks, U.S. bonds, non-U.S. bonds, commodities, and U.S. real estate. Spikes in this index can clearly be seen to coincide with financial turbulent periods. It is also clear to see that the financial crisis of 2007-2008 is by far the most turbulent episode of recent history.
## Empirical Features of Turbulence
Two empirical features of turbulence are especially interesting. First, returns to risk are substantially lower during turbulent periods than during quiet periods, no matter the source of turbulence. Consider the recent financial crisis, which began with a downturn in housing prices and led to a sharp devaluation of mortgage derivatives. What surprised many investors was that the crisis in the mortgage derivatives market coincided with substantial losses in carry strategies. The carry strategy calls for long positions in currency forward contracts that sell at a discount, combined with short positions in currency forward contracts that sell at a premium. Why should a mortgage crisis lead to losses in a currency strategy? As mortgage derivatives fell in value, many investors (especially hedge funds) were required to raise capital… so they turned to the most liquid components of their portfolios: currency positions. **Figure 4** provides evidence that returns to risk are much lower during episodes of financial turbulence.
These differences in return suggest that predicting turbulent periods would be immensely beneficial, which leads to the second empirical feature of turbulent: Financial turbulence is highly persistent. Financial turbulence is a lot like weather turbulence. When you’re on a plane, air turbulence arrives unexpectedly, but once it begins, you know that it will take time to pass through the weather system or for the pilot to find a smoother altitude. Although we may not be able to anticipate the onset of financial turbulence, we know that once it begins it will continue for a period of weeks as markets react. Evidence of turbulence persistence can be seen in Table 1.
## Applications
The statistical measure of financial turbulence has many beneficial applications. Analysts can use it to stress-test portfolios (more reliably than when using conventional methods), structure portfolios that are relatively resilient to turbulent episodes, and scale a strategy’s exposure to risk.
### Stress-testing Portfolios
Typically, a portfolio’s exposure to loss is measured with Value at Risk (VaR), which provides the largest loss a portfolio may experience at a certain level of confidence. The conventional approach for measuring VaR uses the full-sample covariance to compute the portfolio’s standard deviation and considers the probability distribution only at the end of the investment horizon. We can measure exposure to loss more reliably by estimating covariances from the turbulent sub-periods, when losses are more likely to occur, and by accounting for interim losses as well as losses that occur only at the conclusion of the investment horizon.
**Table 2** shows three portfolios (conservative, moderate, and aggressive) with assumptions for their expected returns and two estimates of standard deviation.
**Table 3** shows the VaR, given a 1 percent confidence level for each portfolio as of December 2006.
If we consider the 2007-2008 financial crisis as a once-in-a-century event, Table 3 shows that the conventional approach to measuring exposure to loss badly underestimated the riskiness of these portfolios. The turbulence-based approach, however, anticipated the exposure to loss much more accurately.
### Building Turbulence-Resistant Portfolios
We have demonstrated how analysts can construct portfolios that are conditioned to better withstand turbulent events as well as perform relatively well in various market conditions. Analysts can also modify two methods of optimization – mean-variance optimization and full-scale optimization – to derive turbulence-resistant portfolios.
We modified **mean-variance optimization** by blending the differences between the realized turbulent returns and full-sample returns with equilibrium returns to estimate expected returns. Additionally, we blended the turbulent subsample covariances with the full-sample covariances in proportion to their sample sizes. We applied a modified version of **full-scale optimization** by increasing the representation of the turbulent subsample returns beyond their empirical frequency.
To evaluate the two methods, we performed 1,000 random trials of training and out-of-sample testing. For each trial, we drew a random half from the historical sample to use as training data—the other half was used as testing data. From the training data, we identified a turbulent subsample by calculating the turbulence index, and subsequently selecting the periods with the highest quartile (the highest turbulence index values). Using the full training sample, we build an unconditioned optimal portfolio that did not account for turbulence. Using the turbulent sample, combined with some information from the full training sample, we built a conditioned optimal portfolio that was expected to be more resistant to turbulence than an unconditioned portfolio. We then used the testing data to test both the unconditioned and the conditioned portfolios, and performed two types of testing: tone on the full testing sample and the other on a turbulent subsample within the testing sample.
**Figure 5** compares the performance of these portfolios, and it is clear that the conditioned portfolios substantially outperformed the unconditioned portfolios in the out-of-sample turbulent periods, and only marginally under-performed the unconditioned portfolios, on average, in all market conditions.
The conditioned portfolios also outperformed the unconditioned portfolios much more frequently in the out-of-sample turbulent periods and outperformed almost as often in all market conditions, on average, as shown in **Table 4.**
This evidence strongly suggests that by understanding the conditional behavior of assets, portfolio managers can construct turbulence-resistant portfolios without substantially compromising average performance.
### **Scaling Exposure to Risk**
The differential performance of risky strategies during turbulent and non-turbulent periods, together with the persistence of turbulence, raising the tantalizing prospect that portfolio managers may be able to improve performance by conditioning exposure to risk on the degree of turbulence. A simple scaling rule applied to the carry strategy *does* significantly improve performance.
We measured the 30-day moving average of turbulence each day from the returns of G-10 currencies and recorded whether the level of turbulence that day fell into the first, second, third, fourth, or fifth quintile of turbulence on the basis of a trailing three-year window. We then weighted exposure to the carry strategy in inverse proportion to turbulence as shown in **Table 5.** We assumed a one-day lag for implementation.
We applied the same scaling rule in using other signals of market stress, swap spreads, and yield spreads. **Table 6** shows the performance of the unfiltered carry strategy as well as its filtered performance. The evidence shows that reducing exposure to the carry strategy in proportion substantially improves performance, and by a wider margin, than using any other signal of market stress.
# Mismeasurement of Risk
Investors tend to consider risk as an outcome—how much could be lost at the end of an investment period? Risk is typically measured as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon. This perspective considers only the result at the end of the investment horizon, ignoring what may happen within the portfolio along the way. We argue that exposure to loss *throughout* an investment horizon is important to investors, and propose two new ways of measuring risk: within-horizon probability of loss and continuous value at risk (VaR). Using these risk measures, we reveal that exposure to loss is often substantially greater than investors assume.
## Where is the danger in measuring risk at the end of an investment period?
Financial analysts worry that means and variances used in portfolio construction techniques are estimated with error. These errors bias the resultant portfolio towards asset for which the mean is over-estimated and variance is underestimated, which may lead analysts to invest in the wrong portfolio. Additionally, financial analysts worry that higher moments, such as skewness and kurtosis, are misestimated. In that case, extreme returns occur more frequently in reality than is implied by a lognormal distribution. These estimation errors often cause investors to underestimate the probability of loss, and to overestimate the probability of gain.
Rather than focusing simply on addressing these issues (though we do address them), we focus on what we believe to be a more fundamental cause of financial failure: Investors’ wealth is affected by risk throughout a period in which it is invested, but risk is generally measured only for the termination of the period.
**Figure 1** demonstrates the distinction between risk based on ending outcomes and risk based on outcomes that may occur along the way. Each line represents the path of a hypothetical investment of 100 through four periods. The horizontal line at 90% represents a loss threshold( which equals 10% here). It is clear here than only one of the five paths beaches the loss threshold at the end of the horizon, which may cause some to believe that the likelihood of a 10% loss is 20%. However, four of the five paths breach that loss threshold throughout the investment horizon. So, if we care about the investment’s performance along the way, we will conclude that likelihood of a 10% loss is not 20%… but a whopping 80%.
## Why should we care about interim risk?
Investors care about exposure to loss throughout the investment horizon because there are often thresholds that cannot be breached if the investment is to survive to the end of the horizon. If survival is not the main concern, investors may be motivated to pay attention to within-horizon risk because they could be penalized for breaching a barrier. Consider the following:
1. **Asset management**\
A client has a portfolio with a provision that it should not depreciate more than 10% over a 5-year investment horizon. Should the asset manager assume that the client will only review performance at the end of the investment period? Not likely. The client will review performance throughout the investment horizon… and terminate the manager if the portfolio dips below 90% of its value at inception. To limit the likelihood of termination, the manager should consider within-horizon risk.
2. **Hedge-fund solvency** \
A hedge-fund manager who believes that the likelihood of significant loss at the end of the investment horizon is slim, leverages the portfolio to increase expected return. However, a significant decline from the value of the underlying assets from inception to any point throughout the investment horizon is much more likely than the likelihood implied by the ending distribution of a hedge-fund’s assets. Additionally, significant interim loss could trigger withdrawals that might impair the hedge-fund’s solvency.
3. **Loan agreement**\
A borrower is required to maintain a particular level of reserves as a condition of a loan. If the reserves fall below the required balance, the loan is called.
4. **Securities lending** \
Many institutional investors lend their securities to others who engage in short selling. These investors are required to deposit collateral with the custodian of the securities. The required collateral is typically adjusted on a daily basis, to offset changes in the values of the securities. Suppose the investor wishes to estimate the amount of additional collateral that might be required at a given probability for the duration of the loan. This value depends on the distribution of the securities’ values throughout the term of the loan.
5. **Regulatory requirements**\
A bank is required to maintain a capital account equal to a certain fraction of its loan portfolio. A breach in this requirement will result in a fine. The probability that the bank will need to replenish the capital account to avoid breaching depends on the distribution of the ratio of the capital account to the loan portfolio throughout the planning horizon, not at the end of the horizon or a finite period within the horizon.
These examples are only a few of the many circumstances in which investors should pay attention to probability distributions that span the duration of their investment horizons.
## How do you measure within-horizon exposure to loss?
We describe how to measure within-horizon risk measures in the following document
{% content-ref url="/pages/-MF7DvpBp8PhRG6Y8kZR" %}
[Rethinking Exposure to Loss](/risk-management/rethinking-exposure-to-loss)
{% endcontent-ref %}
The following video presentation visually describes the concept of measuring risk continuously
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Video Presentation: Within-horizon Risk Management
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## How can I apply within-horizon probability of loss and continuous value at risk?
### 1. Currency Hedging
Suppose we allocate a portfolio equally to Japanese stocks and bonds, represented by the MSCI Japan Index and the Solomon Brothers Japanese Government Bond Index. Exhibit 1 shows, based on monthly returns from January 1995 to December 1999, the standard deviations and correlations of these indexes together with the risk parameters of the Japanese yen from a U.S. dollar perspective.
Let us assume further that the underlying portfolio has an expected return of 7.50 percent, hedging costs equal 0.10 percent, and our risk aversion equals 1.00. Based on these assumptions, the optimal exposure to a Japanese yen forward contract is -87.72 percent. The expected return and risk of the unhedged and hedged portfolios are shown in Exhibit 2.
Now, let us estimate the probability of loss for the unhedged and hedged portfolios. Exhibit 3 shows the likelihood of a 10 percent or greater loss over a 10-year horizon at the end of the horizon and at any point from inception throughout the horizon for an unhedged and optimally hedged portfolio of Japanese stocks and bonds.
If we were concerned only with the portfolio’s performance at the end of the investment horizon, we might not be impressed by the advantage offered by hedging. But, if we instead focus on what might happen along the way to the end of the horizon, the advantage of hedging is much more apparent.
Even with the foreknowledge that we are more likely than not at some point to experience a 10 percent cumulative loss, we may consider such a loss tolerable. But what about a loss of 25 percent or greater? Again, calculating the probabilities indicates that, although the impact of hedging on end-of-period outcomes is unremarkable, it vastly reduces the probability of a 25 percent or greater loss during the investment horizon. Although many investment programs might be resilient to a 10 percent depreciation, they are less likely to experience a decline of 25 percent or more without consequences.
Now, let us compare VaR measured conventionally with continuous VaR for the hedged and unhedged portfolios. Table 4 reveals that the improvement from hedging is substantial whether VaR is measured conventionally or continuously. For example, measured conventionally, hedging improves VaR from a 5 percent chance of no worse than a 14.68 percent loss, to a 5 percent chance of no worse than a 26.52 percent *gain*. More important, however, is the substantial difference between VaR measured conventionally and VaR measured continuously (within-horizon). Continuous VaR is more than twice as high as conventional VaR for the unhedged portfolio, and when the portfolio is hedged, continuous VaR shows a substantial loss compared with a substantial gain when it is measured conventionally.
### 2. Leveraged Hedge Fund
Now consider the implications of these risk measures on a hedge fund’s exposure to loss. Supposed we are interested in a hedge fund that uses an overlay strategy, which has an expected incremental standard deviation of 5 percent. This hedge fund also leverages the overlay strategy. Exhibit 5 shows the expected returns and risks of the hedge fund and its components for varying degrees of leverage.
The data in Table 5 assume that the underlying asset is a government note with a maturity equal to the specified three-year investment horizon and that its returns are uncorrelated with the overlay returns. Managers sometimes have a false sense of security because they view risk as an annualized volatility, which diminishes with the duration of the investment horizon, but as we have noted, the fund’s assets may depreciate significantly during the investment horizon. Figure 2 compares the likelihood of a 10 percent loss at the end of the three-year horizon with its likelihood at some point within the three-year horizon for various leverage factors (e.g., 2 to 1). Figure 2 reveals that the chance of a 10 percent loss at the end of the horizon is low, but there is a much higher probability that the fund will experience such a loss at some point along the way, which could trigger withdrawals and threaten the fund’s solvency.
The same issue applies if exposure to loss is perceived as VaR. Figure 3 shows the hedge fund’s VaR for various leverage factors measured conventionally and continuously. Whereas conventional VaR for leverage factors less than 6 to 1 is negative (a gain) and still very low for leverage factors up to 10 to 1, continuous VaR rangers from approximately 10 percent of the portfolio’s value to approximately 40 percent of its value.
## Conclusion
Investors measure risk incorrectly if they focus exclusively on the distribution of outcomes at the end of their investment horizons. This approach to risk measurement ignored intolerable losses that might occur throughout an investment period, either as the result of the accumulation of many small losses or from a significant loss that later (possibly, too late) recovers.
To address this shortcoming, we have introduced two new approaches to measuring risk — within-horizon probability of loss ad continuous VaR. Our applications of these measures in reasonable scenarios illustrates vividly that investors are exposed to far greater risk throughout their expected investment periods than end-of-horizon risk measures indicate.
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We include these concepts in all [our technology solutions](https://www.windhamlabs.com/#features-section).
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# Rethinking Exposure to Loss
Mark Kritzman, June 29, 2016
Investors typically measure risk as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon, ignoring what might happen along the way. Moreover, they base these risk estimates on return histories that fail to distinguish between calm environments, when losses are unlikely, and turbulent environments, when losses occur more commonly.
We propose modifying exposure to loss to account for within-horizon losses as well as the regime-dependent nature of large drawdowns. Because value at risk and probability of loss are two sides of the same coin, we focus our analysis on value at risk.
## Conventional Value at Risk
Simply stated, Value at Risk (VaR) is equal to a portfolio’s initial wealth multiplied by a quantity equal to expected return over a stated horizon minus the portfolio’s volatility multiplied by the standard normal variable\[2] associated with a chosen probability. Unfortunately, this simple description ignores an important complexity. Asset returns are not normally distributed. Because compounding causes positive cumulative returns to drift further above the mean than the distance negative cumulative returns drift below the mean, returns tend to be log-normally distributed.\[3] This means that logarithmic returns, also called continuous returns, are normally distributed. Therefore, we must estimate value at risk in continuous units and then convert these values back to discrete units, as shown below.
$$
\text{VaR}=-(e^{\mu T-Z\sigma\sqrt T}-1)\times W
$$
VaR refers to value at risk, $$e$$ is the base of the natural logarithm, $$\mu$$ equals the annualized expected return in continuous units, $$T$$ equals the number of years in the investor’s horizon, $$Z$$ equals the standard normal variable, $$\sigma$$ equals the annualized standard deviation of continuous returns, and $$W$$ equals initial wealth.
Suppose, for example that a $10 million portfolio has an 8.5% annualized continuous expected return and an annualized standard deviation of continuous returns equal to 10%. Suppose also that the investment horizon equals five years and we are interested in the amount we could expect to lose at the end of five years given a 1% probability. The first percentile return, that is, the return for which there is only a 1 percent chance of breaching, is 2.33 standard deviations below the mean of a normal distribution with a mean of 0 and a standard deviation of 1. If we substitute these values into the value at risk equation, we discover that value at risk equals $907,971. In other words, there is a 1 percent chance that this portfolio could lose as much as 9.08% of our portfolio’s initial value at the end of five years.
## Regimes
This estimate of potential loss assumes that returns come from a single distribution. It is likely the case that there are distinct risk regimes, each of which may be normally distributed but with a unique risk profile. For example, we might assume that returns fit into two regimes, a turbulent regime characterized by above-average volatility and unstable correlations, and a calm regime characterized by below-average volatility and stable correlations.
We can think of a turbulent regime as a period in which the returns across a set of assets behave in an uncharacteristic fashion. One or more assets’ returns, for example, may be unusually high or low, or two assets that are highly positively correlated may move in the opposite direction.
There is persuasive evidence showing that returns to risk are substantially lower when markets are turbulent than when they are calm. This is to be expected, because when markets are turbulent investors become fearful and retreat to safe assets, thus driving down the prices of risky assets. This phenomenon is documented below.\[4]
| | 10% Most Turbulent Days | Other 90% |
| ----------------- | :---------------------: | :-------: |
| Global Equities | -12% | 7% |
| Small Cap Premium | -25% | 4% |
| Hedge Funds | -10% | 9% |
> Source: Kritzman and Li \[2010]
This description of turbulence is captured by a statistic known as the Mahalanobis distance. It is used to determine the contrast in different sets of data. In the case of returns, it captures differences in magnitude and differences in interactions, which can be thought of respectively as volatility and correlation surprise. We compute the Mahalanobis distance, MD, as follows.
$$
\text{MD}=(x-\mu)\Sigma^{-1}(x-\mu)'
$$
## Continuous Value at Risk
Investors typically measure value at risk at the end of their investment horizon, as described by Equation (1). This view of risk ignores what might happen along the way. We argue that investors should perceive risk differently. They should care about exposure to loss throughout their investment horizon and not just at its conclusion.
To account for losses that might occur prior to the conclusion of the investment horizon, we use a statistic called first passage time probability.\[5] This statistic measures the probability of a first occurrence of an event within a finite horizon. It is equal to
$$
Pr\_w=N \left\[ \frac{ ln(1+L) - \mu T } { \sigma \sqrt T } \right] + N \left\[ \frac{ ln(1+L) - \mu T } { \sigma \sqrt T } \right] (1+L)^{2\mu / \sigma^2}
$$
$$Pr\_w$$ equals the probability of a within-horizon loss, $$N$$equals the cumulative normal distribution function, $$ln$$equals the natural logarithm, $$L$$equals the cumulative percentage loss in discrete units, $$\mu$$equals the annualized expected return in continuous units, $$T$$equals the number of years in the investment horizon, and $$\sigma$$ equals the annualized standard deviation of continuous returns. This equation describes the probability that the portfolio will depreciate to a particular value over some horizon if it is monitored continuously.
The first part of this equation, up to the second plus sign, gives the end-of-period probability of loss. It is augmented by another probability multiplied by a constant, and there are no circumstances in which this constant equals zero or is negative. Therefore, the probability of loss throughout an investment horizon must always exceed the probability of loss at the end of the horizon. Moreover, within-horizon probability of loss rises as the investment horizon expands in contrast to end-of-horizon probability of loss, which diminishes with time.
We use the same equation to estimate continuous value at risk. Whereas value at risk measured conventionally gives the worst outcome at a chosen probability at the end of an investment horizon, continuous value at risk gives the worst outcome at a chosen probability from inception to any time during an investment horizon. It is not possible to solve for continuous value at risk analytically. We must resort to numerical methods. We set $$Pr\_w$$ equal to the chosen confidence level and solve iteratively for $$L$$. Continuous value at risk equals $$L$$ multiplied by initial wealth.
## Conventional versus Regime-dependent Continuous Value at Risk
Earlier we assumed that our portfolio had an annualized continuous expected return equal to 8.5% and an annualized standard deviation of continuous returns equal to 10%. Given a confidence level of 1% and based on the entire distribution of returns combining both turbulent and calm regimes, we estimated that this portfolio could lose as much as 9.08% of its initial value at the end of a five-year investment horizon.
Suppose instead that during turbulent regimes, the annualized standard deviation of continuous returns is 12% rather than 10%. If we substitute 12% into Equation (3) for standard deviation, retain our other assumptions, and iteratively solve for the cumulative percentage loss, such that the probability of a such a loss equals 1%, we discover that value at risk as a percentage of the portfolio’s initial value equals 30.34% rather than 9.08%. The table below shows how value at risk varies depending on whether we use volatility estimated over all regimes or the volatility that prevailed during the turbulent sub samples, as well as whether we measure value at risk at the end of the investment horizon or continuously throughout the horizon.
| | Full Sample | Turbulent Sample |
| -------------- | :---------: | :--------------: |
| End-of-Horizon | -9.08% | 18.06% |
| Within-Horizon | -23.04% | 30.34% |
If there are indeed distinct volatility regimes, end-of-horizon value at risk dramatically understates a portfolio’s exposure to loss within an investment horizon. It is more than three times as great (30.34 versus 9.08%) when we focus on the periods throughout history when losses typically occurred and take into account within-horizon drawdowns.
## Video Presentation
The following video presentation describes the risk regime methodology within our [software](https://www.windhamlabs.com/products/windham-portfolio-advisor.html) and advisory services.
{% embed url="" %}
## Endmatter
1. Kritzman, M. and Y. Li. 2010. “Skulls, Financial Turbulence and Risk Management.” *Financial Analysts Journal*, vol. 66, no. 5 (September/October).
2. A standard normal variable is a normally distributed random variable with expected value of zero and a variance of one.
3. For example, a positive 10% return will accumulate to 20% over two periods, whereas a negative 10% return will fall to 19% percent over two periods.
4. These returns are annualized daily returns for the period January 1993 through December 2008. For more information about this study, see Kritzman and Li \[2010]
5. The first passage of time probability is described in Karlin, S. and H. Taylor, *A First Course in Stochastic Processes*, 2nd edition, Academic Press, 1975.
#### Disclaimer
> This material is not intended to provide professional or investment advice and you are advised to seek independent professional advice prior to investing in any products or strategies described herein or recommended by Windham Capital Management, LLC. In addition, this constitutes neither an offer to buy or sell any securities, nor a solicitation of an offer to buy or sell interests or shares in any fund or strategy. Past performance, including any projection or forecast, are not necessarily indicative of future or likely performance of any investment products. No assurance may be given that the strategies’ investment objectives will be achieved. Investments are subject to investment risks including possible loss of principal amount invested.
{% hint style="info" %}
[Get in touch with us](https://www.windhamlabs.com/contact-us.html) to see how Windham Labs can help your practice!
{% endhint %}
# Risk Budgets
There are two common definitions of risk budgets that seem to prevail in the industry.
1. A plan for converting a portfolio’s monetary allocations to various categories into Value at Risk (VaR) assignments
2. The sensitivities of a portfolio’s VaR to a small change in the portfolio’s exposure to each component.
## Efficient Portfolio Allocations
The first perception of a risk budget is appropriate as long as it follows from mean-variance optimization. It would be inefficient to plan the value at risk independently of mean-variance optimization, assuming lognormally distributed returns. Any portfolio that is efficient with respect to VaR must lie on the mean-variance efficient frontier.
The intuition here is that, for any given expected return, a portfolio located on the efficient frontier has the lowest standard deviation. Thus, for any given expected return, the portfolio with the lowest VaR must also lie on the efficient frontier. It follows, therefore, that any risk budget that isn’t mean-variance efficient has a higher portfolio VaR for a given confidence level than a corresponding portfolio that is mean-variance efficient.
We should not regard risk budgeting as a process for determining efficient portfolio allocations. We should instead regard it as a means for converting efficient portfolio allocations into VaR assignments. Portfolio choice based on minimizing VaR implies an improbable attitude towards risk, which often leads to overlooking other factors. Investors should consider all portfolios along the efficient frontier, weigh all factors and all possible outcomes—not just a single threshold.
We should consider risk budgets an *extension* of mean-variance optimization that enables us to decouple allocations from fixed monetary values. Suppose the optimal allocation of a $100 million fund calls for the percentage allocations shown in the table below.
| Asset | Percentage Allocation |
| :---: | :-------------------: |
| 1 | 20.70% |
| 2 | 30.73% |
| 3 | 48.57% |
## An Alternative Approach
The traditional approach for implementing these allocations would be to invest $20.70 million in asset 1, $30.73 million in asset 2, and $48.57 million in asset 3. The risk budget alternative would instead view these allocations as Value at Risk assignments of $4.9567 million for asset 1, $4.5429 million for asset 2, and $4.6664 million for asset 3—the respective amounts that each assignment could lose with a 5% probability over a one-year horizon.
Note that the sum of the individual VaRs is nearly twice as large as the portfolio VaR. This is because the assets are less that perfectly correlated with each other, and therefore introduce diversification to the portfolio.
This interpretation of a portfolio’s allocation offers the flexibility to allocate and leverage a smaller monetary amount to each asset. This leveraged investment would contribute the same marginal utility to the portfolio, which is tantamount to preserving the portfolio’s optimality—if two conditions prevail:
* **The leveraged investment preserves the expected return, volatility, and correlation with the balance of the portfolio as assumed by the original percentage allocation.**
* **The balance of the portfolio preserves the expected return and risk attributes assumed in the original optimization.**
The economic equivalence of a VaR assignment and a monetary allocation prevails for any confidence level used to measure VaR, if returns are lognormally distributed. Thus a risk budget, in effect, simply maps a portfolio’s percentage allocations onto VaR assignments. This offers the benefit of freeing portfolio allocations from monetary constraints. However, a risk budget is efficient *only* if determined by mean-variance optimization.
### Sensitivities to Changes in Exposures
The second definition recognizes that the VaR of the individual categories won’t sum up to the portfolio’s total VaR unless all categories are perfectly positively correlated. Thus, the independent VaRs could mislead an investor. Some investors thus define a risk budget as *the sensitivities of a portfolio’s VaR to a small change in the portfolio’s exposure to each component.*
This definition suffers from a problem of semantics; a budget implies a plan or action. Due to this notion, we propose the label RISK ATTRIBUTION for this definition of risk budget.
### Risk Attribution
Let us once again consider a portfolio of three assets. To calculate its risk attribution, we take the partial derivative of its VaR with respect to each of the assets. First, we write the partial derivative of a portfolio’s percentage loss in continuous units, with respect to exposure to Asset 1 as the sum of the derivatives of the two components of a portfolio’s percentage loss:
$$
\frac{\partial{L\_c}}{\partial{w\_1}}=\frac{\partial{\mu\_{pc}}}{\partial{w\_1}}+\frac{\partial{Z \sigma\_{pc}}}{\partial{w\_1}}
$$
In this equation, $$L\_c$$ is percentage loss in continuous units, $$w\_1$$ is the exposure to Asset 1, $$\mu\_{pc}$$ is the expected portfolio return in continuous units, $$Z$$ is the Normal deviate, and $$\sigma\_{pc}$$ is the expected portfolio standard deviation in continuous units.
Next, we define portfolio expected return and standard deviation measured in continuous units as a function of the portfolio’s exposure to the component assets
$$
\mu\_{pc}=\mu\_{1c} w\_1+\mu\_{2c} w\_2+\mu\_{3c} w\_3
$$
$$
\sigma\_{pc}=\left( \sigma\_{1c}^2w\_1^2+\sigma\_{2c}^2w\_2^2+\sigma\_{3c}^2w\_3^2+2\rho\_{1,2}\sigma\_{1c}w\_1\sigma\_{2c}w\_2+
2\rho\_{1,3}\sigma\_{1c}w\_1\sigma\_{3c}w\_3+2\rho\_{2,3}\sigma\_{2c}w\_2\sigma\_{3c}w\_3 \right)^{1/2}
$$
where $$\mu\_{ic}$$ is the expected return of asset $$i$$ in continuous units, $$\sigma\_{ic}$$ is the expected return of asset $$i$$ in continuous units, and $$\rho\_{i,j}$$ is the correlation of asset $$i$$ and $$j$$ in continuous units.
The derivative of the portfolio expected return with respect to exposure to Asset 1 is straightforward:
$$
\frac{\partial \mu\_{pc}}{\partial w\_1}=\mu\_{1c}
$$
The derivative of $$Z\sigma\_{pc}$$ with respect to exposure to Asset 1 is slightly more complicated. We need to invoke the chain rule, by first taking the partial derivative of $$Z\sigma\_{pc}$$ with respect to portfolio variance. We then need to multiply it by the partial derivative of portfolio variance with respect to exposure to Asset 1:
$$
\frac{Z \sigma\_{pc}}{\partial w1}=\frac{Z \sigma\_{pc}}{\partial \sigma\_{pc}^2} \left( \frac{\partial Z\sigma\_{pc}^2}{\partial w\_1} \right)
$$
We convert this sensitivity of percentage loss measured in continuous units to the sensitivity of the portfolio’s VaR measured in monetary units:
$$
\frac{\partial \text{VaR}}{\partial w\_1}=-e^{\frac{\partial L\_c}{\partial w}}-1
$$
The following table shows the risk attribution of the same portfolio described earlier.
| Asset | Allocation | Value at Risk
(per $100)
| VaR Sensitivity |
| --------- | :--------: | :-----------------------------------: | :-------------: |
| 1 | 20.70% | 4.9567 | 0.1846 |
| 2 | 30.73% | 5.5429 | 0.0866 |
| 3 | 48.57% | 4.46663 | 0.0387 |
| Portfolio | 100.00% | 7.1453 | |
### Features of Risk Attribution
Two interesting features of risk attribution:
1. It is mathematically impossible to partition a portfolio’s total VaR into the fractions associated with the individual components. VaR is a function of standard deviation (which is not additive), and it is impossible to disentangle the interactions of the various components.
2. The ranking of the portfolio’s VaR sensitivities will not necessarily match the ranking of the portfolio’s percentage exposures or the ranking of the individual components’ values at risk. Note that Asset 1 has the smallest percentage allocation, yet has the greatest impact on the portfolio’s VaR. Also, although Asset 2’s VaR is lower than Asset 3’s, an additional $1 allocation to Asset 2 increases portfolio VaR by $0.0866, while the same increase in allocation to Asset 3 raises portfolio VaR by less than half as much ($.0.0387). Although Asset 2 has a lower individual VaR, the higher correlation with the portfolio offsets this advantage.
This risk attribution shows a very important message. If we wish to limit the potential loss of our portfolio, we should focus not on its largest holding. We shouldn’t focus on its most volatile asset, or on the asset with the greatest value at risk. Instead, we should focus on the asset to which the portfolio is most sensitive.
## Summary
Risk budgets have gained widespread attention in the last few years, but are often applied inappropriately or mislabeled. Regarding risk budgets as a plan for assigning VaR to categories independently of mean-variance optimization will likely have inefficient results.
Investors who regard a risk budget as a process to choose a portfolio by minimizing value at risk implicitly care only about a single outcome. These investors are no more averse to losses far greater than VaR than a loss equal to VaR. Portfolio choice, instead, should be based on the consideration of every portfolio on the efficient frontier and their probability distributions. This process produces efficient VaR results, and considers the pleasure or disutility associated with all possible outcomes.
The proper conception of a risk budget is the conversion of optimal percentage allocations from mean-variance optimization into VaR assignments. This conversion preserves the economic exposures of the portfolio and enables investors to achieve these exposures with much greater latitude.
# Risk in the Real World
## The Challenge
G.H. Hardy, the legendary mathematician, once claimed that his greatest disappointment in life was learning that someone had discovered an application for one of his theorems. Although Hardy’s disinterest in practical matters was a bit extreme, it sometimes seems that scholars view the real world as an uninteresting, special case of their models. This disinterest in real world complexity, unfortunately, often brings unpleasant consequences. In this article, we address two simplifications about risk that often lead investors to underestimate their portfolios’ exposure to loss. First, investors typically measure risk as the probability of a given loss, or the amount that can be lost with a given probability, at the end of their investment horizon, ignoring what might occur along the way. Second, they base these risk estimates on return histories that fail to distinguish between calm environments, when losses are rare, and turbulent environments, when losses occur more commonly. We show how to estimate exposure to loss in a way that accounts for within-horizon losses as well as the regime-dependent nature of large drawdowns.
## End-of-Horizon Exposure to Loss
### Probability of Loss
We measure the likelihood that a portfolio will experience a certain percentage loss at the end of a given horizon by computing the standardized difference between the percentage loss and the portfolio’s expected return, and then converting this quantity to a probability by assuming returns are normally distributed. Unfortunately, asset class returns are not normally distributed. Returns tend to be lognormally distributed, because compounding causes positive cumulative returns to drift further above the mean than the distance negative cumulative returns drift below the mean. (See Chapter 18 for more detail about log-normality.) This means that logarithmic returns, also called continuous returns, are more likely to be described by a normal distribution. Therefore, in order to use the normal distribution to estimate probability of loss we must express return and standard deviation in continuous units, as shown in Equation 12.1. We provide the full mathematical procedure for converting returns from discrete to continuous units in Chapter 18.
For example, a positive 10 percent return will accumulate to 20 percent over two periods, whereas a negative 10 percent return will fall to 19 percent percent over two periods.
$$
Pr\_{end}=N \left\[\frac{ln(1+L)-\mu\_cT}{\sigma\_c\sqrt{T}} \right]
$$
The equation above describes the probability of loss at the end of the horizon, where $$N \left\[ \cdot \right]$$ is the cumulative normal distribution function, $$ln$$ is the natural logarithm, $$L$$ equals the cumulative percentage loss in discrete units, $$\mu\_c$$ equals the annualized expected return in continuous units, $$T$$ equals the number of years in the investment horizon, and $$\sigma\_c$$ equals the annualized standard deviation of continuous returns.
### Value at Risk
Value at risk gives us another way to measure a portfolio’s exposure to loss. It is equal to a portfolio’s initial wealth multiplied by a quantity equal to expected return over a stated horizon minus the portfolio’s standard deviation multiplied by the standard normal variable associated with a chosen probability. Again, we express return and standard deviation in continuous units.
$$
\text{VaR}=W \times ( e^{\mu\_c T + N^{-1}\[P\_L] \sigma\_c \sqrt{T}}-1)
$$
As the two equations reveal, probability of loss and value at risk are flip sides of the same coin.
These formulas assume that we only observe our portfolio at the end of the investment horizon and disregard its values throughout the investment horizon. We argue that investors should and do perceive risk differently. They care about exposure to loss throughout their investment horizon and not just at its conclusion.
## Within-Horizon Exposure to Loss
### Within-Horizon Probability of Loss
To account for losses that might occur prior to the conclusion of the investment horizon, we use a statistic called first-passage time probability, which gives the probability that a portfolio will depreciate to a particular value over some horizon if it is monitored continuously. It is equal to
$$
Pr\_w=N \left\[ \frac{ ln(1+L) - \mu T } { \sigma \sqrt T } \right] + N \left\[ \frac{ ln(1+L) - \mu T } { \sigma \sqrt T } \right] (1+L)^{2\mu / \sigma^2}
$$
The first part of this equation, up to the second plus sign, gives the end-of-horizon probability of loss, as shown previously. It is augmented by another probability multiplied by a constant, and there are no circumstances in which this constant equals zero or is negative. Therefore, the probability of loss throughout an investment horizon must always exceed the probability of loss at the end of the horizon. Moreover, within-horizon probability of loss rises as the investment horizon expands in contrast to end-of-horizon probability of loss, which diminishes with time.
### Within-Horizon Value at Risk
We use the same first-passage time equation to estimate within-horizon value at risk. Whereas value at risk measured conventionally gives the worst outcome at a chosen probability at the end of an investment horizon, within-horizon value at risk gives the worst outcome at a chosen probability from inception to any time throughout an investment horizon. It is not possible to solve for within-horizon value at risk analytically. We must resort to a numerical method. We set the equation for within-horizon probability of loss equal to the chosen confidence level and solve iteratively for $$L$$. Within-horizon value at risk equals $$L$$ multiplied by initial wealth.
These two measures of within-horizon exposure to loss bring us closer to the real world because they recognize that investors care about drawdowns that might occur throughout the investment horizon. But they ignore another real world complexity, to which we now turn.
## Regimes
Thus far we have assumed implicitly that returns come from a single distribution. It is more likely that there are distinct risk regimes, each of which may be normally distributed but with a unique risk profile. For example, we might assume that returns fit into two regimes, a calm regime characterized by below-average volatility and stable correlations, and a turbulent regime characterized by above-average volatility and unstable correlations. The returns within a turbulent regime are likely to be event driven, whereas the returns within a quiet regime perhaps reflect the simple fact that prices are noisy.
We detect a turbulent regime by observing whether or not returns across a set of asset classes behave in an uncharacteristic fashion, given their historical pattern of behavior. One or more asset class returns, for example, may be unusually high or low, or two asset classes that are highly positively correlated may move in the opposite direction.
There is persuasive evidence showing that returns to risk are substantially lower when markets are turbulent than when they are calm. This is to be expected, because when markets are turbulent investors become fearful and retreat to safe asset classes, thus driving down the prices of risky asset classes. This phenomenon is documented below.
| | 10% Most Turbulent Months | Other 90% |
| --------------------------------- | :-----------------------: | :-------: |
| U.S. Equities | -5.5% | 13.7% |
| Foreign Developed Market Equities | -10.0% | 13.1% |
| Emerging Market Equities | -43.0% | 20.4% |
| Commodities | -12.5% | 8.2% |
This description of turbulence is captured by a statistic known as the Mahalanobis distance. It is used to determine the contrast in different sets of data. In the case of returns, it captures differences in magnitude and differences in interactions, which can be thought of respectively as volatility and correlation surprise.
$$
\text{Turbulence}\_t=\frac{1}{N}(x\_t-\mu)' \Sigma (x\_t-\mu)
$$
The term $$(x\_t-\mu)$$ captures extreme price moves. By multiplying this term by the inverse of the covariance matrix, we capture the interaction of the returns, and we render the measure scale independent, as well. We multiply by $$\frac{1}{N}$$ so that the average turbulence score across the data set equals 1. We illustrate this concept with a scatter plot of U.S. and foreign developed market equities shown below.
Each dot represents the returns of stocks and bonds for a particular period, such as a day or a month. The center of the ellipse represents the average of the joint returns of stocks and bonds. The observations within the ellipse represent return combinations associated with calm periods, because the observations are not particularly unusual. The observations outside the ellipse are statistically unusual and therefore likely to characterize turbulent periods. Notice that some returns just outside the narrow part of the ellipse are closer to the ellipse’s center than some returns within the ellipse at either end. This illustrates the notion that some periods qualify as unusual not because one or more of the returns was unusually high or low, but instead because the returns moved in the opposite direction that period despite the fact that the asset classes are positively correlated, as evidenced by the positive slope of the scatter plot.
This measure of turbulence is scale independent in the following sense. Observations that lie on a particular ellipse all have the same Mahalanobis distance from the center of the scatter plot, even though they have different Euclidean distances.
We suggest that investors measure probability of loss and value at risk not based on the entire sample of returns, but rather on the returns that prevailed during the turbulent subsamples, when losses occur more commonly. This distinction is especially important if investors care about losses that might occur throughout their investment horizon, and not only at its conclusion.
## The Bottom Line
Investors dramatically underestimate their portfolios' exposure to loss, because they focus on the distribution of returns at the end of the investment horizon and disregard losses that might occur along the way.
Moreover, investors base their estimates of exposure to loss on full-sample standard deviations, which obscure episodes of higher risk that prevail during turbulent periods. It is during these periods that losses are likely to occur.
{% hint style="danger" %}
Complexity is inconvenient but not always unimportant.
{% endhint %}
## Video Presentations
### Within-horizon Risk
This video describes several shortcomings of end of horizon risk measures and defines two innovative risk measures that consider risk throughout the investment period, as summarized in this article.
{% embed url="" %}
Within-horizon Exposure to Loss
{% endembed %}
### Risk Regimes
This video introduces a method to partition historical returns into those that are associated with quiet periods and those that reflect market turbulence.
{% embed url="" %}
Risk Regimes
{% endembed %}
# Time Diversification
As mentioned by Windham CEO [Mark Kritzman in Episode #51](https://hwcdn.libsyn.com/p/1/9/d/19d2a7acd6bec647/Ep._51_-_The_Meb_Faber_Show.mp3?c_id=15232086\&cs_id=15232086\&expiration=1598923988\&hwt=00202fc9f6d30e61e16ffd4d730a6e45) of the Meb Faber Show, time diversification is the common assumption that investing over the long-term is safer than investing over shorter periods. For example, suppose you were going to buy a house in three months and needed to pay $100,000 in cash. In the meantime, would you be more inclined to invest that amount in a riskless asset, such as a Treasury bill, or in a risk asset, such as an S\&P 500 index fund? Alternatively, suppose you wanted to buy that house in 10 years. How would you invest in the meantime?
Typical investors would choose the riskless investment for the three-month horizon, and the riskier investment for the 10 year horizon. Keep in mind that *the only difference between these scenarios is the length of the investment horizon.*
## The Argument for Time Diversification
Time diversification is the notion that above-average returns tend to offset below-average returns over long investment horizons. If returns are independent from one year to the next, the standard deviation of annualized returns diminishes with time. Consequently, the distribution of annualized returns converges as the investment horizon increases.
Figure 1 shows a 95% confidence interval of annualized returns as a function of investment horizon, assuming that the expected return is 10% and the standard deviation of returns is 15%. These confidence intervals are based on the assumption that the returns are lognormally distributed, so the standard deviation measures the dispersion of the logarithms of one plus the returns. Figure 1 confirms that the distribution of annualized returns converges as the investment horizon lengthens.
Another way to consider time diversification is from the perspective of *losing* money. We can determine the likelihood of a negative return by measuring the difference in standard deviation units between a 0% return and the expected return. If we assume that the S\&P 500’s expected return is 10% and its standard deviation equals 15%, the expected return is 0.64 standard deviation above a 0% return, given a one-year horizon. This value corresponds to a 26% probability that the S\&P 500 will generate a negative return in any one year.
However, the outcome changes with a longer investment horizon. With a 10-year investment horizon, the annualized expected return is 2.01 standard deviations above 0.0%. So, there is only a 2.2% that the S\&P 500 will produce a negative return, on average, over 10 years. This does not imply that it is just as improbable to lose money in any one of these 10 years, but merely reflects the tendency of above-average returns to cancel out below-average returns.
## Time Diversification Refuted
Many financial professionals argue that time diversification is inaccurate because, while it is true that the annualized dispersion of returns converges toward the expected return with the passage of time, the dispersion of terminal wealth also diverges from the expected terminal wealth as the investment horizon expands. This implies that, although you are less likely to lose money over a long horizon than over a short horizon, the magnitude of your potential loss actually increases with the duration of your investment horizon. According to critics of time diversification, if you choose the riskless alternative when you are faced with a three-month horizon, you should also select that investment option for all investment horizons (10-year, 20-year, etc.).
This critique applies to cross-sectional diversification as well as temporal diversification. Suppose you have an opportunity to invest $10,000 in a risky venture, but you decline because you think it is too risky. Would you any be less averse to investing in 10 independent ventures that had the same levels of risk as the one you initially declined?
You are clearly less likely to lose money by investing in 10 equally risky, but independent, ventures than by investing in just one. The amount you could conceivable lose, however, is 10 times as great.
**Now consider a third investment option. Suppose you are offered a chance to invest a total of $10,000 in 10 independent but equally risky ventures. In this case, you would invest only $1,000 in each venture. This investment opportunity diversifies your risk around the 10 ventures without increasing your total exposure.**
Perhaps you are unpersuaded by these arguments. You reason as follows: although it is true that the dispersion of terminal wealth increases with the passage of time, or, with the number of risky opportunities, the expected wealth of the risky venture also increases. The dispersion of wealth thus expands around a growing mean as the investment horizon lengthens, or as the number of independent risky ventures increases.
Consider again the choice of investing in an S\&P500 index fund versus a riskless asset. Suppose the riskless asset has a certain 3% annual return compared with the S\&P’s 10% expected return and 15% standard deviation. **Table 1** compares the dispersion of wealth for these two options.
After 1 year, the terminal wealth of an initial $100,000 investment in the S\&P index fund ranges from $81,980 to $147,596, while the riskless investment grows with certainty to $103,000 (bearing in mind the confidence level of 95% for the S\&P investment). After 10 years, the spread in the S\&P investment’s terminal wealth expands from $65,616 to $554,829, but it surrounds a higher expected wealth. Thus, the lower boundary of the 95% confidence interval is greater than the initial investment. If the investment horizon is extended to 20 years, the lower boundary of the 95% confidence interval actually exceeds the terminal wealth of the riskless investment.
Although this line of reasoning might strike you as a credible challenge to the critics of time diversification, in the limit it fails to resurrect the validity of time diversification. Even though it is true that the lower boundary of a 95% confidence interval of the S\&P investment exceeds the terminal wealth of the riskless investment after 20 years, the lower boundary of a 99% confidence interval falls below the riskless investment, and the lower boundary of a 99.9% confidence interval is even worse. The growing improbability of a loss is offset by the increasing magnitude of potential losses.
It is an indisputable mathematical fact that if you prefer a riskless asset to a risky asset given a three-month horizon, you should also prefer a riskless asset to a risky one given a 10-year horizon, given that the following conditions are satisfied:
1. **Your risk aversion is invariant to changes in your wealth.**
2. **You believe that risky returns are random.**
3. **Your future wealth depends only on investment results.**
Risk aversion implies that the satisfaction you derive from increments to your wealth is not linearly related to increases in your wealth. Rather, your satisfaction increases at a decreasing rate as your wealth increases. You thus derive more satisfaction when your wealth grows from $100,000 to $150,000 than when it grows from $150,000 to $200,000. It also follows that a decrease in your wealth conveys more disutility than the utility that comes from an equal increase in your wealth.
Most financial literature assumes that the typical investor has a utility function equal to the logarithm of wealth. Based on this assumption, let’s explore the following numeric demonstration of why it is that your investment horizon is *irrelevant* to your choice of a riskless versus a risky asset.
Suppose you have $100.00. This $100.00 conveys 4.60517 units of utility \[ln(100.00) =4.60517]. Now, consider an investment opportunity that has a 50% chance of a 1/3 gain and a 50% chance of a 1/4 loss. A $100.00 investment in this risky venture has an expected terminal wealth equal to $104.17, but it also conveys 4.60517 units of utility \[50% x ln(133.33) + 50% x ln(75.00) = 4.60517]. Therefore, if your utility function is defined by the logarithm of wealth, you should be indifferent between holding onto your $100.00 or investing it in this risky venture. In this example, $100.00 is the certainty equivalent of the risky venture because it conveys the same utility as the riskless venture.
Now suppose you are offered an opportunity to invest in this risky venture over two periods, and the same odds prevail. Your initial $100.00 can either increase by 1/3 or decrease by 1/4 (both with 50% probability). Over two periods, the expected terminal wealth increases to $108.51, but the utility of the investment opportunity remains the same. You should therefore remain indifferent between keeping your $100.00 and investing it over two independent periods.
This remains true regardless of the investment horizon. The expected utility of the risky venture will always remain 4.60517, implying that you derive no additional satisfaction by diversifying your risk across time. This result holds, even though the standard deviation of returns increases approximately with the square root of time, while the expected terminal wealth increases almost linearly with time.
**Table 2** shows the possible outcomes of this opportunity after one, two, and three periods, along with the expected wealth and expected utility after each period. The possible wealth values are computed by linking all possible sequences of return. Expected wealth equals the probability-weighted sum of each possible outcome, while expected utility equals the probability-weighted sum of the logarithm of each possible wealth outcome.
The result does not require that you have a log wealth utility function. Suppose, instead, that your utility function is defined by minus the reciprocal of wealth. This utility function implies greater risk aversion than a log wealth utility function. You would thus prefer to hold onto your $100.00, given the opportunity to invest in a risky venture that has an equal chance of increasing by 1/3 or decreasing by 1/4. You would, however, be indifferent between a certain $100.00 and a risky venture that offers an equal chance of increasing by 1/3 or decreasing by 1/5.
**Table 3** shows that the expected utility of this risky venture remains constant as a function of investment horizon, even though the expected terminal wealth grows at a faster pace than it does in the previous example. Again, time diversification would not induce you to favor the risky venture over a multi-period horizon if you did not prefer it for a single-period horizon.
## Time Diversification Resurrected
Now that you have been exposed to the incontrovertible truth that time does not diversify risk, would you truly invest the same in your youth as you would in your retirement? There are several reasons why you might still condition your posture on your investment horizon, even though you accept the mathematical truth about time diversification.
First, you may not believe that risky asset returns are random. Perhaps investment returns follow a mean-reverting pattern. If returns revert to their mean, then the dispersion of terminal wealth increases at a slower rate than implied by a lognormal distribution (the distribution that results from random returns). If you are more averse to risk than the degree of risk version implicit in a log wealth utility function, then a mean-reverting process will lead you to favor risky assets over a long horizon, even if you are indifferent between a riskless and a risky asset over a short horizon.
Suppose, for example, that returns are not random. Instead, the risky venture in Table 3 has a 60% chance of reversing direction, and, therefore only a 40% chance of repeating its prior return. **Table 4** reveals that expected utility rises from -0.010 over a single period to -0.00988 over two periods and to -0.00978 over three periods. Thus, if you believe in mean reversion and you are more risk averse than a log wealth investor, you would rationally increase your exposure to risk and your investment horizon expands.
Second, you might believe that the extremely bad outcomes required to justify the irrelevancy of time diversification would result from events or conditions that would have equally dire consequences for the so-called riskless asset, especially if you measure wealth in consumption units.
Third, even if you believe that returns are random, you might still choose to accept more risk over longer horizons than over shorter horizons because you have more discretion to adjust your consumption and work habits. If a risky investment performs poorly at the beginning of a short horizon, there is not much you can do to compensate for this loss in wealth. On the other hand, if a risky investment performs at the beginning of a long horizon, you can postpone consumption or work harder to achieve your financial goals. **The argument against time diversification assumes implicitly that your terminal wealth depends only on investment performance.**
Fourth, you may have a discontinuous utility function. Consider, for example, a situation in which you require a minimum level of wealth to maintain a certain standard of living. Your lifestyle might change drastically if you penetrate this threshold, but further reductions in wealth are less meaningful. You might be more likely to penetrate the threshold given a risky investment over a short horizon than you would be if you invested in the same risky asset over the long run.
Moreover, even if you are not confronted with a real threshold, you might still behave as though you have a discontinuous utility function. Perhaps we can only process a finite set of possible outcomes, or perhaps human nature leads us to ignore terrible outcomes that are extremely remote. Only time will reveal whether or not such behavior is prudent.
Finally, you are irrational. This does not make you a bad person, but rather implies that you behave inconsistently.