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.

**Financial turbulence** - a condition in which asset prices, given their historical patterns of behavior, behave in an uncharacteristic fashion.

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):

Using this statistical measure, turbulence can be estimated for any set of assets, rather than only for assets with liquid option markets

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.

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.

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.

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.

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.

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.