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.
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 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. 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.
VaR refers to value at risk, is the base of the natural logarithm, equals the annualized expected return in continuous units, equals the number of years in the investor’s horizon, equals the standard normal variable, equals the annualized standard deviation of continuous returns, and 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.
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.
10% Most Turbulent Days
Small Cap Premium
Source: Kritzman and Li 
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.
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. This statistic measures the probability of a first occurrence of an event within a finite horizon. It is equal to
equals the probability of a within-horizon loss, equals the cumulative normal distribution function, equals the natural logarithm, equals the cumulative percentage loss in discrete units, equals the annualized expected return in continuous units, equals the number of years in the investment horizon, and 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 equal to the chosen confidence level and solve iteratively for . Continuous value at risk equals multiplied by initial wealth.
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.
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.
The following video presentation describes the risk regime methodology within our software and advisory services.
Kritzman, M. and Y. Li. 2010. “Skulls, Financial Turbulence and Risk Management.” Financial Analysts Journal, vol. 66, no. 5 (September/October).
A standard normal variable is a normally distributed random variable with expected value of zero and a variance of one.
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.
These returns are annualized daily returns for the period January 1993 through December 2008. For more information about this study, see Kritzman and Li 
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.
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