In 1952 **Harry Markowitz** showed that the most revelant risk for an investor is generally not the risk of any one investment by itself but rather it is that of the aggregate portfolio of investments. As a result of this seminal work the standard deviation of this aggregate portfolio's return, or of its excess return relative to a benchmark, was for many years the principle measure of investment risk. It was an essential component of the Nobel prize-winning theory of asset pricing by **William Sharpe**. It was also a key building block of **Stephen Ross**' arbitrage pricing theory. Value at Risk or VaR has recently supplanted the standard deviation as the preferred measure of risk. VaR can often be interpreted as a multiple of the standard deviation assuming that a portfolio's returns follow a normal, or log normal, probability distribution.

Derivatives dealers pioneered the use of VaR in the eighties and early nineties. In 1993 the Group of Thirty, a global derivatives study group headed by **Paul Volcker**, an ex-chairman of the **U.S. Federal Reserve Board**, recommended that Value at Risk or VaR be the basis of risk management for derivatives positions. This highly influential report defined VaR as "The expected loss from an adverse market movement with a specified probability over a particular time period." Since then VaR has been applied to all kinds of assets, not just to derivatives.

**Limitations Of VaR**

VaR has become the single best known investment risk management tool. Nonetheless, it has its limitations and these have become increasingly obvious over time. For example the chief executive of a major dealer in the aftermath of the August 1998 Russian debt default crisis said, "I used to sleep easy at night with my VaR model."

The Group of Thirty recognized that VaR needed to be supplemented with risk measurement tools such as 'scenario analysis' to deal with extreme market events. VaR assumes that investment returns are well behaved statistically. Extreme market moves often violate this assumption. Their magnitude and frequency differ greatly from what would be true if investment returns follow a normal or log-normal distribution.

The chart illustrates this for daily stock market returns over the last 12 months ending Nov 6, 2000. The first bar shows the expected number of daily returns to the S&P500 index (SPX) that should be lower than the 97.5% one-day VaR level given the one-year estimated standard deviation of the returns. This expected number is 6.3 observations of daily returns. The second bar shows the actual number of returns that fell below the VaR level. The actual number, seven, was above the expected number. The chart also shows the actual number of observations of returns that were more negative than the VaR level consistently exceeded the predicted number for the EURO STOXX 50 index (SX5E), the Nasdaq (CCMP), and the OMX index. These indices were chosen because the first two reflect the behavior of larger old economy stocks' exposure while the last two have new economy stock exposure.

Last year VaR consistently under-estimated the risk of extreme events. Interestingly the under-estimation is greater for more volatile Nasdaq and OMX indices than it is for the more stable indices. Over relatively stable periods VaR may be an excellent estimator of risk, but then measuring risk may be of less importance in stable periods. In fact, VaR often is least effective in truly extreme market dislocations such as the 1987 stock market break. On the Oct 19, 1987 the S&P 500 index fell by nearly 23%. An estimate of one-day 97.5% VaR for an S&P 500 index portfolio would probably have been a decline in value of only 2.2%. A VaR based estimate of risk would have hugely underestimated risk in the extreme environment of the 1987 market break. Generally, VaR is too conservative during normal market conditions levels and may underestimate risk in extreme conditions.

**The VaR Conundrum**

Another less obvious problem of VaR is that it ignores all risks beyond the VaR level regardless of severity. This leads to a paradox in certain cases where VaR estimates the risk level of a diversified portfolio of risky assets is greater than that of a single risky asset.

Consider a stock that has a 2% risk of going bankrupt. Suppose the stock has a standard deviation of return equal to 50% annualized. The daily VaR (97.5%) of a single stock portfolio that owns just this stock will approximately equal 8.1%. This VaR does not take into account the possible loss due to bankruptcy because this loss has a probability smaller than 2.5%. A 97.5% VaR level ignores risks that have a probability smaller than 2.5%.

In contrast suppose an investor buys an equally weighted portfolio consisting of two stocks. Assume the risk of bankruptcy for each stock is identical to the risk of the single stock described above. Also assume the returns of the two stocks are uncorrelated. The probability that either one or both of the two stocks will go bankrupt is 3.96%. If one stock only goes bankrupt the portfolio will lose approximately 50% of its original value, assuming that in bankruptcy a stock is worthless. The resulting VaR of the diversified portfolio is greater than 50%. But this VaR is much greater than the VaR of an undiversified portfolio that consists of only one stock. The risk of the diversified portfolio is actually smaller than that of the single stock, but the VaR analysis does not show this. The VaR analysis is inaccurate in this case because it ignores the huge loss that can occur in a bankruptcy. The probability of this event is less than the cutoff probability for the VaR level.

**Conditional VaR**

Another risk measure recently advocated as an alternative to VaR overcomes the limitations outlined above. This risk measure is called conditional VaR, or expected shortfall. The starting point is to estimating the probability of an extreme event. Suppose this probability is 2.5%. Using a VaR approach it is possible to estimate the loss that corresponds to this probability level. This is the standard VaR level, VaR 97.5% except that instead of assuming a normal, or log normal distribution, the probability of the extreme event is calculated using more realistic probability distribution that is based on realized experience with extreme events. Next calculate the expected loss over and above the VaR 97.5% level if and only if an extreme event occurs. The sum of the VaR level and this conditional expected loss is the conditional VaR.

Suppose that the portfolio VaR 97.5% is 5%. Suppose further that if an extreme event occurs, then the average loss in excess of VaR on a portfolio is 2%. Then the conditional VaR is 2% + 5%, or 7%.

Conditional VaR overcomes the first limitation of VaR because the unusual non-normal behavior of portfolio returns at extremes is explicitly addressed. Of course the estimated probability will not be perfect, but at least it can be constructed to eliminate a consistent bias.

Perhaps more importantly conditional VaR explicitly takes into account the potential size of extreme events, even of those that are highly unlikely. For example it would take into account the size of loss associated with bankruptcy because the expected loss if an extreme event occurs, will have to include the loss from bankruptcy by definition.

**Conclusion**

VaR has proven to be a useful way to evaluate risk, but it has significant limitations. Firstly the normal, or log normal, distribution that is generally used to compute VaR significantly underestimates the risk of extreme events. Secondly VaR ignores, by construction, losses, however large, that have a probability of occurring that is less than the cut-off probability level chosen for the VaR calculation. Both of these limitations are overcome by an alternative risk measure for extreme events called conditional VaR.

*This week's Learning Curve was written by* *Andrew Harmstone**, European head of equity derivatives and quantitative research at* *Lehman Brothers**in London, and* *Roberto Torresetti**at quantitative analyst in the same team.*

**Comparison Of Theoretical Number Of Returns And Actual Number Of Returns Below 97.5%. One-Day VaR Level From Nov 6, 1999 to Nov 6, 2000.**

*Source:**Bloomberg***,***Lehman Brothers*