risk factors, and then to estimate the covariance structure-that is, the volatilities and correlations-of those different risk factors. Depending on whether or not the stress exposures are linear, different methods are available for computing the portfolio volatility. Intuitively, however, the basic idea is that the covariance structure creates a probability distribution for risk factors, and the stress tests provide a basis for valuing the portfolio with respect to each risk factor outcome. Thus, a distribution is implied for portfolio valuations, and we can measure the volatility of that distribution. The strength of volatility as a measure of risk is that it summarizes many possible outcomes in one number. The weaknesses of volatility as a measure of risk are many, but the most important is that it tries to capture risk, which is generally a multidimensional concept, in a single number. Only in special cases, such as when returns are known to have a normal distribution, does volatility alone provide enough information to measure the likelihood of most events of interest. Another weakness of volatility as a measure of risk is that it does not distinguish upside risk from downside risk-all deviations from the expected value create risk. This weakness is mitigated for portfolios because the distributions tend to be approximately symmetric. Finally, the volatility measure provides no insight into the sources of risk. Despite these shortcomings, and despite the fact that for all these reasons volatility has been discredited as a measure of risk in the securities and banking industry, volatility is still the most common measure of risk in investment portfolios. This is not, however, necessarily a weakness. It certainly is the case that in the typical investment context, most of the limitations of volatility are less important. For example, over longer periods of time the aggregation of independent returns is likely to create more normally shaped distributions. Investors are less likely to use options or other derivatives that create significant nonlinear responses to market moves. Moreover, in most situations it is very difficult to estimate precise measures of the shape of return distributions. In many contexts a one-dimensional measure is adequate and the primary interest is in whether and to what extent portfolio changes impact the basic shape of the distribution of portfolio returns. For this purpose volatility is the preferred measure. Thus, while it is important to understand the limitations of this statistical measure, it is likely to remain an important tool in the management of risk in investment portfolios. Economists have struggled for centuries with the problem of measuring investor's utility and how it changes as a function of wealth. There is general agreement on very little other than that this function is concave-that is, that utility increases with wealth, but that the rate of increase gets smaller as wealth increases. When utility has this concave shape it is said to exhibit risk aversion. An investor will prefer a known level of wealth to a distribution of outcomes with the same expected value. Modern portfolio theory has developed a very elegant set of insights based on a simple utility function, which in turn is based on the idea that utility increases with higher expected returns and decreases with increased volatility. We can write this utility function as: U{rp) = E{rp)-.5.k.tf{rp) (3-1)