when the market move is scaled up or down, the gain or loss is scaled the same amount. When exposures are linear, it suffices to measure the response to an event of any given size. All other sized events can be extrapolated from the one. More generally, when exposures are not linear, then we need to measure the response to events of different sizes. Nonlinear exposures most commonly arise from options and other derivatives. Stress tests are relatively simple to perform and provide a relatively straightforward set of signals of what types of shocks could create portfolio losses. The limitations of stress tests are important to recognize, however. Because the stress test provides no guidance about the likelihood of shocks of different sizes, or the likelihood that different markets will move together or offset each other, it is difficult to measure overall portfolio risk. In order to make sense of stress tests alone, the investor has to have a good intuition about the volatilities and correlations of all the different risk factors. Another simple risk management tool is the scenario analysis. A scenario is like a stress test, except that generally a number of different risk factors are stressed at the same time. In fact, a stress test can be thought of as one particularly simple version of a scenario analysis. What makes the scenario analysis useful, and conceptually different from a stress test, is that the scenario is generally constructed to represent an event that is likely to constitute a particularly significant risk to the portfolio. For example, a common scenario to analyze is a global recession and the expected impacts on equity, real estate, credit, bond, and currency markets around the world. Such a scenario would most likely include the different impact on cyclically sensitive industries relative to more stable sectors, and it might also include secondary impacts such as increased likelihood of defaults, monetary policy changes, changes in wages and rents, and so on. The strength of a scenario analysis is that it is an excellent tool for preparing oneself for a particular outcome. Two weaknesses of scenario analysis as a risk management tool, however, are that it is hard to know which scenarios to analyze and how to react. Portfolio managers often try to put probabilities on different scenarios, but it is very difficult to approximate all possible outcomes with a few scenarios, and even more difficult to reasonably put probabilities on such scenarios. The standard statistical measure of risk is volatility, which measures the size of a typical outcome's deviation relative to its expected value. When quantifying the volatility of portfolios, the volatility is generally measured in terms of percent per year. A balanced portfolio with equities and bonds might, for example, have an annualized volatility of 9 percent. If such a portfolio has an expected return of 10 percent with a 9 percent volatility, that implies that the portfolio returns will typically-that is, about two-thirds of the time-fall between 1 percent and 19 percent. There are many approaches to measuring volatility. Most such measures rely on extrapolating past behavior into the future. Perhaps the simplest approach, when the portfolio has not changed recently, is to measure the historical volatility of the portfolio returns directly. When the portfolio itself has changed, or when the volatility or correlations of different components of the portfolio have changed, then a more disaggregated approach must be taken. In this case the usual approach