concern in the calculation of VaR is what happens in these rare, short-term events. This concern is especially relevant with respect to portfolios that incorporate options, since these and other derivatives allow the level of exposure to increase rapidly with changes in the levels of markets. Investors, in contrast, do not usually focus on rare, short-term events. Investors tend to have much longer horizons, and they have used different risk measures, which reflect that longer focus. The two most common measures of risk in the investment world are annualized volatility and annualized tracking error. Annualized volatility is simply the volatility of portfolio returns over a one-year horizon. Tracking error measures the volatility, measured in percent or basis points-that is, hundredths of a percent-of active risk relative to a benchmark over a one-year horizon. These different measures of risk, VaR in the case of financial institutions and tracking error or annualized volatility in the case of portfolio managers, are but one reflection of the different needs and concerns of these two different communities. There has been, though, a very beneficial cross-fertilization of ideas. Because the risk management effort grew very rapidly recently in financial institutions, many practitioners with a securities firm background have tried to take the concepts, the language, and even the software of the financial institutions and apply them to the investment world. Despite the occasional confusion and resistance that this transfer has sometimes caused, a positive effect has been the rapid advances in availability of risk management tools in the investment community. There is a common unifying principle that runs through all financial risk management: In financial planning one needs to recognize and to be prepared for dealing with all possible future outcomes. This principle, as applied to portfolios, implies that the investor needs to have a realistic understanding of potential changes in market levels and valuations of individual securities, and an understanding of how those changes will impact his or her portfolio valuation. Thus, the fundamental focus of risk management is the understanding of this distribution of potential future outcomes. Given this distribution, and comparing it with the distributions of future outcomes associated with other portfolios, the investor can make informed decisions about asset allocation and the risk budget. In practice there are many complexities to risk management. In general, there is no one characteristic or measure that can summarize the distribution of potential outcomes adequately. Many characteristics of the distribution may affect decisions. For most investors the primary focus is on reducing the probability of bad outcomes. Portfolio decisions are generally driven by the inability to sustain losses above a certain size. While much of the science of modern portfolio theory focuses on the mean and volatility of the distribution of outcomes, these two statistics may not be adequate for the purposes of many investors whose focus is on particular downside events. Another issue that arises in assessing risk is that picking the appropriate time horizon for decision making is not always obvious, nor inconsequential. On the one hand, decisions can always be revised with new information, suggesting a relatively shorter horizon may be adequate. On the other hand, focusing on a short horizon can have very important, and generally negative, consequences for investment decisions. Avoiding bad outcomes clearly requires either reducing risk or buying securities that