Value at Risk

This article is a self-contained introduction to the concept and methodology of value at risk (VAR), a recently developed tool for measuring an entity's exposure to market risk. We explain the concept of VAR and then describe in detail the three methods for computing it—historical simulation, the delta-normal method, and Monte Carlo simulation. We also discuss the advantages and disadvantages of the three methods for computing VAR. Finally, we briefly describe stress testing and two alternative measures of market risk. We provide an introduction to the concept and methodology of value at risk (VAR), a recently developed tool for measuring an entity's exposure to market risk. We explain the concept of VAR, describe and compare the three methods for computing it, and describe two alternative concepts. The need for VAR stems from the past few decades' tremendous volatility in exchange rates, interest rates, and commodity prices and its proliferation of derivative instruments for managing the risks of changes in market rates and prices. Increased trading of cash instruments and securities and the growth of financing opportunities accompanied the proliferation of derivatives. As a result, many companies have portfolios that include large numbers of (sometimes complex) cash and derivative instruments. Moreover, the magnitudes of the risks in companies' portfolios often are not obvious. The result is increasing demand for a portfolio-level quantitative measure of market risk. VAR is a single, summary statistical measure of possible portfolio losses. Using a probability of x percent and a holding period of t days, an entity's VAR is the loss that is expected to be exceeded with a probability of only x percent during the next t-day holding period. Loosely, it is the loss that is expected to be exceeded during x percent of t-day holding periods. For example, if the VAR computed for a one-day holding period at a probability of 5 percent is $90,000, then the loss in the mark-to-market value of the portfolio will exceed $90,000 with a probability of 5 percent. Thus, the VAR is $90,000. Subject to the simplifying assumptions used in its calculation, VAR aggregates all of the risks in a portfolio into a single number that is suitable for use in the boardroom, reporting to regulators, or disclosure in an annual report. Of the three main methods for computing VAR, historical simulation is the simplest. The approach begins by constructing the distribution of possible profits and losses. The distribution is constructed by taking the current portfolio and subjecting it to the actual changes in the market factors to which the portfolio is exposed that were experienced during each of the last N periods, usually days. That is, N sets of hypothetical market factors are constructed by using their current values and the changes experienced during the last N periods. Using these hypothetical values of the market factors, N hypothetical mark-to-market portfolio values are computed. This step allows one to compute N hypothetical mark-to-market profits and losses on the portfolio as compared with the current mark-to-market portfolio value. VAR is the magnitude of loss that is exceeded by only x percent of the market losses. The delta-normal approach is based on two key ideas. First, the assumption is made that changes in the underlying market factors that drive the changes in the value of the portfolio have a multivariate normal distribution. Second, the approach is to replace the actual portfolio held by the company or institution with a simpler portfolio that has approximately the same risk and then compute the VAR of the simpler portfolio. In particular, the approach involves identifying a set of k market factors (which might be yields on zero-coupon bonds) that account for most of the changes in value of the portfolio and, therefore, capture its risks. Then, for each market factor, one identifies a standardized position (e.g., a zero-coupon bond) that is exposed only to the risk of one market factor. The simpler approximating portfolio consists of a portfolio of the standardized positions, which has the same deltas or exposures to the basic market factors (and thus the same risks) as the original portfolio. Using these ideas, one can determine the distribution of mark-to-market portfolio profits and losses, which is also normal. Once this distribution has been obtained, one uses the standard mathematical properties of the normal distribution to determine the loss that will be equaled or exceeded x percent of the time (i.e., the VAR). The Monte Carlo simulation methodology is similar to historical simulation. The main difference is that one chooses a statistical distribution that is believed to adequately capture or approximate the possible changes in the market factors. Then, a pseudo-random number generator is used to generate thousands of hypothetical changes in the market factors. These results are then used to construct thousands of hypothetical portfolio profits and losses on the current portfolio and the distribution of possible portfolio profit or loss. Finally, the VAR is determined from this distribution; that is, the VAR is the portfolio loss that is exceeded only x percent of the time. We briefly describe some alternatives to VAR but focus on the advantages and disadvantages of the three methods for computing VAR. The methods differ in ability to capture the risks of options and option-like instruments, ease of implementation, ease of explanation to senior managers, flexibility in analyzing the effect of changes in the assumptions, and reliability of the results. The best choice will be determined by which dimensions the risk manager finds most important.