CHAPTER 10 Value at Risk M any times one wants to know for planning purposes what is the ‘‘worst that can happen.’’ In many situations, the worst that can happen is to lose one’s entire investment; however, this usually has an extremely low probability of occurrence. The concept of Value at Risk (VaR) was devised to obtain a risk measure that associates a severe loss with a probability level of reasonable interest to the decision maker, such as 1 percent or 5 percent. See Jorion (2001) for more about VaR. In this chapter, we see how to use Crystal Ball to find VaR and a related measure, Conditional VaR (CVaR). VAR In practice, we can think of a potential loss L as the worst that can happen if the probability of losing L or more during a selected time period is a specified amount such as 5 percent. In that case, L is called the ‘‘5 percent Value at Risk (VaR).’’ More precisely, let R denote the total return (in dollars) on an investment, I,andlet c denote the α percentile of the distribution of R. Then the α percent VaR is defined as L = I −c. Figure 10.1 shows a segment of the one-year Crystal Ball model in PortfolioVaR. xls, which is adapted from the file Portfolio.xls described in Chapter 9. The potential loss from investing in the portfolio, I −R, is measured directly in cell B11 with the Excel formula =A4-A11, which is simply the difference between the initial investment and the final value of the portfolio. A copy of the forecast window for this quantity is shown at the bottom of the spreadsheet segment in Figure 10.1. Because the certainty is 95 percent that portfolio loss is between −Infinity and $792.47, we say that the 5 percent VaR for one year is $792.47. Note that when we find the α percent VaR from the loss (I −R) distribution, we use the (1 − α) percentile in the upper tail rather than the α percentile, c, in the lower tail of the distribution of R. VaR is used by regulators to compute capital requirements for financial insti- tutions, and by managers as an input to risk-management decisions. VaR can also be used by managers to assess the quality of their models. For example, if a model provides that there is a 5 percent chance that a bank’s trading operations will lose $1 million over a 1-day horizon, then on average the trading operation should lose 140 Value at Risk 141 FIGURE 10.1 Spreadsheet segment from model to illustrate the concept of Value at Risk (VaR). $1 million or more on 5 percent of the days in a randomly selected period. If there are many more losses, it implies that the model assigns too little risk to the situation. If there are many fewer losses, it implies that the model assigns too much risk. In this sense, VaR can therefore be used to check the validity of a model. Some regulators have adopted VaR as part of their risk management guidelines, but critics of this measure have pointed out some of its shortcomings, which are discussed in the next section. 142 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL SHORTCOMINGS OF VAR VaR provides no information about the extent of losses that might occur beyond the threshold level. In that sense it is very optimistic because it gives a lower bound on potential loss at the α percent level. Further, it is not always subadditive, which means that the VaR of the combination of two or more investments can exceed the sum of the individual VaR for each investment. This is contrary to the basic principle of diversification, which holds that risk will decrease when more assets are held, not increase. This failure to reward diversification is perceived as the greatest shortcoming of VaR. An alternate risk measure, Conditional Value at Risk (CVaR), overcomes these shortcomings. CVAR Conditional Value at Risk (CVaR) is the expected value of losses beyond the threshold level. Figure 10.2 shows a forecast window for the portfolio loss in PortfolioVaR.xls. During the definition of cell C11 as a forecast, a filter was set on the forecast values to exclude values in the range −Infinity up to $792.47. That is why ‘‘500 Trials’’ appears in the upper left part of the forecast window, even though 10,000 trials were run. The mean of these 500 largest forecast values is $1,192.56. In general, the α percent CVaR is the expected value of losses that exceed the α percent VaR level. Figure 10.3 depicts the 5 percent VaR and CVaR on the loss distribution for a one-year holding period for the portfolio in PortfolioVaR.xls. FIGURE 10.2 Filtered forecast window for Portfolio Loss in Figure 10.1. Value at Risk 143 FIGURE 10.3 Forecast window for Portfolio Loss in Figure 10.1 showing the 5 percent VaR over one year of $1,967 and the corresponding CVaR of $2,794. Investments with high CVaR will necessarily have high VaR as well. CVaR, which is also called Conditional Tail Expectation, Expected Tail Loss, Mean Excess Loss, Mean Shortfall,orTail VaR, is considered to be a more ‘‘coherent’’ measure of risk than VaR. Artzner, Delbaen, Eber, and Heath (1999) describe subadditivity and other coherent measures of risk in detail. Uryasev (2000) and Hardy (2006) are good references on the basics of CVaR. CVaR.xls The file CVaR.xls, a segment of which is shown in Figure 10.4, contains a simulation model of a three-asset portfolio. Assets 1, 2, and 3 have mean returns of 10 percent, 12 percent, and 13 percent, respectively, with variance-covariance matrix 0.10 0.04 0.03 0.04 0.20 −0.04 0.03 −0.04 0.30 . The variance-covariance matrix is contained in cells B9:D11, and the corresponding Pearson correlation matrix is computed in cells B14:D16. As correlation matrices are symmetric, it appears as a lower triangular matrix. The variance-covariance matrix is used in an Excel array formula in this example (cell C32), so the entire symmetric matrix is in the file. The Crystal Ball model simulates returns from a portfolio with weights 0.30, 0.25, and 0.45, invested in Assets 1, 2, and 3, respectively. Asset rates of return are 144 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 10.4 Crystal Ball model to find VaR and CVaR. normally distributed, but truncated at −100 percent. The model has four forecast cells: Asset1 Value, Asset2 Value, Asset3 Value, Total Value. To reproduce the results tabulated below, run 10,000 Trials with LHS and Seed = 813. To get the 1 percent VaR and CVaR values from Crystal Ball, first find the 1st percentile of each forecast using using the command =CB.GetForePercentFN(Range,Percent) as shown in cells B25:E25.Thenuse Preferences→Forecast →Filter Tab to set a filter on the forecast values to include values in the range −Infinity to the 1st percentile (entered as a number) of each forecast. The means of the filtered values are used to compute the α = 1 percent CVaR for each asset and the portfolio as shown in Table 10.1. Again, CVaR is preferred by some analysts because it is subadditive, which means that the CVaR of the portfolio is always less than or equal to the sum Value at Risk 145 TABLE 10.1 VaR and CVaR for three assets and portfolio modeled in CVaR.xls. 1st Percentile Investment VaR CVaR Asset1 Value $ 10.98 $ 30.00 $ 19.02 $ 21.86 Asset2 Value $ 4.03 $ 25.00 $ 20.97 $ 22.72 Asset3 Value $ 4.20 $ 45.00 $ 40.80 $ 42.76 Portfolio Value $ 49.05 $ 100.00 $ 50.95 $ 56.75 of the individual CVaRs of the portfolio components. VaR is not always subadditive (although it is so in this example), which means that the risk of a portfolio can be larger than the sum of the stand-alone risks of its components when measured by VaR. The CVaR is also less sensitive to changes in the defining percentile α than is VaR. CVaRSubadditivity.xls The model in Figure 10.5 is a Crystal Ball simulation of an analytical result presented by Tasche (2002). The model simulates investments in each of two independent FIGURE 10.5 Crystal Ball model to demonstrate subadditivity of CVaR in a situation for which VaR is not subadditive. 146 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL TABLE 10.2 VaR and CVaR for independent investments modeled in CVaRSubadditivity.xls. 1% VaR 1% CVaR Total Loss $ 209.51 $ 4,687 Loss1 $ 98.22 $ 2,392 Loss2 $ 99.27 $ 2,407 opportunities with potential losses X 1 and X 2 .BothX 1 and X 2 loss distributions have Pareto distributions with Location Parameter =1andShape Parameter = 1. Cell D4 is a Crystal Ball forecast representing the loss on the first opportunity, Loss1 = X 1 .CellD5 is a Crystal Ball forecast representing the loss on the second opportunity, Loss2 = X 2 .CellD6 is a Crystal Ball forecast representing the total loss, Loss1 +Loss2. The VaR and CVaR of the Loss1, Loss2, and Total Loss are shown in Table 10.2. The table shows that CVaR is subadditive because the CVaR for Total Loss, $4687 is less than the sum of the CVaRs for Loss1 and Loss2, $4799 = $2392 + $2407. However, VaR is not subadditive because the VaR for the Total Loss, $209.51, is greater than the sum of the VaRs for Loss1 and Loss2, $197.49 = $98.22 +$99.27. In other words, the 1 percent VaR for the portfolio is greater than the sum of the 1 percent VaRs of the investments considered individually. The principle of diversification holds that risk is lower when two or more assets are combined into a portfolio. As VaR indicates in this example that the risk of holding the portfolio is greater than the sum of the risks of holding each portfolio component individually, VaR is not a satisfactory measure of risk in this example. However, CVaR indicates correctly that the risk of holding the portfolio is lower than the sum of the component risks, and so it is a satisfactory measure of risk from this point of view. . and 3, respectively. Asset rates of return are 144 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL FIGURE 10. 4 Crystal Ball model to find VaR and CVaR. normally distributed, but truncated at 100 . demonstrate subadditivity of CVaR in a situation for which VaR is not subadditive. 146 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL TABLE 10. 2 VaR and CVaR for independent investments modeled in CVaRSubadditivity.xls. 1%. tabulated below, run 10, 000 Trials with LHS and Seed = 813. To get the 1 percent VaR and CVaR values from Crystal Ball, first find the 1st percentile of each forecast using using the command =CB.GetForePercentFN(Range,Percent) as