Chapter 26

1. Chapter 26 Value at Risk

2. Introduction • Risk assessment is the evaluation of distributions of possible outcomes, with a focus on the worst that might happen • Insurance companies, for example, assess the likelihood of insured events, and the resulting possible losses for the insurer • Financial institutions must understand their portfolio risks in order to determine the capital buffer needed to support their business

3. Value at Risk • Value at risk (VaR) is one way to perform risk assessment for complex portfolios • In general, computing value at risk means finding the value of a portfolio such that there is a specified probability that the portfolio will be worth at least this much over a given horizon • The choice of horizon and probability will depend on how VaR is to be used

4. Value at Risk (Cont’d) • There are at least three uses of value at risk • Regulators can use VaR to compute capital requirements for financial institutions • Managers can use VaR as an input in making risk-taking and risk-management decisions • Managers can also use VaR to assess the quality of the bank’s models

5. Suppose is the dollar return on a portfolio over the horizon h, and f (x, h) is the distribution of returns Define the value at risk of the portfolio as the return, xh(c), such that Suppose a portfolio consists of a single stock and we wish to compute value at risk over the horizon h Value at Risk for One Stock

6. If we pick a stock price and the distribution of the stock price after h periods, Sh, is lognormal, then (26.5) Value at Risk for One Stock (cont’d)

7. In practice, it is common to simplify the VaR calculation by assuming a normal return rather than a lognormal return. A normal approximation is (26.7) We could further simplify by ignoring the mean Mean is hard to estimate precisely For short horizons, the mean is less important than the diffusion term in an Itô process (26.8) Both equations become less reasonable as h grows Value at Risk for One Stock (cont’d)

8. Value at Risk for One Stock (cont’d) • Comparison of three models—lognormal, normal with mean, and normal without mean

9. VaR for Two or More Stocks • When we consider a portfolio having two or more stocks, the distribution of the future portfolio value is the sum of lognormally distributed random variables and is therefore not lognormal • Since the distribution is no longer lognormal, we can use the normal approximation

10. Let the annual mean of the return on stock i, , be i The standard deviation of the return on stock i isi The correlation between stocks i and j is ij The dollar investment in stock i is Wi The value of a portfolio containing n stocks is VaR for Two or More Stocks (cont’d)

11. If there are n assets, the VaR calculation requires that we specify the standard deviation for each stock, along with all pairwise correlations The return on the portfolio over the horizon h, Rh, is Assuming normality, the annualized distribution of the portfolio return is (26.9) VaR for Two or More Stocks (cont’d)

12. Var for Nonlinear Portfolios • If a portfolio contains options as well as stocks, it is more complicated to compute the distribution of returns • The sum of the lognormally distributed stock prices is not lognormal • The option price distribution is complicated • There are two approaches to handling nonlinearity • Delta approximation: we can create a linear approximation to the option price by using the option delta • Monte Carlo simulation: we can value the option using an appropriate option pricing formula and then perform Monte Carlo simulation to obtain the return distribution

13. If the return on stock i is , we can approximate the return on the option as , where i is the option delta The expected return on the stock and option portfolio over the horizon h is then (26.10) The term i + Nii measures the exposure to stock i The variance of the return is (26.11) With this mean and variance, we can mimic the n-stock analysis Delta Approximation

14. Delta Approximation (cont’d) • Comparison of exact portfolio value with a delta approximation

15. Monte Carlo Simulation • Monte Carlo simulation works well in situations where we need a two-tailed approach to VaR (e.g., straddle) • Simulation produces the distribution of portfolio values • To use Monte Carlo simulation • We randomly draw a set of stock prices • Once we have the portfolio values corresponding to each draw of random prices, we sort the resulting portfolio values in ascending order • The 5% lower tail of portfolio values, for example, is used to compute the 95% value at risk

16. Monte Carlo Simulation (cont’d) • Example 26.6 • Consider the 1-week 95% value at risk of an at-the-money written straddle on 100,000 shares of a single stock • Assume that S = $100, K = $100,  = 30%, r = 8%, T = 30 days, and  = 0 • The initial value of the straddle is $685,776

17. First, we randomly draw a set of z ~ N(0,1), and construct the stock price as (26.13) Next, we compute the Black-Scholes call and put prices using each stock price, which gives us a distribution of straddle values We then sort the resulting straddle values in ascending order The 5% value is used to compute the 95% value at risk Monte Carlo Simulation (cont’d)

18. Monte Carlo Simulation (cont’d) • Histogram of values resulting from 100,000 random simulations of the value of the straddle • The 95% value at risk is $943,028  ($685,776) = $257,252

19. Monte Carlo Simulation (cont’d) • Note that the value of the portfolio never exceeds about $597,000 • If a call and put are written on the same stock, stock price moves can never induce the two to appreciate together. The same effect limits a loss • When options are written on different stocks, it is possible for both to gain or lose simultaneously. As a result, the distribution of prices has a greater variance and increased value at risk

20. Monte Carlo Simulation (cont’d) • Example 26.7: Histogram of values of a portfolio that contains a written put and call having different, correlated underlying stocks

21. VaR for Bonds • The risk of a bond and other interest-rate sensitive claims can be measured as the risk of a portfolio of zero-coupon bonds • Suppose a zero-coupon bond matures at time T, has price P(T), and that the annualized yield volatility of the bond is T • For a zero-coupon bond, duration equals maturity. Thus, if the yield changes by , the percentage change in the bond price will be approximately T • Using this linear approximation based on duration, over the horizon h the bond has a 95% chance of being worth more than

22. VaR for Bonds (cont’d) • Now suppose that instead of a single bond we have a portfolio of zero-coupon bonds • As with a portfolio of stocks, we can use the delta approximation, only instead of correlated stock returns we have correlated bond yields

23. VaR for Bonds (cont’d) • In general, if we are analyzing the risk of an instrument with multiple cash flows, the first step is to find the equivalent portfolio of zero-coupon bonds • For example, a 10-year bond with semiannual coupons = a portfolio of 20 zero-coupon bonds • Every interest rate claim is decomposed in this way into interest rate “buckets” containing the claim’s constituent zero-coupon bonds • A set of bonds and swaps reduces to a portfolio of long and short positions in zero-coupon bonds

24. VaR for Bonds (cont’d) • We need volatilities and correlations for all the bonds • Volatility and yields are tracked only at certain benchmark maturities • The goal is to find an interpolation procedure to express any hypothetical zero-coupon bond in terms of the benchmark zero-coupon bonds • This procedure in which cash flows are allocated to benchmark claims is called cashflow mapping

25. VaR for Bonds (cont’d) • Suppose that we wish to assess the risk of a 12-year zero-coupon bond, given information on the 10-year and 15-year zero-coupon bonds • We can use simple linear interpolation to obtain the yield and yield volatility for the 12-year bond from those of the 10-year and 15-year bonds • If the yield and volatility of the t-year bond are yt and t, the (26.14) (26.15) • These interpolations enable us to determine the price and volatility of $1 paid in year 12. Note, however, that they do not provide correlations between the 12-year zero and the adjacent benchmark bonds

26. VaR for Bonds (cont’d) • The price is • To find the combination of the 10- and 15-year zero-coupon bonds have the same volatility as the hypothetical 12-year bond, we must solve (26.16) • where  equals the fraction allocated to the 10-year bond • Since this is a quadratic equation, there are two solutions for . Typically, only one of the two solutions will be economically appealing • Given the weights, value at risk for the 12-year bond can be computed in the same way as VaR for a bond portfolio • Similarly, mapping can be applied to any claim with multiple cash flows

27. Estimating Volatility • Volatility is the key input in any VaR calculation • In most examples, return volatility is assumed to be constant and returns are independent over time • We require correlation estimates in order to compute volatilities of portfolios • Assessing return correlation is complicated • Over horizons as short as a day, returns may be negatively correlated due to factors such as bid-ask bounce • With commodities, return independence is not reasonable for long horizons due to supply and demand responses

28. Bootstrapping Return Distributions • It is possible to use observed past returns to create an empirical probability distribution of returns that can then be used for simulation. This procedure is called bootstrapping • The idea of bootstrapping is to sample, with replacement, from observed historical returns under the assumption that future returns will be drawn from the same distribution • Advantages and disadvantages of bootstrapping • The advantage is that, since bootstrapping is not based on a particular assumed distribution, it is consistent with any distribution of returns • The disadvantage is that key features of the data might be lost when the data are reshuffled. There is also the question of how to bootstrap multiple series in such a way that correlations are preserved

29. Issues With VaR • One of the problems with VaR is that small changes in the VaR probability can cause VaR to change by a large amount • Suppose you are comparing activity C, which generates a $1 loss with a 1.1% probability, with activity D, which generates a $1m loss with a 0.9% probability • Any reasonable rule would require more capital for activity D, but a 1% VaR would be greater for C than for D

30. Issues With VaR (cont’d) • One approach to improving VaR is to compute the average loss conditional upon the VaR loss being exceeded. This is called the Tail VaR • The 1% Tail VaR for activity C would be $1, and the 1% Tail VaR for activity D would be $900,000 (instead of the VaR of $0)

31. Issues With VaR (cont’d) • It is argued that a reasonable risk measure should have certain properties, among them subadditivity • If (X) is the risk measure associated with activity X, then  is subadditive if for two activities X and Y (26.21) • This says that the risk measure (i.e., the capital required) for the two activities combined should be less than for the two activities separately • VaR is not subadditive