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Chapter 26 Value at Risk 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-2 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-3 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-4 Value at Risk for One Stock • Suppose ~xh 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 Pr( xh xh (c)) c • Suppose a portfolio consists of a single stock and we wish to compute value at risk over the horizon h © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-5 Value at Risk for One Stock (cont’d) • If we pick a stock price S h and the distribution of the stock price after h periods, Sh, is lognormal, then Sh (c) S0e ( 0.5 2 ) h hN 1 ( c ) (26.5) © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-6 Value at Risk for One Stock (cont’d) • In practice, it is common to simplify the VaR calculation by assuming a normal return rather than a lognormal return. A normal approximation is Sh S0 (1 h z h ) (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 Sh S0 (1 z h ) (26.8) • Both equations become less reasonable as h grows © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-7 Value at Risk for One Stock (cont’d) • Comparison of three models—lognormal, normal with mean, and normal without mean © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-8 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-9 VaR for Two or More Stocks (cont’d) • Let the annual mean of the return on stock i, 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 n W Wi i 1 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-10 VaR for Two or More Stocks (cont’d) • 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 n 1 Rh W i 1 Wi i ,h – Assuming normality, the annualized distribution of the portfolio return is 1 Rh ~ N W n hW i 1 i i, 1 n n hWW i j 2 i j ij W i1 j 1 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. (26.9) 26-11 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-12 Delta Approximation ~ , we can approximate the • If the return on stock i is i return on the option as i~i , where i is the option delta • The expected return on the stock and option portfolio over the horizon h is then n 1 Rp W S ( N ) i 1 i i i i i (26.10) – The term i + Nii measures the exposure to stock i • The variance of the return is 1 n n 2 p 2 Si S j (i Ni i )( j N j j ) i j ij W i1 j 1 (26.11) • With this mean and variance, we can mimic the n-stock analysis © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-13 Delta Approximation (cont’d) • Comparison of exact portfolio value with a delta approximation © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-14 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-15 Monte Carlo Simulation (cont’d) • Example 26.6 – Consider the 1-week 95% value at risk of an atthe-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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-16 Monte Carlo Simulation (cont’d) • First, we randomly draw a set of z ~ N(0,1), and construct the stock price as S h S0e ( 0.5 2 ) h h z (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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-17 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-18 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-19 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-20 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 P(T ) 1 T T h (1.645) © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-21 VaR for Bonds (cont’d) • Now suppose that instead of a single bond we have a portfolio of zerocoupon bonds – As with a portfolio of stocks, we can use the delta approximation, only instead of correlated stock returns we have correlated bond yields © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-22 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-23 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-24 VaR for Bonds (cont’d) • Suppose that we wish to assess the risk of a 12year 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 y12 (0.6 y10 ) (0.4 y15 ) (26.14) 12 (0.6 10 ) (0.4 15 ) (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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-25 VaR for Bonds (cont’d) • The price is e y12 12 • 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 212 ( 2 210 ) [(1 )2 210 ] [210.15 (1 )1015] (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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-26 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-27 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-28 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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-29 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) © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-30 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 (X Y ) (X) (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 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-31 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-32 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-33 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-34 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-35 © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 26-36