1. Market Risk VaR: Historical Simulation Approach N. Gershun

2. Historical Simulation Collect data on the daily movements in all market variables. The first simulation trial assumes that the percentage changes in all market variables are as on the first day The second simulation trial assumes that the percentage changes in all market variables are as on the second day and so on

3. Historical Simulation continued Suppose we use n days of historical data with today being day n Let v i be the value of a variable on day i There are n-1 simulation trials Translate the historical experience of the market factors into percentage changes The ith trial assumes that the value of the market variable tomorrow (i.e., on day n+1) is

4. Rank the n-1 resulting values VaR is the required percentile rank Historical Simulation continued

5. Example of Historical Simulation Assume a one-day holding period and 5% probability Suppose that a portfolio has two assets, a one- year T-bill and a 30-year T-bond First, gather the 100 days of market info DateT-Bond Value% ChangeT-Bill Value% Change 12/31/10102-97- 12/30/101002.00%98-1.02% 12/29/10973.09%980.00% ::::: ::::: 9/12/10103-2.91%962.08% 9/11/101030.00%97-1.03%

6. Example of Historical Simulation cont. Apply all changes to the current value of assets in the portfolio T-bond value = 102 x % change T-bill value = 97 x % change T-BondModeledT-BillModeledPortfolio Date% ChangeValue% ChangeValue 12/31/102.00%104.04-1.02%96.01200.05 12/30/103.09%105.150.00%97.00202.15 :::::: :::::: 9/12/10-2.91%99.032.08%99.02198.05 9/11/100.00%102.00-1.03%96.00198.00

7. Example of Historical Simulation cont. Rank the resulting 100 portfolio values The 5th lowest portfolio value is the VaR Rank 1 2 3 4 5 : 99 100 Date 11/12/10 12/1/10 10/17/10 10/13/10 9/11/10 : 12/8/10 9/25/10 Value 195.45 196.24 197.13 197.60 198.00 : 202.15 203.00

8. Notes on Historical Simulation Historical simulation is relatively easy to do: Only requires knowing the market factors and having the historical information Correlations between the market factors are implicit in this method because we are using historical information In our example, short bonds and long bonds would typically move in the same direction

9. Accuracy Suppose that x is the qth quantile of the loss distribution when it is estimated from n observations. The standard error of x is where f(x) is an estimate of the probability density of the loss at the qth quantile calculated by assuming a probability distribution for the loss

10. Example We are interested in estimating the 99 percentile from 500 observations We estimated f(x) by approximating the actual empirical distribution with a normal distribution mean zero and standard deviation $10 million Using Excel, the 99 percentile of the approximating distribution is NORMINV(0.99,0,10) = 23.26 and the value of f(x) is NORMDIST(23.26,0,10,FALSE)=0.0027 The estimate of the standard error is therefore

11. Example (cont.) Suppose that we estimated the 99 th percentile using historical simulation as $25M Using our estimate of standard error, the 95% confidence interval is: 25-1.96×1.67<VaR<25+1.96×1.67 That is: Prob($21.7 $28.3) = 95%

12. Extension 1

13. Extension 2 Use a volatility updating scheme and adjust the percentage change observed on day i for a market variable for the differences between volatility on day i and current volatility Value of market variable under ith scenario becomes –Where  n+1 is the current estimate of the volatility of the market variable and  i is the volatility estimated at the end of day i-1

14. Extreme Value Theory Extreme value theory can be used to investigate the properties of the right tail of the empirical distribution of a variable x. (If we are interested in the left tail we consider the variable –x.) We then use Gnedenko’s result which shows that the tails of a wide class of distributions share common properties.

15. Extreme Value Theory Suppose F(*) is a the cumulative distribution function of the losses on a portfolio. We first choose a level u in the right tail of the distribution of losses on the portfolio The probability that the particular loss lies between u and u +y (y>0) is F(u+y) – F(u) The probability that the loss is greater than u is: 1-F(u)

16. Extreme Value Theory

17. Extreme Value Theory Gnedenko’s result shows that for a wide class of distributions, F u (y) coverges a Generalized Pareto Distribution 17

18. Generalized Pareto Distribution (GPD) GDP has two parameters  (the shape parameter) and  (the scale parameter) The cumulative distribution is The probability density function

19. f x (x) //  =+0.5  =-0.5 0 Generalized Pareto Distribution  = 0 if the underlying variable is normal  increases as tails of the distribution become heavier For most financial data  >0 and is between 0.1 and 0.4

20. Generalized Pareto Distribution (cont). G.P.D. is appropriate distribution for independent observations of excesses over defined thresholds GPD can be used to predict extreme portfolio losses

21. Maximum Likelihood Estimator 21 The observations, i, are sorted in descending order. Suppose that there are n u observations greater than u We choose  and  to maximize

22. Tail Probabilities Our estimator for the cumulative probability that the variable is greater than x is Extreme Value Theory therefore explains why the power law holds so widely

23. Estimating VaR Using Extreme Value Theory The estimate of VaR at the confidence level q is obtained by solving

24. Estimating Expected Shortfall Using Extreme Value Theory The estimate of ES, provided that the losses exceed the VaR, at the confidence level q, is given by:

25. Example Consider an example in the beginning of the lecture. Suppose that u= 4 and n u = 20. That is there are 20 scenarios out of total of 100 where the loss is greater than 4. Suppose that the maximum likelihood estimation results in  = 34 and  = 0.39 The VaR with the 99% confidence limit is

26. Example The VaR with the 99% confidence limit is