1. Advanced Risk Management I Lecture 5 Value at Risk & co.

2. Map of the exposures Equity –Country –Sector Bond –Currency and bucket –Issuer (rating class) and bucket Foreign exchange –Value of the exposures in foreign currency

3. Profit and loss Define, at time t, for a given market, –A set of maturities t 1,t 2,…t n –A set of nominal cash-flows c 1,c 2,…c n –A set of discount factors P(t,t 1 ),P(t,t 2 )…P(t,t n ) The mark-to-market value at time t is V(t) = c 1 P(t,t 1 )+ c 2 P(t,t 2 )+ …+c n P(t,t n ) At time t +, te mark-to-market is P(t +,t i )=(1+r i ) P(t,t i ) for every i, so that V(t + )-V(t) = c 1 r 1 P(t,t 1 )+ c 2 r 2 P(t,t 2 )+ …+c n r n P(t,t n )

4. Risk measurement The key problem for the construction of a risk measurement system is then the joint distribution of the percentage changes of value r 1, r 2,…r n. The simplest hypothesis is a multivariate normal distribution. The RiskMetrics™ approach is consistent with a model of “locally” normal distribution, consistent with a GARCH model.

5. Risk measurement methodologies Parametric approach: assume a distribution conditionally normal (EWMA model ) and is based on volatility and correlation parameters Monte Carlo simulation: risk factors scenarios are simulated from a given distributon, the position is revaluated and the empirical distribution of losses is computed Historical simulation: risk factors scenarios are simulated from market history, the position is revaluated, and the empirical distribution of losses is computed.

6. Value-at-Risk Define X i = r i c i P(t,t i ) the profit and loss on bucket i. The loss is then given by –X i. A risk measure is a function  (X i ). Value-at-Risk: VaR(X i ) = q  (–X i ) = inf(x: Prob(–X i  x) >  ) The function q  (.) is the  level quantile of the distribution of losses  (X i ).

7. VaR as “margin” Value-at-Risk is the corresponding concept of “margin” in the futures market. In futures markets, positions are marked-to-market every day, and for each position a margin (a cash deposit) is posted by both the buyer and the seller, to ensure enough capital is available to absorb the losses within a trading day. Likewise, a VaR is the amount of capital allocated to a given risk to absorb losses within a holding period horizon (unwinding period).

8. VaR as “capital” It is easy to see that VaR can also be seen as the amount of capital that must be allocated to a risk position to limit the probability of loss to a given confidence level. VaR(X i ) = q  (–X i ) = inf(x: Prob(–X i  x) >  ) = inf(x: Prob(x + X i > 0) >  ) = = inf(x: Prob(x + X i  0)  1 –  )

9. VaR and distribution Call F X the distribution of X i. Notice that F X (–VaR(X i )) = Prob(X i  –VaR(X i )) = Prob(– X i >VaR(X i )) = Prob(– X i > F –X –1 (  )) = Prob(F –X (– X i ) >  ) = 1 –  So, we may conclude Prob(X i  –VaR(X i )) = 1 – 

10. VaR methodologies Parametric: assume profit and losses to be (locally) normally distributed. Monte Carlo: assumes the probability distribution to be known, but the pay-off is not linear (i.e options) Historical simulation: no assumption about profit and losses distribution.

11. VaR in a parametric approach p i =c i P(t,t i ) marking-to-market of cash flow i r i, percentage daily change of i-th factor X i, profits and losses p i r i Example: r i has normal distribution with mean  i and volatility  i, Take  = 99% Prob(r i   i –  i 2.33) = 1% If  i = 0, Prob(X i = r i p i  –  i p i 2.33) = 1% VaR i =  i p i 2.33 = Maximum probable loss (1%)

12. Volatility estimation Volatility estimation is the key issue in the parametric approach Choice of the information: implied and historical Measurement risk Model risk

13. Volatility information Historical volatility –Pros: historical info available for a large set of markets –Cons: history never repeats itself in the same way Implied vol –Pros: forward looking –Cons: available for a limited number of markets

14. Measurement risk Estimation risk of volatility can be reduced using more information on –Opening and closing prices –Maximum and minimum price in the perido Estimators: i) Garman and Klass; ii) Parkinson; iii) Rogers and Satchell; iv) Yang and Zhang

15. Estimation risk (1) O i and C i are opening and closing prices of day i respectively H i and L i are the highest and lowest prices of day i. Parkinson: GK

16. Estimation risk (2) Define: o i = O i – C i-1, h i = H i – O i, l i = L i – O i, c i = C i – O i. Moreover,  2 o and  2 c are variances computed with opening and closing prices respectively Rogers-Satchell : Yang-Zang:

17. Estimation risk (3) Parkinson: 5 times more efficient –Mean return = 0; “opening jump” f = 0 Garman and Klass: 6 times more efficient –Mean return = 0; “opening jump” f  0 Rogers and Satchell: –Mean return  0; “opening jump” f = 0 Yang and Zhang: –Mean return  0; “opening jump” f  0

18. Example: Italian blue chips

19. Results

20. Model risk Beyond estimation risk, it may happen that volatility itself may change in time, making the distribution non normal. Garch models: shocks reaching the return change the volatility of the nexgt period return. Stochastic volatility models: volatility may depend on other variables than the return itself.

21. Blue chips volatility

22. Garch(p,q) models Conditional distribution of the returns is normal, but volatility changes in time following an autoregressive process of the ARMA(p,q) kind. For example, the Garch(1,1) model is:

23. Garch: ABC… In a Garch model the unconditional distribution NON of returns is not normal, and in particular is leptokurtic (“fat-tails”): extreme events are more likely than under the normal distribution In a Garch model the future variance is forecasted recursively with the formula The degree of persistence is given by  1 +  1  1

24. A special Garch… Assume:  = 0 and  1 +  1 = 1. This is integrated Garch (Igarch) without drift: – i) volatility is persistent: every shock remains in the history of volatility forever –ii) the best predictor of time t + i volatility is t + i – 1 volatility –iii) time t volatility is given by (   1 )

25. …called EWMA Notice that IGarch(1,1) with  = 0 is the same as a model in which volatility is updated with a moving average with exponentially decaying weights (EWMA). The model, with parameter = 0.94, is employed by RiskMetrics™ to evaluate volatility and correlations. The model corresponds to an estimate of volatility that weights the more recent observations (the parameter corresponds to giving positive weights to the last 75 observations)

26. Volatility estimates

27. Ghost feature Tuning the weights of the EWMA option allows to reduce the relevance of a phenomenon called the ghost feature. Ghost feature: a shock continues to affect with the same weight the VaR estimate for all the period it remains in the sample, and when it exits from the sample, the VaR estimate changes with no apparent motivation.

28. Cross-section aggregation Once the Value-at-Risk is computed for every factor and position, the measure is aggregated across factors or across different business units. Aggregation is performed according to two methods –Undiversified VaR: the algebraic sum of individual VaR values –Diversified VaR: quadratic sum computed with correlation matrix C.

29. Cross-section aggregation: diversified VaR

30. VaR: temporal aggregation Aggregating VaR for different uwinding periods require assumption concerning the dynamic process describing losses The relationship is Notice: the relationship is based on the assumption that: –i) shocks are not serially correlated –ii) the portfolio does not change during the unwinding period

31. Example Position: 1 mil. euros on Italian equity and 0.5 mil. euros on US equity. Stocks on the US market are denominated in dollars. Exposure: 1 000 000 Euro Italy equity 500 000 Euro US equity 500 000 Euro US/Euro exchange rate risk

33. Value-at-Risk validation Once one has built a system for the computation of VaR, how to test its effectiveness? A possible strategy is to verify how many times in past history losses have been higher than the VaR measure computed for the corresponding periods. These are called validation procedures (or backtesting)

34. What P&L is used for validation? Notice again the difference between market price and marked-to-market price Since VaR has the goal of evaluating the marked-to- market loss of a position, the validation procedure must be carried out on the same concept of value Since market prices are determined by other elements than the mark-to-market (liquidity factors and others), it would be mistake to evaluate the VaR measure directly on market losses.

35. Kupiec test A statistical test, suggested by Kupiec, is based on the hypothesis that losses exceeding VaR be indipendent. Based on this hypothesis one may compare the number x of episodes of exceeding losses out of a sample on N cases, and the binomial distribution with probability 

36. Likelihood ratio The test is simply given by the ratio of the probability to extract x excess losses from the binomial distribution with respect to the theoretical probability. The test, which is distributed as chi-square with one degree of freedom, is

37. Example In applications one typically takes one year of data and a 1% confidence interval If we assume to count 4 excess losses in one year, Since the value of the chi-square distribution with one degree of freedom is 6.6349, the hypothesis of accuracy of the VaR measure is not rejected ( p- value of 0.77 è 38,02%).

38. Christoffersen extension A flaw of Kupiec test isnbased on the hypothesis of independent excess losses. Christoffersen proposed an extension taking into account serial dependence. It is a joint test of the two hypotheses. The joint test may be written as LR cc = LR un + LR ind where LR un is the unconditional test and LR ind is that of indipendence. It is distributed as a chis- square with 2 degrees of freedom.