1. Advanced Risk Management I Lecture 7

2. 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%).

3. 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.

4. Value-at-Risk criticisms The issue of coherent risk measures (aximoatic approach to risk measures) Alternative techniques (or complementary): expected shorfall, stress testing. Liquidity risk

5. Coherent risk measures In 1999 Artzner, Delbaen-Eber-Heath addressed the following problems “Which features must a risk measure have to be considered well defined?” Risk measure axioms:  Positive homogeneity:  ( X) = (X)  Translation invariance:  (X +  ) =  (X) –   Subadditivity:  (X 1 + X 2 )   (X 1 ) +  (X 2 )

6. Convex risk measures The hypothesis of positive homogeneity has been criticized on the grounds that market illiquidity may imply that the risk increases with the dimension of the position For this reason, under the theory of convex risk measures, the axioms of positive homogeneity and sub-additivity were substituted with that of convexity  ( X 1 + (1 – ) X 2 )   (X 1 ) + (1 – )  (X 2 )

7. Discussion It is diversification a property of the measure? VaR is not sub-additive. Does it mean that information in a super-additive measure is irrelevant? Assume that one merges two businesses for which VaR is not sub-additive. He uses a measure that is sub-additive by definition. Does he lose some information that may be useful for his choice?

8. Expected shortfall Value-at-Risk is the quantile corresponding to a probability level. Critiques: –VaR does not give any information on the shape of the distribution of losses in the tail –VaR of two businesses can be super-additive (merging two businesses, the VaR of the aggregated business may increase –In general, the problem of finding the optimal portfolio with VaR constraint is extremely complex.

9. Expected shortfall Expected shortfall is the expected loss beyond the VaR level. Notice however that, like VaR, the measure is referred to the distribution of losses. Expected shortfall is replacing VaR in many applications, and it is also substituting VaR in regulation (Base III). Consider a position X, the extected shortfall is defined as ES = E(X: X  VaR)

10. Expected shortfall: pros and cons Pros: i) it is a measure of the shape of the distribution: ii) it is sub-additive, iii) it is easily used as a constraint for portfolio optimization Cons: does not give information on the fact that merging two businesses may increase the probability of default.

11. Stress testing Stress testing techniques allow to evaluate the riskiness of the position to specific events The choice can be made –Collecting infotmation on particular events or market situations –Using implied expectations in financial instruments, i.e. futures, options, etc… Scenario construction must be consistent with the correlation structure of data

12. Stress testing How to generate consistent scenarios Cholesky decomposition –The shock assumed on a given market and/or bucket propagates to others via the Cholesky matrix Black and Litterman –The scenario selected for a given market and/or bucket is weighted and merged with historical info by a Bayesian technique.

13. Multivariate Normal Variables Cholesky Decomposition –Denote with X a vector of independent random variables each one of which is ditributed acccording to a standard normal, so that the variance- covariance matrix of X is the n  n identity matrix Assume one wants to use these variables to generate a second set of variables, that will be denoted Y, that will be correlated with variance-covariance matrix given . –The new system of random variables can be found as linear combination of the independent variables –The problem is reduced to determining a matrix A of dimension n  n such that

14. Cholescky Decomposition –The solution of the previous problem is not unique meaning that there exost many matrices A that, multiplied by their transposed, give  as a result. If matrix  is positive definite, the most efficient method to solve the problem consists in Cholescky decomposition. Multivariate Normal Variables –The key point consists in looking for A in the shape of a lower triangular matrix.

15. Cholesky Decomposition –It may be verified that the elements of A can be recoverd by a set of iterative formulas –In the simple two-variable case we have Multivariate Normal Variables

16. Black and Litterman The technique proposed in Black and Litterman and largely used in asset management can be used to make the scenarios consistent. Information sources –Historical (time series of prices) –Implied (cross-section info from derivatives) –Private (produced “in house”)

17. Views Assume that “in house” someone proposes a “view” on the performance of market 1 and a “view” on that of market 3 with respect to market 2. Both “views” have error margins  i with covariance matrix  e 1 ' r = q 1 +  1 e 3 ' r - e 2 ' r = q 2 +  2 The dynamics of percentage price changes r must be “condizioned” on views “view” q i.

18. Conditioning scenarios to “views” Let us report the “views” in matrixform and compute the joint distribution ~

19. Conditional distribution The conditional distribution of r with respect to q is then and noticed that this may be interpreted as a GLS regression model (generalised least squares)

20. Esempio: costruzione di uno scenario Assumiamo di costruire uno scenario sulla curva dei tassi a 1, 10 e 30 anni. I valori di media, deviazione standard e correlazione sono dati da

21. A shock to the term structure

22. Stress testing analysis (1) The short rate increases to 6% (0.1% sd)

23. Stress testing analysis (1) The short rate increases to 6%(1% sd)