SENSITIVITY ANALYSIS OF VALUES AT RISK

Transcription

1 SENSITIVITY ANALYSIS OF VALUES AT RISK Cristian GOURIEROUX CREST and CEPREMAP Jean-Paul LAURENT ISFA, University of Lyon and CREST Olivier SCAILLET IAG/Economics Dpt - UCL May 2 Paper publised in te Journal of Empirical Finance, Vol.7 (3-4), , 2. We will also use some results from Nonparametric estimation and sensitivity analysis of expected sortfall by O. Scaillet.

2 Agenda. Motivation ² Risk Management / Portfolio Selection 2. Main Teoretical Results ² Analytical Expressions for st and 2nd Derivatives of Value at Risk (VaR) w.r.t. Portfolio Allocation ² Analytical Expressions of rst derivatives of expected sortfall 3. Some Examples ² Gaussian Distribution wit/witout Unobserved Heterogeneity 4. Application ² VaR E cient Portfolios 5. Estimation Procedure ² Nonparametric Approac (Kernel) 6. Empirical Illustration ² Frenc Stock Data

3 . Motivation ² Framework regulatory environment proprietary Risk Measurement Models VaR = Syntetic Measure of Risk Portfolio ² Risk Management risk analysis risk control portfolio optimization ² Need to go furter tan a single number ² Wat are te risk drivers?. VaR of subportfolios ex-post analysis aggregation issues VaR is neiger additive or subadditive 2. Statistical analysis of risk scenarios 3. Incremental VaR (sensitivity analysis) ex-ante management of risk limits Large Portolios preclude online computations =) Sensitivity of VaR w.r.t. portfolio allocation

4 ² Wy determine convexity of VaR w.r.t. portfolio allocation? ² Portfolio Selection under VaR Constraints Determination of Optimal Portfolios Need to ceck tat te set of portfolios satisfying VaR constraint is convex ² VaR may not be subadditive (i.e. a coerent measure of risk) No bonus for diversi cation Poor internal risk management subadditivity can be statistically tested... and depends on nancial intermediaries policies... Credit risk, Out of te money options

5 2. Main Teoretical Results ² VaR de nition: n Financial Assets wit Prices p i;t at Date t X Portfolio Value : W t (a) = n a ip i;t = a p t i= P t [W t+ (a) W t (a)+var t (a; ) < ] = Upper Quantile ( )... since wit y t+ = p t+ p t P t [ a y t+ >VaR t (a; )] = ² First properties VaR is degree one positively omogeneous: VaR t ( a; ) = VaR t (a; ) ar t(a; = VaR t (a; ) (Euler) Risk contribution of i: a ar t(a; i VaR is not always subadditive: VaR t (a + b; ) VaR t (a; )+VaR t (b; )? subadditivity + positive omogeneity ) convexity of VaR(a; ) (w.r.t portfolio allocation a)

6 ² Sensitivities of VaR i) st derivative of ar t (a; = E t [y t+ ja y t+ = VaR t (a; )] ii) 2nd derivative of 2 VaR t (a; ) log g a;t ( t (a; ))V t [y t+ ja y t+ = VaR t (a; ( V t[y t+ja y t+ = z]) z=var t (a; ) ; wit g a;t te conditional p.d.f of a y t+

7 ² Expected Sortfall de nition: m t (a; ) =E t [ a y t+ j a y t+ >Var t (a; )] Also known as Mean Excess Loss, astailvar or as Lower Partial Moment. Anoter commonly used measure of risk Expected Sortfall is subadditive i) st derivative of Expected t (a; = E t [y t+ j a y t+ >VaR t (a; )] Similar result to tat obtained on sensitivity of VaR

8 3. Some Examples ) First example: Gaussian Distribution ² y t+» N (¹ t ; t ) ² VaR t (a; ) = a ¹ t +(a t a) =2 z ² z quantile of level of Gaussian distribution. Remark: ² (a t a) =2 : standard dev. of portfolio absolute returns ² subadditivity of standard deviations: ² ((a + b) t (a + b)) =2 (a t a) =2 +(b t b) =2 ²)In te Gaussian case, VaR is always subadditive i) st derivative of ar t (a; = ¹ t + t a (a t a) =2z = ¹ t + ta a t a (VaR t(a; )+a ¹ t ) = E t [y t+ ja y t+ = VaR t (a; )]

9 ii) 2nd derivative of 2 VaR t (a; = 2 z 4 (a t a) =2 t taa t a t a 3 2 VaR t (a; = z (a t a) =2V t[y t+ ja y t+ = VaR t (a; )] Remark log g a;t ( VaR t (a; )) = VaR t(a; )+a ¹ a t a = z (a t a) =2 2nd term = (conditional omoscedasticity) ² Expected Sorfall m t (a; ) ² m t (a; ) = a ¹ t +(a t a) =2'(z ) ' : Gaussian density i) st derivative of Expected t (a; = ¹ t + t a '(z ) (a t a) =2

10 ) Second example: Gaussian wit Unobserved Heterogeneity ² y t+ j u» N (; t (u)) ² wit eterogeneity factor u wit distribution ² Gaussian Random Walk wit Stocastic Volatility Ceck of VaR log g ( VaR t (a; )) = VaR t (a; )E ~ 4 a 7 5 > : V t[y t+ ja y t+ = V 4 z t(u)a a t (u)a 2 6 t (u)a =+2zV 4 a t (u)a wic is nonnegative for z = VaR t (a; )! ;

11 4. Application : VaR E cient Portfolios Budget w allocated among n Risky Assets and Riskfree Asset (r) 8 >< max a a E t y t+ >: s:t: V ar t (a; ) VaR o w( + r) = VaR g o VaR o = Bound for Autorized Risk (CAD) First Order Conditions : 8 >< >: E t y t+ = t VaR t (a t ; ar (a t ; ) VaR g o Proportionality between Global and Local Expectations : " E t y t+ = t E t yt+ ja t y t+ = VaR g # o

12 5. Estimation Procedure Nonparametric approac (kernel) : i.i.d. returns ² Estimation of VaR P [ a y t+ >VaR(a; )] = estimated by Gaussian Kernel T t= a y t VaR d C A = Gauss-Newton Algoritm : var (p+) = var (p) + T t= T a y t var (p) C A a y t + var (p) B A t= ' Starting values : Gaussian VaR or Empirical Quantile

13 ² Convexity of VaR VaR(a; If, for negative z values positive semide log g a;t > [y t+ja y t+ = À Estimator of p.d.f. of portfolio value ^g a (z) = T t= ' a y t z C A Estimator of conditional variance ^V [y t+ ja y t+ = z] = t= t= y t a y t z a y t z A A t= y t a y t z 2 4 t= ' A a y t z a y t z 32 A5 A

14 ² Estimation of VaR e cient portfolio Simple forms of st and 2nd derivatives of VaR =) Gauss-Newton Algoritm a (p+) = a (p) VaR (a (p) ; ar (a(p) ; ) wit ( g VaR o VaR(a (p) ; )) + Q(a (p) ; ) Ey (a (p) ; )] Ey t+ (a (p) ; )] Ey t =2 Q(a (p) ; (a (p) ; (a (p) ; (a(p) ; ) Teoretical recursion replaced by empirical counterpart

15 6. Empirical Illustration Frenc Stock Data from CAC 4 - Tompson-CSF (electronic devices) - L Oréal (cosmetics) Daily Returns : 4//997 to 5/4/999 (546 obs.) Empirical Results : Standard Normal VaR underestimate (Skew. and Kurt.) Smooter patterns for Kernel estimates Nonmonotonicity of Sensitivities Ceck for Convexity fails for some allocations VaR Symmetry lost VaR E cient Portfolios = Tangency Points of a ^¹ + a 2^¹ 2 = cst wit IsoVar curve of level g VaR o