On the L p -quantiles and the Student t distribution

Transcription

1 1 On the L p -quantiles and the Student t distribution Valeria Bignozzi based on joint works with Mauro Bernardi, Luca Merlo and Lea Petrella MEMOTEF Department, Sapienza University of Rome Workshop Recent Advances in Quantile and M-quantile Regression, Pisa, July 15, 2016

2 2 Motivation Basel III (pillar II) Financial institutions (banks/insurance companies) are required to hold risk capital requirement for their financial losses B Y represents a financial loss B Value-at-Risk (VaR), Expected Shortfall (ES)

3 3 Risk measures B VaR (Y )=q (Y ), large Quantile q, 2 (0, 1) (Koenker and Bassett, 1978) q (Y )=argmin m2r E[ ((Y m) +)+(1 )((Y m) )] y + =max(y, 0) and y =max( y, 0) B Expected Shortfall ES, 2 (0, 1] ES (Y )= 1 1 Z 1 q u (Y )du I ES is a coherent risk measure (Artzner et al. 1999)

4 4 Anewriskmeasure... EVaR (Y )=e (Y ) (Aigner et al. 1976, Newey and Powell, 1987) Expectiles e, 2 (0, 1) e (Y )=argmin m2r E ((Y m) + ) 2 +(1 )((Y m) ) 2 Unique solution to the equation E[(Y m) + ]=(1 )E [(Y m) ] B EVaR 1/2 (Y )=E[Y] B For 1/2Expectilesaretheonlyelicitable coherent risk measure (Ziegel 2014, Bellini and B. 2015, Delbaen, Bellini, B., Ziegel 2015)

5 5 Estimating the risk measures VaR and EVaR are elicitable risk measures (Y )=argmin m2r E[L(Y m)] B ES is NOT elicitable B Elicitability is useful for backtesting (Gneiting, 2011) 1 n nx L(y i m i ) i=1 I (y1,...,y n ) are observed outcomes of the financial asset I (m1,...,m n ) are forecasts of the risk measure (Y )

6 5 Estimating the risk measures VaR and EVaR are elicitable risk measures (Y )=argmin m2r E[L(Y m)] B ES is NOT elicitable B Elicitability is useful for backtesting (Gneiting, 2011) 1 n nx L(y i m i ) i=1 I (y1,...,y n ) are observed outcomes of the financial asset I (m1,...,m n ) are forecasts of the risk measure (Y ) B Regression (Mauro!!)

7 6 L p -quantiles B Introduced by Chen (1996) L p -quantiles, p,, p 2, 2 (0, 1) p, (Y )=argmin m2r E[ ((Y m) +) p +(1 )((Y m) ) p ] Unique solution to the equation E ((Y m) + ) p 1 =(1 )E ((Y m) ) p 1 B Elicitable risk measures B The L 2 -quantile corresponds to the expectile B Bellini et al. (2014)

8 7 Computing L p -quantiles L p -quantiles generally are not available in closed form solution: B Solve numerically E ((Y m) + ) p 1 =(1 )E ((Y m) ) p 1 L p -quantiles can be interpreted as quantiles of another distribution: B Jones (1994) B Y F E[((Y m) ) p 1 ] = E[((Y m) + ) p 1 ]+E[((Y m) ) p 1 ] = G(m) The L p -quantile of a distribution F correspond to the quantile of a distribution G

9 8 L p -quantiles and quantiles Is there a distribution F such that for Y F E[((Y m) ) p 1 ] = E[((Y m) + ) p 1 ]+E[((Y m) ) p 1 ] = F (m) that is p, (Y )=q (Y )forany 2 (0, 1)?

10 8 L p -quantiles and quantiles Is there a distribution F such that for Y F E[((Y m) ) p 1 ] = E[((Y m) + ) p 1 ]+E[((Y m) ) p 1 ] = F (m) that is p, (Y )=q (Y )forany 2 (0, 1)? B Koenker (1992) Is there a distribution F such that 2, (Y )=q (Y ) for any 2 (0, 1)?

11 9 L p -quantiles and quantiles (Cont ed) B Koenker (1993) Yes : 8 >< F (y) = >: q y q y if y 0 if y<0 I For c = p 2ifY t(2), then cy F B Zou (2014) characterises distribution functions for which 2,!( ) (Y )=q (Y ) for monotone functions!( )

12 10 General case What can we say about the general L p -quantiles and quantiles?

13 10 General case What can we say about the general L p -quantiles and quantiles? Theorem (Bernardi, B., Petrella) Let Y be a random variable with Student t distribution with p degrees of freedom, then p, (Y )=q (Y ) for any 2 (0, 1) B Di erent proof for p even or odd B The proof is rather long and involves concepts from combinatorial analysis B It requires a recursive formula for the truncated moments of the Student t distribution

14 11 Proof (Only a sketch!) From the first order condition E ((Y m) + ) p 1 =(1 )E ((Y m) ) p 1

15 11 Proof (Only a sketch!) From the first order condition E ((Y m) + ) p 1 =(1 )E ((Y m) ) p 1 B Let p be an odd number, then p 1 X p 1 ( m) k E[Y p 1 k ] G p 1 k,y (m) =0 k k=0

16 11 Proof (Only a sketch!) From the first order condition E ((Y m) + ) p 1 =(1 )E ((Y m) ) p 1 B Let p be an odd number, then p 1 X p 1 ( m) k E[Y p 1 k ] G p 1 k,y (m) =0 k k=0 B Let p be an even number, then p 1 X p 1 ( m) k E[Y p 1 k ]+(1 2 )G p 1 k,y (m) =0 k k=0 where G p 1 k,y (m) = Z m 1 y p 1 k df (y)

17 12 Truncated moments of the Student t Truncated moments: G j,y (m) = Z m 1 y j df (y) B For the Student t distribution with p degrees of freedom G j,y (m) = Yi+1 k=1 C p 1+ m2 p 1 p 2 b j 1 2 c X i=0 1 p j 2+2k + F Y (m)e[y j ], for 0 <japple p 1andG 0,Y (m) =F Y (m) B C p is the normalising constant m j 1 2i p i+1 (j 1)!! (j 1 2i)!!

18 Student t distribution with 3 degrees of freedom ξ p ξ 3 expectile quantile ES τ

19 Student t distribution with 4 degrees of freedom ξ p ξ 4 ξ 3 expectile quantile ES τ

20 Student t distribution with 5 degrees of freedom ξ p ξ 5 ξ 4 ξ 3 expectile quantile ES τ

21 16 Student t symmetry Theorem/Conjecture (B., Merlo, Petrella) Given a random variable Y with a Student t distribution with p degrees of freedom, the L p i+1 -quantile coincides with the L i -quantile where i =1,...,p: p i+1, (Y )= i, (Y ) B Proof is almost completed B Di erent cases depending on whether p and i are even or odds

22 17 Student t symmetry (Cont ed) Consequences Given a random variable Y with Student t distribution with p =4degrees of freedom, the analytical form for 2, (Y )= 3, (Y )is: 8 >< 2, (Y )= >: r r 2 2 p (1 p (1 ) 2 for apple 1 2 ) 1 for 2 2 B It is (up to our knowledge!) the only example of explicit expression for the expectile of a distribution (a part from the uniform distribution) B Work in progress to extend this result

23 18 Thank you for your kind attention!

24 19 References I Aigner, D. J., Amemiya, T. and Poirier, D. J. (1976). On the estimation of production frontiers: Maximum Likelihood Estimation of the parameters of a discontinuous density function. Internat. Econom. Rev., 17, Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance, 9(3), Bellini, F. and Bignozzi, V. (2015). On elicitable risk measures. Quant. Finance, 15(5), Bernardi, M., Gayraud, G. and Petrella, L. (2015). Bayesian tail risk interdependence using quantile regression. Bayesian Anal., 10(3),

25 20 References II Chen, Z. (2001). Conditional L p -quantiles and their application to the testing of symmetry in non-parametric regression. Statist. Probab. Lett, 29, Delbaen, F., Bellini, F., Bignozzi, V., Ziegel, J. F. (2015). On convex risk measures with the CxLS property. Finance Stoch., forthcoming. Gerlach, R. H., Chen W. S. and Lin, L. (2012). Bayesian semi-parametric expected shortfall forecasting in financial markets. Business Analytics Working Paper Series. Gneiting, T. (2011). Making and evaluating point forecasts. J. Amer. Statist. Assoc., 106(494),

26 21 References III Newey, W. and Powell, J. (1987). Asymmetric least squares estimation and testing. Econometrica, 55, Koenker, R. and Basset, G. (1978). Regression Quantiles. Econometrica, 46, Yu, K. and Moyeed, R. A. (2001). Bayesian Quantile Regression. Statist. Probab. Lett, 54, Zhu, D. and Zinde-Walsh, V. (2009). Properties and estimation of asymmetric exponential power distribution. J. Econometrics, 148(1), Ziegel, J. (2014). Coherence and elicitability. Math. Finance, forthcoming.