Copulas: a Review and Recent Developments (2007)

Z¾ C;à (s) =dC(F X1 (x 1 );::: ;F Xn (x n ));fÃ(X 1 ;::: ;X n )·sg½ C;à (s) =inf¡ ¡x 1 ;::: ;x n¡1 2(¡1;1) Cdual F X1 (x 1 );::: ;F Xn¡1 (x n¡1 );F Xn á1n¡1 (s)¢¢ ;where à n¡1(s) ¡1 is the left continuous (generalized) inverse function of à n¡1 (s).Theorem (best possible bounds, Embrechts et al. (2003a)). Under the abovenotations if a copula C associated to (X 1 ;:::;X n ) satis¯es C ¸ C 0 and C dual · C1dualfor some given n-copulas C 0 and C 1 , then¿ C0 ;Ã(s) · ¾ C;à (s) · ½ C1 ;Ã(s):Note that, ¾ C;à (s) =P (Ã(X 1 ;:::;X n ) · s) =¤ à (s). Also, the bounds ¿ C0 ;Ã(s)and ½ C1 ;Ã(s) are themselves distribution functions. Since there is not necessarily aunique copula satisfying the inequalities in the last theorem, it is assumed additionallythat there exists a copula C such that C ¸ C 0 and C dual · C1 dual . The copulas C 0and C 1 represent the partial information available about the dependence structure of(X 1 ;:::;X n ). In Embrechts et al. (2003b) is proposed a methodology in order to ¯ndthe bounds for VaR p (¤ à (s)), recently re¯ned by Embrechts and Puccetti (2006).In most cases, the bounds in last theorem does not have closed form. In general,one has to use numerical approximations as proposed by Williamson and Downs(1990). Their algorithm is based on the discretization of ¿ C0 ;Ã(s), ½ C1 ;Ã(s) andonaduality principle due to Frank et al. (1987).Some problems related to the control and optimization of VaRin a portfolio withnon-normal or non-log-normal returns have taken the researchers to look for anotherquantile measures. Such a measure is called Tail Value-at-Risk at level p, andisde¯ned byZ 1TVaR p (X) = 1 VaR p (X)dq; p 2 (0; 1):1 ¡ p pIt is the arithmetic average of the VaR p (X), from p on. Therefore, TVaRis a measureof risk more suitable than VaR because it takes into account the losses beyond theVaR level. Note that always TVaR p (X) > VaR p (X), TVaR 0 (X) = E(X) andTVaR p (X) is a non-decreasing function of p. The fundamental di®erences betweenVaRand TVaR canbesummarizedasfollows: VaRis the \optimistic" lower boundof the tail loses and the TVaR is the expected value of tail losses. Therefore, TVaRcan be interpreted as a \conservative" risk measure.Rockfellar and Uryasev (2000, 2002) use CV aR instead of TVaR and provide analternative de¯nition of CV aR as a coherent measure (i.e. satisfy the sub-additivity,monotonicity, homogeneity and translation invariance axioms, see Artzner et al.18¥