Conditional Value at Risk and Partial Moments for the Metalog Distributions
aa r X i v : . [ q -f i n . R M ] F e b Conditional Value at Risk and Partial Moments for theMetalog Distributions
Valentyn KhokhlovNovember 2019
Abstract
The metalog distributions represent a convenient way to approach many practicalapplications. Their distinctive feature is simple closed-form expressions for quantilefunctions. This paper contributes to further development of the metalog distributionsby deriving the closed-form expressions for the Conditional Value at Risk, a risk mea-sure that is closely related to the tail conditional expectations. It also addressed thederivation of the first-order partial moments and shows that they are convex withrespect to the vector of the metalog distribution parameters.
The metalog distribution family was developed by Keelin (2016) and represents a system ofcontinuous univariate probability distributions suitable for practical applications of fittinga distribution to the vector of known quantile values. As any metalog distribution quantilefunction has the closed-form expression that is linear with respect to the vector of thedistribution parameters, it is easy to find those parameters when the set of quantile valuesis given (e.g. from the set of observations).Keelin (2016) defined the metalog distribution family using its quantile function asfollows: M n ( α, a ) = a + a ln α − α for n = 2 ,M ( α, a ) + a ( α − .
5) ln α − α for n = 3 ,M ( α, a ) + a ( α − .
5) for n = 4 ,M n − ( α, a ) + a n ( α − . ( n − / for odd n ≥ ,M n − ( α, a ) + a n ( α − . ( n/ − ln α − α for even n ≥ , The closed-form expression for some of the metalog distribution moments were derivedin Keelin (2016). However, modern finance and risk management applications are largelybased on the tail risk measures, especially after the coherent risk measures concept was1ntroduced by Artzner et al. (1999). One of the most popular coherent risk measure is thethe Conditional Value at Risk (CVaR), also knows as the tail conditional expectation. Inthe paper we first focus on CVaR, and provide its closed-form expressions for the metalogdistributions. After that, we derive the expressions for the first partial moments and showthat they are convex with respect to the vector of the metalog distribution parameter.Their convexity allows applying a wide range of the optimization techniques.In this paper we use basically the same mathematical notation and format of expres-sions as in Norton et al. (2019) in order to make the derived closed-form CVaR expressioncompatible with the comprehensive set of the CVaR formulas derived for many continuousprobability distributions.
Let the loss be represented by a real valued random variable X that follows a metalogdistribution with the quantile function q n ( p, X ) = M n ( p, a ). The Conditional Value atRisk (CVaR), or the superquantile, at confidence level α is equal to the expected lossexceeding M n ( α, a ), given by¯ q n ( α, X ) = E [ X | X > M n ( α, a )] = 11 − α Z ∞ M n ( α, a ) xf n ( x, a ) dx = 11 − α Z α M n ( p, a ) dp where f n ( x, a ) is the Probability Density Function (PDF) of X . Notice that we can repre-sent the superquantile as an integral of the PDF or the quantile as x = M n ( p, a ). Proposition 1.
Assume X ∼ M etalog n ( a ) , then its superquantile equals to ¯ q n ( α, X ) = a − a ln h (1 − α ) α α − α i for n = 2 , ¯ q ( α, X ) + a [ αln α − α + 1] for n = 3 , ¯ q ( α, X ) + a α for n = 4 , ¯ q n − ( α, X ) + a n − α n +1 h . n +12 − ( α − . n +12 i for odd n ≥ , ¯ q n − ( α, X ) + a n − α . k +1 k +1 h Ψ (cid:16) k (cid:17) + γ + 2 ln i −− a n − α ( α − . k +1 ( k +1) h F (1 , k + 1; k + 2; 1 − α ) −− F (1 , k + 1; k + 2; 2 α −
1) + ( k + 1) ln α − α i for even n ≥ , where Ψ is the digamma function, F is the hypergeometric function, γ is the Euler–Mascheroniconstant, and for even n ≥ , k = n − . roof. For n = 2,¯ q ( α, X ) = 11 − α Z α M ( p, a ) dp = 11 − α Z α [ a + a ln p − p ] dp == a − α Z α dp + a − α Z α ln p − p dp == a + a − α (cid:20) ln (1 − p ) + p ln p − p (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α , and as lim p → − h ln(1 − p ) + p ln p − p i = 0,¯ q ( α, X ) = a + a − α (cid:20) − ln (1 − α ) − α ln α − α (cid:21) = a − a − α ln h (1 − α ) − α α α i == a − a − α [(1 − α ) ln (1 − α ) + α ln α ] = a − a ln h (1 − α ) α α − α i . For n = 3,¯ q ( α, X ) = 11 − α Z α M ( p, a ) dp = 11 − α Z α h M ( p ; a ) + a (cid:16) p − (cid:17) ln p − p i dp == ¯ q ( α, X ) + a − α Z α p ln p − p dp − . a − α Z α ln p − p dp == ¯ q ( α, X ) + 0 . a − α (cid:2) (1 − α ) ln (1 − α ) + α ln α (cid:3) ++ 0 . a − α (cid:20) − α ln α − α + (1 − α ) − ln (1 − α ) (cid:21) == ¯ q ( α, X ) + 0 . a − α (cid:2) (1 − α ) ln(1 − α ) + α ln α − α ln α + α ln(1 − α )++ (1 − α ) − ln(1 − α ) (cid:3) == ¯ q ( α, X ) + 0 . a − α (cid:2) − α (1 − α ) ln (1 − α ) + α (1 − α ) ln α + (1 − α ) (cid:3) == ¯ q ( α, X ) + a (cid:20) α ln α − α + 1 (cid:21) . n = 4,¯ q ( α, X ) = 11 − α Z α M ( p ; a ) dp = 11 − α Z α (cid:20) M ( p ; a ) + a (cid:18) p − (cid:19)(cid:21) dp == ¯ q ( α, X ) + a − α Z α pdp − . a − α Z α dp == ¯ q ( α, X ) + 0 . a − α (cid:0) − α (cid:1) − . a − α (1 − α ) == ¯ q ( α, X ) + 0 . a − α α (1 − α ) = ¯ q ( α, X ) + a α. For odd n ≥ n = 2 k + 1, k = 2 , , ... , and¯ q n ( α, X ) = 11 − α Z α M n ( p ; a ) dp = 11 − α Z α " M n − ( p ; a ) + a n (cid:18) p − (cid:19) k dp == ¯ q n − ( α, X ) + a n − α Z α (cid:18) p − (cid:19) k dp = ¯ q n ( α, X ) + a n − α ( p − . k +1 k + 1 (cid:12)(cid:12)(cid:12) α == ¯ q n − ( α, X ) + a n − α n + 1 h . n +12 − ( α − . n +12 i . For even n ≥ n = 2 k + 2, k = 2 , , ... , and¯ q n ( α, X ) = 11 − α Z α M n ( p ; a ) dp = 11 − α Z α " M n − ( p ; a ) + a n (cid:18) p − (cid:19) k ln p − p dp. Substituting r = p − . q n ( α, X ) = ¯ q n − ( α, X ) + a n − α Z . α − . r k ln 0 . r . − r dr == ¯ q n − ( α, X ) + a n − α r k +1 ( k + 1) " F (1 , k + 1; k + 2; − r ) − F (1 , k + 1; k + 2; 2 r )++ ( k + 1) ln 1 + 2 r − r . α − . . For r = 0 . " F (1 , k + 1; k + 2; − r ) − F (1 , k + 1; k + 2; 2 r ) + ( k + 1) ln 1 + 2 r − r r =0 . == 1 + k (cid:18) Ψ (cid:18) k (cid:19) − Ψ (cid:18) k (cid:19)(cid:19) + (1 + k ) ( γ + ln 2 + Ψ (1 + k )) . k ) = Ψ (cid:0) k (cid:1) + Ψ (cid:0) k (cid:1) + ln 2,1 + k (cid:18) Ψ (cid:18) k (cid:19) − Ψ (cid:18) k (cid:19)(cid:19) + (1 + k ) ( γ + ln 2 + Ψ (1 + k )) == 1 + k (cid:18) Ψ (cid:18) k (cid:19) − Ψ (cid:18) k (cid:19)(cid:19) + (1 + k ) (cid:18) γ + 2 ln 2 + 12 Ψ (cid:18) k (cid:19) + 12 Ψ (cid:18) k (cid:19)(cid:19) == 1 + k (cid:20) (cid:18) k (cid:19) + 2 γ + 4 ln 2 (cid:21) = (1 + k ) (cid:20) Ψ (cid:18) k (cid:19) + γ + 2 ln 2 (cid:21) . Plugging this sub-expression into the formula for the superquantile,¯ q n ( α, X ) = ¯ q n − ( α, X ) + a n − α . k +1 k + 1 (cid:20) Ψ (cid:18) k (cid:19) + γ + 2 ln 2 (cid:21) − a n − α ( α − . k +1 ( k + 1) ×× (cid:20) F (1 , k + 1; k + 2; 1 − α ) − F (1 , k + 1; k + 2; 2 α −
1) + ( k + 1) ln α − α (cid:21) . Corollary 1. If X ∼ M etalog ( a ) , its superquantile can be simplified to ¯ q ( α, X ) = ¯ q ( α, X ) − a − α " α (cid:16) α − α (cid:17) ln α − α + α − ! + ln (1 − α )12 Assume that X is a real valued random variable that follows a metalog distribution. Letsdefine α w = P r { X ≤ w } . By definition, F X ( w ) = α w . Proposition 2.
The first-order partial moments at w for a metalog distribution can beexpressed as µ +1 ( w ) = (1 − α w )(¯ q n ( α w , X ) − w ) ,µ − ( w ) = wα w − E [ X ] + (1 − α w )¯ q n ( α w , X ) . Proof.
The upper partial moment, by definition, µ +1 ( w ) = R + ∞ w ( x − w ) f ( x ) dx . By substi-tuting p = F ( x ), dp = f ( x ) dx , x = F − ( p ) = M n ( p, a ), and considering that F (+ ∞ ) = 1and F ( w ) = α w , µ +1 ( w ) = Z + ∞ w ( x − w ) f ( x ) dx = Z + ∞ w xf ( x ) dx − w Z + ∞ w f ( x ) dx == Z α w M n ( p, a ) dp − w Z α w dp = (1 − α w )¯ q n ( α w , X ) − w (1 − α w ) == (1 − α w )(¯ q n ( α w , X ) − w ) . µ − ( w ) = R w −∞ ( w − x ) f ( x ) dx . Performing thesame substitutions, µ − ( w ) = Z w −∞ ( w − x ) f ( x ) dx = w Z w −∞ f ( x ) dx − Z w −∞ xf ( x ) dx == w Z w −∞ f ( x ) dx − (cid:18)Z + ∞−∞ xf ( x ) dx − Z + ∞ w xf ( x ) dx (cid:19) == w Z α w dp − (cid:18) E [ X ] − Z α w M n ( p, a ) dp (cid:19) == wα w − E [ X ] + (1 − α w )¯ q n ( α w , X ) . Lets define the upper quantile function M + n ( w ; p, a ) = max { M n ( p, a ) − w, } and thelower quantile function M − n ( w ; p, a ) = max { w − M n ( p, a ) , } . As the quantile function isnon-decreasing, M n ( p, a ) ≤ w ∀ p < α w and M n ( p, a ) ≥ w ∀ p ≥ α w . Proposition 3.
The first-order partial moments at w for a metalog distribution can beexpressed as µ +1 ( w ) = Z M + n ( w ; p, a ) dp,µ − ( w ) = Z M − n ( w ; p, a ) dp, Proof.
The upper partial moment:Considering that M + n ( w ; p, a ) = 0 ∀ p < α w and M + n ( w ; p, a ) = M n ( p, a ) − w ∀ p ≥ α w , Z M + n ( w ; p, a ) dp = Z α w M n ( p, a ) dp − Z α w wdp = (1 − α w )¯ q n ( α w , X ) − (1 − α w ) w = µ +1 ( w ) . The lower partial moment:Considering that M − n ( w ; p, a ) = 0 ∀ p > α w and M − n ( w ; p, a ) = w − M n ( p, a ) ∀ p ≤ α , Z M − n ( w ; p, a ) dp = Z α wdp − Z α M n ( p, a ) dp = wα w − (cid:18)Z M n ( p, a ) dp − Z α M n ( p, a ) dp (cid:19) == wα w − E [ X ] + (1 − α w )¯ q n ( α w , X ) = µ − ( w ) . Corollary 2.
The upper and the lower partial moments at w for any metalog distributionare convex with respect to a . roof. The metalog quantile function is linear with respect to a , so ∀ i, j ∂ M n ( p, a ) ∂a i = 0 and ∂ M n ( p, a ) ∂a i ∂a j = 0, so the Hessian matrix is zero and thus it is positive semi-definite, whichimplies M n ( p, a ) is convex with respect to a . The same is actually true for − M n ( p, a ), asits Hessian matrix is also zero and thus positive semi-definite. Being a maximum of twoconvex functions, both M + n ( w ; p, a ) and M − n ( w ; p, a ) are convex with respect to a . As anintegral of a convex function is also convex if it exists, µ +1 ( w ) and µ − ( w ) are convex withrespect to a . References
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