Generalized quantiles as risk measures

Bellini, Klar, Muller, Rosazza Gianin

December 1, 2014Vorisek Jan

Introduction

Quantiles qα of a random variable X can be defined as theminimizers of a piecewise linear loss function:

qα(X ) = arg minx∈R

{α E

[(X − x)+

]+ (1− α) E

[(X − x)−

]}.

This property lies at the heart of quantile regression (Koenker,2005) and has been used by Rockafellar and Uryasev (2002) for thecomputation of the CVaR.

Generalized quantiles

have been introduced, by considering more general loss functions:

I expectiles (Newey, Powell; 1987)

I M-quantiles ( Breckling, Chambers; 1988)

I Lp-quantiles (Chen; 1996)

General asymmetric loss function:

πα(X , x) = α E[Φ1((X − x)+

)]+ (1− α) E

[Φ2((X − x)−

)],

where Φ1, Φ2 are convex. Minimizer

x∗α ∈ arg minx∈R

πα(X , x)

is called generalized quantile.

M-quantilesGiven a sample {x1, . . . , xn} of univariate observations, withempirical distribution function Fn(x), the sample lower quartileq̂1/4 is obtained as the solution of∫

ψ1/4(x − q̂1/4)Fn(dx) = 0,

where

ψ1/4(x) =

{34 sgn(x) (x < 0),14 sgn(x) otherwise.

For arbitrary α (0 < α < 1) and any standard influence functionψ(x), is reasonable to define the influence function of the αthM-quantile θα as follows:

ψα(x) =

{(1− α)ψ(x) (x < 0),αψ(x) otherwise,

which leads to the estimating equation∫ψα(x − θ̂α)Fn(dx) = 0.

Lp-quantilesLet X be a random variable with cumulative distribution functionF (x). For 0 < α < 1, 1 ≤ p

Elictability

For evaluation of point forecasts, Gneiting (2011) introduced thenotion of elicitability for a functional that is defined by means ofloss minimization process. All generalized quantiles are elicitable.The relevance of elictability for backtesting was addressed byEmbrechts and Hofert (2013). It is well known how to backtestVaR, while backtesting of CVaR (that is not elicitable) is not asstraightforward.The connections between elicitability and coherence are alsoinvestigated in Ziegel (2013), who shows that expectiles are theonly elicitable law-invariant coherent risk measures.

Law-invariant risk measures

Law invariance means that the risk assessment only depends onthe distribution of the random variable under the given probabilitymeasure, without regard to the financial context.

Definition: A monetary risk measure ρ on L∞(Ω;F ;P) is calledlaw-invariant if ρ(X ) only depends on the distribution of X underP, i. e. ρ(X ) = ρ(Y ) whenever X and Y have the samedistribution under P.

Luxemburg norm

Given Φ : [0,+∞)→ [0,+∞) convex, strictly increasing functionsatisfying Φ(0) = 0,Φ(1) = 1, a probability space (Ω,F ,P) andthe space L0 of all r.v. X on (Ω,F ,P) the Orlicz heart

MΦ :=

{X ∈ L0 : E

[Φ

(|X |a

)]< +∞, for every a > 0

}is a Banach space w.r.t. the Luxemburg norm ‖.‖Φ, defined as

‖Y ‖Φ := inf{a > 0 : E

[Φ

(|X |a

)]≤ 1}.

The case Φ(x) = xp corresponds to the usual Lp spaces.

Properties of generalized quantiles

Since minimization problem of πα(X , x) is convex, generalizedquantiles can be characterized by means of first-order condition.

Proposition 1. Have πα(X , x) and Φi as earlier. LetX ∈ MΦ1 ∩MΦ2 and α ∈ (0, 1).

(a) πα(X , x) is finite, non-negative, convex and satisfieslimx→±∞ πα(X , x) = +∞;

(b) the set of minimizers is a closed interval:arg min πα(X , x) := [x

∗−α , x

∗+α ] ;

(c) x∗α ∈ arg min πα(X , x) iff (f.o.c.)α E

[I{X>x∗α}Φ

′1−(δ+X)]≤ (1− α) E

[I{X≤x∗α}Φ

′2+

(δ−X)]

α E[I{X≥x∗α}Φ

′1+

(δ+X)]≥ (1− α) E

[I{X

First order condition

Example 2. For Φ1(x) = Φ2(x) = x , generalized quantiles reduceto the usual quantiles and f.o.c. becomes

α E[I{X>x∗α}

]≤ (1− α) E

[I{X≤x∗α}

]α E

[I{X≥x∗α}

]≥ (1− α) E

[I{X

Connection to shortfall risk measures

When the f.o.c. is given by an equation, generalized quantiles mayalso be defined as the unique solutions of the equation

E [ψ(X − x∗α)] = 0,

where

ψ(t) =

{−(1− α)Φ′2(−t) t < 0

αΦ′1(t) t ≥ 0

is nondecreasing with ψ(0) = 0.This shows that generalized quantiles can be seen as special casesof zero utility premium principles, also known as shortfall riskmeasures or u-mean certainty equivalents (see Deprez, Gerber,1985; Follmer, Schied, 2002; Ben-Tal, Teboulle, 2007).

Zero utility premium

Suppose that u(.) is the insurers utility function, with the usualproperties (increasing, concave), and that z represents the insurersfortune without the new policy. Then the premium P = H(X ) isdetermined from the condition that

E [u(z + P − X )] = u(z),

which is the requirement that the premium should be fair in termsof utility. In the case of exponential utility,

u(x) =1

a(1− e−ax),

with parameter a > 0, equation has an explicit solution; one findsthat

P =1

alnE (eaX ),

which is called the exponential principle.

u-mean certainty equivalent & shortfall risk measure

One of certainty equivalents based on utility functions is theso-called u-mean, Mu(·), defined for any random variable X byMu(·) satisfying

E [u(X −Mu(X ))] = 0.

This equation is also known as the principle of zero utility.

As an example, the u-mean Mu(·) is closely related to the riskmeasure called shortfall risk introduced by Follmer and Schied(2002), and defined by

ρFS(X ) = inf{m ∈ R : E [u(X −m)] ≤ x0}.

For a strictly increasing utility and x0 = 0, ρFS(X ) = Mu(X ).

Properties of generalized quantiles

Proposition 6. Let Φi : [0,+∞)→ [0,+∞) be strictly convexand differentiable with Φi (0) = Φ

′i+(0) = 0 and Φi (1) = 1. Let

α ∈ (0, 1) and

x∗α(X ) = arg minx∈R

{α E

[Φ1((X − x)+

)]+ (1− α) E

[Φ2((X − x)−

)]}.

(a) x∗α(X ) is positively homogeneous iff Φi (x) = xβ, with β > 1.

(b) x∗α(X ) is convex (concave) iff the function ψ : R→ R is convex(concave).

(c) x∗α(X ) is coherent iff Φi (x) = x2 and α ≥ 12 .

Thus expectiles with α ≥ 12 are the only generalized quantiles thatare coherent risk measure.

Expectiles eα(X )

The f.o.c. could be written in several equivalent ways:

α E[(X − eα(X ))+

]= (1− α) E

[(X − eα(X ))−

],

eα(X )− E [X ] =2α− 11− α

E[(X − eα(X ))+

],

α =E[(X − eα(X ))−

]E [|X − eα(X )|]

.

The latter shows, that expectiles can be seen as the usual quantilesof a transformed distribution (Jones, 1994).Proposition 7. Let X ,Y ∈ L1, then:(a) X ≤ Y P-a.s. and P(X < Y ) > 0 imply that eα(X ) < eα(Y )(strong monotonicity);(b) if α ≤ 12 , then eα(X + Y ) ≥ eα(X ) + eα(Y );(c) eα(X ) = −e1−α(−X ).

Dual representation

as maximal expected value over a set of scenarios.

Proposition 8. Let X ∈ L1, α ∈ (0, 1) and let eα(X ) be theα-expectile of X . Then:

eα(X ) =

{maxϕ∈Mα E [ϕX ] α ≥ 12minϕ∈Mα E [ϕX ] α ≤ 12 ,

where

Mα ={ϕ ∈ L∞, ϕ ≤ 0 a.s.,EP [ϕ] = 1,

ess supϕ

ess inf ϕ≤ β

},

with β = max{

α1−α ,

1−αα

}. The optimal scenario is

ϕ̄ :=αI{X>eα} + (1− α)I{X≤eα}

E[αI{X>eα} + (1− α)I{X≤eα}

] .

Kusuoka representation

From the dual representation it is possible to derive Kusuoka(2001) representation, which is the representation of law invariantcoherent risk measure as a supremum of convex combinations ofCVaR.

Proposition 9. Let X ∈ L1, α ∈ [ 12 , 1) and β =α

1−α , then

eα(X ) = maxγ∈[ 1

β,1]

{(1− γ)CVaR βγ−1

γ(β−1)+ γE [X ]

}.

In particular,

eα(X ) ≥E [X ]

2α+

(1− 1

2α

)CVaRα(X ).

Robustness

In robust statistics, the notion of qualitative robustness of astatistical functional corresponds essentially to the continuity withrespect to weak convergence. Coherent risk measures are notrobust in statistical sense. Stahl et al. (2012) suggest that a betternotion of robustness might be continuity with respect to theWasserstein distance, defined as

dW (P,Q) := inf {E [|X − Y |] : X ∼ P,Y ∼ Q} .

Convergence in the Wasserstein distance is stronger than weakconvergence:

dW (Xn,X )→ 0⇔ Xn→ X in distribution and E [Xn]→ E [X ].

Theorem 10. For all X ,Y ∈ L1 and all α ∈ (0, 1) holds that|eα(X )− eα(Y )| ≤ β dW (X ,Y ), where β = max

{α

1−α ,1−αα

}.

Comparing expectiles with quantiles

Koenker (1993) provided an example of a distribution with infinitevariance for which expectiles eα(X ) and quantiles qα(X ) coincidefor all α ∈ (0, 1). This distribution is Pareto-like with tail indexβ = 2.Theorem 11. Assume that X has a Pareto-like distribution withtail index β > 1. Then

F̄ (eα(X ))

β − 1∼ 1− α ∼ F̄ (qα(X )) as α→ 1.

If β < 2, then there exists some α0 < 1 such that for all α > α0eα(X ) > qα(X ) holds; if β > 2, the reverse inequality applies.So for high α expectiles are more conservative than the quantilesfor distributions with heavy tails (infinite variance).

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