On a family of risk measures based on proportional hazards models and tail probabilities

Insurance: Mathematics and Economics 86 (2019) 232–240

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

On a family of risk measures based on proportional hazards models and tail probabilities ∗

Georgios Psarrakos a , , Miguel A. Sordo b a b

Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece Department of Statistics and Operation Research, University of Cadiz, 11510 Puerto Real, Cadiz, Spain

article

info

Article history: Received June 2018 Received in revised form March 2019 Accepted 18 March 2019 Available online 26 March 2019 Keywords: Proportional hazards model Variability measures Gini mean difference Residual lifetime Dispersive order Premium principle Cumulative residual entropy

a b s t r a c t In this paper, we explore a class of tail variability measures based on distances among proportional hazards models. Tail versions of some well-known variability measures, such as the Gini mean difference, the Wang right tail deviation and the cumulative residual entropy are, up to a scale factor, in this class. These tail variability measures are combined with tail conditional expectation to generate premium principles that are especially useful to price heavy-tailed risks. We study their properties, including stochastic consistency and bounds, as well as the coherence of the associated premium principles. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In actuarial science, right tail risk analysis refers to the study of large losses that occur with very small probability (see, for example Wang, 1998). In this framework, the value-at-risk (VaR) is one of the more popular risk measures, still widely used by insurance companies and financial institutions due to its conceptual simplicity. However, VaR has been criticized because it is not (in general) subadditive and hence it is not coherent. Another criticism is that VaR does not account for losses beyond a certain quantile (which is precisely, the VaR). An approach that overcomes these drawbacks is to use the tail conditional expectation (TCE), which is the conditional expectation of the losses above the VaR. This measure is coherent when restricted to loss random variables with a continuous distribution function. TCE is the expected size of losses exceeding the VaR. Since tail events are subject to variability, there is also a concern regarding variability of losses above the VaR and some tail variability measures have been considered to complement TCE. The approach of combining tail-loss measures (such as VaR and TCE) and tail variability measures to obtain a more solid protection has received attention in the actuarial and financial literature. Valdez (2005) and Furman and Landsman (2006a,b) combine TCE and different versions of tail variances into a single measure ∗ Corresponding author. E-mail addresses: [email protected] (G. Psarrakos), [email protected] (M.A. Sordo). https://doi.org/10.1016/j.insmatheco.2019.03.005 0167-6687/© 2019 Elsevier B.V. All rights reserved.

and derive explicit expressions in the framework of multivariate elliptical distributions. Sordo (2009) studies the consistency of these measures with respect to some stochastic orderings and considers additional tail variability measures, such as the tail Gini mean difference, which is further explored and combined with TCE in Furman et al. (2017) to produce the so-called Gini shortfall. Righi and Ceretta (2016) combine TCE and the shortfall-deviation (a deviation measure in the sense of Rockafellar et al., 2006) to obtain a new risk measure. Following this approach, we explore in this paper a wide class of tail variability measures based on distances among proportional hazards models. This family, besides including some of the previously cited measures, also contains tail versions of other variability measures that sometimes perform better, such as the Wang’s right tail deviation (Wang, 1998) and the cumulative residual entropy (Rao et al., 2004). We study these measures and combine them with TCE to evaluate and price risks when the variability along the right tail is crucial (for example, in the case of heavy-tailed loss distributions). One way to evaluate and price risks is to use premium principles. In general, for a risk X with distribution function F , tail function F = 1 − F and quantile function (or value-at-risk) F −1 (q) = inf {x | FX (x) ≥ q} for 0 < q < 1 , a premium principle typically equals the sum of the expected loss E [X ] plus a risk loading proportional to a specific variability measure D (X ), taking the form T (X ) = E [X ] + λD (X ) , for some λ ≥ 0.

(1)

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

Following Bickel and Lehmann (1979), here and throughout this paper, a functional D (X ) is a variability measure if it satisfies the next intuitive properties: (P1) D (X + c ) = D (X ) for all constant c , (P2) D (cX ) = cD (X ) for all c > 0, (P3) D (c ) = 0 for any degenerate random variable at c , (P4) D (X ) ≥ 0 for all risk X , (P5) X ≤disp Y implies D (X ) ≤ D (Y ) . In (P5), X ≤disp Y stands for the dispersive order, which is defined as follows (see Shaked and Shanthikumar, 2007). Definition 1.1. Let X and Y be two random variables with respective distribution functions F and G. Then, X is said to be smaller than Y in the dispersive order (denoted by X ≤disp Y ) if F

−1

(p) − F

−1

−1

(q) ≤ G

(p) − G

−1

(q), for all 0 < q < p < 1.

(2)

Clearly, the order ≤disp corresponds to a comparison of two risks X and Y by variability because it requires the difference between any two quantiles of X to be smaller than the corresponding quantiles of Y . Variability measures are used in this paper to determine the safety loading for premium principles. Therefore, the loading parameter λ and the variability measure D(X ) in (1) must be chosen to provide T (X ) with suitable properties, such as monotonicity (Pr [X < Y ] = 1 implies T (X ) ≤ T (Y )), translation invariance (T (X + c ) = T (X ) + c for all constant c), positive homogeneity (T (cX ) = cT (X ) for all c > 0) and subadditivity (T (X + Y ) ≤ T (X ) + T (Y ) for all risks X and Y ). A premium principle T (X ) satisfying these four properties is an example of a coherent risk measure (see Artzner et al., 1999). Another interesting property for a variability measure D (X ) involved in a premium principle of the form (1) is comonotonic additivity, which means that D (X + Y ) = D (X )+ D (Y ) for X and Y comonotonic. Recall that two risks X and Y are comonotonic if there exist a random variable Z and increasing functions t1 , t2 such that (X1 , X2 ) =st (t1 (Z ), t2 (Z )) (roughly speaking, two comonotonic risks ‘move together’). If D (X ) is comonotonic additive, so is T (X ) in (1). See Goovaerts et al. (1984, 2010), Young (2004) and Föllmer and Schied (2011), for an extensive overview and historical developments on premium principles. Premium principles of the form (1) are sometimes inadequate for capturing tail risks. For example, if the standard deviation or the variance is used as the variability measure D(X ) in (1), then two risks with equal first and second moments are charged the same premium. This may be questionable because even though they have the same mean and variance, the probability of very large losses can be significantly different. Moreover, the standard deviation, as well as the variance, consider high and low losses that are the same distance of the mean as equally undesirable. This is a drawback because large losses represent, of course, a more important source or risk (see Sections 2.4 and 4.1 in Ramsay (1993, for discussion). The Gini mean semidifference can also be inadequate: it has been shown by Wang (1998, see Section 8.1) that the ranking by this measure for several skewed risk models (such as Pareto, Lognormal or Weibull with shape parameter less than 1) with the same mean is not in agreement with our knowledge of their relative tail thickness. They may also fail to determine the size of the risk above a threshold value, which is of interest, for example, in the case of insurance policies including deductibles. To overcome these drawbacks, some authors have suggested to replace E[X ] and D(X ) respectively, in (1), ]by the [ tail conditional expectation TCE(X , p) = E X |X > F −1 (p) and a certain tail-variability characteristic D(X , p), where p ∈ (0, 1) is a convenient safety level (a specific definition of D(X , p) is given in Section 2). Under this approach, a typical premium principle takes the form T (X , p) = E X |X > F −1 (p) + λD(X , p), for some λ ≥ 0.

[

]

(3)

233

Examples of tail-variability measures are the tail variance (Furman and Landsman, 2006a,b) given by TV (X , p) = Var X |X > F −1 (p)

]

[

(4)

[ ] = E (X − TCE(X , p))2 |X > F −1 (p) , and the tail standard deviation, given by

√

SD(X , p) =

TV (X , p).

Unfortunately, TV (X , p) and SD(X , p) require finite second moment of X and therefore they are inadequate in the case of heavy-tailed distributions with E(X 2 ) = ∞. An alternative is the tail Gini mean difference (see Sordo, 2009) defined by GMD(X , p) = GMD X |X > F −1 (p) , 0 < p < 1,

)

(

(5)

where ⏐] ) is the mean Gini difference given by GMD(X ) = [⏐ GMD(X E ⏐X − X ′ ⏐ , with X ′ being an independent copy of X (see Yitzhaki, 2003). GMD(X , p) is a variability measure that does not require the existence of a second moment. Furman et al. (2017) study in depth the properties of GMD(X , p) under the representation

E ⏐X − X ′ ⏐ |X > F −1 (p) , X ′ > F −1 (p) , 0 < p < 1,

⏐

[⏐

]

where X ′ is an independent copy of X . For more details see also Liu and Wang (2016). The growing interest in modeling tail variability suggests to explore tail versions of other variability measures that have been discussed in the actuarial literature, such as the right tail deviation, which outperforms the standard deviation and the Gini mean difference for actuarial applications (see Wang, 1998), given by ∞

∫

√ F (x) dx − E [X ]

W (X ) =

(6)

0

and the cumulative residual entropy (Rao et al., 2004) given by ∞

∫

F (x) log F (x) dx,

E (X ) = −

(7)

0

(see Sordo et al. (2016a) and Psarrakos and Toomaj (2017) for actuarial applications of E (X )). The corresponding tail versions, as well as GMD(X , p), belong (up to scalar factor) to a larger class of variability measures ΨX based on distances among proportional hazards models. To introduce this class, we think of the loss X as a lifetime with survival function F , something usually done in risk theory (see, for example Wang, 1995). Provided that F is f (t) absolutely continuous with continuous density f , let rF (t) = F (t) be the hazard (or failure) rate of F and let

µF (t) = E [X − t |X > t] , t ≥ 0, be the corresponding mean residual lifetime. For α > 0, we consider the random variable Xα with distribution function Fα (t) = Pr(Xα ≤ t) = 1 − [F (t)]α , t ≥ 0,

(8)

and hazard rate rFα (t) = α rF (t), t ≥ 0. The rest of the paper is structured as follows. In Section 2 we introduce a class ΨX of tail variability measures that can be combined with TCE to evaluate tail risks. Members of ΨX will be denoted Dα,β (X , p), where α, β > 0 are two parameters and p ∈ (0, 1) is the safety level. We provide different representations for these measures and highlight some of them. In Section 3 we provide upper bounds for every Dα,β (X , p) in terms of the tail variance (whenever exists). In Section 4, we consider the associated premium principle Tα,β,λ (X , p) = E X |X > F −1 (p) + λDα,β (X , p), λ ≥ 0,

[

]

234

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

and study conditions on λ, α and β to ensure coherence of Tα,β,λ (X , p), for all p ∈ (0, 1). In Section 5 we compare the performance of different premium principles in ΨX . In particular, we give an example to show that the tail version of the Wang right tail deviation can be more appropriate for pricing heavytailed risks than the Gini mean difference. Section 6 contains conclusions. Throughout the paper when we say that a function g is increasing (decreasing), we mean that it is non-decreasing (nonincreasing), that is, g(x) ≤ g(y) (≥) for all x ≤ y. We assume that all random variables have a continuous distribution function. 2. A class of tail variability measures Consider a non-negative random variable X˜ α,β defined by the condition

[

]

X˜ α,β |Xβ = s ≡st [Xα − s|Xα > s] , for all s > 0,

(9)

with survival function

[

F α,β (x) = Pr X˜ α,β > x ∞

∫

where X ′ is an independent copy of X and Y ′ is an independent copy of Y . Our aim is to define and study a family ΨX of tail-variability measures. In view of Theorem 2.1, the expectation of X˜ α,β (or any positive scale transformation) is a good candidate to measure variability. In order to include a wider variety of measures in ΨX and capture the variability of the risk beyond a threshold F −1 (p), we prefer to study the expectation 1 E[X˜ α,β |Xβ > F −1 (p)] β where α, β > 0 and p ∈ (0, 1); this makes the measure Dα,β (X , p) =

symmetric with respect to the parameters. In the rest of the paper, we focus on the class of measures

} { ΨX = Dα,β (X , p) , α > 0, β > 0, p ∈ (0, 1) . The limiting case when p → 0 is denoted Dα,β (X ). Theorem 2.2. Let X be a random variable with distribution function F . Given α, β > 0 and p ∈ (0, 1), we have

]

[

]

Pr X˜ α,β > x|Xβ = s dF β (s)

=−

∫0 ∞ =− 0

∞

∫

Dα,β (X , p) =

Pr [Xα − s > x|Xα > s] dF β (s)

[

=−

F (x + s)

]α

⎪ ⎪ ⎩ ∫∞ − F −1 (p)

Theorem 2.1. Let X and Y be two non-negative random variables with distribution functions F and G, respectively. If X ≤disp Y , then for α, β > 0 we have X˜ α,β ≤st Y˜α,β .

]α

F (s)

0

∞[

∫ ≤

β [F (s)]β−1 dF (s)

G(x + s)

]α

G(s)

0

β [G(s)]β−1 dG(s), x > 0

]α

1−p

0

∫ 1[

[

1

β

≤

1−p

0

β−1

(1 − p)

In particular, by taking α = 1, β = 1, we have the following corollary, which should be compared to Theorem 3.B.42 of Shaked and Shanthikumar where a similar result involving the ⏐ ⏐ (2007), ⏐ ⏐ variables ⏐X − X ′ ⏐ and ⏐Y − Y ′ ⏐ is given. Corollary 2.1. Let X and Y be two non-negative random variables. If X ≤disp Y , then X − X |X > X ≤st Y − Y |Y > Y , ′

′

]

[

′

′

]

∫

1

∞

[F (x)]β−1 µFα (x) dF (x) ∫ ∞ ∫ ∞ 1 [F (x)]β−α−1 [F (y)]α dy dF (x) = (1 − p)β F −1 (p) x ∫ ∞ ∫ y 1 = [F (x)]β−α−1 [F (y)]α dF (x) dy (1 − p)β F −1 (p) F −1 (p)

β

(1 −

p)β

F −1 (p)

E[X˜ α,β | Xβ > F −1 (p)]

(∫ y ( )′ ) α β−α [ F (y) ] [ F (x) ] dx dt (α − β ) (1 − p)β F −1 (p) F −1 (p) ( ) ∫ ∞ 1 α β−α β−α = [ F (y) ] [ F (y) ] − [ F (t) ] dt (α − β )(1 − p)β F −1 (p) ∫ ∫ ∞ ( ∞ ) [F (y)]β dy [F (y)]α dy 1 F −1 (p) F −1 (p) − = α−β (1 − p)β (1 − p)α ∫

1

=

dp, x > 0.

If follows from (3.B.4) in Shaked and Shanthikumar (2007) that X ≤disp Y implies (10). □

[

]

E[X˜ α,β | Xβ > F −1 (p)]

=

(10)

]α

α = β.

Since dFβ (x) = β [F (x)]β−1 dF (x) and F β (x) = [F (x)]β , we have

(1 − p)β−1 dp

G(x + G−1 (p))

dx ,

α ̸= β,

E X˜ α,β |Xβ = x = µFα (x), for all x > 0.

1

or, equivalently, (making the changes s = F (p) in the left side integral and s = G−1 (p) in the right one), if and only if, F (x + F −1 (p))

,

We distinguish two cases (i) α ̸ = β and (ii) α = β . (i) For α ̸ = β we have

−1

∫ 1[

(1−p)α

)

Proof. Recall from (9) that

Proof. Given α, β > 0, the condition X˜ α,β ≤st Y˜α,β holds if and only if F (x + s)

F (x) 1−p

[F (x)]α dx

(11)

pendent copy of X , a random variable considered by Kapodistria and Psarrakos (2012). The following result shows that the stochastic size of the random variable X˜ α,β is an indicator of variability of X .

∞[

log

F −1 (p)

β

[ ] Observe that ˜ X1,1 reduces to X ′ − X |X ′ > X , where X ′ is an inde-

∫

[F (x)]α (1−p)α

∫∞

dF (s) .

F (s)

0

(∫∞ ⎧ [F (x)]β dx ⎪ F −1 (p) 1 ⎪ − ⎨ α−β (1−p)β

∞

(ii) For α = β we obtain 1

α

E[X˜ α,β | Xβ > F −1 (p)]

= =

1

∫

(1 − p)α 1 (1 − p)α

∞ F −1 (p)

∫

∞

∫

y

[F (y)]α rF (x) dx dy

F −1 (p)

[ ] F (y) dy [F (y)] − log 1−p F −1 (p) α

which completes the proof. □

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

It is easy to verify that Dα,β (X , p) Dα,β (X , p) = Dα,α (X , p) and Dα,β (X , p) =

1−β

α−β

D1,β (X , p) +

= Dβ,α (X , p), limβ→α

Some particular members of this family have been recently considered in the literature. For example, for α = β = 1, D1,1 (X , p) equals the dynamic cumulative residual quantile entropy (Kang and Yan, 2016) given by

) ( E (X , p) = E X |X > F −1 (p) = −

∫

∞

F (x)

F −1 (p)

1−p

log

F (x) 1−p

dx (12)

The limiting case when p → 0 gives the cumulative residual entropy (CRE) proposed in reliability theory by Rao et al. (2004). Note that E (X , p) is the quantile version of the dynamic CRE (see Asadi and Zohrevand, 2007) considered, among others, by Navarro et al. (2010), Psarrakos and Navarro (2013), Sordo and Psarrakos (2017) and Psarrakos and Di Crescenzo (2018). For α = 1, β = 2 we have D1,2 (X , p) =

∞

∫

1 (1 − p)

F (x) dx − F −1 (p)

∞

∫

1

F −1 (p)

= TCE(X , p) − E[min(X , X ′ |X > F −1 (p), X ′ > F −1 (p))], with X ′ being an independent copy of X . Using that

]

(

[

1 2 1

E[|X − X ′ | |X > F −1 (p), X ′ > F −1 (p)]

(13)

D1,0.5 (X , p) = 2W (X , p),

(14)

where W (X , p) = W X |X > F −1 (p)

)

(

= √

1

1−p

∫

∞

√ F (x) dx −

F −1 (p)

1 1−p

∫

∞

F (x) dx

Theorem 2.3. Let X be a random variable with distribution function F and let α, β > 0. For p ∈ (0, 1) we have

⎧ ( [( ) ( )β ]) α ∫ 1 −1 ⎪ 1−x 1−x 1 ⎪ F (x)d − , ⎪ p 1−p 1−p ⎨ α−β

α ̸= β,

⎪ [( )α ( )] ⎪ ∫ ⎪ ⎩− 1 F −1 (x)d 1−x log 1−x , p 1−p 1−p

α = β. (15)

Fβ−1 ,

(1 − p)β

α−β

∫∞

F −1 (p)

−

[F (x)]α dx

)

(1 − p)α

E[Xβ |Xβ > F −1 (p)] − E[Xα |Xα > F −1 (p)]

(

)

(

(16)

where, in the third equality, we have used that Fα−1 (p) = F −1 1 − (1 − p)1/α , 0 < p < 1,

(

)

(17)

and in the last one we have used that for a random variable X with distribution function F ,

E X |X > F −1 (p) =

[

]

1

∫

1 1−p

F −1 (t)dt ,

p ∈ (0, 1).

p

The result for α = β follows from (16) by the monotone convergence theorem applied to the sequence of non-negative functions 1

[(

α−β

1−x 1−p

)α

( −

1−x

)β ]

1−p

Representation (15) is very useful, because empirical estimators of quantities with this form can be obtained using the theory of L -statistics, which are linear combinations of order statistics; see Jones and Zitikis (2003, 2007), Jones et al. (2006) and Sordo et al. (2016b), for details. Our next purpose is to show that the class of tail-variability measures Dα,β (X , p) satisfies properties (P1) to (P5). Moreover, for some values of the parameters α and β , we will show that (P5) can be strengthened in terms of the excess-wealth order, which is defined as follows. Definition 2.1. Let X and Y be two random variables with distribution functions F and G, respectively. We say that X is smaller [ than Y in the ] excess [ wealth order ] (denoted by X ≤ew Y ) if E (X − F −1 (p))+ ≤ E (Y − G−1 (p))+ for all p ∈ (0, 1), where x+ = max{x, 0}. It is well-known (see Shaked and Shanthikumar, 2007) that

F −1 (p)

is the tail version of the Wang right tail deviation W (X ) given by (6). The limiting case is D1,0.5 (X ) = 2W (X ). In general, Dα,β (X , p), with α = 1 and β > 0, has been considered in Sunoj et al. (2018, see eq. (17)) under the name of quantile-based dynamic cumulative residual Tsallis entropy of order α. The next result provides an alternative representation for Dα,β (X , p) in terms of quantile integrals.

Dα,β (X , p) =

[F (x)]β dx

using that limβ→α Dα,β (X , p) = Dα,α (X , p). □

GMD(X , p), 2 where GMD(X , p) is the tail Gini mean difference. The limiting case when p → 0 gives D1,2 (X ) = 12 GMD(X ), where GMD(X ) is the classical Gini mean difference. For α = 1, β = 0.5 we have

=

1

F −1 (p)

E[Xβ |Xβ > Fβ−1 (1 − (1 − p)β )] α−β ) −E[Xα |Xα > Fα−1 (1 − (1 − p)α )] (∫ [( )α ( )β ]) 1 1−x 1−x 1 −1 F (x)d . − = α−β 1−p 1−p p

=

])

(see Dorfman, 1979), we see from the above equation that D1,2 (X , p) =

1

α−β

hβ (x) =

E |X − X ′ | = 2 E [X ] − E min(X , X ′ )

[

=

(∫ ∞

1

2

[F (x)] dx

(1 − p)2

we have Dα,β (X , p) =

α−1 D1,α (X , p) . α−β

235

Proof. Denote by Fα−1 and respectively, the quantile functions associated to the distribution functions Fα and Fβ . For α ̸ = β

X ≤disp Y H⇒ X ≤ew Y . Before proving the theorem, we require a lemma regarding the dispersive and the excess wealth order. Lemma 2.1. Let X and Y be two random variables with distributions functions F and G [ , respectively.]Then, [ ] (a) X ≤disp Y implies X |X > F −1 (p) ≤disp Y |Y > G−1 (p) for all p ∈ (0, 1) , [ ] [ ] (b) X ≤ew Y implies X |X > F −1 (p) ≤ew Y |Y > G−1 (p) for all p ∈ (0, 1) . Proof. The proof is a direct 2.9 in Navarro [ application ]of Theorem [ ] et al. (2013) using that X |X > F −1 (p) and Y |Y > G−1 (p) are two distorted random variables, for any p ∈ (0, 1) , obtained from{ X and}Y , respectively, by the concave distortion1 h(t) = min

t 1−p

,1 . □

1 A distortion function h is an increasing function from [0, 1] to [0, 1] such that h(0) = 0 and h(1) = 1. A continuous distortion h induces [ ] a distorted random variable associated to the survival function F h (x) = h F (x) (for details see Sordo and Suárez-Llorens, 2011).

236

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

Theorem 2.4. For any random variable X and α, β > 0, p ∈ (0, 1), Dα,β (X , p) satisfies the following properties: (a) Dα,β (X + c , p) = Dα,β (X , p) for all c, (b) Dα,β (cX , p) = cDα,β (X , p) for all c > 0, (c) Dα,β (c , p) = 0 for any degenerate random variable at c, (d) Dα,β (X , p) ≥ 0, (e) If X ≤disp Y , then Dα,β (X , p) ≤ Dα,β (Y , p), (f) Dα,β (X , p) = Dα,β (X , p) + Dα,β (Y , p), for X and Y comonotonic. In particular, these properties hold for Dα,β (X ). Proof. We first consider the case α > β . For a fixed p ∈ (0, 1), using representation (15), we can write 1

∫

(α − β )Dα,β (X , p) =

F

−1

1

∫ (t)dA(t) −

0

F

−1

(t)dB(t)

(18)

d

[

F (x + F −1 (p))

]α (1+δ)

1−p

|δ=0

dδ

1−

0,

(

1−t 1−p

)β

t
,

{ B(t) =

t≥p

1−

(0, )α 1−t 1−p

,

α Dα,α (X , p) = −

(

∞ F −1 (p)

F (x)

)α

1−p

( log

F (x)

= E ([X |X > F −1 (p)]α )

is the cumulative residual entropy with respect to survival function [F p (·)]α . From this expression, properties (a) to (d) are easy to verify. To prove (e), note that if X ≤disp Y then Xα ≤disp Yα for all α > 0 (this is verified by direct application of the definition of dispersive order). Combining this property and Lemma 2.1 it follows that X ≤disp Y implies [X |X > F −1 (p)]α ≤disp [Y |Y > G−1 (p)]α for all α > 0. Since the cumulative residual entropy is consistent with the dispersive order (see Section 2 in Sordo et al. (2016a)) it follows that E ([X |X > F −1 (p)]α ) ≤ E ([Y |Y > G−1 (p)]α ), that is, Dα,α (X , p) ≤ Dα,α (Y , p) for all p ∈ (0, 1). Finally, to prove (f), assume that X and Y are comonotonic. We can write Dα,α (X + Y , p) = lim Dα,β (X + Y , p) β→α

( ) = lim Dα,β (X , p) + Dα,β (Y , p) β→α

∞[

∫

F (x + F −1 (p)) ∞[

∫

rFβ (x, p) = (1 + δ ) rFα (x, p),

F (x + F −1 (p))

F (x + F −1 (p)) 1−p

F (x + F −1 (p)) 1−p

log

F (x + F −1 (p)) 1−p

.

dx

dx δ,

(21)

Theorem 2.5. Let X be a random variable and let α, β > 0 and p ∈ (0, 1). Then, X ≤ew Y implies Dα,β (X , p) ≤ Dα,β (Y , p) for α, β ∈ (0, 1]. Proof. Suppose 0 < β < α ≤ 1. Then, the functional (15) belongs to the class C2 in Sordo (2008) and the result follows from Theorem 8(ii) in that paper. The result also holds if 0 < α < β < 1 by using that Dα,β (X , p) = Dβ,α (X , p). □ 3. Further properties and bounds When the second moment of X exists, we can give an upper bound for Dα,β (X , p) in terms of the tail variance of Xα , namely TV (Xα , p) (see (4)). In the following result we use that

]

1 1−p

∫

∞ F −1 (p)

(µF (t))2 dF (t), p ∈ (0, 1) ,

where µF (t) is the mean residual life of X at t, a result that Hall and Wellner (1981) attributes to Pyke (1965) (see also p. 21 in Jeong, 2014). Theorem 3.1. Let α and β be two positive numbers. Assume that E(Xα2 ) < ∞ and E(Xβ2 ) < ∞. Given p ∈ (0, 1) we have: (i) For β ≥ α ,

√ Dα,β (X , p) ≤

1

αβ

TV (Xα , 1 − (1 − p)α ).

(ii) For β ≤ α ,

]α (1+δ)

]α

From (20) and (21) we see that E ([X | X > F −1 (p)]α ) measures the change in the value of E([X |X > F −1 (p)]α ) as a function of α under a proportional small variation of the hazard rate. The use of perturbation analysis of hazard rate and its relation with entropy was considered originally by Leser (1955), in order to measure the elasticity of life expectancy. An application to the cost of life annuities was given by Haberman et al. (2011); for more details see also the references therein.

√ =

1−p

0

]α

1−p

F (x + F −1 (p))

≈ −E ([X | X > F −1 (p)]α ) δ .

or equivalently

[

,

(22)

Remark 2.2. Formula (19) provides the following interpretation for E ([X | X > F −1 (p)]α ). First, observe that [X | X > F −1 (p)]α ≡st [Xα | Xα > F −1 (p)]. Let rFα (x, p) and rFβ (x, p) be the hazard rates of random variables [X | X > F −1 (p)]α and [X | X > F −1 (p)]β , respectively. We consider the proportional hazards model

]β

∞[

∫ dx −

□

Remark 2.1. We note that Theorem 2.4, except part (e), can also be proved from the fact that Dα,β (X , p) has a signed Choquet integral representation and from corresponding results on these integrals (see Wang et al., 2018, for details).

[

]β

1−p

0

[

where we have used that (f) holds when α ̸ = β .

1−p

we have the following approximation

Var X |X > F −1 (p) =

= Dα,α (X , p) + Dα,α (Y , p),

]α

F (x + F −1 (p))

E([X |X > F −1 (p)]β ) − E([X |X > F −1 (p)]α )

(19)

as a function

1−p

t≥p

dx

]α (1+δ)

F (x + F −1 (p))

or equivalently,

)α

1−p

[

t
are two distortion functions such that AB−1 (u) = 1 −(1 − u)β/α is convex. Therefore, (α −β )Dα,β (X , p) satisfies the conditions of the class C1 in Sordo (2008) and all properties follow from Theorems 1 and 8(i) in Sordo (2008). The result also holds for α < β using that Dα,β (X , p) = Dβ,α (X , p). Now we prove the case α = β . Given p ∈ (0, 1) we denote by F p the survival function associated to the conditional random variable [X |X > F −1 (p)]. Observe that

∫

=α

× log

0

{

F (x+F −1 (p)) 1−p

of δ in a neighborhood of δ = 0, and keeping in mind that

≈α

0

where A(t) =

[ Applying the Taylor expansion for

(20)

Dα,β (X , p) ≤

1

αβ

TV (Xβ , 1 − (1 − p)β ).

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

237

Proof. We only prove (i). The proof of (ii) follows directly from the symmetry of Dα,β (X , p) with respect to the parameters α and β . Consider the random variable X˜ α,β defined by (9) and recall that, with the notation of Theorem 2.2, we have

A DFR distribution has a heavier tail than an IFR one. Examples of DFR distributions include gamma and Weibull distributions with shape parameters less than one and Pareto distribution. The DFR class of distributions is a subclass of the IMRL class.

β Dα,β (X , p) = E[X˜ α,β |Xβ > F −1 (p)].

Corollary 3.1. Let X be a non-negative random variable with distribution function F . Then: (a) If F is DFR, then Dα,β (X , p) is increasing in p ∈ (0, 1) for all α, β > 0. (b) If F is IMRL, then Dα,β (X , p) is increasing in p ∈ (0, 1) for all α, β ∈ (0, 1].

(23)

Since dFβ (x) =

β [F (x)]β−α dFα (x) , α

we get

] E (X˜ α,β )2 | Xβ > Fα−1 (p) = =

β

∫ β

α (1 − p) α

∞ Fα−1 (p)

(1 −

∞

∫

1

[

β p) α

Fα−1 (p)

[µFα (x)]2 dFβ (x)

[µFα (x)]2 [F (x)]β−α dFα (x),

(24)

where we have used (17). Since β ≥ α , the function [F (x)]β−α is decreasing in x and (24) yields

]

[

E (X˜ α,β )2 | Xβ > Fα−1 (p) ≤

=

β α (1 − p) [ β α

∫

∞

Fα−1 (p)

[µFα (x)]2 dFα (x)

Var Xα | Xα > Fα−1 (p) ,

]

where we have used the formula (22) for the random variable Xα . From the Jensen inequality we have

Proof. If F is DFR, then [X |X > F −1 (p)] ≤disp [X |X > F −1 (q)] for all 0 < p < q < 1 (see Theorem 3.B.24 in Shaked and Shanthikumar, 2007). Therefore, using that Dα,β (X , p) = Dα,β (X |X > F −1 (p)) part (a) follows from Theorem 2.4(e). Similarly, part (b) follows from Theorem 2.5 using that if F is IMRL, then [X |X > F −1 (p)] ≤ew [X |X > F −1 (q)] for all 0 < p < q < 1 (see Theorem 3.C.13 in Shaked and Shanthikumar, 2007). □ We can also give sharp bounds for Dα,β (X , p) when the underlying risk distribution is IFR or DFR. The next result is a generalization of Corollary 3.3.1 in Rajesh and Sunoj (2016). Theorem 3.2. Let α, β > 0 such that E[µFα (Xβ )] < ∞. If F is DFR (IFR) then 1

∞

∫

[F (x)]α dx, p ∈ (0, 1). 1−p

( [

Dα,β (X , p) ≥ (≤)

Therefore, using (17) and (23) it follows

Proof. Suppose that F is DFR. Then, Fα is DFR (since rFα (·) = α rF (·)) and also IMRL (since DFR implies IMRL), which means that µFα (x) is increasing in x. From the proof of Theorem 2.2 we know

])2 ≤ E((X˜ α,β )2 | Xβ > Fα−1 (p)) E X˜ α,β | Xβ > Fα−1 (p) [ ] β ≤ Var Xα | Xα > Fα−1 (p) . α [

]

[

E X˜ α,β | Xβ > Fα−1 (p) = E X˜ α,β | Xβ > F −1 1 − (1 − p)1/α

(

Dα,β (X , p) =

≥

From the last inequality, the change of variable u = 1 − (1 − p) proves the result. □

Example 3.1. As an example of application of Theorem 3.1, we provide upper bounds for the dynamic cumulative residual quantile entropy and the tail Gini functional in terms of the tail variance of X . Given p ∈ (0, 1) and by taking α = β = 1 in Theorem 3.1, we have TV (X , p).

The choice α = 1, β = 2 yields to GMD(X , p) ≤

√

2 TV (X , p).

Intuitively, if X is heavy-tailed, a tail variability measure is expected to increase as the level p increases. The following result formalizes this idea, since some heavy-tailed distributions with applications in insurance are DFR. We recall the definition of DFR and other ageing classes (see Barlow and Proschan, 1981). Definition 3.1. A distribution function F with non-negative support is said to be: (i) Decreasing (increasing) failure rate or DFR (IFR) if F (x + y)/F (x) is increasing (decreasing) in x for any y ≥ 0. If F is absolutely continuous, then it is DFR (IFR) when the failure rate rF (x) is decreasing (increasing) in x. (ii) Increasing (decreasing) mean residual lifetime or IMRL (DMRL) if µF (x) is increasing (decreasing) in x.

∫

1 (1 − p)β 1 (1 − p)β

∞

[F (x)]β−1 µFα (x) dF (x) ∫ ∞ −1 [F (x)]β−1 dF (x) µFα (F (p)) F −1 (p)

F −1 (p)

µFα (F (p)) , β −1

1/α

√

F −1 (p)

that

)]

= β Dα,β (X , 1 − (1 − p)1/α ) √ β ≤ Var(Xα | Xα > Fα−1 (p)) α

E (X , p) ≤

β

=

and the result follows. The proof when F is IFR is similar. □ Example 3.2. As an example of application of Theorem 3.2, we provide bounds for the dynamic cumulative residual quantile entropy and the tail Gini functional in the case when F is DFR or IFR. By taking α = β = 1 in Theorem 3.2, we have that if F is DFR (IFR), then

[ ] E (X , p) ≥ (≤) E X − F −1 (p) | X > F −1 (p) , p ∈ (0, 1). In particular, if F is DFR (which implies that the second term in the above inequality is increasing) with support (0, ∞), we have E (X , p) ≥ E[X ], p ∈ (0, 1).

Similarly, if F is DFR (IFR), the choice α = 1, β = 2 yields to GMD(X , p) ≥ (≤) E X − F −1 (p) | X > F −1 (p) , p ∈ (0, 1),

[

]

and if F is DFR GMD(X , p) ≥ E[X ], p ∈ (0, 1). 4. Coherence of the associated tail premium principles Furman et al. (2017) have shown that the premium principle

E X |X > F −1 (p) + λGMD (X , p) , λ > 0,

[

]

238

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240

is coherent for λ ≤ 1/2. In this section, we extend this study to premium principles of the form

where g1 (t) = g1 (t ; α, λ, p)

Tα,β,λ (X , p) = E X |X > F −1 (p) + λDα,β (X , p), λ ≥ 0,

]

[

where Dα,β (X , p) is given by (15). To investigate the coherence of this family of premium principles we use that any functional of the form

=

F −1 (t)dg(t),

1≥λ

0

where g is an increasing function defined on [0, 1] , with g(0) = 0 and g(1) = 1, is coherent if and only if g is convex (see Acerbi (2002) or Jones and Zitikis (2003)).

t −p 1−p

+λ

[(

1−t 1−p

α

t

(

1−t 1−p

)]

,

p ≤ t ≤ 1.

Clearly, g1 (t) = 0 for 0 ≤ t ≤ p and g1 (1) = 1. Differentiating we see that g1 (t) is increasing on [p, 1] if and only if

1

∫

0),

{

(

)α−1 [ ( ) ] 1−t α log + 1 , ∀ t ∈ (p, 1], 1−p 1−p 1−t

or, equivalently, if and only if 1 ≥ λxα−1 [α log x + 1] , ∀ x ∈ (0, 1].

(27)

′′

For the convexity, observe that g1 (t) ≥ 0 if and only if

Theorem 4.1. For α, β, λ > 0, the premium principle Tα,β,λ (X , p) is coherent if and only if one of the following holds: (i) (α − 1) (1 − β) ≥ 0 and 0 ≤ λ ≤ 1, (ii) 1/2 ≤ α = β ≤ 1 and 0 ≤ λ ≤ 1.

α (α − 1) log

Proof. Case (i): Assume that α > β . We can write

α (α − 1) log x + 2α − 1 ≥ 0, ∀x ∈ (0, 1].

Tα,β,λ (X , p) =

F −1 (t)dg0 (t)

where g0 (t) = g0 (t ; α, β, λ, p)

⎧ ⎨ ⎩

t −p 1−p

+

λ α−β

[( 0,) α 1−t 1−p

−

(

1−t 1−p

)β ]

t
,

p ≤ t ≤ 1.

Clearly, g0 (t) = 0 for 0 ≤ t ≤ p and g0 (1) = 1. Straightforward manipulations show that g0 (t) is increasing on [p, 1] if and only if

[ ( )β−1 ( )α−1 ] 1−t 1−t −α ≥ β − α, ∀ t ∈ [p, 1] , λ β 1−p 1−p or, equivalently (by writing x =

β xβ−1 − α xα−1 ≥

1−t ), 1−p

if

β −α , ∀ x ∈ [0, 1] . λ

(25)

Moreover, g0′′ (t) ≥ 0 if and only if

α (α − 1) (1 − t )α−2 β (β − 1) (1 − t )β−2 ≥ . α (1 − p) (1 − p)β

1−t

)α−β

1−p

(26)

β (β − 1) α (α − 1)

≥

which holds, for all t ∈ [0, 1] , if and only if β ≤ 1. Now observe that the function ϕ (x) = β xβ−1 −α xα−1 is decreasing for β ≤ 1 ≤ α. Consequently, the inequality (25) holds if ϕ (1) = β−α ≥ β−α , λ which occurs when 0 < λ ≤ 1. To resume, Tα,β,λ (X , p), with α > β, is coherent if α > 1 ≥ β or α = 1 > β . The case α < β follows immediately by using that T is symmetric with respect to α and β. That is, Tα,β,λ (X , p), with α < β , is coherent if β > 1 ≥ α or β = 1 > α. Case (ii): Assume that α = β. Then, we can write Tα,β,λ (X , p) =

∫ 0

1−p

+ 2α − 1 ≥ 0, ∀ t ∈ (p, 1],

or, equivalently, if and only if (28)

Corollary 4.1. The following premium principles are coherent for p ∈ (0, 1) and 0 ≤ λ [ ≤1: ] (a) T1,1,λ (X , p) = E [X |X > F −1 (p)] + λE (X , p), λ −1 (b) T1,2,λ (X , p) = E X [ |X > F −1(p) ]+ 2 GMD(X , p), (c) T1,0.5,λ (X , p) = E X |X > F (p) + 2λW (X , p). Part (b) of Corollary 4.1 was first proved in Furman et al. (2017). 5. A numerical example In this example we compare the performance of the premium principles that appear in Corollary 4.1 when they are applied to certain skewed distributions. As in Wang (1998, Section 8.1), we consider the following two loss distributions: X is Pareto with tail function F (x) = 8 (x + 2)−3 , x ≥ 0

Therefore, g0 (t) is convex for α = 1 > β. For α > 1, (26) is the same as

(

)

In the following corollary, E (X , p) is the dynamic cumulative residual quantile entropy given by (12), GMD(X , p) is the Gini mean semidifference given by (13) and W (X , p) is the tail version of Wang deviation given by (14).

0

=

1−t

The solution for both (27) and (28) is 1/2 ≤ α ≤ 1 and λ ≤ 1. □

1

∫

(

and Y is Weibull with tail function G(x) = e−1.26957 x

0.607248

, x ≥ 0.

Both distributions have the same mean (E(X ) = E(Y ) = 1) and variance (Var(X ) = Var(Y ) = 3). Therefore, we look for a more appropriate variability measure to reflect into the premium our knowledge of their relative tail thickness (Pareto is heavier-tailed than Weibull distribution, see Embrechts and Veraverbeke (1982) and Panjer and Willmot (1992, p. 350)). We focus on measures of the form Dα,β (X ) where, for simplicity, we take α = 1 and β ∈ [0.5, 2]. For p → 0, Fig. 1 plots D1,β (X ) (solid line) and D1,β (Y ) (dashed line) for β ∈ [0.5, 2]. The two lines intersect for β = 0.733611. In particular, D1,β (X ) > D1,β (Y ), for β ∈ [0.5, 0.733611), D1,0.733611 (X ) = D1,0.733611 (Y ) = 2.49827 > and

1

F −1 (t)dg1 (t)

D1,β (X ) < D1,β (Y ), for β ∈ (0.733611, 2].

√

Var(X ).

G. Psarrakos and M.A. Sordo / Insurance: Mathematics and Economics 86 (2019) 232–240 Table 1 The functions xp , T1,2,1 (X , p), T1,0.5,1 (X , p) and T1,1,1 (X , p) for various values of p. p 0.01 0.05 0.10 0.15 0.25 0.50 0.75 0.90 0.95 0.99

xp = F −1 (p) 0.0067115 0.0344895 0.0714883 0.1113344 0.2012848 0.5198421 1.1748021 2.3088694 3.4288352 7.2831777

T1,1,1 (X , p) 2.51510 2.57760 2.66085 2.75050 2.95289 3.66964 5.14330 7.69496 10.2149 18.8871

T1,2,1 (X , p) 1.61208 1.66208 1.72868 1.80040 1.96231 2.53572 3.71464 5.75596 7.77190 14.7097

239

Table 4 The functions TCE(Y , p), E (Y , p), GMD(Y , p) and W (Y , p) for various values of p.

T1,0.5,1 (X , p)

p

TCE(Y , p)

E (Y , p)

GMD(Y , p)

W (Y , p)

7.03020 7.15520 7.32170 7.50100 7.90578 9.33929 12.2866 17.3899 22.4298 39.7743

0.01 0.05 0.10 0.15 0.25 0.50 0.75 0.90 0.95 0.99

1.01010 1.05253 1.11043 1.17429 1.32311 1.88184 3.07467 5.05145 6.81754 11.7250

1.65326 1.67955 1.71338 1.74846 1.82319 2.05043 2.40325 2.82487 3.12018 3.75087

1.36853 1.39776 1.43514 1.47368 1.55523 1.79987 2.17347 2.61405 2.92014 3.56933

2.13700 2.16009 2.18994 2.22104 2.28773 2.49323 2.81799 3.21222 3.49120 4.09273

Table 2 The functions yp , T1,2,1 (Y , p), T1,0.5,1 (Y , p) and T1,1,1 (Y , p) for various values of p. p

yp = G−1 (p)

T1,1,1 (Y , p)

T1,2,1 (Y , p)

T1,0.5,1 (Y , p)

0.01 0.05 0.10 0.15 0.25 0.50 0.75 0.90 0.95 0.99

0.0003462 0.0050706 0.0165910 0.0338718 0.0867467 0.3691276 1.1558588 2.6655584 4.1114360 8.3467320

2.66336 2.73209 2.82381 2.92275 3.14630 3.93228 5.47792 7.87633 9.93772 15.4759

1.69436 1.75141 1.82800 1.91114 2.10073 2.78178 4.16141 6.35848 8.27762 13.5097

5.28410 5.37271 5.49031 5.61638 5.89856 6.86830 8.71065 11.4759 13.7999 19.9105

Table 3 The functions TCE(X , p), E (X , p), GMD(X , p) and W (X , p) for various values of p. p

TCE(X , p)

E (X , p)

GMD(X , p)

W (X , p)

0.01 0.05 0.10 0.15 0.25 0.50 0.75 0.90 0.95 0.99

1.01007 1.05173 1.10723 1.16700 1.30193 1.77976 2.76220 4.46330 6.14325 11.9248

1.50503 1.52587 1.55362 1.58350 1.65096 1.88988 2.38110 3.23165 4.07163 6.96238

1.20403 1.22069 1.24289 1.26680 1.32077 1.51191 1.90488 2.58532 3.25730 5.56991

3.01007 3.05173 3.10723 3.16700 3.30193 3.77976 4.76220 6.46330 8.14325 13.9248

The ranking for β ∈ (0.733611, 2] disagrees with the perception that X is more dangerous risk than Y . In particular, D1,0.5 seems to be more appropriate than D1,1 and D1,2 to determine the safety loading for these risks. The corresponding unconditional (p = 0) premium principles for λ = 1 are given by T1,2,1 (X ) = 1.6, T1,0.5,1 (X ) = 7 and T1,1,1 (X ) = 2.5 and T1,2,1 (Y ) = 1.68065, T1,0.5,1 (Y ) = 5.26265 and T1,1,1 (Y ) = 2.64677. As we can see in Tables 1 and 2, the premiums can be increased substantially by conditioning on values along the tails. For T1,0.5,1 the rankings for each p are in agreement with the perceived ranking of tail thickness (this is not the case, however, of T1,1,1 and T1,2,1 ). Finally, Tables 3 and 4 compare the tail variability measures W (X , p), GMD(X , p), E (X , p) and W (Y , p), GMD(Y , p), E (Y , p) for different values of p ∈ (0, 1). Observe that for W and E the rankings for each p agree with the perception that X is more dangerous than Y . However, for GMD, the ranking only agrees for p > 0.95. These results support that W (X , p) (and even E (X , p)) can be more appropriate than GMD(X , p) to measure the right tail risk.

Fig. 1. The measures Dα,β (X ) and Dα,β (Y ) of X ∼ Pareto(3,2) (solid line) and Y ∼ Weibull(0.607248, 1.26957) (dashed line), for α = 1 and 0.5 ≤ β ≤ 2.

6. Conclusions This paper was mainly concerned with measuring variability of losses above the VaR. To address this issue, we have studied a class of tail variability measures based on distances among proportional hazards models. The class contains, among others, tail versions of some well-known variability measures, such as the Wang’s right tail deviation, the cumulative residual entropy and the Gini mean difference. As in Furman and Landsman (2006a,b) and, more recently, Furman et al. (2017), we have combined these tail variability measures with TCE to obtain premium principles that can be used to price heavy-tailed risks. These premium principles share the good properties of the Gini shortfall introduced by Furman et al. (2017). In particular, they do not require the finiteness of the second moment of the underlying risks and they are coherent for some choices of the loading parameter. Moreover, we have shown with an example that some of the tail premiums studied in this paper could be more appropriate that the Gini shortfall to price heavy-tailed risks. Finally, it is worth mentioning that some results in this paper hold for general strictly increasing distortion functions (not necessarily based on proportional hazard models). Theorem 2.1, for example, can be stated in terms of random variables X˜ hg and Y˜hg , where X˜ hg is defined by the condition

[

]

X˜ hg |Xg = s ≡st [Xh − s|Xh > s] , for all s > 0,

where Xh and Xg are distorted random variables induced from X by strictly increasing distortion functions h and g (Y˜hg is similarly defined). An advantage of the proportional hazard model framework is the mathematical tractability that allows to obtain the bounds in Section 3. Moreover, this framework provides a nice interpretation of the cumulative residual entropy as a limiting case of the class of variability measures under study.

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