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Journal of Probability and Statistics
Volume 2010 (2010), Article ID 357321, 19 pages
http://dx.doi.org/10.1155/2010/357321
Research Article

On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models

1Actuarial Research Center, University of Haifa, Haifa 31905, Israel
2Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3

Received 13 October 2009; Revised 21 March 2010; Accepted 10 April 2010

Academic Editor: Johanna Nešlehová

Copyright © 2010 Olga Furman and Edward Furman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Layer-based counterparts of a number of well-known risk measures have been proposed and studied. Namely, some motivations and elementary properties have been discussed, and the analytic tractability has been demonstrated by developing closed-form expressions in the general framework of exponential dispersion models.

1. Introduction

Denote by 𝒳 the set of (actuarial) risks, and let 0𝑋𝒳 be a random variable (rv) with cumulative distribution function (cdf) 𝐹(𝑥), decumulative distribution function (ddf) 𝐹(𝑥)=1𝐹(𝑥), and probability density function (pdf) 𝑓(𝑥). The functional 𝐻𝒳[0,] is then referred to as a risk measure, and it is interpreted as the measure of risk inherent in 𝑋. Naturally, a quite significant number of risk measuring functionals have been proposed and studied, starting with the arguably oldest Value-at-Risk or VaR (cf. [1]), and up to the distorted (cf. [25]) and weighted (cf. [6, 7]) classes of risk measures.

More specifically, the Value-at-Risk risk measure is formulated, for every 0<𝑞<1, asVaR𝑞[𝑋]=inf𝑥𝐹𝑋,(𝑥)𝑞(1.1) which thus refers to the well-studied notion of the 𝑞th quantile. Then the family of distorted risk measures is defined with the help of an increasing and concave function 𝑔[0,1][0,1], such that 𝑔(0)=0 and 𝑔(1)=1, as the following Choquet integral:𝐻𝑔[𝑋]=𝐑+𝑔𝐹(𝑥)𝑑𝑥.(1.2) Last but not least, for an increasing nonnegative function 𝑤[0,)[0,) and the so-called weighted ddf 𝐹𝑤(𝑥)=𝐄[𝟏{𝑋>𝑥}𝑤(𝑋)]/𝐄[𝑤(𝑋)] the class of weighted risk measures is given by𝐻𝑤[𝑋]=𝐑+𝐹𝑤(𝑥)0𝑥0200𝑑𝑑𝑥.(1.3) Note that for at least once differentiable distortion function, we have that the weighted class contains the distorted one as a special case, that is, 𝐻𝑔[𝑋]=𝐄[𝑋𝑔(𝐹(𝑋))] is a weighted risk measure with a dependent on 𝐹 weight function.

Interestingly, probably in the view of the latter economic developments, the so-called “tail events” have been drawing increasing attention of insurance and general finance experts. Naturally therefore, tail-based risk measures have become quite popular, with the tail conditional expectation (TCE) risk measure being a quite remarkable example. For 0<𝑞<1 and thus 𝐹(VaR𝑞[𝑋])0, the TCE risk measure is formulated asTCE𝑞[𝑋]=1𝐹VaR𝑞[𝑋]VaR𝑞[𝑋]𝑥𝑑𝐹(𝑥).(1.4) Importantly, the TCE belongs in the class of distorted risk measures with the distortion function𝑔𝑥(𝑥)=𝟏1𝑞(𝑥<1𝑞)+𝟏(𝑥1𝑞),(1.5) where 𝟏 denotes the indicator function (cf., e.g., [8]), as well as in the class of weighted risk measures with the weight function𝑤(𝑥)=𝟏𝑥VaR𝑞[𝑋](1.6) (cf., e.g., [6, 7]). The TCE risk measure is often referred to as the expected shortfall (ES) and the conditional Value-at-Risk (CVaR) when the pdf of 𝑋 is continuous (cf., e.g., [9]).

Functional (1.4) can be considered a tail-based extension of the net premium 𝐻[𝑋]=𝐄[𝑋]. Furman and Landsman [10] introduced and studied a tail-based counterpart of the standard deviation premium calculation principle, which, for 0<𝑞<1, the tail varianceTV𝑞[𝑋]=𝐕𝐚𝐫𝑋𝑋>VaR𝑞[𝑋],(1.7) and a constant 𝛼0, is defined asTSD𝑞[𝑋]=TCE𝑞[𝑋]+𝛼TV𝑞1/2[𝑋].(1.8) For a discussion of various properties of the TSD risk measure, we refer to Furman and Landsman [10]. We note in passing that for 𝑞0, we have that TSD𝑞[𝑋]SD[𝑋]=𝐄[𝑋]+𝛼𝐕𝐚𝐫1/2[𝑋].

The rest of the paper is organized as follows. In the next section we introduce and motivate layer-based extensions of functionals (1.4) and (1.8). Then in Sections 3 and 4 we analyze the aforementioned layer-based risk measures as well as their limiting cases in the general context of the exponential dispersion models (EDMs), that are to this end briefly reviewed in the appendix. Section 5 concludes the paper.

2. The Limited TCE and TSD Risk Measures

Let 0<𝑞<𝑝<1 and let 𝑋𝒳 have a continuous and strictly increasing cdf. In many practical situations the degree of riskiness of the layer (VaR𝑞[𝑋],VaR𝑝[𝑋]) of an insurance contract is to be measured (certainly the layer width VaR𝑝[𝑋]VaR𝑞[𝑋]=Δ𝑞,𝑝>0). Indeed, the number of deductibles in a policy is often more than one, and/or there can be several reinsurance companies covering the same insured object. Also, there is the so-called “limited capacity” within the insurance industry to absorb losses resulting from, for example, terrorist attacks and catastrophes. In the context of the aforementioned events, the unpredictable nature of the threat and the size of the losses make it unlikely that the insurance industry can add enough capacity to cover them. In these and other cases neither (1.4) nor (1.8) can be applied since (1) both TCE and TSD are defined for one threshold, only, and (2) the aforementioned pair of risk measures is useless when, say, the expectations of the underlying risks are infinite, which can definitely be the case in the situations mentioned above.

Note. As noticed by a referee, the risk measure 𝐻𝒳[0,] is often used to price (insurance) contracts. Naturally therefore, the limited TCE and TSD proposed and studied herein can serve as pricing functionals for policies with coverage modifications, such as, for example, policies with deductibles, retention levels, and so forth (cf., [11, Chapter 8]).

Next, we formally define the risk measures of interest.

Definition 2.1. Let 𝑥𝑞=VaR𝑞[𝑋] and 𝑥𝑝=VaR𝑝[𝑋], for 0<𝑞<𝑝<1. Then the limited TCE and TSD risk measures are formulated as LTCE𝑞,𝑝[𝑋]=𝐄𝑋𝑥𝑞<𝑋𝑥𝑝,(2.1) and LTSD𝑞,𝑝[𝑋]=𝐄𝑋𝑥𝑞<𝑋𝑥𝑝+𝛼𝐕𝐚𝐫1/2𝑋𝑥𝑞<𝑋𝑥𝑝,(2.2) respectively.

Clearly, the TCE and TSD are particular cases of their limited counterparts. We note in passing that the former pair of risk measures need not be finite for heavy tailed distributions, and they are thus not applicable. In this respect, limited variants (2.1) and (2.2) can provide a partial resolution. Indeed, for 𝑘=1,2,, we have that 𝐄𝑋𝑘𝑥𝑞<𝑋𝑥𝑝=𝐹𝑥𝑝𝐄𝑋𝑘𝑋𝑥𝑝𝑥𝐹𝑞𝐄𝑋𝑘𝑋𝑥𝑞𝐹𝑥𝑝𝑥𝐹𝑞<,(2.3) regardless of the distribution of 𝑋.

We further enumerate some properties of the LTSD risk measure, which is our main object of study.(1)Translation Invariance. For any constant 𝑐0, we have that LTSD𝑞,𝑝[]𝑋+𝑐=LTSD𝑞,𝑝[𝑋]+𝑐.(2.4)(2)Positive Homogeneity. For any constant 𝑑>0, we have that LTSD𝑞,𝑝[]𝑑𝑋=𝑑LTSD𝑞,𝑝[𝑋].(2.5)(3)Layer Parity. We call 𝑋𝒳 and 𝑌𝒳 layer equivalent if for some 0<𝑞<𝑝<1, such that 𝑥𝑞=𝑦𝑞,𝑥𝑝=𝑦𝑝, and for every pair {(𝑡1,𝑡2)𝑞<𝑡1<𝑡2<𝑝}, it holds that 𝐏[𝑥𝑡1<𝑋𝑥𝑡2]=𝐏[𝑦𝑡1<𝑌𝑦𝑡2]. In such a case, we have that LTSD𝑡1,𝑡2[𝑋]=LTSD𝑡1,𝑡2[𝑌].(2.6) Literally, this property states that the LTSD risk measure for an arbitrary layer is only dependent on the cdf of that layer. Parity of the ddfs implies equality of LTSDs.

Although looking for original ways to assess the degree of (actuarial) riskiness is a very important task, subsequent applications of various theoretical approaches to a real-world data are not less essential. A significant number of papers have been devoted to deriving explicit formulas for some tail-based risk measures in the context of various loss distributions. The incomplete list of works discussing the TCE risk measure consists of, for example, Hürlimann [12] and Furman and Landsman [13], gamma distributions; Panjer [14], normal family; Landsman and Valdez [15], elliptical distributions; Landsman and Valdez [16], and Furman and Landsman [17], exponential dispersion models; and Chiragiev and Landsman [18], Vernic [19], Asimit et al. [20], Pareto distributions of the second kind.

As we have already noticed, the “unlimited” tail standard deviation risk measure has been studied in the framework of the elliptical distributions by Furman and Landsman [10]. Unfortunately, all members of the elliptical class are symmetric, while insurance risks are generally modeled by nonnegative and positively skewed random variables. These peculiarities can be fairly well addressed employing an alternative class of distribution laws. The exponential dispersion models include many well-known distributions such as normal, gamma, and inverse Gaussian, which, except for the normal, are nonsymmetric, have nonnegative supports, and can serve as adequate models for describing insurance risks’ behavior. In this paper we therefore find it appropriate to apply both TSD and LTSD to EDM distributed risks.

3. The Limited Tail Standard Deviation Risk Measure for Exponential Dispersion Models

An early development of the exponential dispersion models is often attributed to Tweedie [21], however a more substantial and systematic investigation of this class of distributions was documented by Jørgensen [22, 23]. In his Theory of dispersion models, Jørgensen [24] writes that the main raison d’étre for the dispersion models is to serve as error distributions for generalized linear models, introduced by Nelder and Wedderburn [25]. Nowadays, EDMs play a prominent role in actuarial science and financial mathematics. This can be explained by the high level of generality that they enable in the context of statistical inference for widely popular distribution functions, such as normal, gamma, inverse Gaussian, stable, and many others. The specificity characterizing statistical modeling of actuarial subjects is that the underlying distributions mostly have nonnegative support, and many EDM members possess this important phenomenon, (for a formal definition of the EDMs, as well as for a brief review of some technical facts used in the sequel, cf., the appendix).

We are now in a position to evaluate the limited TSD risk measure in the framework of the EDMs. Recall that, for 0<𝑞<𝑝<1, we denote by (𝑥𝑞,𝑥𝑝) an arbitrary layer having “attachment point” 𝑥𝑞 and width Δ𝑞,𝑝. Also, let𝑥𝑞,𝑥𝑝=𝜕;𝜃,𝜆𝐹𝑥𝜕𝜃log𝑝𝑥;𝜃,𝜆𝐹𝑞;𝜃,𝜆(3.1) denote the generalized layer-based hazard function, such that𝑥𝑞,𝑥1=𝜕;𝜃,𝜆𝜕𝜃log𝐹𝑥𝑞𝑥;𝜃,𝜆=𝑞,𝑥;𝜃,𝜆0,𝑥𝑝𝜕;𝜃,𝜆=𝜕𝜃log𝐹𝑥𝑝𝑥;𝜃,𝜆=𝑝,;𝜃,𝜆(3.2) and thus𝑥𝑞,𝑥𝑝=;𝜃,𝜆𝐹𝑥𝑞;𝜃,𝜆𝐹𝑥𝑞;𝜃,𝜆𝐹𝑥𝑝𝑥;𝜃,𝜆𝑞;𝜃,𝜆𝐹𝑥𝑝;𝜃,𝜆𝐹𝑥𝑞;𝜃,𝜆𝐹𝑥𝑝𝑥;𝜃,𝜆𝑝.;𝜃,𝜆(3.3)

The next theorem derives expressions for the limited TCE risk measure, which is a natural precursor to deriving the limited TSD.

Theorem 3.1. Assume that the natural exponential family (NEF) which generates EDM is regular or at least steep (cf. [24, page 48]). Then the limited TCE risk measure (i)for the reproductive EDM𝑌ED(𝜇,𝜎2) is given by LTCE𝑞,𝑝[𝑌]=𝜇+𝜎2𝑥𝑞,𝑥𝑝;𝜃,𝜆(3.4) and(ii)for the additive EDM𝑋ED(𝜃,𝜆) is given by LTCE𝑞,𝑝[𝑋]=𝜆𝜅𝑥(𝜃)+𝑞,𝑥𝑝.;𝜃,𝜆(3.5)

Proof. We prove the reproductive case only, since the additive case follows in a similar fashion. By the definition of the limited TCE, we have that LTCE𝑞,𝑝[𝑌]=𝐹𝑦𝑞𝐄𝑌𝑌>𝑦𝑞𝐹𝑦𝑝𝐄𝑌𝑌>𝑦𝑝𝐹𝑦𝑝𝑦𝐹𝑞.(3.6) Further, following Landsman and Valdez [16], it can be shown that for every 0<𝑞<1, we have that 𝐄𝑌𝑌>𝑦𝑞=𝜇+𝜎2𝑦𝑞,;𝜃,𝜆(3.7) which then, employing (3.1) and (3.3), yields LTCE𝑞,𝑝[𝑌]=𝐹𝑦𝑞;𝜃,𝜆𝜇+𝜎2𝑦𝑞;𝜃,𝜆𝐹𝑦𝑝;𝜃,𝜆𝜇𝜎2𝑦𝑝;𝜃,𝜆𝐹𝑦𝑞;𝜃,𝜆𝐹𝑦𝑝;𝜃,𝜆=𝜇+𝜎2𝑦𝑞,𝑦𝑝;𝜃,𝜆(3.8) and hence completes the proof.

In the sequel, we sometimes write LTCE𝑞,𝑝[𝑌;𝜃,𝜆] in order to emphasize the dependence on 𝜃 and 𝜆.

Note. To obtain the results of Landsman and Valdez [16], we put p1, and then, for instance, in the reproductive case, we end up with lim𝑝1LTCE𝑞,𝑝[𝑌]=𝜇+𝜎2𝑦𝑞;𝜃,𝜆=TCE𝑞[𝑌],(3.9) subject to the existence of the limit.

Next theorem provides explicit expressions for the limited TSD risk measure for both reproductive and additive EDMs.

Theorem 3.2. Assume that the NEF which generates EDM is regular or at least steep. Then the limited TSD risk measure (i)for the reproductive EDM𝑌ED(𝜇,𝜎2) is given by LTSD𝑞,𝑝[𝑌]=LTCE𝑞,𝑝[𝑌]+𝛼𝜎2𝜕𝜕𝜃LTCE𝑞,𝑝[]𝑌;𝜃,𝜆(3.10) and (ii)for the additive EDM𝑋ED(𝜃,𝜆) is given by LTSD𝑞,𝑝[𝑋]=LTCE𝑞,𝑝[𝑋]+𝛼𝜕𝜕𝜃LTCE𝑞,𝑝[].𝑋;𝜃,𝜆(3.11)

Proof. We again prove the reproductive case, only. Note that it has been assumed that 𝜅(𝜃) is a differentiable function, and thus we can differentiate the following probability integral in 𝜃 under the integral sign (cf., the appendix): 𝐏𝑦𝑞<𝑌𝑦𝑝=𝑦𝑝𝑦𝑞𝑒𝜆(𝜃𝑦𝜅(𝜃))𝑑𝜈𝜆(𝑦),(3.12) and hence, using Definition 2.1, we have that 𝜕𝜕𝜃LTCE𝑞,𝑝[]𝐹𝑦𝑌;𝜃,𝜆𝑝𝑦;𝜃,𝜆𝐹𝑞=;𝜃,𝜆𝑦𝑝𝑦𝑞𝜕𝜕𝜃𝑦𝑒𝜆(𝜃𝑦𝜅(𝜃))𝑑𝜈𝜆(𝑦)=𝜆𝑦𝑝𝑦𝑞𝑦2𝑒𝜆(𝜃𝑦𝜅(𝜃))𝑦𝜅(𝜃)𝑒𝜆(𝜃𝑦𝜅(𝜃))𝑑𝜈𝜆(𝑦)=𝜎2𝐄𝑌2𝑦𝟏𝑞<𝑌𝑦𝑝𝑦𝜇(𝜃)𝐄𝑌𝟏𝑞<𝑌𝑦𝑝,(3.13) with the last line following from the appendix. Further, by simple rearrangement and straightforward calculations, we obtain that 𝐄𝑌2𝑦𝑞<𝑌𝑦𝑝=𝑦𝑝𝑦𝑞𝑦2𝑒𝜆(𝜃𝑦𝜅(𝜃))𝑑𝜈𝜆(𝑦)𝐹𝑦𝑝𝑦;𝜃,𝜆𝐹𝑞;𝜃,𝜆=𝜇LTCE𝑞,𝑝[𝑌]+𝜎2(𝜕/𝜕𝜃)LTCE𝑞,𝑝[]𝐹𝑦𝑌;𝜃,𝜆𝑝𝑦;𝜃,𝜆𝐹𝑞;𝜃,𝜆𝐹𝑦𝑝𝑦;𝜃,𝜆𝐹𝑞;𝜃,𝜆=𝜎2𝜕𝜕𝜃LTCE𝑞,𝑝[]𝑌;𝜃,𝜆+LTCE𝑞,𝑝[𝑌]𝜇+𝜎2𝜕𝐹𝑦𝜕𝜃log𝑝𝑦;𝜃,𝜆𝐹𝑞;𝜃,𝜆=𝜎2𝜕𝜕𝜃LTCE𝑞,𝑝[]+𝑌;𝜃,𝜆LTCE𝑞,𝑝[]𝑌;𝜃,𝜆2,(3.14) which along with the definition of the limited TSD risk measure completes the proof.

We further consider two examples to elaborate on Theorem 3.2. We start with the normal distribution, which occupies a central role in statistical theory, and its position in statistical analysis of insurance problems is very difficult to underestimate, for example, due to the law of large numbers.

Example 3.3. Let 𝑌𝑁(𝜇,𝜎2) be a normal random variable with mean 𝜇 and variance 𝜎2, then we can write the pdf of 𝑌 as 1𝑓(𝑦)=12𝜋𝜎exp2𝑦𝜇𝜎2=112𝜋𝜎exp2𝜎2𝑦21exp𝜎21𝜇𝑦2𝜇2,𝑦𝐑.(3.15) If we take 𝜃=𝜇 and 𝜆=1/𝜎2, we see that the normal distribution is a reproductive EDM with cumulant function 𝜅(𝜃)=𝜃2/2. Denote by 𝜑() and Φ() the pdf and the cdf, respectively, of the standardized normal random variable. Then using Theorem 3.1, we obtain the following expression for the limited TCE risk measure for the risk 𝑌: LTCE𝑞,𝑝[𝑌]𝜑𝜎=𝜇+𝜎1𝑦𝑞𝜎𝜇𝜑1𝑦𝑝𝜇Φ𝜎1𝑦𝑝𝜎𝜇Φ1𝑦𝑞.𝜇(3.16) If we put 𝑝1, then the latter equation reduces to the result of Landsman and Valdez [16]. Namely, we have that lim𝑝1LTCE𝑞,𝑝[𝑌]𝜑𝜎=𝜇+𝜎1𝑦𝑞𝜇𝜎1Φ1𝑦𝑞𝜇=TCE𝑞[𝑌].(3.17) Further, let 𝑧𝑞=(𝑦𝑞𝜇)/𝜎 and 𝑧𝑝=(𝑦𝑝𝜇)/𝜎. Then 𝜎2𝜕𝜕𝜃LTCE𝑞,𝑝[]𝑌;𝜃,𝜆=𝜎2𝜑𝑧1+𝑞𝑧𝑞𝑧𝜑𝑝𝑧𝑝Φ𝑧𝑝𝑧Φ𝑞𝜑𝑧𝑞𝑧𝜑𝑝Φ𝑧𝑝𝑧Φ𝑞2.(3.18) Consequently, the limited TSD risk measure is as follows: LTSD𝑞,𝑝[𝑌]𝜑𝑧=𝜇+𝜎𝑞𝑧𝜑𝑝Φ𝑧𝑝𝑧Φ𝑞+𝛼𝜎2𝜑𝑧1+𝑞𝑧𝑞𝑧𝜑𝑝𝑧𝑝Φ𝑧𝑝𝑧Φ𝑞𝜑𝑧𝑞𝑧𝜑𝑝Φ𝑧𝑝𝑧Φ𝑞2.(3.19)

We proceed with the gamma distributions, which have been widely applied in various fields of actuarial science. It should be noted that these distribution functions possess positive support and positive skewness, which is important for modeling insurance losses. In addition, gamma rvs have been well-studied, and they share many tractable mathematical properties which facilitate their use. There are numerous examples of applying gamma distributions for modeling insurance portfolios (cf., e.g., [12, 13, 26, 27]).

Example 3.4. Let 𝑋Ga(𝛾,𝛽) be a gamma rv with shape and rate parameters equal 𝛾 and 𝛽, correspondingly. The pdf of 𝑋 is 1𝑓(𝑥)=Γ𝑒(𝛾)𝛽𝑥𝑥𝛾1𝛽𝛾=1Γ𝑥(𝛾)𝛾1exp(𝛽𝑥+𝛾log(𝛽)),𝑥>0.(3.20) Hence the gamma rv can be represented as an additive EDM with the following pdf: 1𝑓(𝑥)=Γ𝑥(𝜆)𝜆1exp(𝜃𝑥+𝜆log(𝜃)),(3.21) where 𝑥>0 and 𝜃<0.The mean and variance of 𝑋 are 𝐄[𝑋]=𝜆/𝜃 and 𝐕𝐚𝐫[𝑋]=𝜆/𝜃2. Also, 𝜃=𝛽, 𝜆=𝛾, and 𝜅(𝜃)=log(𝜃). According to Theorem 3.1, the limited tail conditional expectation is LTCE𝑞,𝑝[𝑋]𝜆=𝜃𝐹𝑥𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞;𝜃,𝜆.(3.22) Putting 𝑝1 we obtain that lim𝑝1𝜆𝜃𝐹𝑥𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞𝜆;𝜃,𝜆=𝜃𝐹𝑥𝑞;𝜃,𝜆+1𝐹𝑥𝑞;𝜃,𝜆=TCE𝑞[𝑋],(3.23) which coincides with [13, page 643]. To derive an expression for the limited TSD risk measure, we use Theorem 3.2, that is, 𝜕𝜕𝜃LTCE𝑞,𝑝[]=𝜕𝑋;𝜃,𝜆𝜆𝜕𝜃𝜃𝐹𝑥𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞=𝜆;𝜃,𝜆𝜃2𝐹𝑥𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞𝜆;𝜃,𝜆𝜃𝜕𝐹𝑥𝜕𝜃𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞.;𝜃,𝜆(3.24) Further, since for 𝑛=1,2,,𝜕𝐹𝑥𝜕𝜃𝑝𝑥;𝜃,𝜆+𝑛𝐹𝑞=;𝜃,𝜆+𝑛𝑥𝑝𝑥𝑞𝜕1𝜕𝜃𝑥Γ(𝜆+𝑛)𝜆+𝑛1=exp(𝜃𝑥+(𝜆+𝑛)log(𝜃))𝑑𝑥𝑥𝑝𝑥𝑞𝑓(𝑥;𝜃,𝜆+𝑛)𝑥+𝜆+𝑛𝜃𝑑𝑥=𝜆+𝑛𝜃𝑥𝑝𝑥𝑞𝑓(𝑥;𝜃,𝜆+𝑛+1)𝑑𝑥𝑥𝑝𝑥𝑞,𝑓(𝑥;𝜃,𝜆+𝑛)𝑑𝑥(3.25) the limited TSD risk measure for gamma is given by LTSD𝑞,𝑝[𝑋]=𝜆𝜃𝐹𝑥𝑝𝑥;𝜃,𝜆+1𝐹𝑞;𝜃,𝜆+1𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞;𝜃,𝜆+𝛼𝜆𝜃2𝐹𝑥(𝜆+1)𝑝𝑥;𝜃,𝜆+2𝐹𝑞;𝜃,𝜆+2𝐹𝑥𝑝𝑥;𝜃,𝜆𝐹𝑞;𝜃,𝜆𝜆𝐹(𝑥𝑝;𝜃,𝜆+1)𝐹(𝑥𝑞;𝜃,𝜆+1)𝐹(𝑥𝑝;𝜃,𝜆)𝐹(𝑥𝑞;𝜃,𝜆)2.(3.26)

In the sequel, we consider gamma and normal risks with equal means and variances, and we explore them on the interval (𝑡,350], with 50<𝑡<350. Figure 1 depicts the results. Note that both LTCE and LTSD imply that the normal distribution is riskier than gamma for lower attachment points and vice-versa, that is quite natural bearing in mind the tail behavior of the two.

fig1
Figure 1: LTCE and LTSD for normal and gamma risks with means 150 and standard deviations 100, alpha = 2.

Although the EDMs are of pivotal importance in actuarial mathematics, they fail to appropriately describe heavy-tailed (insurance) losses. To elucidate on the applicability of the layer-based risk measures in the context of the probability distributions possessing heavy tails, we conclude this section with a simple example.

Example 3.5. Let 𝑋Pa(𝛾,𝛽) be a Pareto rv with the pdf 𝑓(𝑥)=𝛾𝛽𝛾𝑥𝛾+1,𝑥>𝛽>0,(3.27) and 𝛾>0. Certainly, the Pareto rv is not a member of the EDMs, though it belongs to the log-exponential family (LEF) of distributions (cf. [7]). The LEF is defined by the differential equation 𝐹(𝑑𝑥;𝜆,𝜈)=exp{𝜆log(𝑥)𝜅(𝜆)}𝜈(𝑑𝑥),(3.28) where 𝜆 is a parameter, 𝜈 is a measure, and 𝜅(𝜆)=log0𝑥𝜆𝜈(𝑑𝑥) is a normalizing constant (the parameters should not be confused with the ones used in the context of the EDMs). Then 𝑋 is easily seen to belong in LEF with the help of the reparameterization 𝜈(𝑑𝑥)=𝑥1𝑑𝑥, and 𝜆=𝛾.
In this context, it is straightforward to see that 𝐄[𝑋] is infinite for 𝛾1, which thus implies infiniteness of the TCE risk measure. We can however readily obtain the limited variant as follows: LTCE𝑞,𝑝[𝑋]=1𝐏𝑥𝑞<𝑋𝑥𝑝𝑥𝑝𝑥𝑞𝛾𝛽𝛾𝑥𝛾𝑑𝑥=𝛾𝑥𝑝𝑥𝑞𝑥𝛾1𝑝𝛾1𝑥𝑞𝛾1𝑥𝛾𝑝𝑥𝛾𝑞,(3.29) that is finite for any 𝛾>0. Also, since, for example, for 𝛾<1, we have that 𝑥𝑝𝛾1𝑥𝑞𝛾1<0, the limited TCE risk measure is positive, as expected. The same is true for 𝛾1.
We note in passing that, for 𝛾>1 and 𝑝1 and thus 𝑥𝑝, we have that TCE𝑞[𝑋]=lim𝑝1𝛾𝑥𝑝𝑥𝑞𝑥𝛾1𝑝𝛾1𝑥𝑞𝛾1𝑥𝛾𝑝𝑥𝛾𝑞=𝛾𝑥𝛾1𝑞,(3.30) which confirms the corresponding expression in Furman and Landsman [8].

Except for the Pareto distribution, the LEF consists of, for example, the log-normal and inverse-gamma distributions, for which expressions similar to (3.29) can be developed in the context of the limited TCE and limited TSD risk measures, thus providing a partial solution to the heavy-tailness phenomenon.

4. The Tail Standard Deviation Risk Measure for Exponential Dispersion Models

The tail standard deviation risk measure was proposed in [10] as a possible quantifier of the so-called tail riskiness of the loss distribution. The above-mentioned authors applied this risk measure to elliptical class of distributions, which consists of such well-known pdfs as normal and student-𝑡. Although the elliptical family is very useful in finance, insurance industry imposes its own restrictions. More specifically, insurance claims are always positive and mostly positively skewed. In this section we apply the TSD risk measure to EDMs.

The following corollary develops formulas for the TSD risk measure both in the reproductive and additive EDMs cases. Recall that we denote the ddf of say 𝑋 by 𝐹(;𝜃,𝜆) to emphasize the parameters 𝜃 and 𝜆, and we assume that lim𝑝1LTSD𝑞,𝑝[𝑋]<.(4.1) The proof of the next corollary is left to the reader.

Corollary 4.1. Under the conditions in Theorem 3.1, the tail standard deviation risk measure is TSD𝑞[𝑌]=TCE𝑞[𝑌]+𝛼𝜎2𝜕𝜕𝜃TCE𝑞[]𝑌;𝜃,𝜆(4.2) in the context of the reproductive EDMs, and TSD𝑞[𝑋]=TCE𝑞[𝑋]+𝛼𝜕𝜕𝜃TCE𝑞[]𝑋;𝜃,𝜆(4.3) in the context of the additive EDMs.

We further explore the TSD risk measure in some particular cases of EDMs, which seem to be of practical importance.

Example 4.2. Let 𝑌𝑁(𝜇,𝜎2) be again some normal rv with mean 𝜇 and variance 𝜎2. Then we easily evaluate the TSD risk measure using Corollary 4.1 and Example 3.3 as follows: TSD𝑞[𝑋]𝜑𝑧=𝜇+𝜎𝑞𝑧1Φ𝑞+𝛼𝜎2𝜑𝑧1+𝑞𝑧1Φ𝑞𝑧𝑞𝜑𝑧𝑞𝑧1Φ𝑞2,(4.4) which coincides with [10].

Example 4.3. Let 𝑋Ga(𝛾,𝛽) be a gamma rv with shape and scale parameters equal 𝛾 and 𝛽, correspondingly. Taking into account Example 3.4 and Corollary 4.1 leads to TSD𝑞[𝑋]𝜆=𝜃𝐹𝑥𝑞;𝜃,𝜆+1𝐹𝑥𝑞;𝜃,𝜆+𝛼𝜆𝜃2(𝜆+1)𝐹𝑥𝑞;𝜃,𝜆+2𝐹𝑥𝑞;𝜃,𝜆𝜆𝐹(𝑥𝑞;𝜃,𝜆+1)𝐹(𝑥𝑞;𝜃,𝜆)2=𝛾𝛽𝐹𝑥𝑞;𝛾+1,𝛽𝐹𝑥𝑞;𝛾,𝛽+𝛼𝛾𝛽2(𝛾+1)𝐹𝑥𝑞;𝛾+2,𝛽𝐹𝑥𝑞;𝛾,𝛽𝛾𝐹(𝑥𝑞;𝛾+1,𝛽)𝐹(𝑥𝑞;𝛾,𝛽)2,(4.5) where the latter equation follows because of the reparameterization 𝜃=𝛽 and 𝜆=𝛾.

We further discuss the inverse Gaussian distribution, which possesses heavier tails than, say, gamma distribution, and therefore it is somewhat more tolerant to large losses.

Example 4.4. Let 𝑌IG(𝜇,𝜆) be an inverse Gaussian rv. We then can write its pdf as 𝑓(𝑦)=𝜆2𝜋𝑦3𝜆𝑦exp2𝜇21+12𝑦𝜇[,𝑦0,),(4.6) (cf. [24]), which means that 𝑌 belongs to the reproductive EDMs, with 𝜃=1/(2𝜇2)and 𝜅(𝜃)=1/𝜇=2𝜃. Further, due to Corollary 4.1 we need to calculate 𝜕𝜕𝜃TCE𝑞[]=𝜕𝑌;𝜃,𝜆𝜕𝜃𝜇(𝜃)+𝜎2𝜕𝜕𝜃log𝐹𝑌𝑦𝑞;𝜃,𝜆=𝜇(𝜃)+𝜎2𝜕(𝜕𝜃𝜕/𝜕𝜃)𝐹𝑌𝑦𝑞;𝜃,𝜆𝐹𝑌𝑦𝑞;𝜃,𝜆.(4.7) To this end, note that the ddf of 𝑌 is 𝐹𝑦𝑞=;𝜇(𝜃),𝜆Φ𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)1𝑒2𝜆/𝜇(𝜃)Φ𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)+1(4.8) (cf., e.g., [28]), where Φ() is the ddf of the standardized normal random variable. Hence, by simple differentiation and noticing that 𝜇(𝜃)=(2𝜃)3/2=𝜇(𝜃)3,(4.9) we obtain that 𝜕𝜕𝜃𝐹𝑦𝑞;𝜇(𝜃),𝜆=𝜇(𝜃)𝜆𝑦𝑞𝜑𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)1𝑒2𝜆/𝜇(𝜃)𝜆𝑦𝑞𝜑𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)+1+2𝜆𝜇(𝜃)𝑒2𝜆/𝜇(𝜃)Φ𝜆𝑦𝑞𝑦𝑞.𝜇(𝜃)+1(4.10) Notably, 𝜆𝑦𝑞𝜑𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)1=𝑒2𝜆/𝜇(𝜃)𝜆𝑦𝑞𝜑𝜆𝑦𝑞𝑦𝑞,𝜇(𝜃)+1(4.11) and therefore (4.10) results in 𝜕𝜕𝜃𝐹𝑦𝑞;𝜇(𝜃),𝜆=2𝜆𝜇(𝜃)𝑒2𝜆/𝜇(𝜃)Φ𝜆𝑦𝑞𝑦𝑞.𝜇(𝜃)+1(4.12) Consequently, the expression for the TCE risk measure, obtained by Landsman and Valdez [16], is simplified to TCE𝑞[]𝑌;𝜃,𝜆=𝜇(𝜃)+2𝜇(𝜃)𝐹𝑦𝑞𝑒;𝜇(𝜃),𝜆2𝜆/𝜇(𝜃)Φ𝜆𝑦𝑞𝑦𝑞.𝜇(𝜃)+1(4.13) In order to derive the TSD risk measure we need to differentiate again, that is, 𝜕𝜕𝜃TCE𝑞[]=𝜕𝑌;𝜃,𝜆𝜇𝜕𝜃(𝜃)+2𝜇(𝜃)𝐹𝑦𝑞𝑒;𝜇(𝜃),𝜆2𝜆/𝜇(𝜃)Φ𝜆𝑦𝑞𝑦𝑞𝜇(𝜃)+1=𝜇(𝜃)3𝜕1+𝜕𝜃2𝜇(𝜃)𝑒2𝜆/𝜇(𝜃)Φ𝜆/𝑦𝑞𝑦𝑞/𝜇(𝜃)+1𝐹𝑦𝑞,;𝜇(𝜃),𝜆(4.14) where we use 𝜇(𝜃)=𝜇(𝜃)3. Further, we have that 𝜕𝜕𝜃2𝜇(𝜃)𝑒2𝜆/𝜇(𝜃)Φ𝜆/𝑦𝑞𝑦𝑞/𝜇(𝜃)+1𝐹𝑦𝑞;𝜇(𝜃),𝜆=2𝜇(𝜃)3𝑒2𝜆/𝜇(𝜃)Φ̃𝑦𝑞(12𝜆/𝜇(𝜃))+𝜆𝑦𝑞𝜑/𝜇(𝜃)̃𝑦𝑞𝐹𝑦𝑞𝜆;𝜇(𝜃),𝜆2𝜇(𝜃)𝑒2𝜆/𝜇(𝜃)Φ̃𝑦𝑞2𝐹𝑦𝑞;𝜇(𝜃),𝜆2,(4.15) where ̃𝑦𝑞=𝜆/𝑦𝑞(𝑦𝑞/𝜇(𝜃)+1). Therefore TSD𝑞[𝑌]Φ=𝜇1+̃𝑦𝑞𝐹𝑦𝑞;𝜇,𝜆2𝑒2𝜆/𝜇+𝛼𝜇3𝜆𝑒1+2𝜆/𝜇Φ̃𝑦𝑞(12𝜆/𝜇)+𝜆𝑦𝑞𝜑/𝜇̃𝑦𝑞𝐹𝑦𝑞𝜆𝑒;𝜇,𝜆2𝜆/𝜇Φ̃𝑦𝑞2𝜇𝐹𝑦𝑞;𝜇,𝜆2(4.16) subject to 𝐕𝐚𝐫[𝑌]=𝜇3/𝜆.

5. Concluding Comments

In this work we have considered certain layer-based risk measuring functionals in the context of the exponential dispersion models. Although we have made an accent on the absolutely continuous EDMs, similar results can be developed for the discrete members of the class. Indeed, distributions with discrete supports often serve as frequency models in actuarial mathematics. Primarily in expository purposes, we further consider a very simple frequency distribution, and we evaluate the TSD risk measure for it. More encompassing formulas can however be developed with some effort for other EDM members of, say, the (𝑎,𝑏,0) class (cf., [11, Chapter 6]) as well as for limited TCE/TSD risk measures.

Example 5.1. Let 𝑋Poisson(𝜇) be a Poisson rv with the mean parameter 𝜇. Then the probability mass function of 𝑋 is written as 1𝑝(𝑥)=𝜇𝑥!𝑥𝑒𝜇=1𝑥!exp(𝑥log(𝜇)𝜇),𝑥=0,1,,(5.1) which belongs to the additive EDMs in view of the reparametrization 𝜃=log(𝜇),𝜆=1, and 𝜅(𝜃)=𝑒𝜃.
Motivated by Corollary 4.1, we differentiate (cf. [16], for the formula for the TCE risk measure) 𝜕𝜕𝜃TCE𝑞𝜕(𝑋;𝜃,𝜆)=𝑒𝜕𝜃𝜃𝑝𝑥1+𝑞;𝜃,1𝐹𝑥𝑞;𝜃,1=𝑒𝜃𝑝𝑥1+𝑞;𝜃,1𝐹𝑥𝑞+𝑝𝑥;𝜃,1𝑞;𝜃,1𝐹𝑥𝑞𝑥;𝜃,1𝑞𝑒𝜃𝑒𝜃𝑝(𝑥𝑞;𝜃,1)𝐹(𝑥𝑞;𝜃,1)2=𝑒𝜃𝐹𝑥𝑞1;𝜃,1𝐹𝑥𝑞+𝑝𝑥;𝜃,1𝑞;𝜃,1𝐹𝑥𝑞𝑥;𝜃,1𝑞𝑒𝜃𝑒𝜃𝑝(𝑥𝑞;𝜃,1)𝐹(𝑥𝑞;𝜃,1)2,(5.2) where the latter equation follows because 𝐹𝑥𝑞𝑥;𝜃,1+𝑝𝑞=;𝜃,1𝐹𝑥𝑞.1;𝜃,1(5.3) The formula for the TSD risk measure is then TSD𝑞(𝑋)=𝑒𝜃𝑝𝑥1+𝑞;𝜃,1𝐹𝑥𝑞;𝜃,1+𝛼𝑒𝜃𝐹𝑥𝑞1;𝜃,1𝐹𝑥𝑞+𝑝𝑥;𝜃,1𝑞;𝜃,1𝐹𝑥𝑞𝑧;𝜃,1𝑞𝑒𝜃𝑝(𝑥𝑞;𝜃,1)𝐹(𝑥𝑞;𝜃,1)2,(5.4) where 𝐄[𝑋]=𝐕𝐚𝐫[𝑋]=𝑒𝜃 and 𝑧𝑞=𝑥𝑞𝑒𝜃.

Appendix

A. Exponential Dispersion Models

Consider a 𝜎-finite measure 𝜈 on 𝐑 and assume that 𝜈 is nondegenerate. Next definition is based on [24].

Definition A.1. The family of distributions of 𝑋ED(𝜃,𝜆) for (𝜃,𝜆)Θ×Λ is called the additive exponential dispersion model generated by 𝜈. The corresponding family of distributions of 𝑌=𝑋/𝜆ED(𝜇,𝜎2), where 𝜇=𝜏(𝜃) and 𝜎2=1/𝜆 are the mean value and the dispersion parameters, respectively, is called the reproductive exponential dispersion model generated by 𝜈. Moreover, given some measure 𝜈𝜆 the representation of 𝑋ED(𝜃,𝜆) is as follows: exp(𝜃𝑥𝜆𝜅(𝜃))𝜈𝜆(𝑑𝑥).(A.1) If in addition the measure 𝜈𝜆 has density 𝑐(𝑥;𝜆) with respect to some fixed measure (typically Lebesgue measure or counting measure), the density for the additive model is 𝑓(𝑥;𝜃,𝜆)=𝑐(𝑥;𝜆)exp(𝜃𝑥𝜆𝜅(𝜃)),𝑥𝐑.(A.2) Similarly, we obtain the following representation of 𝑌ED(𝜇,𝜎2) as exp(𝜆(𝑦𝜃𝜅(𝜃)))𝜈𝜆(𝑑𝑦),(A.3) where 𝜈𝜆 denotes 𝜈𝜆 transformed by the duality transformation 𝑋=𝑌/𝜎2. Again if the measure 𝜈𝜆 has density 𝑐(𝑦;𝜆) with respect to a fixed measure, the reproductive model has the following pdf: 𝑓(𝑦;𝜃,𝜆)=𝑐(𝑦;𝜆)exp(𝜆(𝜃𝑦𝜅(𝜃))),𝑦𝐑.(A.4)

Note that 𝜃 and 𝜆 are called canonical and index parameters, Θ={𝜃𝐑𝜅(𝜃)<} for some function 𝜅(𝜃) called the cumulant, and Λ is the index set. Throughout the paper, we use 𝑋ED(𝜇,𝜎2) and 𝑋ED(𝜃,𝜆) for the additive form with parameters 𝜇 and 𝜎2 and the reproductive form with parameters 𝜃 and 𝜆, correspondingly, depending on which notation is more convenient.

We further briefly review some properties of the EDMs related to this work. Consider the reproductive form first, that is, 𝑌ED(𝜇,𝜎2), then (i)the cumulant generating function (cgf) of 𝑌 is, for 𝜃=𝜃+𝑡/𝜆, 𝑒𝐾(𝑡)=log𝐄𝑡𝑌=log𝐑𝜆𝑦𝑡exp𝜃+𝜆𝑑𝜅(𝜃)𝜈𝜆𝜆𝜅𝑡(𝑦)=logexp𝜃+𝜆𝜅(𝜃)𝐑𝜆𝜃exp𝜃𝑦𝜅𝑑𝜈𝜆𝜅𝑡(𝑦)=𝜆𝜃+𝜆,𝜅(𝜃)(A.5)(ii)the moment generating function (mgf) of 𝑌 is given by 𝜆𝜅𝑀(𝑡)=exp𝜃+𝑡𝜆,𝜅(𝜃)(A.6)(iii)the expectation of 𝑌 is 𝐄[𝑌]=𝜕𝐾(𝑡)|||𝜕𝑡𝑡=0=𝜅(𝜃)=𝜇,(A.7)(iv)the variance of 𝑌 is [𝑌]=𝜕𝐕𝐚𝐫2𝐾(𝑡)𝜕𝑡2||||𝑡=0=𝜎2𝜅(2)(𝜃).(A.8)

Consider next an rv 𝑋 following an additive EDM, that is, 𝑋ED(𝜃,𝜆). Then, in a similar fashion, (i)the cgf of 𝑋 is 𝐾(𝑡)=𝜆(𝜅(𝜃+𝑡)𝜅(𝜃)),(A.9)(ii)the mgf of 𝑋 is 𝑀(𝑡)=exp(𝜆(𝜅(𝜃+𝑡)𝜅(𝜃))),(A.10)(iii)the expectation of 𝑋 is 𝐄[𝑋]=𝜆𝜅(𝜃),(A.11)(iv)the variance of 𝑋 is [𝑋]𝐕𝐚𝐫=𝜆𝜅(2)(𝜃).(A.12)

For valuable examples of various distributions belonging in the EDMs we refer to Jørgensen [24].

Acknowledgments

This is a concluding part of the authors’ research supported by the Zimmerman Foundation of Banking and Finance, Haifa, Israel. In addition, Edward Furman acknowledges the support of his research by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Also, the authors are grateful to two anonymous referees and the editor, Professor Johanna Nešlehová, for constructive criticism and suggestions that helped them to revise the paper.

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