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Actuarial and Financial Risks: Models, Statistical Inference, and Case Studies

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Volume 2010 |Article ID 357321 | https://doi.org/10.1155/2010/357321

Olga Furman, Edward Furman, "On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models", Journal of Probability and Statistics, vol. 2010, Article ID 357321, 19 pages, 2010. https://doi.org/10.1155/2010/357321

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

Academic Editor: Johanna NeΕ‘lehovΓ‘
Received13 Oct 2009
Revised21 Mar 2010
Accepted10 Apr 2010
Published28 Jun 2010

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 0β‰€π‘‹βˆˆπ’³ be a random variable (rv) with cumulative distribution function (cdf) 𝐹(π‘₯), decumulative distribution function (ddf) 𝐹(π‘₯)=1βˆ’πΉ(π‘₯), and probability density function (pdf) 𝑓(π‘₯). The functional π»βˆΆπ’³β†’[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. [2–5]) and weighted (cf. [6, 7]) classes of risk measures.

More specifically, the Value-at-Risk risk measure is formulated, for every 0<π‘ž<1, asVaRπ‘ž[𝑋]ξ€½=infπ‘₯βˆΆπΉπ‘‹ξ€Ύ,(π‘₯)β‰₯π‘ž(1.1) which thus refers to the well-studied notion of the π‘žth quantile. Then the family of distorted risk measures is defined with the help of an increasing and concave function π‘”βˆΆ[0,1]β†’[0,1], such that 𝑔(0)=0 and 𝑔(1)=1, as the following Choquet integral:𝐻𝑔[𝑋]=ξ€œπ‘+𝑔𝐹(π‘₯)𝑑π‘₯.(1.2) Last but not least, for an increasing nonnegative function π‘€βˆΆ[0,∞)β†’[0,∞) and the so-called weighted ddf 𝐹𝑀(π‘₯)=𝐄[𝟏{𝑋>π‘₯}𝑀(𝑋)]/𝐄[𝑀(𝑋)] the class of weighted risk measures is given by𝐻𝑀[𝑋]=ξ€œπ‘+𝐹𝑀(π‘₯)0π‘₯0200𝑑𝑑π‘₯.(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[π‘‹π‘”ξ…ž(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 β€œtail 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<π‘ž<1 and thus 𝐹(VaRπ‘ž[𝑋])β‰ 0, the TCE risk measure is formulated asTCEπ‘ž[𝑋]=1𝐹VaRπ‘ž[𝑋]ξ€Έξ€œβˆžVaRπ‘ž[𝑋]π‘₯𝑑𝐹(π‘₯).(1.4) Importantly, the TCE belongs in the class of distorted risk measures with the distortion function𝑔π‘₯(π‘₯)=𝟏1βˆ’π‘ž(π‘₯<1βˆ’π‘ž)+𝟏(π‘₯β‰₯1βˆ’π‘ž),(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 function𝑀(π‘₯)=𝟏π‘₯β‰₯VaRπ‘ž[𝑋]ξ€Ύ(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<π‘ž<1, the tail varianceTVπ‘ž[𝑋]ξ€Ί=π•πšπ«π‘‹βˆ£π‘‹>VaRπ‘ž[𝑋]ξ€»,(1.7) and a constant 𝛼β‰₯0, is defined asTSDπ‘ž[𝑋]=TCEπ‘ž[𝑋]+𝛼⋅TVπ‘ž1/2[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 π‘žβ†“0, we have that TSDπ‘ž[𝑋]β†’SD[𝑋]=𝐄[𝑋]+π›Όβ‹…π•πšπ«1/2[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<π‘ž<𝑝<1 and let π‘‹βˆˆπ’³ have a continuous and strictly increasing cdf. In many practical situations the degree of riskiness of the layer (VaRπ‘ž[𝑋],VaR𝑝[𝑋]) of an insurance contract is to be measured (certainly the layer width VaR𝑝[𝑋]βˆ’VaRπ‘ž[𝑋]=Ξ”π‘ž,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 β€œlimited 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 π»βˆΆπ’³β†’[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 π‘₯π‘ž=VaRπ‘ž[𝑋] and π‘₯𝑝=VaR𝑝[𝑋], for 0<π‘ž<𝑝<1. Then the limited TCE and TSD risk measures are formulated as LTCEπ‘ž,𝑝[𝑋]ξ€Ί=π„π‘‹βˆ£π‘₯π‘ž<𝑋≀π‘₯𝑝,(2.1) and LTSDπ‘ž,𝑝[𝑋]ξ€Ί=π„π‘‹βˆ£π‘₯π‘ž<𝑋≀π‘₯𝑝+π›Όβ‹…π•πšπ«1/2ξ€Ίπ‘‹βˆ£π‘₯π‘ž<𝑋≀π‘₯𝑝,(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 π‘˜=1,2,…, we have that π„ξ€Ίπ‘‹π‘˜βˆ£π‘₯π‘ž<𝑋≀π‘₯𝑝=𝐹π‘₯π‘ξ€Έπ„ξ€Ίπ‘‹π‘˜βˆ£π‘‹β‰€π‘₯𝑝π‘₯βˆ’πΉπ‘žξ€Έπ„ξ€Ίπ‘‹π‘˜βˆ£π‘‹β‰€π‘₯π‘žξ€»πΉξ€·π‘₯𝑝π‘₯βˆ’πΉπ‘žξ€Έ<∞,(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 𝑐β‰₯0, we have that LTSDπ‘ž,𝑝[]𝑋+𝑐=LTSDπ‘ž,𝑝[𝑋]+𝑐.(2.4)(2)Positive Homogeneity. For any constant 𝑑>0, we have that LTSDπ‘ž,𝑝[]𝑑⋅𝑋=𝑑⋅LTSDπ‘ž,𝑝[𝑋].(2.5)(3)Layer Parity. We call π‘‹βˆˆπ’³ and π‘Œβˆˆπ’³ layer equivalent if for some 0<π‘ž<𝑝<1, such that π‘₯π‘ž=π‘¦π‘ž,π‘₯𝑝=𝑦𝑝, and for every pair {(𝑑1,𝑑2)βˆΆπ‘ž<𝑑1<𝑑2<𝑝}, it holds that 𝐏[π‘₯𝑑1<𝑋≀π‘₯𝑑2]=𝐏[𝑦𝑑1<π‘Œβ‰€π‘¦π‘‘2]. In such a case, we have that LTSD𝑑1,𝑑2[𝑋]=LTSD𝑑1,𝑑2[π‘Œ].(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, HΓΌrlimann [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 β€œunlimited” 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 JΓΈrgensen [22, 23]. In his Theory of dispersion models, JΓΈrgensen [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<π‘ž<𝑝<1, we denote by (π‘₯π‘ž,π‘₯𝑝) an arbitrary layer having β€œattachment point” π‘₯π‘ž and width Ξ”π‘ž,𝑝. Also, letβ„Žξ€·π‘₯π‘ž,π‘₯𝑝=πœ•;πœƒ,πœ†ξ€·πΉξ€·π‘₯πœ•πœƒlog𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘ž;πœƒ,πœ†ξ€Έξ€Έ(3.1) denote the generalized layer-based hazard function, such thatβ„Žξ€·π‘₯π‘ž,π‘₯1ξ€Έ=πœ•;πœƒ,πœ†ξ‚€πœ•πœƒlog𝐹π‘₯π‘žξ€Έξ‚ξ€·π‘₯;πœƒ,πœ†=β„Žπ‘žξ€Έ,β„Žξ€·π‘₯;πœƒ,πœ†0,π‘₯π‘ξ€Έπœ•;πœƒ,πœ†=βˆ’ξ‚€πœ•πœƒlog𝐹π‘₯𝑝π‘₯;πœƒ,πœ†=βˆ’β„Žπ‘ξ€Έ,;πœƒ,πœ†(3.2) and thusβ„Žξ€·π‘₯π‘ž,π‘₯𝑝=;πœƒ,πœ†πΉξ€·π‘₯π‘žξ€Έ;πœƒ,πœ†πΉξ€·π‘₯π‘žξ€Έβˆ’;πœƒ,πœ†πΉξ€·π‘₯π‘ξ€Έβ„Žξ€·π‘₯;πœƒ,πœ†π‘žξ€Έβˆ’;πœƒ,πœ†πΉξ€·π‘₯𝑝;πœƒ,πœ†πΉξ€·π‘₯π‘žξ€Έβˆ’;πœƒ,πœ†πΉξ€·π‘₯π‘ξ€Έβ„Žξ€·π‘₯;πœƒ,πœ†π‘ξ€Έ.;πœƒ,πœ†(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 EDMπ‘Œβˆ½ED(πœ‡,𝜎2) is given by LTCEπ‘ž,𝑝[π‘Œ]=πœ‡+𝜎2ξ€·π‘₯β‹…β„Žπ‘ž,π‘₯𝑝;πœƒ,πœ†(3.4) and(ii)for the additive EDMπ‘‹βˆ½EDβˆ—(πœƒ,πœ†) is given by LTCEπ‘ž,𝑝[𝑋]=πœ†πœ…ξ…žξ€·π‘₯(πœƒ)+β„Žπ‘ž,π‘₯𝑝.;πœƒ,πœ†(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 LTCEπ‘ž,𝑝[π‘Œ]=πΉξ€·π‘¦π‘žξ€Έπ„ξ€Ίπ‘Œβˆ£π‘Œ>π‘¦π‘žξ€»βˆ’πΉξ€·π‘¦π‘ξ€Έπ„ξ€Ίπ‘Œβˆ£π‘Œ>π‘¦π‘ξ€»πΉξ€·π‘¦π‘ξ€Έξ€·π‘¦βˆ’πΉπ‘žξ€Έ.(3.6) Further, following Landsman and Valdez [16], it can be shown that for every 0<π‘ž<1, we have that π„ξ€Ίπ‘Œβˆ£π‘Œ>π‘¦π‘žξ€»=πœ‡+𝜎2ξ€·π‘¦β‹…β„Žπ‘žξ€Έ,;πœƒ,πœ†(3.7) which then, employing (3.1) and (3.3), yields LTCEπ‘ž,𝑝[π‘Œ]=πΉξ€·π‘¦π‘ž;πœƒ,πœ†ξ€Έξ€·πœ‡+𝜎2ξ€·π‘¦β‹…β„Žπ‘žβˆ’;πœƒ,πœ†ξ€Έξ€ΈπΉξ€·π‘¦π‘;πœƒ,πœ†ξ€Έξ€·πœ‡βˆ’πœŽ2ξ€·π‘¦β‹…β„Žπ‘;πœƒ,πœ†ξ€Έξ€ΈπΉξ€·π‘¦π‘žξ€Έβˆ’;πœƒ,πœ†πΉξ€·π‘¦π‘ξ€Έ;πœƒ,πœ†=πœ‡+𝜎2ξ€·π‘¦β‹…β„Žπ‘ž,𝑦𝑝;πœƒ,πœ†(3.8) and hence completes the proof.

In the sequel, we sometimes write LTCEπ‘ž,𝑝[π‘Œ;πœƒ,πœ†] in order to emphasize the dependence on πœƒ and πœ†.

Note. To obtain the results of Landsman and Valdez [16], we put p↑1, and then, for instance, in the reproductive case, we end up with lim𝑝↑1LTCEπ‘ž,𝑝[π‘Œ]=πœ‡+𝜎2ξ€·π‘¦β‹…β„Žπ‘žξ€Έ;πœƒ,πœ†=TCEπ‘ž[π‘Œ],(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 EDMπ‘Œβˆ½ED(πœ‡,𝜎2) is given by LTSDπ‘ž,𝑝[π‘Œ]=LTCEπ‘ž,𝑝[π‘Œ]ξ‚™+π›Όβ‹…πœŽ2πœ•πœ•πœƒLTCEπ‘ž,𝑝[]π‘Œ;πœƒ,πœ†(3.10) and (ii)for the additive EDMπ‘‹βˆ½EDβˆ—(πœƒ,πœ†) is given by LTSDπ‘ž,𝑝[𝑋]=LTCEπ‘ž,𝑝[𝑋]ξ‚™+π›Όβ‹…πœ•πœ•πœƒLTCEπ‘ž,𝑝[].𝑋;πœƒ,πœ†(3.11)

Proof. We again prove the reproductive case, only. Note that it has been assumed that πœ…(πœƒ) is a differentiable function, and thus we can differentiate the following probability integral in πœƒ under the integral sign (cf., the appendix): πξ€·π‘¦π‘ž<π‘Œβ‰€π‘¦π‘ξ€Έ=ξ€œπ‘¦π‘π‘¦π‘žπ‘’πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))π‘‘πœˆπœ†(𝑦),(3.12) and hence, using Definition 2.1, we have that πœ•ξ€·πœ•πœƒLTCEπ‘ž,𝑝[]ξ€·πΉξ€·π‘¦π‘Œ;πœƒ,πœ†π‘ξ€Έξ€·π‘¦;πœƒ,πœ†βˆ’πΉπ‘ž=ξ€œ;πœƒ,πœ†ξ€Έξ€Έξ€Έπ‘¦π‘π‘¦π‘žπœ•πœ•πœƒπ‘¦π‘’πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))π‘‘πœˆπœ†ξ€œ(𝑦)=πœ†π‘¦π‘π‘¦π‘žξ€·π‘¦2π‘’πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))βˆ’π‘¦πœ…ξ…ž(πœƒ)π‘’πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))ξ€Έπ‘‘πœˆπœ†(𝑦)=πœŽβˆ’2ξ€·π„ξ€Ίπ‘Œ2ξ€½π‘¦βˆ£πŸπ‘ž<π‘Œβ‰€π‘¦π‘ξ€Ίξ€½π‘¦ξ€Ύξ€»βˆ’πœ‡(πœƒ)β‹…π„π‘Œβˆ£πŸπ‘ž<π‘Œβ‰€π‘¦π‘,ξ€Ύξ€»ξ€Έ(3.13) with the last line following from the appendix. Further, by simple rearrangement and straightforward calculations, we obtain that π„ξ€Ίπ‘Œ2βˆ£π‘¦π‘ž<π‘Œβ‰€π‘¦π‘ξ€»=βˆ«π‘¦π‘π‘¦π‘žπ‘¦2π‘’πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))π‘‘πœˆπœ†(𝑦)𝐹𝑦𝑝𝑦;πœƒ,πœ†βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†=πœ‡β‹…LTCEπ‘ž,𝑝[π‘Œ]+𝜎2(πœ•/πœ•πœƒ)LTCEπ‘ž,𝑝[]ξ€·πΉξ€·π‘¦π‘Œ;πœƒ,πœ†π‘ξ€Έξ€·π‘¦;πœƒ,πœ†βˆ’πΉπ‘ž;πœƒ,πœ†ξ€Έξ€ΈπΉξ€·π‘¦π‘ξ€Έξ€·π‘¦;πœƒ,πœ†βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†=𝜎2πœ•πœ•πœƒLTCEπ‘ž,𝑝[]π‘Œ;πœƒ,πœ†+LTCEπ‘ž,𝑝[π‘Œ]ξ‚€πœ‡+𝜎2πœ•ξ€·πΉξ€·π‘¦πœ•πœƒlog𝑝𝑦;πœƒ,πœ†βˆ’πΉπ‘žξ‚;πœƒ,πœ†ξ€Έξ€Έ=𝜎2πœ•πœ•πœƒLTCEπ‘ž,𝑝[]+ξ€·π‘Œ;πœƒ,πœ†LTCEπ‘ž,𝑝[]ξ€Έπ‘Œ;πœƒ,πœ†2,(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 π‘Œβˆ½π‘(πœ‡,𝜎2) be a normal random variable with mean πœ‡ and variance 𝜎2, then we can write the pdf of π‘Œ as 1𝑓(𝑦)=βˆšξ‚΅βˆ’12πœ‹πœŽexp2ξ‚€π‘¦βˆ’πœ‡πœŽξ‚2ξ‚Ά=1βˆšξ‚€βˆ’12πœ‹πœŽexp2𝜎2𝑦21exp𝜎2ξ‚€1πœ‡π‘¦βˆ’2πœ‡2,π‘¦βˆˆπ‘.(3.15) If we take πœƒ=πœ‡ and πœ†=1/𝜎2, we see that the normal distribution is a reproductive EDM with cumulant function πœ…(πœƒ)=πœƒ2/2. Denote by πœ‘(β‹…) and Ξ¦(β‹…) 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 π‘Œ: LTCEπ‘ž,𝑝[π‘Œ]πœ‘ξ€·πœŽ=πœ‡+πœŽβˆ’1ξ€·π‘¦π‘žξ€·πœŽβˆ’πœ‡ξ€Έξ€Έβˆ’πœ‘βˆ’1ξ€·π‘¦π‘βˆ’πœ‡ξ€Έξ€ΈΞ¦ξ€·πœŽβˆ’1ξ€·π‘¦π‘ξ€·πœŽβˆ’πœ‡ξ€Έξ€Έβˆ’Ξ¦βˆ’1ξ€·π‘¦π‘ž.βˆ’πœ‡ξ€Έξ€Έ(3.16) If we put 𝑝↑1, then the latter equation reduces to the result of Landsman and Valdez [16]. Namely, we have that lim𝑝↑1LTCEπ‘ž,𝑝[π‘Œ]πœ‘ξ€·πœŽ=πœ‡+πœŽβˆ’1ξ€·π‘¦π‘žβˆ’πœ‡ξ€Έξ€Έξ€·πœŽ1βˆ’Ξ¦βˆ’1ξ€·π‘¦π‘žβˆ’πœ‡ξ€Έξ€Έ=TCEπ‘ž[π‘Œ].(3.17) Further, let π‘§π‘ž=(π‘¦π‘žβˆ’πœ‡)/𝜎 and 𝑧𝑝=(π‘¦π‘βˆ’πœ‡)/𝜎. Then 𝜎2πœ•πœ•πœƒLTCEπ‘ž,𝑝[]π‘Œ;πœƒ,πœ†=𝜎2βŽ›βŽœβŽœβŽπœ‘ξ€·π‘§1+π‘žξ€Έπ‘§π‘žξ€·π‘§βˆ’πœ‘π‘ξ€Έπ‘§π‘Ξ¦ξ€·π‘§π‘ξ€Έξ€·π‘§βˆ’Ξ¦π‘žξ€Έβˆ’ξƒ©πœ‘ξ€·π‘§π‘žξ€Έξ€·π‘§βˆ’πœ‘π‘ξ€ΈΞ¦ξ€·π‘§π‘ξ€Έξ€·π‘§βˆ’Ξ¦π‘žξ€Έξƒͺ2⎞⎟⎟⎠.(3.18) Consequently, the limited TSD risk measure is as follows: LTSDπ‘ž,𝑝[π‘Œ]πœ‘ξ€·π‘§=πœ‡+πœŽπ‘žξ€Έξ€·π‘§βˆ’πœ‘π‘ξ€ΈΞ¦ξ€·π‘§π‘ξ€Έξ€·π‘§βˆ’Ξ¦π‘žξ€Έξ„Άξ„΅ξ„΅ξ„΅βŽ·+π›ΌπœŽ2βŽ›βŽœβŽœβŽπœ‘ξ€·π‘§1+π‘žξ€Έπ‘§π‘žξ€·π‘§βˆ’πœ‘π‘ξ€Έπ‘§π‘Ξ¦ξ€·π‘§π‘ξ€Έξ€·π‘§βˆ’Ξ¦π‘žξ€Έβˆ’ξƒ©πœ‘ξ€·π‘§π‘žξ€Έξ€·π‘§βˆ’πœ‘π‘ξ€ΈΞ¦ξ€·π‘§π‘ξ€Έξ€·π‘§βˆ’Ξ¦π‘žξ€Έξƒͺ2⎞⎟⎟⎠.(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 π‘‹βˆ½Ga(𝛾,𝛽) be a gamma rv with shape and rate parameters equal 𝛾 and 𝛽, correspondingly. The pdf of 𝑋 is 1𝑓(π‘₯)=Γ𝑒(𝛾)βˆ’π›½π‘₯π‘₯π›Ύβˆ’1𝛽𝛾=1Ξ“π‘₯(𝛾)π›Ύβˆ’1exp(βˆ’π›½π‘₯+𝛾log(𝛽)),π‘₯>0.(3.20) Hence the gamma rv can be represented as an additive EDM with the following pdf: 1𝑓(π‘₯)=Ξ“π‘₯(πœ†)πœ†βˆ’1exp(πœƒπ‘₯+πœ†log(βˆ’πœƒ)),(3.21) where π‘₯>0 and πœƒ<0.The mean and variance of 𝑋 are 𝐄[𝑋]=βˆ’πœ†/πœƒ and π•πšπ«[𝑋]=πœ†/πœƒ2. Also, πœƒ=βˆ’π›½, πœ†=𝛾, and πœ…(πœƒ)=βˆ’log(βˆ’πœƒ). According to Theorem 3.1, the limited tail conditional expectation is LTCEπ‘ž,𝑝[𝑋]πœ†=βˆ’πœƒπΉξ€·π‘₯𝑝π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†.(3.22) Putting 𝑝↑1 we obtain that lim𝑝↑1ξ‚€βˆ’πœ†πœƒξ‚πΉξ€·π‘₯𝑝π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έπœ†;πœƒ,πœ†=βˆ’πœƒπΉξ€·π‘₯π‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯π‘žξ€Έ;πœƒ,πœ†=TCEπ‘ž[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, πœ•πœ•πœƒLTCEπ‘ž,𝑝[]=πœ•π‘‹;πœƒ,πœ†ξƒ©βˆ’πœ†πœ•πœƒπœƒπΉξ€·π‘₯𝑝π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έξƒͺ=πœ†;πœƒ,πœ†πœƒ2𝐹π‘₯𝑝π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έβˆ’πœ†;πœƒ,πœ†πœƒξƒ©πœ•πΉξ€·π‘₯πœ•πœƒπ‘ξ€Έξ€·π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έξƒͺ.;πœƒ,πœ†(3.24) Further, since for 𝑛=1,2,…,πœ•ξ€·πΉξ€·π‘₯πœ•πœƒπ‘ξ€Έξ€·π‘₯;πœƒ,πœ†+π‘›βˆ’πΉπ‘ž=ξ€œ;πœƒ,πœ†+𝑛π‘₯𝑝π‘₯π‘žπœ•ξ‚΅1πœ•πœƒπ‘₯Ξ“(πœ†+𝑛)πœ†+π‘›βˆ’1ξ‚Ά=ξ€œexp(πœƒπ‘₯+(πœ†+𝑛)log(βˆ’πœƒ))𝑑π‘₯π‘₯𝑝π‘₯π‘žξ‚€π‘“(π‘₯;πœƒ,πœ†+𝑛)π‘₯+πœ†+π‘›πœƒξ‚π‘‘π‘₯=βˆ’πœ†+π‘›πœƒξƒ©ξ€œπ‘₯𝑝π‘₯π‘žξ€œπ‘“(π‘₯;πœƒ,πœ†+𝑛+1)𝑑π‘₯βˆ’π‘₯𝑝π‘₯π‘žξƒͺ,𝑓(π‘₯;πœƒ,πœ†+𝑛)𝑑π‘₯(3.25) the limited TSD risk measure for gamma is given by LTSDπ‘ž,𝑝[𝑋]=ξ‚€βˆ’πœ†πœƒξ‚πΉξ€·π‘₯𝑝π‘₯;πœƒ,πœ†+1βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έξ„Άξ„΅ξ„΅βŽ·;πœƒ,πœ†+π›Όπœ†πœƒ2𝐹π‘₯(πœ†+1)𝑝π‘₯;πœƒ,πœ†+2βˆ’πΉπ‘žξ€Έ;πœƒ,πœ†+2𝐹π‘₯𝑝π‘₯;πœƒ,πœ†βˆ’πΉπ‘žξ€Έξ‚΅;πœƒ,πœ†βˆ’πœ†πΉ(π‘₯𝑝;πœƒ,πœ†+1)βˆ’πΉ(π‘₯π‘ž;πœƒ,πœ†+1)𝐹(π‘₯𝑝;πœƒ,πœ†)βˆ’πΉ(π‘₯π‘žξ‚Ά;πœƒ,πœ†)2ξƒͺ.(3.26)

In the sequel, we consider gamma and normal risks with equal means and variances, and we explore them on the interval (𝑑,350], with 50<𝑑<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.

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 π‘‹βˆ½Pa(𝛾,𝛽) be a Pareto rv with the pdf 𝑓(π‘₯)=𝛾𝛽𝛾π‘₯𝛾+1,π‘₯>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(𝑑π‘₯;πœ†,𝜈)=exp{πœ†log(π‘₯)βˆ’πœ…(πœ†)}𝜈(𝑑π‘₯),(3.28) where πœ† is a parameter, 𝜈 is a measure, and βˆ«πœ…(πœ†)=log∞0π‘₯πœ†πœˆ(𝑑π‘₯) 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(𝑑π‘₯)=π‘₯βˆ’1𝑑π‘₯, and πœ†=βˆ’π›Ύ.
In this context, it is straightforward to see that 𝐄[𝑋] is infinite for 𝛾≀1, which thus implies infiniteness of the TCE risk measure. We can however readily obtain the limited variant as follows: LTCEπ‘ž,𝑝[𝑋]=1𝐏π‘₯π‘ž<𝑋≀π‘₯π‘ξ€»ξ€œπ‘₯𝑝π‘₯π‘žπ›Ύπ›½π›Ύπ‘₯𝛾𝑑π‘₯=𝛾π‘₯𝑝π‘₯π‘žξƒ©π‘₯π›Ύβˆ’1π‘π›Ύβˆ’1βˆ’π‘₯π‘žπ›Ύβˆ’1π‘₯π›Ύπ‘βˆ’π‘₯π›Ύπ‘žξƒͺ,(3.29) that is finite for any 𝛾>0. Also, since, for example, for 𝛾<1, we have that π‘₯π‘π›Ύβˆ’1βˆ’π‘₯π‘žπ›Ύβˆ’1<0, the limited TCE risk measure is positive, as expected. The same is true for 𝛾β‰₯1.
We note in passing that, for 𝛾>1 and 𝑝↑1 and thus π‘₯π‘β†’βˆž, we have that TCEπ‘ž[𝑋]=lim𝑝↑1𝛾π‘₯𝑝π‘₯π‘žξƒ©π‘₯π›Ύβˆ’1π‘π›Ύβˆ’1βˆ’π‘₯π‘žπ›Ύβˆ’1π‘₯π›Ύπ‘βˆ’π‘₯π›Ύπ‘žξƒͺ=𝛾π‘₯π›Ύβˆ’1π‘ž,(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-𝑑. 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(β‹…;πœƒ,πœ†) to emphasize the parameters πœƒ and πœ†, and we assume that lim𝑝↑1LTSDπ‘ž,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 TSDπ‘ž[π‘Œ]=TCEπ‘ž[π‘Œ]ξ‚™+π›ΌπœŽ2πœ•πœ•πœƒTCEπ‘ž[]π‘Œ;πœƒ,πœ†(4.2) in the context of the reproductive EDMs, and TSDπ‘ž[𝑋]=TCEπ‘ž[𝑋]ξ‚™+π›Όπœ•πœ•πœƒTCEπ‘ž[]𝑋;πœƒ,πœ†(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 π‘Œβˆ½π‘(πœ‡,𝜎2) be again some normal rv with mean πœ‡ and variance 𝜎2. Then we easily evaluate the TSD risk measure using Corollary 4.1 and Example 3.3 as follows: TSDπ‘ž[𝑋]πœ‘ξ€·π‘§=πœ‡+πœŽπ‘žξ€Έξ€·π‘§1βˆ’Ξ¦π‘žξ€Έξ„Άξ„΅ξ„΅ξ„΅βŽ·+π›ΌπœŽ2βŽ›βŽœβŽœβŽπœ‘ξ€·π‘§1+π‘žξ€Έξ€·π‘§1βˆ’Ξ¦π‘žξ€Έπ‘§π‘žβˆ’ξƒ©πœ‘ξ€·π‘§π‘žξ€Έξ€·π‘§1βˆ’Ξ¦π‘žξ€Έξƒͺ2⎞⎟⎟⎠,(4.4) which coincides with [10].

Example 4.3. Let π‘‹βˆ½Ga(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 TSDπ‘ž[𝑋]πœ†=βˆ’πœƒπΉξ€·π‘₯π‘žξ€Έ;πœƒ,πœ†+1𝐹π‘₯π‘žξ€Έξ„Άξ„΅ξ„΅ξ„΅βŽ·;πœƒ,πœ†+π›Όπœ†πœƒ2βŽ›βŽœβŽœβŽ(πœ†+1)𝐹π‘₯π‘žξ€Έ;πœƒ,πœ†+2𝐹π‘₯π‘žξ€Έξƒ©;πœƒ,πœ†βˆ’πœ†πΉ(π‘₯π‘ž;πœƒ,πœ†+1)𝐹(π‘₯π‘žξƒͺ;πœƒ,πœ†)2⎞⎟⎟⎠=𝛾𝛽𝐹π‘₯π‘žξ€Έ;𝛾+1,𝛽𝐹π‘₯π‘žξ€Έξ„Άξ„΅ξ„΅ξ„΅βŽ·;𝛾,𝛽+𝛼𝛾𝛽2βŽ›βŽœβŽœβŽ(𝛾+1)𝐹π‘₯π‘žξ€Έ;𝛾+2,𝛽𝐹π‘₯π‘žξ€Έξƒ©;𝛾,π›½βˆ’π›ΎπΉ(π‘₯π‘ž;𝛾+1,𝛽)𝐹(π‘₯π‘žξƒͺ;𝛾,𝛽)2⎞⎟⎟⎠,(4.5) where the latter equation follows because of the reparameterization πœƒ=βˆ’π›½ and πœ†=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 π‘Œβˆ½IG(πœ‡,πœ†) be an inverse Gaussian rv. We then can write its pdf as 𝑓(𝑦)=πœ†2πœ‹π‘¦3ξ‚΅πœ†ξ‚΅βˆ’π‘¦exp2πœ‡2βˆ’1+12π‘¦πœ‡[ξ‚Άξ‚Ά,π‘¦βˆˆ0,∞),(4.6) (cf. [24]), which means that π‘Œ belongs to the reproductive EDMs, with πœƒ=βˆ’1/(2πœ‡2)and βˆšπœ…(πœƒ)=βˆ’1/πœ‡=βˆ’βˆ’2πœƒ. Further, due to Corollary 4.1 we need to calculate πœ•πœ•πœƒTCEπ‘ž[]=πœ•π‘Œ;πœƒ,πœ†ξ‚€πœ•πœƒπœ‡(πœƒ)+𝜎2πœ•πœ•πœƒlogπΉπ‘Œξ€·π‘¦π‘žξ€Έξ‚;πœƒ,πœ†=πœ‡ξ…ž(πœƒ)+𝜎2πœ•(πœ•πœƒπœ•/πœ•πœƒ)πΉπ‘Œξ€·π‘¦π‘žξ€Έ;πœƒ,πœ†πΉπ‘Œξ€·π‘¦π‘žξ€Έ;πœƒ,πœ†.(4.7) To this end, note that the ddf of π‘Œ is πΉξ€·π‘¦π‘žξ€Έ=;πœ‡(πœƒ),πœ†Ξ¦ξƒ©ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺπœ‡(πœƒ)βˆ’1βˆ’π‘’2πœ†/πœ‡(πœƒ)Ξ¦ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺπœ‡(πœƒ)+1(4.8) (cf., e.g., [28]), where Ξ¦(β‹…) is the ddf of the standardized normal random variable. Hence, by simple differentiation and noticing that πœ‡ξ…ž(πœƒ)=(βˆ’2πœƒ)βˆ’3/2=πœ‡(πœƒ)3,(4.9) we obtain that πœ•πœ•πœƒπΉξ€·π‘¦π‘žξ€Έξƒ©ξ”;πœ‡(πœƒ),πœ†=πœ‡(πœƒ)πœ†π‘¦π‘žπœ‘ξƒ©ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺπœ‡(πœƒ)βˆ’1βˆ’π‘’2πœ†/πœ‡(πœƒ)ξ”πœ†π‘¦π‘žπœ‘ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άπœ‡(πœƒ)+1ξƒͺξƒͺ+2πœ†πœ‡(πœƒ)𝑒2πœ†/πœ‡(πœƒ)Ξ¦ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺ.πœ‡(πœƒ)+1(4.10) Notably, ξ”πœ†π‘¦π‘žπœ‘ξƒ©ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺπœ‡(πœƒ)βˆ’1=𝑒2πœ†/πœ‡(πœƒ)ξ”πœ†π‘¦π‘žπœ‘ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺ,πœ‡(πœƒ)+1(4.11) and therefore (4.10) results in πœ•πœ•πœƒπΉξ€·π‘¦π‘žξ€Έ;πœ‡(πœƒ),πœ†=2πœ†πœ‡(πœƒ)𝑒2πœ†/πœ‡(πœƒ)Ξ¦ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺ.πœ‡(πœƒ)+1(4.12) Consequently, the expression for the TCE risk measure, obtained by Landsman and Valdez [16], is simplified to TCEπ‘ž[]π‘Œ;πœƒ,πœ†=πœ‡(πœƒ)+2πœ‡(πœƒ)πΉξ€·π‘¦π‘žξ€Έπ‘’;πœ‡(πœƒ),πœ†2πœ†/πœ‡(πœƒ)Ξ¦ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άξƒͺ.πœ‡(πœƒ)+1(4.13) In order to derive the TSD risk measure we need to differentiate again, that is, πœ•πœ•πœƒTCEπ‘ž[]=πœ•π‘Œ;πœƒ,πœ†ξƒ©πœ‡πœ•πœƒ(πœƒ)+2πœ‡(πœƒ)πΉξ€·π‘¦π‘žξ€Έπ‘’;πœ‡(πœƒ),πœ†2πœ†/πœ‡(πœƒ)Ξ¦ξƒ©βˆ’ξƒŽπœ†π‘¦π‘žξ‚΅π‘¦π‘žξ‚Άπœ‡(πœƒ)+1ξƒͺξƒͺ=πœ‡(πœƒ)3ξƒ©πœ•1+πœ•πœƒ2πœ‡(πœƒ)𝑒2πœ†/πœ‡(πœƒ)Ξ¦ξ€·βˆ’βˆšπœ†/π‘¦π‘žξ€·π‘¦π‘ž/πœ‡(πœƒ)+1ξ€Έξ€ΈπΉξ€·π‘¦π‘žξ€Έξƒͺ,;πœ‡(πœƒ),πœ†(4.14) where we use πœ‡ξ…ž(πœƒ)=πœ‡(πœƒ)3. Further, we have that πœ•πœ•πœƒ2πœ‡(πœƒ)𝑒2πœ†/πœ‡(πœƒ)Ξ¦ξ€·βˆ’βˆšπœ†/π‘¦π‘žξ€·π‘¦π‘ž/πœ‡(πœƒ)+1ξ€Έξ€ΈπΉξ€·π‘¦π‘žξ€Έ;πœ‡(πœƒ),πœ†=2πœ‡(πœƒ)3𝑒2πœ†/πœ‡(πœƒ)ξ€·Ξ¦ξ€·Μƒπ‘¦π‘žξ€Έξ€·βˆš(1βˆ’2πœ†/πœ‡(πœƒ))+πœ†π‘¦π‘žξ€Έπœ‘ξ€·/πœ‡(πœƒ)Μƒπ‘¦π‘žξ€Έξ€ΈπΉξ€·π‘¦π‘žξ€Έβˆ’πœ†ξ€·;πœ‡(πœƒ),πœ†2πœ‡(πœƒ)𝑒2πœ†/πœ‡(πœƒ)Ξ¦ξ€·Μƒπ‘¦π‘žξ€Έξ€Έ2πΉξ€·π‘¦π‘žξ€Έ;πœ‡(πœƒ),πœ†2,(4.15) where Μƒπ‘¦π‘žβˆš=βˆ’πœ†/π‘¦π‘ž(π‘¦π‘ž/πœ‡(πœƒ)+1). Therefore TSDπ‘ž[π‘Œ]Φ=πœ‡1+Μƒπ‘¦π‘žξ€ΈπΉξ€·π‘¦π‘žξ€Έ;πœ‡,πœ†2𝑒2πœ†/πœ‡ξƒͺξ„Άξ„΅ξ„΅ξ„΅βŽ·+π›Όπœ‡3πœ†ξƒ©π‘’1+2πœ†/πœ‡ξ€·Ξ¦ξ€·Μƒπ‘¦π‘žξ€Έξ€·βˆš(1βˆ’2πœ†/πœ‡)+πœ†π‘¦π‘žξ€Έπœ‘ξ€·/πœ‡Μƒπ‘¦π‘žξ€Έξ€ΈπΉξ€·π‘¦π‘žξ€Έβˆ’πœ†ξ€·π‘’;πœ‡,πœ†2πœ†/πœ‡Ξ¦ξ€·Μƒπ‘¦π‘žξ€Έξ€Έ2πœ‡πΉξ€·π‘¦π‘žξ€Έ;πœ‡,πœ†2ξƒͺ(4.16) subject to π•πšπ«[π‘Œ]=πœ‡3/πœ†.

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 (π‘Ž,𝑏,0) class (cf., [11, Chapter 6]) as well as for limited TCE/TSD risk measures.

Example 5.1. Let π‘‹βˆ½Poisson(πœ‡) be a Poisson rv with the mean parameter πœ‡. Then the probability mass function of 𝑋 is written as 1𝑝(π‘₯)=πœ‡π‘₯!π‘₯π‘’βˆ’πœ‡=1π‘₯!exp(π‘₯log(πœ‡)βˆ’πœ‡),π‘₯=0,1,…,(5.1) which belongs to the additive EDMs in view of the reparametrization πœƒ=log(πœ‡),πœ†=1, and πœ…(πœƒ)=π‘’πœƒ.
Motivated by Corollary 4.1, we differentiate (cf. [16], for the formula for the TCE risk measure) πœ•πœ•πœƒTCEπ‘žπœ•(𝑋;πœƒ,πœ†)=ξƒ©π‘’πœ•πœƒπœƒξƒ©π‘ξ€·π‘₯1+π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έ;πœƒ,1ξƒͺξƒͺ=π‘’πœƒβŽ›βŽœβŽœβŽπ‘ξ€·π‘₯1+π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έ+𝑝π‘₯;πœƒ,1π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έξ€·π‘₯;πœƒ,1π‘žβˆ’π‘’πœƒξ€Έβˆ’π‘’πœƒξƒ©π‘(π‘₯π‘ž;πœƒ,1)𝐹(π‘₯π‘žξƒͺ;πœƒ,1)2⎞⎟⎟⎠=π‘’πœƒβŽ›βŽœβŽœβŽπΉξ€·π‘₯π‘žξ€Έβˆ’1;πœƒ,1𝐹π‘₯π‘žξ€Έ+𝑝π‘₯;πœƒ,1π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έξ€·π‘₯;πœƒ,1π‘žβˆ’π‘’πœƒξ€Έβˆ’π‘’πœƒξƒ©π‘(π‘₯π‘ž;πœƒ,1)𝐹(π‘₯π‘žξƒͺ;πœƒ,1)2⎞⎟⎟⎠,(5.2) where the latter equation follows because 𝐹π‘₯π‘žξ€Έξ€·π‘₯;πœƒ,1+π‘π‘žξ€Έ=;πœƒ,1𝐹π‘₯π‘žξ€Έ.βˆ’1;πœƒ,1(5.3) The formula for the TSD risk measure is then TSDπ‘ž(𝑋)=π‘’πœƒξƒ©π‘ξ€·π‘₯1+π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έξƒͺξ„Άξ„΅ξ„΅ξ„΅βŽ·;πœƒ,1+π›Όπ‘’πœƒβŽ›βŽœβŽœβŽπΉξ€·π‘₯π‘žξ€Έβˆ’1;πœƒ,1𝐹π‘₯π‘žξ€Έ+𝑝π‘₯;πœƒ,1π‘žξ€Έ;πœƒ,1𝐹π‘₯π‘žξ€Έπ‘§;πœƒ,1π‘žβˆ’π‘’πœƒξƒ©π‘(π‘₯π‘ž;πœƒ,1)𝐹(π‘₯π‘žξƒͺ;πœƒ,1)2⎞⎟⎟⎠,(5.4) where 𝐄[𝑋]=π•πšπ«[𝑋]=π‘’πœƒ and π‘§π‘ž=π‘₯π‘žβˆ’π‘’πœƒ.

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 π‘‹βˆ½EDβˆ—(πœƒ,πœ†) for (πœƒ,πœ†)βˆˆΞ˜Γ—Ξ› is called the additive exponential dispersion model generated by 𝜈. The corresponding family of distributions of π‘Œ=𝑋/πœ†βˆ½ED(πœ‡,𝜎2), where πœ‡=𝜏(πœƒ) and 𝜎2=1/πœ† are the mean value and the dispersion parameters, respectively, is called the reproductive exponential dispersion model generated by 𝜈. Moreover, given some measure πœˆπœ† the representation of π‘‹βˆ½EDβˆ—(πœƒ,πœ†) is as follows: exp(πœƒπ‘₯βˆ’πœ†πœ…(πœƒ))πœˆπœ†(𝑑π‘₯).(A.1) If in addition the measure πœˆπœ† has density π‘βˆ—(π‘₯;πœ†) with respect to some fixed measure (typically Lebesgue measure or counting measure), the density for the additive model is π‘“βˆ—(π‘₯;πœƒ,πœ†)=π‘βˆ—(π‘₯;πœ†)exp(πœƒπ‘₯βˆ’πœ†πœ…(πœƒ)),π‘₯βˆˆπ‘.(A.2) Similarly, we obtain the following representation of π‘Œβˆ½ED(πœ‡,𝜎2) as exp(πœ†(π‘¦πœƒβˆ’πœ…(πœƒ)))πœˆπœ†(𝑑𝑦),(A.3) where πœˆπœ† denotes πœˆπœ† transformed by the duality transformation 𝑋=π‘Œ/𝜎2. Again if the measure πœˆπœ† has density 𝑐(𝑦;πœ†) with respect to a fixed measure, the reproductive model has the following pdf: 𝑓(𝑦;πœƒ,πœ†)=𝑐(𝑦;πœ†)exp(πœ†(πœƒπ‘¦βˆ’πœ…(πœƒ))),π‘¦βˆˆπ‘.(A.4)

Note that πœƒ and πœ† are called canonical and index parameters, Θ={πœƒβˆˆπ‘βˆΆπœ…(πœƒ)<∞} for some function πœ…(πœƒ) called the cumulant, and Ξ› is the index set. Throughout the paper, we use π‘‹βˆ½EDβˆ—(πœ‡,𝜎2) and π‘‹βˆ½ED(πœƒ,πœ†) for the additive form with parameters πœ‡ and 𝜎2 and the reproductive form with parameters πœƒ and πœ†, 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, π‘Œβˆ½ED(πœ‡,𝜎2), then (i)the cumulant generating function (cgf) of π‘Œ is, for πœƒβ€²=πœƒ+𝑑/πœ†, 𝑒𝐾(𝑑)=logπ„π‘‘π‘Œξ€»ξ‚΅ξ€œ=logπ‘ξ‚€πœ†ξ‚€π‘¦ξ‚€π‘‘expπœƒ+πœ†ξ‚π‘‘βˆ’πœ…(πœƒ)ξ‚ξ‚πœˆπœ†ξ‚Άξ‚΅ξ‚€πœ†ξ‚€πœ…ξ‚€π‘‘(𝑦)=logexpπœƒ+πœ†ξ‚ξ€œβˆ’πœ…(πœƒ)ξ‚ξ‚π‘ξ€·πœ†ξ€Ίπœƒexpξ…žξ€·πœƒπ‘¦βˆ’πœ…ξ…žπ‘‘ξ€Έξ€»ξ€Έπœˆπœ†ξ‚Άξ‚€πœ…ξ‚€π‘‘(𝑦)=πœ†πœƒ+πœ†ξ‚ξ‚,βˆ’πœ…(πœƒ)(A.5)(ii)the moment generating function (mgf) of π‘Œ is given by ξ‚€πœ†ξ‚€πœ…ξ‚€π‘€(𝑑)=expπœƒ+π‘‘πœ†ξ‚,βˆ’πœ…(πœƒ)(A.6)(iii)the expectation of π‘Œ is 𝐄[π‘Œ]=πœ•πΎ(𝑑)|||πœ•π‘‘π‘‘=0=πœ…ξ…ž(πœƒ)=πœ‡,(A.7)(iv)the variance of π‘Œ is [π‘Œ]=πœ•π•πšπ«2𝐾(𝑑)πœ•π‘‘2||||𝑑=0=𝜎2πœ…(2)(πœƒ).(A.8)

Consider next an rv 𝑋 following an additive EDM, that is, π‘‹βˆ½EDβˆ—(πœƒ,πœ†). Then, in a similar fashion, (i)the cgf of 𝑋 is 𝐾(𝑑)=πœ†(πœ…(πœƒ+𝑑)βˆ’πœ…(πœƒ)),(A.9)(ii)the mgf of 𝑋 is 𝑀(𝑑)=exp(πœ†(πœ…(πœƒ+𝑑)βˆ’πœ…(πœƒ))),(A.10)(iii)the expectation of 𝑋 is 𝐄[𝑋]=πœ†πœ…ξ…ž(πœƒ),(A.11)(iv)the variance of 𝑋 is [𝑋]π•πšπ«=πœ†πœ…(2)(πœƒ).(A.12)

For valuable examples of various distributions belonging in the EDMs we refer to JΓΈrgensen [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 NeΕ‘lehovΓ‘, for constructive criticism and suggestions that helped them to revise the paper.

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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.


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