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

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.


Introduction
Denote by X the set of (actuarial) risks, and let 0 ≤ X ∈ X be a random variable (rv) with cumulative distribution function (cdf) F (x), decumulative distribution function (ddf) F (x) = 1 − F (x), and probability density function (pdf) f (x). The functional H : X → [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. Leavens, 1945), and up to the distorted (cf. Denneberg, 1994;Wang, 1995Wang, , 1996Wang et al., 1997) and weighted (cf. Furman & Zitikis, 2008, 2009) classes of risk measures.
More specifically, the Value-at-Risk risk measure is formulated, for every 0 < q < 1, as Note that for at least once differentiable distortion function, we have that the weighted class contains the distorted one as a special case, i.e., H g [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 (V aR q [X]) = 0, the TCE risk measure is formulated as [X] xdF (x) .
(1.4) Importantly, the TCE belongs to the class of distorted risk measures with the distortion function where 1 denotes the indicator function (cf., e.g., Furman & Landsman, 2006a), as well as in the class of weighted risk measures with the weight function (cf., e.g., Furman & Zitikis, 2008, 2009 and a constant α ≥ 0, is defined as For a discussion of various properties of the TSD risk measure, we refer to Furman and Landsman (2006b). We note in passing that for q ↓ 0, we have that 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 (1.6). 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 (EDM's), that are to this end briefly reviewed in Appendix 5. Section 5 concludes the paper.

The limited TCE and TSD risk measures
Let 0 < q < p < 1, and X ∈ X have a continuous and strictly increasing cdf. In and 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 regardless of the distribution of X.
We further enumerate some properties of the LTSD risk measure, which is our main object of study: 3. Layer parity. We call X ∈ X and Y ∈ X layer equivalent if for some 0 < q < p < 1, such that x q = y q , x p = y p , and for every pair In such a case, we have that 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 ddf's implies equality of LTSD's.
Although looking for original ways to assess the degree of (actuarial) riskiness is a 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 (2006b). Unfortunately, all members of the elliptical class are symmetric, while insurance risks are generally modeled by non-negative 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 non-negative 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 (1947), however a more substantial and systematic investigation of this class of distributions was documented by Jorgensen (1986Jorgensen ( , 1987. In his Theory of dispersion models, Jorgensen (1997) 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 (1972). Nowadays, EDM's 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 non-negative support and many EDM members possess this important phenomenon.
We are now in a position to evaluate the limited TSD risk measure in the framework of the EDM's. Recall that, for 0 < q < p < 1, we denote by (x q , x p ) an arbitrary layer having 'attachment point' x q and width ∆ q, p . Also, let denote the generalized layer-based hazard function, such that and thus 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. Jorgensen, 1997, page 48). Then the limited TCE risk measure • for the reproductive EDM Y ED(µ, σ 2 ) is given by

5)
and • for the additive EDM X ED * (θ, λ) is given by 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 Further, following Landsman and Valdez (2005), it can be shown that for every 0 < q < 1, which then, employing (3.1) and (3.4), yields and hence completes the proof.
In the sequel, we sometimes write LT CE q, p [Y ; θ, λ] in order to emphasize the dependence on θ and λ.
Note 3.1. To obtain the results of Landsman and Valdez (2005), we put p ↑ 1 and then, for instance, in the reproductive case, we end up with subject to the existence of the limit.
Next theorem provides explicit expressions for the limited TSD risk measure for both reproductive and additive EDM's.
Theorem 3.2. Assume that the NEF which generates EDM is regular or at least steep.
Then the limited TSD risk measure

7)
and • for the additive EDM X ED * (θ, λ) is given by 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., Appendix 5), and hence, using Definition 2.1, we have that with the last line following from Appendix 5. Further, by simple re-arrangement and straightforward calculations, we obtain that 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 3.1. Let Y N (µ, σ 2 ) be a normal random variable with mean µ and variance σ 2 , then we can write the pdf of Y as If we take θ = µ and λ = 1/σ 2 , we see that the normal distribution is a reproductive EDM with cumulant function κ (θ) = θ 2 /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 , ) .
If we put p ↑ 1, then the latter equation reduces to the result of Landsman and Valdez (2005). Namely, we have that Further, let z q = (y q − µ) /σ and z p = (y p − µ) /σ. Then Consequently, the limited TSD risk measure is as following 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 rv's 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., Hürlimann, 2001;Rioux and Klugman, 2004;Furman andLandsman, 2005, Furman 2008).
Example 3.2. Let X Ga (γ, β) be a gamma rv with shape and rate parameters equal γ and β, correspondingly. The pdf of X is Hence the gamma rv can be represented as an additive EDM with the following pdf Putting p ↑ 1 we obtain that which coincides with Furman and Landsman (2005) Further, since for n = 1, 2, . . . , the limited TSD risk measure for gamma is given by 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. Next figures depict 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 EDM's 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 3.3. Let X P a(γ, β) be a Pareto rv with the pdf and γ > 0. Certainly, the Pareto rv is not a member of the EDM's, though it belongs to the log-exponential family of distributions (LEF) (cf. Furman and Zitikis, 2009). The LEF is defined by the differential equation where λ is a parameter, ν is a measure, and κ(λ) = log ∞ 0 x λ ν(dx) is a normalizing constant (the parameters should not be confused with the ones used in the context of the EDM's). Then X is easily seen to belong in LEF with the help of the reparameterization ν(dx) = x −1 dx, and λ = −γ.
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 that is finite for any γ > 0. Also, since, e.g., for γ < 1, we have that the limited TCE risk measure is positive, as expected. The same is true for γ ≥ 1.
We note in passing that, for γ > 1 and p ↑ 1 and thus x p → ∞, we have that which confirms the corresponding expression in Furman and Landsman (2006a).
Except for the Pareto distribution, the LEF consists of, e.g., the log-normal and inversegamma distributions, for which expressions similar to (3.12) 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 Furman and Landsman (2006b) 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 pdf's 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 EDM's.
The following corollary develops formulas for the TSD risk measure both in the reproductive and additive EDM's 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 The proof is left to the reader.
in the context of the reproductive EDM's, and in the context of the additive EDM's.
We further explore the TSD risk measure in some particular cases of EDM's, which seem to be of practical importance.
Example 4.2. Let X Ga (γ, β) be a gamma rv with shape and scale parameters equal γ and β, correspondingly. Taking into account Example 3.2 and Corollary 4.1 leads to where the latter equation follows because of the reparametrization 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 4.3. Let Y IG (µ, λ) be an inverse Gaussian rv. We then can write its pdf as Jorgensen, 1997), which means that Y belongs to the reproductive EDM's, with To this end, note that the ddf of Y is cf., e.g., Chhikara and Folks, 1974), where Φ (·) is the ddf of the standardized normal random variable. Hence, by simple differentiation and noticing that Notably, Consequently, the expression for the TCE risk measure, obtained in Landsman and Valdez (2005), simplifies to In order to derive the TSD risk measure we need to differentiate again, i.e., where we use that µ (θ) = µ(θ) 3 . Further, we have that where y q = − λ yq y q µ(θ) + 1 . Therefore

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 EDM's, 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. Klugman et al., 2008, Chapter 6), as well as for limited TCE/TSD risk measures.
Example 5.1. Let X P oisson (µ) be a Poisson rv with the mean parameter µ. Then the probability mass function of X is written as which belongs to the additive EDM's in view of the reparametrization θ = log (µ) , λ = 1 and κ (θ) = e θ .

The formula for the TSD risk measure is then
where E[X] = Var[X] = e θ and z q = x q − e θ .