Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses

Abstract

The conditional tail expectation (CTE) is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Abdelhakim Necir 1 and Abdelaziz Rassoul 2 and Ricardas Zitikis 3

Recommended by Edward Furman

1, Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, 07000, Algeria
2, Ecole Nationale Superieure d'Hydraulique, Guerouaou, BP 31, Blida, 09000, Algeria
3, Department of Statistical and Actuarial Sciences, University of Western Ontario, London, ON, N6A5B7, Canada

Received 21 October 2009; Accepted 20 January 2010

1. Introduction

One of the most important actuarial risk measures is the conditional tail expectation (CTE) (see, e.g., [1]), which is the average amount of loss given that the loss exceeds a specified quantile. Hence, the CTE provides a measure of the capital needed due to the exposure to the loss, and thus serves as a risk measure. Not surprisingly, therefore, the CTE continues to receive increased attention in the actuarial and financial literature, where we also find its numerous extensions and generalizations (see, e.g., [2-8], and references therein). We next present basic notation and definitions.

Let X be a loss random variable with cumulative distribution function (cdf) F . Usually, the cdf F is assumed to be continuous and defined on the entire real line, with negative loss interpreted as gain. We also assume the continuity of F throughout the present paper. The CTE of the risk or loss X is then defined, for every t∈(0,1) , by

[figure omitted; refer to PDF] where Q(t)=inf {x:F(x)≥t} is the quantile function corresponding to the cdf F . Since the cdf F is continuous, we easily check that

[figure omitted; refer to PDF]

Naturally, the CTE is unknown since the cdf F is unknown. Hence, it is desirable to establish statistical inferential results such as confidence intervals for CT_EF (t) with specified confidence levels and margins of error. We shall next show how to accomplish this task, initially assuming the classical moment assumption E[^X2 ]<∞ . Namely, suppose that we have independent random variables _X1 , _X2 , ... , each with the cdf F , and let _X1:n <...<_Xn:n denote the order statistics of _X1 ,...,_Xn . It is natural to define an empirical estimator of CT_EF (t) by the formula

[figure omitted; refer to PDF] where _Qn (s) is the empirical quantile function, which is equal to the ith order statistic _Xi:n for all s∈((i-1)/n,i/n] , and for all i=1,...,n . The asymptotic behavior of the estimator _{CTE ...n} (t) has been studied by Brazauskas et al. [9], and we next formulate their most relevant result for our paper as a theorem.

Theorem 1.1.

Assume that E[^X2 ]<∞ . Then for every t∈(0,1) , we have the asymptotic normality statement [figure omitted; refer to PDF] when n[arrow right]∞ , where the asymptotic variance ^σ2 (t) is given by the formula [figure omitted; refer to PDF]

The assumption E[^X2 ]<∞ is, however, quite restrictive as the following example shows. Suppose that F is the Pareto cdf with index γ>0 , that is, 1-F(x)=^x-1/γ for all x≥1 . Let us focus on the case γ<1 , because when γ≥1 , then CT_EF (t)=+∞ for every t∈(0,1) . Theorem 1.1 covers only the values γ∈(0,1/2) in view of the assumption E[^X2 ]<∞ . When γ∈[1/2,1) , we have E[^X2 ]=∞ but, nevertheless, CT_EF (t) is well defined and finite since E[X]<∞ . Analogous remarks hold for other distributions with Pareto-like tails, an we shall indeed work with such general distributions in this paper.

Namely, recall that the cdf F is regularly varying at infinity with index (-1/γ)<0 if

[figure omitted; refer to PDF] for every x>0 . This class includes a number of popular distributions such as Pareto, generalized Pareto, Burr, Fréchet, Student, and so forth, which are known to be appropriate models for fitting large insurance claims, fluctuations of prices, log-returns, and so forth (see, e.g., [10]). In the remainder of this paper, therefore, we restrict ourselves to this class of distributions. For more information on the topic and, generally, on extreme value models and their manifold applications, we refer to the monographs by Beirlant et al. [11], Castillo et al. [12], de Haan and Ferreira [13], Resnick [14].

The rest of the paper is organized as follows. In Section 2 we construct an alternative, called "new", CTE estimator by utilizing an extreme value approach. In Section 3 we establish the asymptotic normality of the new CTE estimator and illustrate its performance with a little simulation study. The main result, which is Theorem 3.1 stated in Section 3, is proved in Section 4.

2. Construction of a New CTE Estimator

We have already noted that the "old" estimator _{CTE ...n} (t) does not yield the asymptotic normality (in the classical sense) beyond the condition E[^X2 ]<∞ . Indeed, this follows by setting t=0 , in which case _{CTE ...n} (t) becomes the sample mean of _X1 ,...,_Xn , and thus the asymptotic normality of _{CTE ...n} (0) is equivalent to the classical Central Limit Theorem (CLT). Similar arguments show that the finite second moment is necessary for having the asymptotic normality (in the classical sense) of _{CTE ...n} (t) at any fixed "level" t∈(0,1) . Indeed, note that the asymptotic variance ^σ2 (t) in Theorem 1.1 is finite only if E[^X2 ]<∞ .

For this reason, we next construct an alternative CTE estimator, which takes into account different asymptotic properties of moderate and high quantiles in the case of heavy-tailed distributions. Hence, from now on we assume that γ∈(1/2,1) . Before indulging ourselves into construction details, we first formulate the new CTE estimator:

[figure omitted; refer to PDF] where we use the simplest yet useful and powerful Hill's [15] estimator

[figure omitted; refer to PDF] of the tail index γ∈(1/2,1) . Integers k=_kn ∈{1,...,n} are such that k[arrow right]∞ and k/n[arrow right]0 when n[arrow right]∞ , and we note at the outset that their choices present a challenging task. In Figures 1 and 2, we illustrate the performance of the new estimator _{CTE ...n} (t) with respect to the sample size n≥1 , with the integers k=_kn chosen according to the method proposed by Cheng and Peng [16]. Note that when t increases through the values 0.25 , 0.50 , 0.75 , and 0.90 (panels (a)-(d), resp.), the vertical axes of the panels also increase, which reflects the fact that the larger the t gets, the more erratic the "new" and "old" estimators become. Note also that the empirical (i.e., "old") estimator underestimates the theoretical CT_EF (t) , which is a well known phenomenon (see [17]).

Values of the CTE estimator _{CTE ...n} (t) (vertical axis) versus sample sizes n (horizontal axis) evaluated at the levels t=0.25 , t=0.50 , t=0.75 , and t=0.90 (panels (a)-(d), resp.) in the Pareto case with the tail index γ=2/3 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

Values of the CTE estimator _{CTE ...n} (t) (vertical axis) versus sample sizes n (horizontal axis) evaluated at the levels t=0.25 , t=0.50 , t=0.75 , and t=0.90 (panels (a)-(d), resp.) in the Pareto case with the tail index γ=3/4 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

We have based the construction of _{CTE ...n} (t) on the recognition that one should estimate moderate and high quantiles differently when the underlying distribution is heavy-tailed. For this, we first recall that the high quantile _qs is, by definition, equal to Q(1-s) for sufficiently small s . For an estimation theory of high quantiles in the case of heavy-tailed distributions we refer to, for example, Weissman [18], Dekkers and de Haan [19], Matthys and Beirlant [20], Gomes et al. [21], and references therein. We shall use the Weissman estimator

[figure omitted; refer to PDF] of the high quantile _qs . Then we write CT_EF (t) as the sum CT_E1,n (t)+CT_E2,n (t) with the two summands defined together with their respective empirical estimators _{CTE ...1,n} (t) and _{CTE ...2,n} (t) as follows:

[figure omitted; refer to PDF] Simple integration gives the formula

[figure omitted; refer to PDF] Consequently, the sum _{CTE ...1,n} (t)+_{CTE ...2,n} (t) is an estimator of CT_EF (t) , and this is exactly the estimator _{CTE ...n} (t) introduced above. We shall investigate asymptotic normality of the new estimator in the next section, accompanied with an illustrative simulation study.

3. Main Theorem and Its Practical Implementation

We start this section by noting that Hill's estimator _γ...n has been thoroughly studied, improved, and generalized in the literature. For example, weak consistency of _γ...n has been established by Mason [22] assuming only that the underlying distribution is regularly varying at infinity. Asymptotic normality of γ... has been investigated under various conditions by a number of researchers, including Csörgo and Mason [23], Beirlant and Teugels [24], Dekkers et al. [25], see also references therein.

The main theoretical result of this paper, which is Theorem 3.1 below, establishes asymptotic normality of the new CTE estimator _{CTE ...n} (t) . To formulate the theorem, we need to introduce an assumption that ensures the asymptotic normality of Hill's estimator _γ...n . Namely, the cdf F satisfies the generalized second-order regular variation condition with second-order parameter ρ≤0 (see [26, 27]) if there exists a function a(t) which does not change its sign in a neighbourhood of infinity and is such that, for every x>0 ,

[figure omitted; refer to PDF] When ρ=0 , then the ratio on the right-hand side of (3.1) is interpreted as log x . For statistical inference concerning the second-order parameter ρ , we refer, for example, to Peng and Qi [28], Gomes et al. [21], Gomes and Pestana [29]. Furthermore, in the formulation of Theorem 3.1, we shall also use the function A(z)=^γ2 a(...(z)) , where ...(z)=Q(1-1/z) .

Theorem 3.1.

Assume that the cdf F satisfies condition (3.1) with γ∈(1/2,1) . Then for any sequence of integers k=_kn [arrow right]∞ such that k/n[arrow right]0 and ^k1/2 A(n/k)[arrow right]0 when n[arrow right]∞ , we have that for any fixed t∈(0,1) , [figure omitted; refer to PDF] where the asymptotic variance ^σγ2 is given by the formula [figure omitted; refer to PDF]

The asymptotic variance ^σγ2 does not depend on t , unlike the variance in Theorem 1.1. This is not surprising because the heaviness of the right-most tail of F makes the asymptotic behaviour of ^∫t1 (_Qn (s)-Q(s))ds "heavier" than the classical CLT-type behaviour of ^∫0t (_Qn (s)-Q(s))ds , for any fixed t . This in turn implies that under the conditions of Theorem 3.1, statement (3.2) is equivalent to the same statement in the case t=0 . The latter statement concerns estimating the mean E[X] of a heavy-tailed distribution. Therefore, we can view Theorem 3.1 as a consequence of Peng [30], and at the same time we can view results of Peng [30] as a consequence of Theorem 3.1 by setting t=0 in it. Despite this equivalence, in Section 4 we give a proof of Theorem 3.1 for the sake of completeness. Our proof, however, is crucially based on a powerful technique called the Vervaat process (see [31-33], for details and references).

To discuss practical implementation of Theorem 3.1, we first fix a significance level [varsigma]∈(0,1) and use the classical notation _{z[varsigma]/2} for the (1-[varsigma]/2) -quantile of the standard normal distribution ...A9;(0,1) . Given a realization of the random variables _X1 ,...,_Xn (e.g., claim amounts), which follow a cdf F satisfying the conditions of Theorem 3.1, we construct a level 1-[varsigma] confidence interval for CT_EF (t) as follows. First, we choose an appropriate number k of extreme values. Since Hill's estimator has in general a substantial variance for small k and a considerable bias for large k , we search for a k that balances between the two shortcomings, which is indeed a well-known hurdle when estimating the tail index. To resolve this issue, several procedures have been suggested in the literature, and we refer to, for example, Dekkers and de Haan [34], Drees and Kaufmann [35], Danielsson et al. [36], Cheng and Peng [16], Neves and Fraga Alves [37], Gomes et al. [38], and references therein. In our current study, we employ the method of Cheng and Peng [16] for an appropriate value ^k* of the "parameter" k . Having computed Hill's estimator and consequently determined _{Xn-^k* :n} , we then compute the corresponding values of _{CTE ...n} (t) and ^{σ_γ...n 2} , and denote them by ^{CTE ...n*} (t) and ^{σ_γ...n 2*} , respectively. Finally, using Theorem 3.1 we arrive at the following (1-[varsigma]) -confidence interval for CT_EF (t) :

[figure omitted; refer to PDF]

To illustrate the performance of this confidence interval, we have carried out a small-scale simulation study based on the Pareto cdf F(x)=1-^x-1/γ , x≥1 , with the tail index γ set to 2/3 and 3/4 , and the level t set to 0.75 and 0.90 . We have generated 200 independent replicates of three samples of sizes n=1000 , 2000 , and 5000 . For every simulated sample, we have obtained estimates _{CTE ...n} (t) . Then we have calculated the arithmetic averages over the values from the 200 repetitions, with the absolute error (error) and root mean squared error (rmse) of the new estimator _C...n (t) reported in Table 1 (γ=2/3 ) and Table 2 (γ=3/4 ). In the tables, we have also reported 95% -confidence intervals (3.4) with their lower and upper bounds, coverage probabilities, and lengths.

Table 1: Point estimates _{CTE ...n} (t) and 95% confidence intervals for _{CTE F} (t) when γ=2/3 .

t=0.75	CTEF (t)=7.005
n	k*	CTE ...n (t)	error	rmse	lower	upper	cover	length
1000	054	6.876	0.045	0.303	6.356	7.397	0.839	1.041
2000	100	6.831	0.025	0.231	6.463	7.199	0.882	0.736
5000	219	7.119	0.016	0.194	6.881	7.357	0.895	0.476
t=0.90	CTEF (t)=12.533
n	k*	CTE ...n (t)	error	rmse	lower	upper	cover	length
1000	054	12.753	0.017	0.534	12.241	13.269	0.847	1.028
2000	100	12.487	0.003	0.294	12.137	12.838	0.841	0.701
5000	219	12.461	0.005	0.236	12.246	12.676	0.887	0.430

Table 2: Point estimates _{CTE ...n} (t) and 95% confidence intervals for _{CTE F} (t) when γ=3/4 .

t=0.75	CTEF (t)=9.719
n	k*	CTE ...n (t)	error	rmse	lower	upper	cover	length
1000	051	9.543	0.018	0.582	8.589	9.543	0.854	0.954
2000	104	9.808	0.009	0.466	9.150	10.466	0.888	1.316
5000	222	9.789	0.007	0.410	9.363	10.215	0.915	0.852
t=0.90	CTEF (t)=18.494
n	k*	CTE ...n (t)	error	rmse	lower	upper	cover	length
1000	051	18.199	0.015	0.989	17.437	18.960	0.874	1.523
2000	104	18.696	0.011	0.858	18.052	19.340	0.895	1.288
5000	222	18.541	0.002	0.798	18.092	18.990	0.925	0.898

We note emphatically that the above coverage probabilities and lengths of confidence intervals can be improved by employing more precise but, naturally, considerably more complex estimators of the tail index. Such estimators are described in the monographs by Beirlant et al. [11], Castillo et al. [12], de Haan and Ferreira [13], and Resnick [14]. Since the publication of these monographs, numerous journal articles have appeared on the topic. Our aim in this paper, however, is to present a simple yet useful result that highlights how much Actuarial Science and developments in Mathematical Statistics, Probability, and Stochastic Processes are interrelated, and thus benefit from each other.

4. Proof of Theorem 3.1

We start the proof of Theorem 3.1 with the decomposition

[figure omitted; refer to PDF] where

[figure omitted; refer to PDF] We shall show below that there are Brownian bridges _Bn such that

[figure omitted; refer to PDF] [figure omitted; refer to PDF] Assuming for the time being that statements (4.3) and (4.4) hold, we next complete the proof of Theorem 3.1. To simplify the presentation, we use the following notation:

[figure omitted; refer to PDF] Hence, we have the asymptotic representation

[figure omitted; refer to PDF] The sum _W1,n +_W2,n +_W3,n is a centered Gaussian random variable. To calculate its asymptotic variance, we establish the following limits:

[figure omitted; refer to PDF] Summing up the right-hand sides of the above six limits, we obtain ^σγ2 , whose expression in terms of the parameter γ is given in Theorem 3.1. Finally, since _Xn-k:n /Q(1-k/n) converges in probability to 1 (see, e.g., the proof of Corollary in [39]), the classical Sultsky's lemma completes the proof of Theorem 3.1. Of course, we are still left to verify statements (4.3) and (4.4), which make the contents of the following two subsections.

4.1. Proof of Statement (4.3)

If Q were continuously differentiable, then statement (4.3) would follow easily from the proof of Theorem 2 in [39]. We do not assume differentiability of Q and thus a new proof is required, which is crucially based on the Vervaat process (see [31-33], and references therein)

[figure omitted; refer to PDF] Hence, for every t such that 0<t<1-k/n , which is satisfied for all sufficiently large n since t is fixed, we have that

[figure omitted; refer to PDF] It is well known (see [31-33]) that _Vn (t) is nonnegative and does not exceed -(_Fn (Q(t))-t)(_Qn (t)-Q(t)) . Since the cdf F is continuous by assumption, we therefore have that

[figure omitted; refer to PDF] where _en (t) is the uniform empirical process n(_Fn (Q(t))-F(Q(t)) , which for large n looks like the Brownian bridge _Bn (t) . Note also that with the just introduced notation _en , the integral on the right-hand side of (4.9) is equal to ^{∫Q(t)Q(1-k/n)}_en (F(x))dx . Hence,

[figure omitted; refer to PDF] We shall next replace the empirical process _en by an appropriate Brownian bridge _Bn in the first integral on the right-hand side of (4.11) with an error term of magnitude _oP (1) , and we shall also show that the second and third summands on the right-hand side of (4.11) are of the order _oP (1). The replacement of _en by _Bn can be accomplished using, for example, Corollary 2.1 on page 48 of Csörgo et al. [40], which states that on an appropriately constructed probability space and for any 0≤ν<1/4 , we have that

[figure omitted; refer to PDF] This result is applicable in the current situation since we can always place our original problem into the required probability space, because our main results are "in probability". Furthermore, since Q(t)≤x≤Q(1-k/n) , we have that t≤F(x)≤1-k/n . Hence, statement (4.12) implies that

[figure omitted; refer to PDF] Changing the variables of integration and using the property (k/n^)1/2 Q(1-k/n)[arrow right]∞ when n[arrow right]∞ , we obtain that

[figure omitted; refer to PDF] The main term on the right-hand side of (4.14) is _W1,n . We shall next show that the right-most summand of (4.13) converges to 0 when n[arrow right]∞ .

Changing the variable of integration and then integrating by parts, we obtain the bound

[figure omitted; refer to PDF] We want to show that the right-hand side of bound (4.15) converges to 0 when n[arrow right]∞ . For this, we first note that

[figure omitted; refer to PDF] Next, with the notation [varphi](u)=Q(1-u)/^u1/2+ν , we have that

[figure omitted; refer to PDF] when n[arrow right]∞ , where the convergence to 0 follows from Result 1 in the Appendix of Necir and Meraghni [39]. Taking statements (4.15)-(4.17) together, we have that the right-most summand of (4.13) converges to 0 when n[arrow right]∞ .

Consequently, in order to complete the proof of statement (4.3), we are left to show that the second and third summands on the right-hand side of (4.11) are of the order _oP (1) . The third summand is of the order _oP (1) because |_en (t)||_Qn (t)-Q(t)|=_OP (1) and (k/n^)1/2 Q(1-k/n)[arrow right]∞ . Hence, we are only left to show that the second summand on the right-hand side of equation (4.11) is of the order _oP (1) , for which we shall show that

[figure omitted; refer to PDF] To prove statement (4.18), we first note that

[figure omitted; refer to PDF] The first summand on the right-hand side of bound (4.19) is of the order _OP (1) due to statement (4.12) with ν=0 . The second summand on the right-hand side of bound (4.19) is of the order _OP (1) due to a statement on page 49 of Csörgo et al. [40] (see the displayed bound just below statement (2.39 ) therein). Hence, to complete the proof of statement (4.18), we need to check that

[figure omitted; refer to PDF] Observe that, for each n , the distribution of _Qn (1-k/n) is the same as that of Q(^En-1 (1-k/n)) , where ^En-1 is the uniform empirical quantile function. Furthermore, the processes {1-^En-1 (1-s), 0≤s≤1} and {^En-1 (s), 0≤s≤1} are equal in distribution. Hence, statement (4.20) is equivalent to

[figure omitted; refer to PDF] From the Glivenko-Cantelli theorem we have that ^En-1 (k/n)-k/n[arrow right]0 almost surely, which also implies that ^En-1 (k/n)[arrow right]0 since k/n[arrow right]0 by our choice of k . Moreover, we know from Theorem 0 and Remark 1 of Wellner [41] that

[figure omitted; refer to PDF] from which we conclude that

[figure omitted; refer to PDF] Since the function s...Q(1-s) is slowly varying at zero, using Potter's inequality (see the 5th assertion of Proposition B.1.9 on page 367 of de Haan and Ferreira [13],we obtain that

[figure omitted; refer to PDF] for any θ∈(0,γ) . In view of (4.23), the right-hand side of (4.24) is equal to 1+_oP (1) , which implies statement (4.21) and thus finishes the proof of statement (4.3).

4.2. Proof of Statement (4.4)

The proof of statement (4.4) is similar to that of Theorem 2 in Necir et al. [42], though some adjustments are needed since we are now concerned with the CTE risk measure. We therefore present main blocks of the proof together with pinpointed references to Necir et al. [42] for specific technical details.

We start the proof with the function ...(z)=Q(1-1/z) that was already used in the formulation of Theorem 3.1. Hence, if Y is a random variable with the distribution function G(z)=1-1/z , z≥1 , then ...(Y)=Q(G(Y))_=d X because G(Y) is a uniform on the interval [0,1] random variable. Hence,

[figure omitted; refer to PDF] and so we have

[figure omitted; refer to PDF] We next show that the right-most term in (4.26) converges to 0 when n[arrow right]∞ . For this reason, we first rewrite the term as follows:

[figure omitted; refer to PDF] The right-hand side of (4.27) converges to 0 (see notes on page 149 of Necir et al. [42]) due to the second-order condition (3.1), which can equivalently be rewritten as

[figure omitted; refer to PDF] for every s>0 , where A(z)=^γ2 a(...(z)) . Note that kA(n/k)[arrow right]0 when n[arrow right]∞ . Hence, in order to complete the proof of statement (4.4), we need to check that

[figure omitted; refer to PDF] With Hill's estimator written in the form

[figure omitted; refer to PDF] we proceed with the proof of statement (4.29) as follows: [figure omitted; refer to PDF] Furthermore, we have that

[figure omitted; refer to PDF] Arguments on page 156 of Necir et al. [42] imply that the first term on the right-hand side of (4.32) is of the order _OP (kA(_Yn-k:n )) , and a note on page 157 of Necir et al. [42] says that k A(_Yn-k:n )=_oP (1) . Hence, the first term on the right-hand side of (4.32) is of the order _oP (1) . Analogous considerations using bound (2.5) instead of (2.4) on page 156 of Necir et al. [42] imply that the first term on the right-hand side of (4.31) is of the order _oP (1) . Hence, in summary, we have that

[figure omitted; refer to PDF] We now need to connect the right-hand side of (4.33) with Brownian bridges _Bn . To this end, we first convert the Y -based order statistics into U -based (i.e., uniform on [0,1] ) order statistics. For this we recall that the cdf of Y is G , and thus Y is equal in distribution to ^G-1 (U) , which is 1/(1-U) . Consequently,

[figure omitted; refer to PDF] Next we choose a sequence of Brownian bridges _Bn (see pages 158-159 in [42] and references therein) such that the following two asymptotic representations hold:

[figure omitted; refer to PDF] Using these two statements on the right-hand side of (4.34) and also keeping in mind that _γ...n is a consistent estimator of γ (see [22]), we have that

[figure omitted; refer to PDF] Dividing both sides of equation (4.36) by 1-γ , we arrive at (4.29). This completes the proof of statement (4.4) and of Theorem 3.1 as well.

Acknowledgments

Our work on the revision of this paper has been considerably influenced by constructive criticism and suggestions by three anonymous referees and the editor in charge of the manuscript, Edward Furman, and we are indebted to all of them. Results of the paper were first announced at the 44th Actuarial Research Conference at the University of Wisconsin, Madison, Wisconsin, July 30-August 1, 2009. The authors are grateful to participants of this most stimulating conference, organized by the Society of Actuaries, for generous feedback. The research has been partially supported by grants from the Society of Actuaries (SOA) and the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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