Bounds for right tails of deterministic and stochastic sums of random variables.

ABSTRACT

We investigate lower and upper bounds for right tails (stop-loss premiums) of deterministic and stochastic sums of nonindependent random variables. The bounds are derived using the concepts of comonotonicity, convex order, and conditioning. The performance of the presented approximations is investigated numerically for individual life annuity contracts as well as for life annuity portfolios, where mortality is modeled by Makeham's law, whereas investment returns are modeled by a Brownian motion process.

INTRODUCTION

An insurance risk is described by a random variable (rv) X that represents the (discounted) value of future claims of an individual insurance contract or the aggregate claims of an insurance portfolio over a given reference period. One of the main tasks of the actuary is to assess the "dangerousness" of insurance risks, either by determining their distribution functions (df) or by summarizing their characteristics quantitatively by means of one or more risk measures.

An important class of risk measures related to an insurance risk X consists of the expectations E[[(X - d).sub.+]] = E[max (0, X - d)] for different values of d. In a reinsurance context, [(X - d).sub.+] can be interpreted as the liability of the reinsurer in a stop-loss reinsurance contract and E[[(X - d).sub.+]] is the associated net reinsurance premium, called the stop-loss premium at retention d. More generally, the quantity E[[(X - d).sub.+]] can be interpreted as a measure for the right tail of the df of X, beyond outcome d. Intuitively, a rv with larger right tails has more probability mass concentrated in the right tail of the df and hence, is "more dangerous." In the sequel, we will often call the stop-loss premium E[[(X - d).sub.+]] the right tail at level d of (the df of) X.

As applications we consider the problem of measuring the right tail of a single life annuity (cash flows with a stochastic time horizon) and of a diversified portfolio of life annuities (cash flows with a deterministic time horizon). Using our results for compound sums we obtain very precise bounds. We provide a number of numerical illustrations which reveal a significant improvement compared with the bounds obtained by traditional comonotonic approximations.

An rv Y that has uniformly larger right tails than X is said to be "larger in the increasing convex order sense," notation X [less than or equal to] [sub.icx] Y. In an actuarial context, the increasing convex order is also called stop-loss order (see, e.g., Denuit et al., 2005, and the references therein). In terms of utility theory, X [less than or equal to] [sub.icx] Y means that any risk-averse decision maker prefers risk X over risk Y. Hence, the calculation of right tails of insurance risks makes sense in order to evaluate the "dangerousness" of the risks at hand. In case X [less than or equal to] [sub.icx] Y and in addition E[X] = E[Y], one says that X is smaller than Y in the convex order sense, notation X [less than or equal to] [sub.cx] Y.

In practice, it is not always straightforward to compute right tails or stop-loss premiums. In the actuarial literature a lot of attention has been devoted to determine bounds for stop-loss premiums in case only partial information about the distribution is available (see, e.g., De Vylder and Goovaerts, 1982; Jansen, Haezendonck, and Goovaerts, 1986; Hurlimann, 1996, 1998, 2002).

A particular type of problems arises when determining right tails of a sum S = [X.sub.1] + ... + [X.sub.n] when full information about the distributions of the [X.sub.i] is available but the dependence structure between these [X.sub.i] is not known or too cumbersome to work with. In Dhaene et al. (2002a, b) it is shown that in this context the least upper bound of S in the increasing convex order sense is obtained by replacing the unknown copula of the random vector ([X.sub.1], [X.sub.2], ..., [X.sub.n]) by the comonotonic copula. These authors also propose comonotonic lower bounds for the df of S obtained by conditioning. Such an approach allows to determine analytical bounds for right tails or stop-loss premiums, in particular in the multivariate lognormal case.

In practical applications, the comonotonic lower bounds have proved to be very accurate approximations in case the distributions of the random sum is light to moderate heavy tailed (see, e.g., Vanduffel, Hoedemakers, and Dhaene, 2005). In Laeven, Goovaerts, and Hoedemakers (2005), asymptotic results are derived for the tail probabilities of S in the presence of heavy-tailedness conditions.

As opposed to the comonotonic lower bounds, the comonotonic upper bound is only accurate in case of a very strong dependence between summands. Therefore, in this contribution we present a number of techniques which allow to determine improved upper bounds for right tails. Following the ideas presented in Deelstra, Liinev, and Vanmaele (2004) and Vanmaele et al. (2006), we use the method of conditioning as in Curran (1994) and in Rogers and Shi (1995). We compare these bounds with the upper and lower bounds for right tails of sums of dependent rv's as derived in Dhaene et al. (2002a, b).

In this article, we are also especially interested in bounds for right tails of a stochastic sum of rv's, which show up in a natural way both in life and non-life insurances. These bounds follow from the bounds for deterministic sums of rv's, taking into account the tower property for conditional expectations.

In particular, we illustrate our theoretical results by calculating lower and upper bounds for the right tails of the stochastic sum of rv's representing the stochastic present value of future liabilities related with a single life annuity. As a second illustration, we determine lower and upper bounds for the right tails of the deterministic sum of rv's representing the stochastic present value of the future liabilities of a well-diversified portfolio of life annuities.

As the derivation of bounds for a stochastic sum of rv's is based on the study of deterministic sums of rv's, we present different bounds for the latter and do this in the particular situation of sums of lognormal distributed rv's. Such sums are widely encountered in practice, both in actuarial science and in finance. Typical examples are stochastic present values of future cash flows (see, e.g., Dhaene et al., 2002b, 2005), Asian options (see, e.g., Simon, Goovaerts, and Dhaene, 2000; Vanmaele et al., 2006; Reynaerts et al., 2006), basket options (see Deelstra, Liinev, and Vanmaele, 2004; Vanmaele, Deelstra, and Liinev, 2004) and Asian basket options (see Deelstra, Diallo, and Vanmaele, 2008, in press).

Using these bounds, we then propose different bounds for right tails of stochastic sums of dependent rv's in the setting of lognormal distributions, and in particular for our two examples mentioned above, namely a single life annuity (cash flows with a stochastic time horizon) and a diversified portfolio of life annuities (cash flows with a deterministic time horizon). By several numerical experiments, the different bounds are evaluated and compared.

The importance of the proposed bounds lies in the fact that they are analytical and that they are bounds and not only approximations. Monte-Carlo simulations are very time consuming compared to the computation of the analytical expressions of these bounds. Multivariate integration on the other hand would require the knowledge of the dependence structure of the involved multivariate random variable which is however not known. The proposed comonotonicity approach circumvents this problem.

The article is organized as follows. In the "Life Annuities and Right Tails" section we describe the present value of rv's related to (portfolios of) life annuities and present some basic notations and a simple but important theorem for right tails of a stochastic sum of rv's. In the "Bounds Based on Comonotonicity and Conditioning" section we recapitulate the lower and upper bounds of Kaas, Dhaene, and Goovaerts (2000) and Dhaene et al. (2002a), as well as the results of Rogers and Shi (1995). The "Bounds by Conditioning Through Decomposition of the Right Tail" section explains how the upper bounds can be improved by decomposing right tails. In the "Bounds for Compound Sums" section we discuss right tails of compound sums. All presented bounds are illustrated by considering the particular case where all nonindependent components constituting the sum are lognormally distributed. In the "Numerical Illustrations" section we calculate the presented bounds for the life annuity problems described above. Finally, the "Summary and Conclusions" section concludes the article.

LIFE ANNUITIES AND RIGHT TAILS

A life annuity underwritten on a life (x) of age x provides a series of periodic payments, where each payment is due conditional on the survival of (x) at the moment of this payment. We denote the future lifetime of (x) by [T.sub.x]. Thus, x + [T.sub.x] is the age at death of the insured. The future lifetime [T.sub.x] is a rv with df denoted by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We assume that the df of [T.sub.x] is given. We define [K.sub.x] = [??][T.sub.x][??], the number of completed future years lived by (x), or the curtate future lifetime of (x), where [??]x[??] is the largest integer less than or equal to x. The df of the integer-valued rv [K.sub.x] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In our numerical illustrations, we will assume that the distribution of the remaining lifetime belongs to the Gompertz-Makeham family. Hence, the probabilities defined above follow from a lifetable of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

with parameters a > 0,0 < s < 1, 0 < g < 1 and c > 1. (1) See Bowers et al. (1986) for more details.

Consider a whole life annuity on a life (x) which pays an amount of I at the end of each year, provided the insured is still alive at that time. Assume that the discounting is performed with a random return. The stochastic present value of the future payments of this annuity is denoted by [S.sup.policy]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

where the rv's Y(i) are defined by Y(i) = [Y.sub.1] + [Y.sub.2] + ... + [Y.sub.i], where [Y.sub.i] is the random logreturn over the period [i - 1, i] and [e.sup.-Y(i)] is the random discount factor over the period [0, i]. We assume that the curtate lifetime [K.sub.x] is independent of the return variables [Y.sub.i].

The rv defined in (2) is a special case of a compound sum of the form

[S.sub.N] = [N.summation over (i=1)] [X.sub.i], (3)

with N a stochastic number which is independent of the [X.sub.i]. The right tails of [S.sub.N] will be denoted by [pi] ([S.sub.N], d). They can be determined using the next result, which follows immediately from the tower property for conditional expectations.

Theorem 1: The right tails [pi]([S.sub.N], d) = E[[([S.sub.N] - d).sub.+]] of the compound sum [S.sub.N] are given by

[pi]([S.sub.N], d) = [[infinity].summation over (j=1)] Pr(N = j)[pi]([S.sub.j], d), with [S.sub.j] = [j.summation over (i=1)] [X.sub.i]. (4)

For the compound sum [S.sup.policy] defined in (2), we find from Theorem 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

In a second application we will consider a portfolio of [l.sub.x] life annuity contracts with respective future lifetimes of the insureds [T.sup.(1).sub.x]], [T.sup.(2).sub.x]], ..., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which are assumed to be mutually independent. The insurer of this portfolio faces two types of risk: mortality risk and investment risk. The stochastic present value of the portfolio liabilities are given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [L.sub.x+i] denotes the random number of survivors at age x + i. The present value of the liabilities per portfolio are then given by [[summation over].sup.[infinity].sub.i=1] [L.sub.x+i] / [l.sub.x] [e.sup.-Y(i)]. From the Law of Large Numbers it follows that the mortality risk decreases with the number of policies [l.sub.x] whereas the investment risk remains the same as each of the policies is exposed to the same investment risk. Thus, for a sufficiently large number of policies the present value of the liabilities per policy can be approximated by

[S.sup.average] = [[infinity].summation over (i=1)] i [p.sub.x][e.sup.-Y(i)]. (6)

The right tails of the present value of the portfolio liabilities per policy can then be approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that in practice there exists a t such that [sub.t] [p.sub.x] = 0 for all t [greater than or equal to] 0. In this case, the infinite sums that appear in our previous derivations will all transform in a finite sum and the rv's [[??].sub.j] in (5) and [S.sup.average] in (6) are both of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

BOUNDS BASED ON COMONOTONICITY AND CONDITIONING

Lower and Upper Bounds for the df of S

In the life annuity applications mentioned in the previous section, we are confronted with a rv of the type S = [[summation].sup.n.sub.i=1] [X.sub.i] as defined in (7) where the df's of the terms [X.sub.i] are given, but the multivariate df of the random vector ([X.sub.1], [X.sub.2], ..., [X.sub.n]) is not completely specified or too cumbersome to work with. In such cases it may be helpful to find rv's [S.sub.1] and [S.sub.2] such that

[S.sub.1] [less than or equal to] [sub.cx] S [less than or equal to] [sub.cx] [S.sub.2],

which implies by definition of convex order that the corresponding right tails satisfy

E[[([S.sub.1] - d).sub.+]] [less than or equal to] E[[(S - d).sub.+]]

for all d.

Let us first consider the case that the only information that is available concerning the multivariate df of the random vector ([X.sub.1], ..., [X.sub.n]) are the marginal df's of the [X.sub.i]. Let U denote a uniformly distributed rv on the unit interval (0, 1) and let [F.sup.-1.sub.X] be the inverse of the df of X defined as usual as

[F.sup.-1.sub.X](p) = inf{x [member of] R | [F.sub.X](x) [greater than or equal to] p}, p [member of] [0,1],

with inf [??] = + [infinity] by convention. In this case the df of the comonotonic sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is a prudent choice for approximating the unknown df of S because S [less than or equal to] [sub.cx] [S.sup.cub]. It is a supremum in terms of convex order in the sense that in the class of all random sums with given marginals any stop-loss premium of the comonotonic sum dominates the corresponding stop-loss premiums of all other sums. In the sequel, we denote the right tails or stop-loss premiums of [S.sup.cub] by [[pi].sup.cub] (S, d).

From Denneberg (1994) and Landsberger and Meilijson (1994), it is known that the inverse distribution function of a sum of comonotonic rv's is simply the sum of the inverse distribution functions of the marginal distributions. Then it follows that the stop-loss premiums of a sum of comonotonic rv's can be easily obtained as a linear combination of the appropriate stop-loss premiums of the terms (see Dhaene et al., 2002a, 2008). In the case of strictly increasing marginals, the right tails [[pi].sup.cub](S, d) can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with the df [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implicitly determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

For strictly increasing and continuous marginals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is uniquely determined by (8).

Applying this result to the sum S of lognormal rv's as defined in (7), we find the following expression for [[pi].sup.cub](s, d):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be found by solving

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

See Dhaene et al. (2002b) for more details.

We can improve the comonotonic upper bound [[pi].sup.cub](S, d) when there is some additional information available concerning the stochastic nature of ([X.sub.1], ..., [X.sub.n]). More precisely, we assume that there exists a rv [LAMBDA] with a given df and such that we know the conditional df's of the rv [X.sub.i], given [LAMBDA] = [lambda], for all possible values of [lambda]. Kaas, Dhaene, and Goovaerts (2000) define the improved comonotonic upper bound [S.sup.u] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the notation for the rv [f.sub.i](U, [LAMBDA]), with the function [f.sub.i] defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

An expression for the right tails E[[([S.sup.u] - d).sub.+]], further denoted by [[pi].sup.icub](S, d, [LAMBDA]), of the improved comonotonic upper bound [S.sup.u] for S can be found in Kaas, Dhaene, and Goovaerts (2000).

Let us now consider the sum of lognormal rv's S defined in (7) and a conditioning variable A satisfying the following assumption:

Assumption 1: [LAMBDA] is a normally distributed rv for which the ([Z.sub.i], [LAMBDA]), i = 1, ..., n, are bivariate normally distributed.

Under this assumption, we find from Kaas, Dhaene, and Goovaerts (2000) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the correlations [r.sub.i], i = 1, ..., n, are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

and where V = [PHI]([LAMBDA] - E[[LAMBDA] / [[sigma].sub.[LAMBDA]]). The conditional probability [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the df of a sum of n comonotonic rv's and follows, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], implicitly from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Remark that when all [r.sub.i] are zero the improved comonotonic upper bound coincides with the comonotonic upper bound. On the other hand, when the retention becomes very large both comonotonic bounds will converge to one another and tend in the limit to zero.

The additional information described via a rv [LAMBDA], as mentioned earlier, can also be used to construct the convex order lower bound [S.sup.l] = E[S | [LAMBDA]] for S (see, e.g., Rogers and Shi, 1995; Kaas, Dhaene, and Goovaerts, 2000). However in contrast to the upper bound [S.sup.u] which is a comonotonic sum by construction, we now have to make additional assumptions on [LAMBDA] to guarantee that this lower bound is a comonotonic sum.

When the rv [LAMBDA] is such that all E[[X.sub.i] | [LAMBDA]] are nondecreasing (or all are nonincreasing) and continuous functions of [LAMBDA], then [S.sup.l] is a comonotonic sum (see Kaas, Dhaene, and Goovaerts, 2000). Let us assume that the df's of the rv's E[[X.sub.i] | [LAMBDA]] are strictly increasing. In this case the right tails E[[([S.sup.l] - d).sub.+]], denoted by [[pi].sup.lb](S, d, [LAMBDA]), of the lower bound [S.sup.l] for S are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

with [LAMBDA] the conditioning rv in the definition of [S.sup.l] and where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be obtained from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This last equation has a unique solution when the df's of the rv's E[[X.sub.i] | [LAMBDA]] are strictly increasing and continuous.

Remark 1: In case the cumulative df's of the rv's E[[X.sub.i] | [LAMBDA]] are not continuous nor strictly increasing or decreasing functions of [LAMBDA], then the stop-loss premiums of [S.sup.l], which is not comonotonic anymore, can be determined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Let us now again assume that S is a sum of lognormal rv's as defined in (7) and that Assumption 1 holds. Furthermore, we assume that the following assumption holds:

Assumption 2: All correlations [r.sub.i] in (10) are nonnegative, that is, [r.sub.i] [greater than or equal to] 0 for all i.

The case [r.sub.i] [less than or equal to] 0 for all i can be treated in a similar way.

For the right tail [[pi].sup.lb](S, d, [LAMBDA]) of [S.sup.l], which is a lower bound for [pi](S, d), we find the following expression for (12) (see Dhaene et al., 2002b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the unique solution to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Note that in case of perfect correlation, that is, all [r.sub.i] = 1, the comonotonic sums [S.sup.l] and [S.sup.c], as well as their respective right tails coincide.

Upper Bound Based on Lower Bound Plus Error Term

Applying the following general inequality for any rv Y and Z from Rogers and Shi (1995):

0 [less than or equal to] E [E[Y.sub.+] | Z] - E[[Y | Z].sub.+] [less than or equal to] 1/2 E[[square root of Var(Y | Z)]]

to the case of Y being S - d and Z being the conditioning variable A in the definition of [S.sup.l], we obtain by an analogous reasoning as in Nielsen and Sandmann (2003) and Vanmaele et a l. (2006), an upper bound [[pi].sup.eub] (S, d, [LAMBDA]) for the stop-loss premium [pi] (S, d) with retention d based on the lower bound:

[pi](S, d) [less than or equal to] [[pi].sup.lb] (S, d, [LAMBDA]) + [epsilon] =: [[pi].sup.eub] (S, d, [LAMBDA]), (16)

where the error bound [epsilon] equals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Note that the error bound is independent of the retention d. Thus, this upper bound (16) is a constant deviation of the lower bound for any value of the retention d. This implies that for large retentions for which the stop-loss premiums will be smaller than for small retentions, this upper bound will not perform very well.

Specifying that the components [X.sub.i] in S are lognormally distributed according to (7) and imposing Assumption I on the conditioning rv [LAMBDA], we can express the error bound more explicitly:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and where [r.sub.ij] stands for the correlation of this [Z.sub.ij] and [LAMBDA]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting this expression (18) for the error bound and relation (14) for the lower bound into (16) provides an analytical expression for the upper bound [[pi].sup.eub](S, d, [LAMBDA]) in the lognormal case under the Assumptions 1 and 2. This expression is therefore more general than the error obtained in Theorem 5 of Vanmaele et al. (2006), which concentrates upon Asian options in a Black and Scholes framework.

BOUNDS BY CONDITIONING THROUGH DECOMPOSITION OF THE RIGHT TAIL

In this section, we show how to improve the upper bounds introduced in the previous section. The idea is to split off that part of the right tail that can be calculated exactly and to bound the remaining part.

Decomposition of the Right Tail

We condition S on some rv [LAMBDA] and impose one of the following assumptions:

Assumption 3: There exists a [d.sub.[LAMBDA] such that [LAMBDA] [greater than or equal to] [d.sub.[LAMBDA]] implies that S [greater than or equal to] d.

Assumption 4: There exists a [d.sub.[LAMBDA]] such that [LAMBDA] [less than or equal to] [d.sub.[LAMBDA]] implies that S [greater than or equal to] d.

Then the right tail of S can be decomposed in two parts, one of which can either be computed exactly or by using numerical integration, depending on the distribution of the underlying rv. For the remaining part we first derive a lower and an upper bound based on comonotonic risks, and another upper bound equal to that lower bound plus an error term. This decomposition idea goes back at least to Curran (1994).

Theorem 2: Under Assumption 3 the right tail of S with retention d can be decomposed in [I.sub.1] and [I.sub.2], that is [pi](S, d) = [I.sub.1] + [I.sub.2], with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

This result has been derived and used in the proof of Theorem 7 of Vanmaele et al. (2006).

Remark 2:

(1) Under Assumption 4 a similar decomposition holds with the appropriate integration bounds. Also in the following sections, the case of Assumption 4 can be treated in a similar way but will not be mentioned any further.

(2) In practical applications the existence of such a [d.sub.[LAMBDA] depends on the actual form of S and [LAMBDA].

(3) The second integral [I.sub.2] can be written out explicitly if the bivariate distribution of ([X.sub.i], [LAMBDA]) is known for all i.

Lower Bound

Jensen's inequality leads to a lower bound for the first integral [I.sub.1] in (19):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By adding the exact part (20) and using the definition of [S.sup.l], we end up with the lower bound (12).

Thus, when [S.sup.l] is a sum of n comonotonic risks we can apply (12), which holds even when we do not know or find an appropriate [d.sub.[LAMBDA].

When [S.sup.l] is not comonotonic we use under Assumption 3 the decomposition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Upper Bound Based on Lower Bound

Under Assumption 3 we improve the bound (16) by making the error bound [epsilon] (17) dependent on the integration bound [d.sub.[LAMBDA]].

Theorem 3: Under Assumption 3 the upper bound [[pi].sup.deub] for the stop-loss premium [pi](S, d) with retention d is given by

[[pi].sup.deub] (S, d, [LAMBDA]) = [[pi].sup.lb] (S, d, [LAMBDA]) + [epsilon]([d.sub.[LAMBDA]]), (21)

where the error bound [epsilon]([d.sub.[LAMBDA]]) equals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the indicator function, that is, [1.sub.{c}] = 1 if the condition c is true and [1.sub.{c}] = 0 if it is not.

This result can be found in the reasoning leading to Equation (4) in Nielsen and Sandmann (2003).

Remark 3: As an intermediate step in the derivation of the upper bound (21) we find the following upper bound:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

The limiting case of this upper bound where [d.sub.[LAMBDA]] equals infinity is precisely the upper bound (16) which is hence independent of [d.sub.[LAMBDA]. Applying Holders inequality to the integral in (23) yields the result (21). Obviously, the error bound in (23) improves the error bound (17). In practical applications, the additional error introduced by Holders inequality turns out to be much smaller than the difference 1/2 E[[square root of Var(S|[LAMBDA])]] - [epsilon]([d.sub.[LAMBDA]]).

Considering the lognormal case we can write out an explicit expression for the error bound [epsilon]([d.sub.[LAMBDA]]) under Assumptions 1 and 3:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

with [d.sup.*.sub.LAMBDA]] = [d.sub.[LAMBDA]] - E[[LAMBDA]] / [[sigma].sub[LAMBDA]] Combining this expression (24) for the error bound with (14) for the lower bound provides an analytical expression for the upper bound (21) under the additional Assumption 2 for [LAMBDA].

Partially Exact/Comonotonic Upper Bound

Under Assumption 3 another upper bound can be obtained as is explained in Section 2.3 of Vanmaele et al. (2006) for Asian option pricing in the Black and Scholes framework. In general, we bound the first-term [I.sub.1] in (19) by replacing S | [LAMBDA] = [lambda] by its comonotonic upper bound [S.sup.u] (9) (in convex order sense):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Adding (25) to the exact part (20) of the decomposition of the stop-loss premium of S results in the so-called partially exact/comonotonic upper bound for a stop-loss premium. We will use the notation [[pi].sup.pecub](S, d, [LAMBDA]) to indicate this upper bound.

Theorem 4: Under Assumption 3 the partially exact/comonotonic upper bound for [pi](S, d) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 4: It is easily seen that

[[pi].sup.pecub](S, d, [LAMBDA]) [less than or equal to] [[pi].sup.icub](S, d, [LAMBDA]).

Although for two distinct conditioning variables [[LAMBDA].sub.1] and [[LAMBDA].sub.2] it does not necessarily hold that

[[pi].sup.pecub](S, d, [[LAMBDA].sub.1]) [less than or equal to] [[pi].sup.icub](S, d, [[LAMBDA].sub.2]).

For the lognormal case we impose again Assumptions 1 and 3. Then the partially exact/comonotonic upper bound is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [d.sup.*.sub.[LAMBDA]] = [d.sub.[LAMBDA]] - E[[LAMBDA]]/[[sigma].sub.[LAMBDA]] and where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be obtained from (11).

Choice of the Conditioning rv in the Multivariate Lognorma] Case

This subsection is restricted to sums S of lognormal rv's as presented in (7). The conditioning rv [LAMBDA] in the bounds presented above has to be chosen. Obviously, an "optimal" choice will provide a better bound. Different choices for a rv [LAMBDA] of the form

[LAMBDA] = [n.summation over (i=1)] [[gamma].sub.i] [Z.sub.i], (26)

in the multivariate lognormal case are discussed and compared in detail in Vanduffel, Hoedemakers, and Dhaene (2005).

Kaas, Dhaene, and Goovaerts (2000) propose the following choice for the parameters [[gamma].sub.i] when computing the lower bound [S.sup.l]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

This "Taylor-based" choice makes [LAMBDA] a linear transformation of a first-order approximation to S.

In view of (27), a possible decomposition variable [d.sub.[LAMBDA]] for which [LAMBDA] [greater than or equal to] [d.sub.[LAMBDA]] implies that S [greater than or equal to] d is given by

[d.sub.[LAMBDA]] = d - [n.summation over (i=1)] [[gammai].sub.i] (1 - ln [[gamma].sub.i] + ln [[alpha].sub.i]). (28)

Vanduffel, Hoedemakers, and Dhaene (2005) propose to use the conditioning rv [LAMBDA] defined in (26) with the coefficients [[gamma].sub.i] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

This choice can be seen as the one that maximizes a first-order approximation of the variance of [S.sup.l].

For this "maximal variance" conditioning variable a possible choice for [d.sub.[LAMBDA] is also of the form (28).

A third possibility for the choice of the conditioning rv [LAMBDA] is based on the standardized logarithm of the geometric average G = [([[PI].sup.n.sub.i=1] [X.sub.i]).sup.1/n] as in Nielsen and Sandmann (2003)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the fact that the geometric average is not greater than the arithmetic average, a possible decomposition variable so that [LAMBDA] [greater than or equal to] [d.sub.[LAMBDA]] implies that S [greater than or equal to] d is in this case given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the aforementioned decomposition variables [d.sub.[LAMBDA]] depend on the retention d and that when the retention d tends to infinity also these [d.sub.[LAMBDA] do. The error bound [epsilon]([d.sub.[LAMBDA]]) (24) will then converge to a constant so that the upper bound [[pi].sup.deub](S, d, [LAMBDA]) will not converge to zero but to this constant instead. The exact part in the partially exact/comonotonic upper bound [[pi].sup.pecub](S, d, [LAMBDA]) will become zero and hence the upper bound itself will coincide with the improved comonotonic upper bound [[pi].sup.icub](S, d, [LAMBDA]) which will tend to zero. This explains the behavior that for large retentions d the upper bounds [[pi].sup.pecub](S, d, [LAMBDA]) and [[pi].sup.icub]](S, d, [LAMBDA]) will outperform the upper bound [[pi].sup.deub](S, d, [LAMBDA]]).

The three choices above for [LAMBDA] will not lead to a good performance for the improved comonotonic upper bound because those choices will not only cause the first-order approximation of the variance of [S.sup.l] to be large but also the first-order approximation of the variance of [S.sup.u] while we just want the opposite for the latter variance. To determine a good choice for [LAMBDA] we should make the correlation structure of the components in [S.sup.u] as close as possible to the correlation structure of those in S. This is attained by putting [LAMBDA] equal to a [Z.sub.k] for some k [member of] {1, ..., n} such that the difference

[absolute value of [r.sub.i][r.sub.j] + [square root of 1 - [r.sup.2.sub.i]] [square root of 1 - [r.sup.2.sub.j]] - Corr([Z.sub.i], [Z.sub.j])]

is small for most of the i and j, where [r.sub.i] is defined in (10).

In the remainder of this article, the choice of [LAMBDA] will be dependent on the time horizon n. To indicate this dependence, we introduce the notation [[LAMBDA].sub.n] for the used conditioning variable [LAMBDA].

BOUNDS FOR COMPOUND SUMS

Bounds for the right tails (4) of a compound sum [S.sub.N] (3) are obtained by bounding the different right tails composing the sum (4). It is straightforward to obtain a lower bound, denoted as [[pi].sup.lb]([S.sub.N], d, [LAMBDA]), by looking at the combination

[[pi].sup.lb]([S.sub.N], d, [LAMBDA]) = [[infinity].summation over (j=1)]Pr(N = j) [[pi].sup.lb]([S.sub.j], d, [[LAMBDA].sub.j]),

with [LAMBDA] = [[LAMBDA].sub.1], [[LAMBDA].sub.2], ... and [[pi].sup.lb] ([S.sub.j], d, [[LAMBDA].sub.j]) given by (12) for n = j. The same reasoning can be followed for obtaining the comonotonic upper bound [[pi].sup.cub]([S.sub.N], d), the improved comonotonic upper bound [[pi].sup.icub]([S.sub.N], d, [LAMBDA]), and the partially exact/comonotonic upper bound [[pi].sup.pecub]([S.sub.N], d, [LAMBDA]).

For each term [pi]([S.sub.j], d) in the sum (4) we can take the minimum of two or more of the above defined upper bounds. We propose two upper bounds based on this simple idea.

The first bound takes each time the minimum of the error term (17) independent of the retention and the error term (22) dependent on the retention (although in practical applications one notices that the minimum is obtained by the error term (22)). Combining this with the stop-loss premium of the lower bound [S.sup.l] results in the following upper bound

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Calculating for each term the minimum of all presented upper bounds

[[pi].sup.min]([S.sub.N], d, [LAMBDA])

= [[infinity].summation over (j=1)] Pr(N = j) min ([[pi].sup.cub]([S.sub.j], d), [[pi].sup.icub]([S.sub.j], d, [[LAMBDA].sub.j]),[[pi].sup.pecub]([S.sub.j], d, [[LAMBDA].sub.j]), [[pi].sup.emub]([S.sub.j], d, [[LAMBDA].sub.j])),

will of course provide a better upper bound, which can even be improved by using different [[LAMBDA].sub.j] for deriving the different upper bounds for each term [pi]([S.sub.j], d) in the sum (4).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

NUMERICAL ILLUSTRATIONS

In this section, we apply the different lower and upper bounds presented in the previous sections to the life annuity problems discussed in the "Life Annuities and Right Tails" section. We compare the performance of these bounds in a numerical illustration.

Recall that [[??].sub.j] in (5) and [S.sub.average] in (6) are both of the form (7) with [Z.sub.i] = Y(i) = [Y.sub.1] = + ... + [Y.sub.i]. In the numerical illustrations, we assume that these yearly returns [Y.sub.i] are i.i.d, normally distributed with mean [mu] = 0.07 and volatility [sigma] = 0.1.

In order to compute the lower and upper bounds for the stop-loss premiums, we consider as conditioning rv [[LAMBDA].sub.n] = [[summation].sup.n.sub.i=1] [[gamma].sub.i] [Z.sub.i] with [[gamma].sub.i] given by (27) in the "Taylor-based" case and [[gamma].sub.i] given by (29) in the "maximal variance" case, and with corresponding decomposition variable of the form (28). For the numerical illustrations in this section we present each time the one which provides the best result.

Remark that the correlation coefficients [r.sub.i] are given by

[r.sub.i] = [[summation].sup.i.sub.j=1] [[summation].sup.n.sub.k=1] [[gamma].sub.k]/ [[square root of [[summation].sup.n.sub.j=1] ([[summation].sup.n.sub.k=1]] [[gamma].sub.k].sup.2]

As they are positive, Equations (14) and (15) can be applied.

The "maximal variance" conditioning variable (29) seems to perform better far in the tail. So for high values of d the different bounds based on this conditioning variable approximate more accurate the real value of the right tails than the approximations using the "Taylor-based" conditioning variable (27). In our numerical illustrations, the "geometric average" conditioning variable performs in general slightly worse than the two others.

Hereafter, we compare the performance of the following bounds presented in the "Bounds Based on Comonotonicity and Conditioning" and "Bounds by Conditioning Through Decomposition of the Right Tail" sections: the lower bound [[pi].sup.lb] (S, d, [LAMBDA]) (LB), the comonotonic upper bound rrCUb(g, d) (CUB), the improved comonotonic upper bound [[pi].sup.icub](S, d, [LAMBDA]) (ICUB), the upper bound based on the lower bound [[pi].sup.eub] (S, d, [LAMBDA]) (EUB) and [[pi].sup.deub(S, d, [LAMBDA]) (DEUB), and the partially exact/comonotonic upper bound [[pi].sup.pecub](S, d, [LAMBDA]) (PECUB). For applications with a stochastic time horizon N we also consider the bounds from the "Bounds for Compound Sums" section with, in particular, the two combination bounds [[pi].sup.emub]([S.sub.N], d, [LAMBDA]) (EMUB) and [[pi].sup.min]([S.sub.N], d, [LAMBDA]) (MIN).

In addition, we calculate the stop-loss premiums by Monte-Carlo simulation (MC). These MC estimates serve as a benchmark for the different lower and upper bounds. The simulation results are based on generating 50 x 1000 000 paths and antithetic variables are used to reduce the variance of the Monte-Carlo estimates. For each estimate we computed the standard error (se). As is well-known, the (asymptotic) 95 percent confidence interval is given by the estimate plus or minus 1.96 times the standard error. Such a simulation is time-consuming compared to the computation of the analytical expressions for the bounds.

For the life annuity with present value [S.sup.policy] (2), we consider a male person of age 65. The different lower and upper bounds for the right tails of the present value of a whole life annuity due of 1 payable at the end of each survival year (annuity-immediate) are compared in Table 1. For the retentions d = 5, 10, and 15 the upper bound MIN improves the comonotonic and improved comonotonic upper bounds CUB and ICUB, respectively. For larger values of d, all approximations are more or less the same. Up to a retention d=15 the upper bound EMUB performs better than the ICUB and the PECUB bounds, but from d=20 on it is the other way around as was predicted earlier when discussing the limiting behavior of these bounds. Overall, the lower bound LB derived from the methodology proposed in Kaas, Dhaene, and Goovaerts (2000) seems to be the closest approximation for the right tails or stop-loss premiums.

Table 2 shows the results for the right tails E[[([S.sup.average] - d).sub.+]] for different values of d. Again the lower bound approach as proposed in Kaas, Dhaene, and Goovaerts (2000) approximates the exact stop-loss premiums extremely well. The results for the upper bounds are in line with the previous ones. Note that for very high values of d the differences become larger.

Remark that only for EUB the error term is independent of the retention and therefore in both tables all values for d = 0, except these for EUB, are identical and equal to 9.3196. This follows from the fact that in this case the expected value of [S.sup.policy] equals the expected value of [S.sup.average]. Note also that the values in Table I are typically larger than the corresponding values in Table 2. This is not surprising. From Example 1 in Hoedemakers, Darkiewicz, and Goovaerts (2005) it immediately follows that [S.sup.average] [less than or equal to] [sub.cx] [S.sup.policy] and hence for any d > 0 one has

E[[([S.sup.average] - d).sub.+]] [less than or equal to] E[[([S.sup.policy] - d).sub.+]].

SUMMARY AND CONCLUSIONS

In this article, we consider several methodologies for approximating right tails of non-independent rv's. We compare and generalize methods proposed by Curran (1994), Rogers and Shi (1995), Kaas, Dhaene, and Goovaerts (2000), Dhaene et al. (2002a, b), Nielsen and Sandmann (2003), and Vanmaele et al. (2006). In particular, we consider bounds based on comonotonicity as proposed in Kaas, Dhaene, and Goovaerts. Further, we concentrate upon the upper bound obtained by adding an error term to the lower bound of Rogers and Shi. We explain how these bounds can be improved by decomposing an integral formula for the right tails into two parts: one can be easily solved analytically, the other part can be approximated by one of the comonotonic upper bounds. We generalize several existing bounds to the case of compound sums.

All the presented approximations are applied to the stochastic present value of the liabilities associated with a well-diversified average portfolio of life annuities as well as with a single life annuity. In the latter case it is possible to decompose the value of the stop-loss premium by conditioning and apply the best (smallest) upper bound on each of the components separately. We provide a number of numerical illustrations, which show that the decomposition significantly improves the bounds.

DOI: 10.1111/j.1539-6975.2009.01322.x

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(1) In our numerical illustrations, we will use the Belgian analytic lifetable MR for life annuity valuation, with corresponding parameters [l.sub.0] = 1 000 000, a = 1 000 266.63, s = 0.999441703848, g = 0.999733441115, and c = 1.101077536030.

Grzegorz Darkiewicz and Jan Dhaene are in the Department of Applied Economics, Katholieke Universiteit Leuven, Naamsestraat 69,3000 Leuven, Belgium. Jan Dhaene can be contacted via e-mail: Jan.Dhaene@econ.kuleuven.be. Griselda Deelstra is in the Department of Mathematics, ECARES and Solvay Business School, Universite Libre de Bruxelles, Boulevard du Triomphe 2, CP 210, 1050 Brussels, Belgium. Tom Hoedemakers is at the University Center of Statistics, W. de Croylaan 54, 3001 Heverlee, Belgium. Michele Vanmaele is in the Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium. Jan Dhaene acknowledges the financial support of the Onderzoeksfonds K.U. Leuven (GOA/02: Actuariele, financiele en statistische aspecten van afhankelijkheden in verzekeringsen financiele portefeuilles). Michele Vanmaele would like to acknowledge the financial support by the BOF-project 001104599 of the Ghent University. Griselda Deelstra acknowledges financial support of the BNB-contract FC0315D00000.
```TABLE 1
Approximations for Right Tails at Level d of [S.sup.policy]

d = 0 d = 5 d = 10 d = 15

LB 9.3196 4.6191 12,269 0.1737
MC 9.3196 4.6191 1.2304 0.1739
(se x [10.sup.5]) (8.49) (5.48) (0.51)
ICUB 9.3196 4.6238 1.3277 0.2530
CUB 9.3196 4.6244 1.3389 0.2610
EMUB 9.3196 4.6197 1.2400 0.2145
PECUB 9.3196 46,219 1.2839 0.2381
MIN 9.3196 4.6195 1.2385 0.2070

d = 20 d = 25 d = 30

LB 0.0207 0.0026 0.0004
MC 0.0216 0.0026 0.0004
(se x [10.sup.5]) (0.19) (0.01) (0.002)
ICUB 0.0454 0.0088 0.0019
CUB 0.0480 0.0095 0.0021
EMUB 0.0718 0.0545 0.0522
PECUB 0.0451 0.0088 0.0019
MIN 0.0444 0.0088 0.0019

TABLE 2
Approximations for Right Tails with Retention d of
[S.sup.average.sub.x]

d = 0 d = 5 d = 10 d = 15

LB 9.3196 4.3200 0.5533 0.0193
MC 9.3196 4.3200 0.5543 0.0197
(sex [10.sup.5]) (0.37) (0.13) (0.035)
ICUB 9.3196 4.3227 0.7076 0.0523
CUB 9.3196 4.3233 0.7217 0.0559
EUB 9.3751 4.3755 0.6090 0.0749
DEUB 9.3196 4.3202 0.5784 0.0744
PECUB 9.3196 4.3219 0.6515 0.0522
```
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