Individual Life Model Research Articles

On the computation of the aggregate claims distribution in the individual life model with bivariate dependencies

In this paper, we focus on the computation of the aggregate claims distribution in the individual life model when the portfolio is composed of independent pairs of dependent risks. We prove that the bivariate probabilities associated to each couple under any intermediate positive (negative) dependency hypothesis about their mortality can always be written as a convex linear combination between the independent and the comonotonic (mutually exclusive) ones. Considering this structure for the bivariate probabilities, we then obtain two recursive schemes for computing the distribution of the aggregate claims of the portfolio. These recursions greatly facilitate the computation of the aggregate claims distribution of a life insurance (sub)portfolio exclusively composed of dependent couples, and ease the interpretation of the impact of the dependence on the associated stop-loss premiums. Numerical results are given to demonstrate the applicability and efficiency of the method.

The safest dependence structure among risks

In this paper we investigate the dependence in Fréchet spaces containing mutually exclusive risks. It is shown that, under some reasonable assumptions, the safest dependence structure, in the sense of the minimal stop-loss premiums for the aggregate claims involved, is obtained with the Fréchet lower bound and precisely corresponds to the mutually exclusive risks of the Fréchet space. In that respect, the present paper complements some previous studies by Heilmann (1986) [On the impact of independence of risks on stop-loss premiums. Insurance: Mathematics and Economics 5, 197–199], Dhaene and Goovaerts (1996) [Dependency of risks and stop-loss order. ASTIN Bulletin 26, 201–212], Dhaene and Goovaerts (1997) [On the dependency of risks in the individual life model, Insurance: Mathematics and Economics 19, 243–253], Müller (1997) [Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics 21, 219–224], Taizhong and Zhiqiang (1999) [On the dependence of risks and the stop-loss premiums. Insurance: Mathematics and Economics 24, 323–332]. A couple of actuarial applications enhance the interest of the results derived here.

On dependence of risks and stop-loss premiums

In this paper, we investigate the notion of multivariate dependence between individuals and its effect on the related stop-loss premiums. First, we consider the type of negative dependence between individuals in a portfolio that gives rise to the safest aggregate claims where the portfolio consists of m life insurance polices with each policy having a positive face amount during a certain reference period. We then apply the superadditive dependence ordering cursorily studied in the literature to compare the riskiness of portfolios. Part of the results in this paper are generalizations of the results in [Dhaene, J., Goovaerts, M.J., 1996. Dependency of risks and stop-loss order. ASTIN Bulletin 26, 201–212; Dhaene, J., Goovaerts, M.J., 1997. On the dependency of risks in the individual life model. Insurance: Mathematics and Economics 19, 243–253].

On the dependency of risks in the individual life model

The paper considers several types of dependencies between the different risks of a life insurance portfolio. Each policy is assumed to have a positive face amount (or an amount at risk) during a certain reference period. The amount is due if the policy holder dies during the reference period. First, we will look for the type of dependency between individuals that gives rise to the riskiest aggregate claims in the sense that it leads to the largest stop-loss premiums. Further, this result is used to derive results for weaker forms of dependency, where the only non-independent risks of the portfolio are the risks of couples.

Exact calculation of the aggregate claims distribution in the individual life model by use of an n-layer model.

A recursive procedure is suggested for calculating the aggregate claims distribution (stop-loss premium) in the individual life model. The method which is based on the well-known de Pril algorithm results in both a considerably reduction of the number of arithmetic operations to be carried out and the number of data to be kept at each step of iteration. The problem of underflow/ overflow which may arise in case of a large number of policies is avoided by iterating in different layers and by suitably defining the transitions between adjacent layers. Thus the algorithm can be applied to a portfolio with an arbitrary number of policies.

Exact calculation of the aggregate claims distribution in the individual life model by use of an n-layer model : 074022 (M11) Waldmann K.-H., Blätter der Deutschen Gesellschaft für Versicherungsmathematik, Vol. 22, nr. 2, 1995, pp. 279–287

Exact calculation of the aggregate claims distribution in the individual life model by use of an n-layer model : 074022 (M11) Waldmann K.-H., Blätter der Deutschen Gesellschaft für Versicherungsmathematik, Vol. 22, nr. 2, 1995, pp. 279–287

Exact calculation of the aggregate claims distribution in the individual life model by use of an n-layer model

A recursive procedure is suggested for calculating the aggregate claims distribution (stop-loss premium) in the individual life model. The method which is based on the well-known de Pril algorithm results in both a considerably reduction of the number of arithmetic operations to be carried out and the number of data to be kept at each step of iteration. The problem of underflow/ overflow which may arise in case of a large number of policies is avoided by iterating in different layers and by suitably defining the transitions between adjacent layers. Thus the algorithm can be applied to a portfolio with an arbitrary number of policies.

072025 (M11, M20, B11) On the exact calculation of the aggregate claims distribution in the individual life model : Waldmann K.-H., Universität Karlsruhe, Astin Bulletin, Vol. 24, No. 1, 1994, pp. 89–96

072025 (M11, M20, B11) On the exact calculation of the aggregate claims distribution in the individual life model : Waldmann K.-H., Universität Karlsruhe, Astin Bulletin, Vol. 24, No. 1, 1994, pp. 89–96

Recursions for the individual model

Recently, Waldmann considered an algorithm to compute the aggregate claims distribution in the individual life model which is an efficient reformulation of the original exact algorithm of De Pril. In this paper we will show that in practice the approximations as proposed by De Pril are still more efficient than the exact algorithm of Waldmann both in terms of the number of computations required and of the memory occupied by intermediate results. Furthermore we will generalize the algorithm of Waldmann to arbitrary claim amount distributions. We will compare this algorithm with respect to efficiency with the algorithms that were derived by De Pril for this model. It turns out that the approximations of De Pril are most efficient for practical computations.

On the Exact Calculation of the Aggregate Claims Distribution in the Individual Life Model

AbstractAn iteration scheme is derived for calculating the aggregate claims distribution in the individual life model. The (exact) procedure is an efficient reformulation of De Pril's (1986) algorithm, considerably reducing both the number of arithmetic operations to be carried out and the number of data to be kept at each step of iteration. Scaling functions are used to stabilize the algorithm in case of a portfolio with a large number of policies. Some numerical results are displayed to demonstrate the efficiency of the method.

Pseudo Compound Poisson Distributions in Risk Theory

AbstractUsing Laplace transforms and the notion of a pseudo compound Poisson distribution, some risk theoretical results are revisited. A well-known theorem by Feller (1968) and Van Harn (1978) on infinitely divisible distributions is generalized. The result may be used for the efficient evaluation of convolutions for some distributions. In the particular arithmetic case, alternate formulae to those recently proposed by De Pril (1985) are derived and shown more adequate in some cases. The individual model of risk theory is shown to be pseudo compound Poisson. It is thus computable using numerical tools from the theory of integral equations in the continuous case, a formula of Panjer type or the Fast Fourier transform in the arithmetic case. In particular our results contain some of De Pril's (1986/89) recursive formulae for the individual life model with one and multiple causes of decrement. As practical illustration of the continuous case we construct a new two-parametric family of claim size density functions whose corresponding compound Poisson distributions are analytical finite sum expressions. Analytical expressions for the finite and infinite time ruin probabilities are also derived.

The Aggregate Claims Distribution in the Individual Model with Arbitrary Positive Claims

AbstractIn an earlier paper the author derived a recursion formula which permits the exact computation of the aggregate claims distribution in the individual life model. To save computing time he also proposed an approximative procedure based on the exact recursion.In the present contribution the exact recursion formula and the related approximations are generalized to the individual risk theory model with arbitrary positive claims. Error bounds for the approximations are given and it is shown that they are smaller than those of the Kornya-type approximations.

Improved approximations for the aggregate claims distribution of a life insurance portfolio

Abstract In an earlier paper the author derived a recursion formula which permits the exact computation of the aggregrate claims distribution in the individual life model. This exact procedure requires of course more computing time than approximative methods such as Kornya's algorithm, which seemed to be the best compromise between accuracy and computational effort. In the present paper it is shown that, to save time, the exact formula can be used in an approximative way and that the corresponding error bound is smaller than the one of the Kornya-type approximations.

On the Exact Computation of the Aggregate Claims Distribution in the Individual Life Model

AbstractA recursive expression is derived for computing exactly the distribution of aggregate claims of a portfolio of life insurance policies. The recursion generalizes a formula of White and Greville for the claim numbers distribution and improves Kornya's approximation method for the aggregate claims distribution. It can be seen as the counterpart in the individual model of Panjer's recursion formula for the collective model.