منافع و هزینه های زندگی وابسته به ذخیره و قرارداد بیمه سلامت

Premiums and benefits associated with traditional life insurance contracts are usually specified as fixed amounts in policy conditions. However, reserve-dependent surrender values and reserve-dependent expenses are common in insurance practice. The famous Cantelli theorem in life insurance ensures that under appropriate assumptions surrendering can be ignored in reserve calculations provided that the surrender payment equals the accumulated reserve. In this paper, more complex reserve-dependent payment patterns are considered, in line with insurance practice. Explicit formulas are derived for the corresponding reserve.

Multistate models provide a convenient representation for generalized life insurance contracts, including life insurance policies, disability insurance policies and permanent health insurance policies, for instance. Each state represents a particular status for the policyholder. The benefits comprised in the contract are associated to sojourns in, or transitions between states. See, e.g., Chapter 8 in Dickson et al. (2009) for an introduction. Under the Markovian assumption, Thiele’s differential equation describes the dynamics of the accumulated reserve. As it can easily be solved numerically, using Euler’s method for instance, it provides an efficient tool to perform actuarial calculations. The situation becomes nevertheless more difficult when benefits are expressed in terms of the reserves, as in the case of surrendering for instance. The famous Cantelli theorem ensures that under appropriate conditions surrendering can be ignored in the reserve calculations provided that the surrender payment equals the reserve. This is true from a prospective perspective as well as from a retrospective perspective. However, the insurer generally applies a penalty when the policyholder cancels the contract so that this result is of little practical use. In this paper, we consider reserve-dependent payment patterns and we derive explicit expressions for the reserve. Typical examples of reserve-dependent insurance benefits include: • Surrender payments, with the surrender value equal to the accumulated reserve minus a cancellation fee. • Capital management fees proportional to the reserves. • Profit participation, with surplus dividends depending on the accumulated reserve. We show that, under fairly general conditions, one can still apply Cantelli’s theorem to derive an explicit expression for the reserves provided that the structure of the benefits and premiums is appropriately modified. Several examples are discussed to illustrate the applicability of the approach proposed in the present paper. The topic investigated here has already been examined in the literature. For instance, Norberg (1991) studies general multistate life insurance products and points out to the fact that Thiele’s differential equation can also cope with payments depending on the reserves in a linear way. This author derives explicit expressions for the accumulated reserves in two particular cases: (a) a widow’s pension where the retrospective reserve is paid back to the husband in case the wife dies first, and (b) a widow’s pension with administration expenses expressed as a linear function of the reserve. The present paper expands on the ideas of Norberg (1991) and presents explicit expressions for more general contracts. Milbrodt and Helbig (1999) also mention the key role played by Thiele’s equations if benefits are reserve dependent. They discuss an annuity insurance with flexible time of retirement and death benefits and calculate accumulated reserves when surrender payments equal a fixed proportion of the accumulated reserve. Notice that the problem considered here has also been discussed in several textbooks. For instance, in the multiple decrement model, Bowers et al. (1997, Section 11.4) show that as long as the withdrawal benefit in a double decrement model whole life insurance is equal to the reserve under the associated single decrement model, premiums and reserves coincide under the single and double decrement models. The present paper revisits this problem in a more general framework. It is shown that the conclusion drawn e.g. by Bowers et al. (1997) can be generalized, provided that mild conditions are fulfilled. The remainder of this paper is organized as follows. Section 2 briefly recalls the multistate Markovian setting for describing generalized life insurance contracts. In particular, definitions for the prospective and retrospective reserves are provided. Section 3 is devoted to Cantelli’s fundamental theorem, which provides the technical argument used in Section 4 to derive the results in the case of reserve-dependent insurance payments. Section 5 discusses several examples of practical relevance before the final Section 6 concludes the paper.

In this paper, several reserve-dependent payment patterns have been considered and explicit expressions have been derived for the corresponding reserves. The key theoretical argument is that Cantelli’s theorem still applies provided that the benefit payments and the technical basis are modified appropriately. Several examples have been discussed to illustrate the wide applicability of the approach proposed in the present paper. Some extensions are possible by letting the interest rate credited to the reserve vary according to the state occupied by the policyholder. For instance, if instead of applying the same coefficient View the MathML sourcec1ij(t) to both Vi(t)Vi(t) and Vj(t)Vj(t) in (5), transition payments of the form equation(9) View the MathML sourcecij(t)=c0ij(t)−c1ij(t)Vj(t)+c2ij(t)Vi(t) Turn MathJax on are considered then it is still possible to identify the corresponding changes on the technical basis to get rid of reserve-dependent benefits. This specification can be explained as follows. In the case of a transition from ii to jj at time tt, the capital Vi(t)Vi(t) is released and a subsequent reserving of Vj(t)Vj(t) is needed. The factor View the MathML source(1−c2ij) describes the percentage of Vi(t)Vi(t) that is inherited to the insurance portfolio upon leaving the state ii. The factor View the MathML source(1−c1ij) describes the percentage of the capital Vj(t)Vj(t) that has to be raised by the insurance portfolio. Stated differently, View the MathML sourcec2ij(t) is the portion that the policyholder benefits from the released capital Vi(t)Vi(t), and View the MathML sourcec1ij(t) is the portion that the policyholder has to contribute to the subsequent reserving of Vj(t)Vj(t). Reserve-dependent sojourn payments represented as equation(10) View the MathML sourcedBi(t)=dB0i(t)+b1i(t)Vi(t−)dt, Turn MathJax on may also deserve consideration. Clearly, the specifications (9)–(10) can be handled by letting the interest rate View the MathML sourceδ¯t depend on the state occupied by the policyholder. Notice that nonlinear expressions can also be of interest instead of (5). For instance, the policy conditions may specify that the accumulated reserve is paid in the case of death, with a guaranteed minimum. This means that transition benefits of the form equation(11) View the MathML sourcecij(t)=max{c0ij(t),c1ij(t)(Vi(t)−Vj(t))} Turn MathJax on may also deserve interest. However, the specification (11) requires numerical procedures to reach a solution whereas the present paper rather focuses on explicit expressions of the solutions. Despite their theoretical interest, we must acknowledge that the formulas involving lapse rates must be considered with great care as it is extremely difficult to estimate or forecast such rates: cancelling the contract is at the discretion of the policyholder and this decision may depend on individual factors as well as macroeconomic conditions (including current market interest rates compared to the technical guaranteed one). See, e.g., Eling and Kiesenbauer (2013) or Fier and Liebenberg (2013) as well as the references therein. This is why many insurers do not allow for lapses in premium calculation. When lapses are used to reduce the premium, the business is called lapse-supported and this may lead to severe adverse financial consequences due to the systematic risk involved. Instead of reserve-dependent benefits, another popular policy condition is premium refund. In this case, premiums paid until death are refunded without interest or compounded at the technical interest rate if death occurs before maturity. In the case of policy cancellation, the surrender value may be equal to a time-varying percentage of the total premiums paid so far (this time-varying percentage mimicking the evolution of the reserve). This approach avoids much of the technicalities developed in the present paper, while being attractive and transparent for the policyholders