مرگ و میر تصادفی در بیمه عمر: ذخایر بازار و قراردادهای بیمه مرتبط با مرگ و میر

In life insurance, actuaries have traditionally calculated premiums and reserves using a deterministic mortality intensity, which is a function of the age of the insured only. Here, we model the mortality intensity as a stochastic process. This allows us to capture two important features of the mortality intensity: Time dependency and uncertainty of the future development. The advantage of introducing a stochastic mortality intensity is twofold. Firstly, it gives more realistic premiums and reserves, and secondly, it quantifies the risk of the insurance companies associated with the underlying mortality intensity. Having introduced a stochastic mortality intensity, we study possible ways of transferring the systematic mortality risk to other parties. One possibility is to introduce mortality-linked insurance contracts. Here the premiums and/or benefits are linked to the development of the mortality intensity, thereby transferring the systematic mortality risk to the insured. Alternatively, the insurance company can transfer some or all of the systematic mortality risk to agents in the financial market by trading derivatives depending on the mortality intensity.

Traditionally, actuaries have been calculating premiums and reserves using a deterministic mortality intensity, which is a function of the age only, and a constant interest rate (representing the payoff of the investments made by the companies). However, since neither the interest rate nor the mortality intensity is deterministic, life insurance companies are essentially exposed to three types of risk when issuing contracts: Financial risk, systematic mortality risk and unsystematic mortality risk. Here, we distinguish between systematic mortality risk, referring to the future development of the underlying mortality intensity, and unsystematic mortality risk, referring to a possible adverse development of the insured portfolio. So far the life insurance companies have dealt with the financial and (systematic) mortality risks by choosing both the interest rate and the mortality intensity to the safe side, as seen from the insurers’ point of view. When the real mortality intensity and investment payoff are experienced over time, this usually leads to a surplus, which, by the so-called contribution principle, must be redistributed among the insured as bonus, see Norberg (1999). Since insurance contracts often run for 30 years or more, a mortality intensity or interest rate, which seems to be to the safe side at the beginning of the contract, might turn out not to be so. This phenomenon has in particular been observed for the interest rate during recent years, where we have experienced large drops in stock prices and low returns on bonds. However, the systematic mortality risk is of a different character than the financial risk. While the assets on the financial market are very volatile, changes in the mortality intensity seem to occur more slowly. Thus, the financial market poses an immediate problem, whereas the level of the mortality intensity poses a more long term, but also more permanent, problem. This difference could be the reason why emphasis so far has been on the financial markets. We hope to turn some of this attention towards the uncertainty associated with the mortality intensity by modelling it as a stochastic process. In order to obtain a more accurate description of the liabilities of life insurance companies, market reserves have been introduced, see Steffensen (2000) and references therein. Here, the financial uncertainty as well as the uncertainty stemming from the development of an insurance portfolio with known mortality intensity is considered. By modelling the mortality intensity as a stochastic process, market reserves can be further extended to include the uncertainty associated with the future development of the mortality intensity. This should allow for an even more accurate assessment of future liabilities, since possible trends in the mortality intensity and the market attitude towards systematic mortality risk can be taken into account. In addition, a stochastic mortality intensity allows for a quantification of the systematic mortality risk of the insurance companies. Having quantified the systematic mortality risk, we investigate how the insurance companies could manage the risk. As a first possibility, we introduce a new type of contracts called mortality-linked contracts. The basic idea is to link and currently adapt benefits (and/or premiums) to the development of the mortality intensity in general, and thereby transfer the systematic mortality risk from the insurance company to the group of insured. A second possibility is to transfer the systematic mortality risk to other parties in the financial market. Here, the idea is to introduce certain traded assets, which depend on the development of the mortality intensity. This paper is organized as follows: Section 2 contains a review of existing literature on stochastic mortality. Section 3 deals with the modelling of the mortality intensity as a stochastic process, and Section 4 introduces the model considered in the rest of this paper. An expression for the market reserve for a general payment stream is given in Section 5. In Section 6, we introduce the concept of a mortality-linked insurance contracts. Finally, Section 7 includes a discussion of how the systematic mortality risk could be transferred to other agents in the financial market.

As a way to control the mortality risk inherent in an insurance portfolio the company may purchase reinsurance cover. Reinsurance contracts usually consider the specific insurance portfolio of the company, and hence provide coverage for both systematic and unsystematic mortality risk. An example of a mortality dependent reinsurance contract sold in practice is a so-called mortality swap. Prices of reinsurance contracts concerning both systematic and unsystematic mortality risk can be found using the methods already established in Section 5. However it seems that many life insurance companies are hesitant to buy long term reinsurance coverage. One reason could be that the riskiness of the reinsurance business would leave the insurance companies with a substantial credit risk. As an alternative to reinsurance we consider securitization. Here, the company trades contracts on the financial market, which depend on the development of the mortality intensity. An important difference between reinsurance and securitization is, that mortality contracts sold on the financial market depend on the general development of the mortality intensity, and hence only offer protection for the systematic mortality risk. Introducing products contingent on the mortality intensity naturally raises questions regarding the estimation of the mortality intensity. Since, these questions are similar to those in the case of mortality-linked contracts, we refer to the discussion in Section 6. The advantages of securitization over traditional reinsurance is the possible lower cost when standardizing products and the larger capacity of the financial market. More details on securitization of mortality risk can be found in Lin and Cox (2003). For treatments of securitization of catastrophe losses, which seems to be the most developed area of securitization, see Christensen, 2000 and Cox et al., 2000 and references therein. In this section we first derive a PDE for the price process of a wide class of derivatives on the mortality intensity. Then we examine different possibilities for an insurance company, which is interested in hedging a pure endowment, and finally we investigate contracts with a risk premium. 7.1. Pricing mortality derivatives Inspired by Björk (1998, Chapter 7) we consider derivatives of the mortality intensity with a payoff of the form View the MathML sourceΦ(T,μ[x]+T,ΨT1,ΨT2), Turn MathJax on where the processes Ψ i, i =1,2, are given by View the MathML sourceΨti=∫0tqi(τ,μ[x]+τ) dτ, Turn MathJax on for positive functions q i. The notation above indicates that the derivative is payable at time T , and that it may depend on the mortality intensity at expiration time T and on the integral over (0,T ] of two different functions of the mortality intensity. This type of contract covers standard European and Asian options, and thus includes most contracts. Using the independence between the financial market and the mortality intensity, the price process can be written as View the MathML sourceπ(t,St,μ[x]+t,Ψt1,Ψt2)=p(t,T)EQ[Φ(T,μ[x]+T,ΨT1,ΨT2)|It]. Turn MathJax on Given an expression for p (t ,T ) it is thus sufficient to derive a PDE for the Q -martingale ϒ defined by View the MathML sourceϒ(t,μ[x]+t,Ψt1,Ψt2)=EQ[Φ(T,μ[x]+T,ΨT1,ΨT2)|It]. Turn MathJax on Using Itô’s formula and the product rule, we can now find the dynamics of ϒ . Since ϒ is a Q -martingale, the drift term must be 0, such that we get the following PDE on View the MathML source[0,T]×R+3: equation(7.1) View the MathML source0=∂tϒ(t,μ,ψ1,ψ2)+αμ,Q(t,μ)∂μϒ(t,μ,ψ1,ψ2)+q1(t,μ)∂ψ1ϒ(t,μ,ψ1,ψ2)+q2(t,μ)∂ψ2ϒ(t,μ,ψ1,ψ2)+12(σμ(t,μ))2∂μμϒ(t,μ,ψ1,ψ2), Turn MathJax on with boundary condition ϒ(T,μ,ψ1,ψ2)=Φ(T,μ,ψ1,ψ2).ϒ(T,μ,ψ1,ψ2)=Φ(T,μ,ψ1,ψ2). Turn MathJax on 7.2. Possible ways of hedging The fair premium for a pure endowment contract with sum insured K can be written as View the MathML sourceπ0=KEQ[e-∫0T(1+gu)μ[x]+u du]p(0,T). Turn MathJax on In the following we examine some possibilities for hedging/controlling the systematic mortality risk associated with a pure endowment on the financial market. One possibility is to buy a derivative with payout View the MathML sourceKe-∫0Tμ[x]+u du at time T . The price for such a derivative at time 0 is equation(7.2) View the MathML sourceπ(0,μ[x],Ψ0)=KEQ[e-∫0Tμ[x]+u du]p(0,T), Turn MathJax on where the process Ψ =(Ψ t)0≤t≤T is given by Ψ t=∫0tμ [x]+u, i.e. q (t ,μ [x]+t)=μ [x]+t. This derivative hedges the financial risk and systematic mortality risk and leaves the company with the unsystematic mortality risk only. From (7.2), we see that the price of the derivative is larger than the premium obtained from the insured if and only if View the MathML sourceEQ[e-∫0Tμ[x]+u du]>EQ[e-∫0T(1+gu)μ[x]+u du]. Since the companies want a premium in order to carry a risk, the above hedging possibility only becomes interesting if the price of the derivative is less than the premium paid by the insured. Often the companies are interested in carrying parts of the systematic mortality risk themselves. In this case the companies can buy a call option on the survival probability with strike C . The payoff from the call option is given by equation(7.3) Φ(T,μ[x]+T,ΨT)=(e-ΨT-C)+.Φ(T,μ[x]+T,ΨT)=(e-ΨT-C)+. Turn MathJax on Here, as in (7.2), the process Ψ =(Ψ t)0≤t≤T is defined by View the MathML sourceΨt=∫0tμ[x]+u du. The derivative with payoff (7.3) leads to a payment if the real survival probability is above some predefined level C. This leaves the insurance company with the systematic mortality risk up to a certain level. Here, the strike C could be the survival probability calculated by using some known mortality intensity, for example the market forward mortality intensity. The price process π(t,St,μ[x]+t,Ψt) for the call option can be found by solving (7.1) with boundary condition ϒ(T,μ,ψ)=Φ(T,μ,ψ) and multiplying by p(t,T). 7.3. Contracts with a risk premium Assume that the company calculates the premium of a pure endowment with sum insured K using some specified mortality intensity View the MathML source(μ[x]+u*)0≤u≤T, which satisfies View the MathML sourcee-∫0Tμ[x]+u* du>e-∫0Tfμ,Q(0,u) du. Turn MathJax on The mortality intensity View the MathML source(μ[x]+u*)0≤u≤T can be interpreted as the first order mortality intensity used in practice. Using View the MathML source(μ[x]+u*)0≤u≤T the company charges a premium View the MathML sourceπ0*, which is larger than the fair premium π 0, given by the market price under Q . This is similar to charging a risk premium. The systematic surplus generated by pricing with View the MathML source(μ[x]+u*)0≤u≤T instead of (fμ,Q(0,u))0≤u≤T must be returned to the policyholders, and this could be obtained by increasing benefits if the mortality intensity behaves as expected. For example, the company could pay equation(7.4) View the MathML sourceKT=K(1+a(e-∫0Tμ[x]+u* du-e-∫0Tμ[x]+u du)+), Turn MathJax on if the person survives. Here, a∈(0,1) is the proportion of the surplus which is paid to the policyholder. A natural restriction for contracts of the form (7.4) is that they are fair as measured by the market measure. This gives the following equation: View the MathML sourceπ0*=EQ[IT0KT e-∫0Tru du]=EQ[IT0K e-∫0Tru du]+EQ[IT0aK e-∫0Tru du(e-∫0Tμ[x]+u* du-e-∫0Tμ[x]+u du)+]=π0+aKp(0,T)EQEQ[IT0(e-∫0Tμ[x]+u* du-e-∫0Tμ[x]+u du)+|IT]=π0+aKp(0,T)EQ[e-∫0T(1+gu)μ[x]+u du(e-∫0Tμ[x]+u* du-e-∫0Tμ[x]+u du)+]. Turn MathJax on Here, View the MathML sourcep(0,T)EQ[e-∫0T(1+gu)μ[x]+u du(e-∫0Tμ[x]+u* du-e-∫0Tμ[x]+u du)+] Turn MathJax on is the price at time 0, henceforth denoted π(0,S0,μ[x],0,0), for a derivative with the following payoff at time T View the MathML sourceΦ(T,μ[x]+T,ΨT1,ΨT2)=e-ΨT1(e-∫0Tμ[x]+u* du-e-ΨT2)+, Turn MathJax on where View the MathML sourceΨt1=∫0t(1+gu)μ[x]+u du and Ψt2=∫0tμ[x]+u du. Turn MathJax on Hence, the price at time 0 can be found by solving (7.1) with the boundary condition ϒ(T,μ,ψ1,ψ2)=Φ(T,μ,ψ1,ψ2) and multiply by p(0,T). We obtain the following expression for the “fair” value of a View the MathML sourcea=π0*-π0Kπ(0,S0,μ[x],0,0). Turn MathJax on This formula can be interpreted in the following way: The benefit is increased with a number of put options on the survival probability, which corresponds to the excess premium over the fair premium divided by the price of the put option. Acknowledgements Parts of this paper originally appeared in a slightly different version in my Master’s thesis, and I would like to use the opportunity to thank my supervisor Thomas Møller for long and fruitful discussions. Also thanks to Peter Holm Nielsen and an anonymous referee for helpful comments.