مقایسه های خطر قوانین حق بیمه: بهینگی و مطالعه بیمه عمر

Consider a risk Y 1(x ) depending on an observable covariate x which is the outcome of a random variable A with a known distribution, and consider a premium p (x ) of the form p(x)=EY1(x)+ηp1(x)p(x)=EY1(x)+ηp1(x). The corresponding adjustment coefficient γ is the solution of E exp{γ[Y1(A)-p(A)]}=1E exp{γ[Y1(A)-p(A)]}=1, and we characterize the rule for the loading premium p 1(·) which maximizes γ subject to the constraint Ep1(A)=1Ep1(A)=1. In a life insurance study, the optimal View the MathML sourcep1*(·) is compared to other premium principles like the expected value, the variance and the standard deviation principles as well as the practically important rules based on safe mortality rates (i.e., using the first order basis rather than the third order one). The life insurance model incorporates premium reserves, discounting, and interest return on the premium reserve but not on the free reserve. Bonus is not included either.

A main topic in expositions of life insurance mathematics Gerber, 1997 and Bowers et al., 1997 is the calculation of equivalence premiums: given an insurance treaty such as whole life (with single or gradual premium) or life annuities, one is concerned with the identities coming from equating the (discounted) stream of payments from the insured to the company to the stream going the opposite way. Once the equivalence premium is calculated, one must of course add a loading premium. However, this topic is only peripherically touched upon in the mathematical literature on life insurance. For example, all that we could find in Gerber (1997, p. 52) was the statement that ‘Net premiums are nevertheless of utmost importance in insurance practice. Moreover, they are usually calculated on conservative assumptions about future interest and mortality, thus creating an implicit safety loading’. (Here ‘conservative’ should be interpreted as in favor of the company, say overestimating the mortality rates in the case of whole life insurance with a single premium and underestimating them in the case of life annuities; see, e.g., Deis et al. (1993) for a discussion of the practical implementation by the Danish life insurance companies.) This rudimentary treatment of the loading premium contrasts the literature on non-life insurance mathematics. Here any text (e.g., Sundt, 1993) inevitably introduces premium principles based upon say expected value, variance, standard deviation or utility, and go into a discussion of their merits from the point of view of both the insurer and the insured. The present paper has as its aims to formalize a criterion for comparing premium rules in general insurance and go in more depth with the comparison in the special case of life insurance. The comparison is in terms of risk, more precisely the adjustment coefficient; this is a traditional choice in non-life insurance, but in life insurance it is not obvious that an adjustment coefficient exists and we will return to this point later on. We will consider risks Y 1(x ) depending on a background covariate x . In the portfolio, x is the outcome of an r.v. A (for any individual risk, x is observable, as opposed to say the standard setting of credibility theory). In life insurance, x is typically the age of the insured when the contract is signed, whereas in say fire insurance, x could be the (floor) space of the buildings insured. The equivalence premium is p0(x)=EY1(x)p0(x)=EY1(x) and we write Y 0(x )=Y 1(x )−p 0(x ). We write the total premium charged as p (x )=p 0(x )+ηp 1(x ) where η is a fixed loading constant and p 1(·) some arbitrary function. In order to be able to compare different premium rules, we assume a fixed loading: equation(1.1) Ep1(A)=1Ep1(A)=1 Turn MathJax on (the safety loading is then δ=η/Ep0(A)δ=η/Ep0(A)). For example, p1(x) is given by View the MathML sourcep0(x)Ep0(A), σ2(x)Eσ2(A), σ(x)Eσ(A) Turn MathJax on for the expected value principle, the variance principle and the standard deviation principle, respectively, where σ2(x)= Var(Y1(x))=Var(Y0(x)). This list is not exhaustive; further principles are, e.g., the percentile principle and the utility principle which we do not discuss here. Write Y(x)=Y1(x)−p(x)=Y0(x)−ηp1(x) and Y=Y(p)=Y(A) where A is independent of Y(x). Then Y is the total surplus of a typical insurance policy (note that the sign is chosen such that Y>0 means a loss for the company and Y<0 a gain). The problem is to choose p(·) or, equivalently, p1(·) so as to minimize the risk of the company subject to the constraint (1.1). Here risk minimization needs to be defined in some appropriate sense, and we consider two ways, variance minimization and adjustment coefficient maximization. We discuss this in more detail in Section 2; for the moment, it will suffice to be aware that the objective of adjustment coefficient maximization is to choose p(·) to maximize the solution γ=γ(p) of equation(1.2) E eγY(p)=1.E eγY(p)=1. Turn MathJax on One first motivation for this comes from a discrete time random walk model: equation(1.3) Rn=u-V1-⋯-VnRn=u-V1-⋯-Vn Turn MathJax on for the reserve where u is the initial reserve and the Vk are i.i.d., such that Vi conditional upon Ai=x has the same distribution as Y(x), where A1,A2,… are i.i.d. distributed as A. Under the appropriate conditions on existence of exponential moments (which are tacitly assumed throughout the paper), the ruin probability ψ(u) for this model is indeed asymptotically of the form equation(1.4) ψ(u)∼C e-γu,ψ(u)∼C e-γu, Turn MathJax on where γ is the solution of (1.2). This model is certainly a very crude approximation since it ignores the complicated issue of delayed claims settlements as in life insurance. However, we will see later that a similar exponential aymptotics also holds in this case. The constant γ is commonly denoted the adjustment coefficient (often denoted by R rather than γ as here) and is the simplest single measure of risk of the company. E.g., when comparing different reinsurance arrangements, one could look for the one maximizing the adjustment coefficient for a given safety loading (see further Section 2). We now state our main general result. Its interpretation is that the optimal premium rule p*(·)p*(·) spreads the risk evenly among the risk groups (each represented by a fixed covariate x), in the sense that all risk groups have the same adjustment coefficient when considered separately. Let F denote the distribution of A (no specific assumptions are made on the set in which A takes its values) and define View the MathML sourceω0(x)(α)=log E eαY0(x), ω0(α)=∫ω0(x)(α)F(dx). Turn MathJax on Theorem 1.1. Assume that equation(1.5) View the MathML sourceω0(α)α↑∞ as α↑α― for some α―≤∞. Turn MathJax on Then there exists an a.s. unique premium rule View the MathML sourcep*(·)=p0(·)+p1*(·)such that (1.1)holds and the solution γ =γ (x ) of equation(1.6) E eγY(x)=1E eγY(x)=1 Turn MathJax on is independent of x . If γ*γ*denotes the common value , then the solution γ of (1.2)satisfies γ≤γ*γ≤γ*for any other p (·), with strict equality unless p(A)=p*(A)p(A)=p*(A)a.s. (The assumption (1.5) holds typically if the Y 1(x ) have unbounded support, but we will also give a complete result without it, covering the cases of interest in life insurance. See further Section 2, where we also give a constructive description of how to find γ*γ* and p*(x)p*(x).) The proof of Theorem 1.1 is given in Section 2 together with some related discussion. The rest of the paper is then devoted to life insurance. A main problem is how at all to define the adjustment coefficient. The random walk model (1.3) as well as the standard Cramér–Lundberg model based upon Poisson arrivals of claims are typically used in non-life insurance like fire insurance or automobile insurance. In life insurance, risk calculations based upon the adjustment coefficient have hardly been carried out at all. The lack of such discussion is probably due in part to the many features of life insurance which do not fit into the Cramér–Lundberg setting. A crucial one is the role of interest on the reserve and discounting, which when added to the Cramér–Lundberg model leads to non-exponential asymptotics of ψ(u) (cf. Asmussen, 2000, Theorem 3.1, Chapter VII). However, also the fact that life insurance policies run over a number of years and the role of the premium reserve (as opposed to the free reserve) and bonus payments are intrinsically new features. In the present paper, we neglect bonus payments for the sake of simplicity Ramlau-Hansen, 1991 and Norberg, 1999. We treat the premium reserve (the discounted present value of future benefits to the insured) and the free reserve (the remaining part of the reserve) as genuinely separated. The premium reserve is assumed to be deposited on a bank account, and carry interest returns. However, the free reserve carries no such interest return (e.g., returns due to interest or investment of the reserve could be assumed to be paid out as dividends or used in other branches of the company’s business). This is our really crucial assumption and, whereas of course debatable from a practical point of view, it seems indispensable in order to obtain exponential asymptotics of ψ(u). If such exponential asymptotics fails, the concept of the adjustment coefficient does not make sense and risk measures have to be defined in a different and typically more complicated way. A prototype of situations covered by our life insurance model (as specified in more detail in Section 3) and used in our numerical examples is a company issuing a single type of life insurance policies like whole life insurance. Policies are signed at the epochs of a Poisson process, and the ages of the insured at these times are i.i.d. replicates of an r.v. A . An insured of age x (we consider one sex only for the sake of simplicity) pays a premium p (x ) when the policy is signed (we also treat level annual premiums in other examples) and receives a payment of unit size at the time T (x ) of death. The single premium payment is calculated as the equivalence premium p0(x)=EvT(x)p0(x)=EvT(x) where v is a discounting factor plus a loading premium ηp 1(x ). At time t after the policy is signed, the part of the company’s premium reserve coming from this particular insured is E[vT(x)-t|T(x)>t]E[vT(x)-t|T(x)>t] if the insured is still alive and 0 otherwise. In particular, p0(x) is set aside to the premium reserve when the policy is signed. In the practical implementation, we follow the tradition of much of life insurance mathematics literature and use a model where time is discretized. The setup incorporates also other types of policies than whole life, like level premium payments and life annuities. In all situations, we have a number of payment streams as long as the policy runs (often as long as the insured lives): premium payments from the insured to the company; benefit payments from the company to the insured; transfers from the company to the premium reserve; transfers from the premium reserve to the company when the insured dies; the year by year adjustments between company and premium reserve (which could go either way) to be performed each year the insured survives. What we refer to here as the company is roughly identical to the free reserve. The existence of the adjustment coefficient γ in such a model is established in Section 4 as a consequence of a general result of Glynn and Whitt (1994) Duffield and O’Connel, 1995, Bremaud, 2000, Bremaud, 2000 and Asmussen, 2000 relying on large deviations theory, more precisely the Gärtner–Ellis theorem. Given this body of theory, the proof is not hard and applies, as noted in Bremaud (2000), to many situations with delayed settlement of claims. Such models are in general applied probability terms of shot noise type (Butler et al., 1999), and are more specifically close to M /G /∞ queues (Makowski and Parulekar, 1997). The conclusion provided by the large deviations approach is that log ψ(u)/u→-γlog ψ(u)/u→-γ, u→∞, for some γ>0. Though less precise than the asymptotics which is available for the Cramér–Lundberg model, this is sufficient to motivate taking this constant γ as definition of the adjustment coefficient and interpreting it as measure of risk in the same way as in the random walk or Cramér–Lundberg setting. The rest of the paper is devoted to numerical illustrations and discussion of premium principles. In Section 3 the general model is defined. 5 and 5.1 describe how the adjustment coefficient is calculated for some typical premium principles and the numerical findings are presented in Section 5.2. Finally some concluding remarks and additional references are given in Section 6.

Our numerical examples indicate that for loading factors of realistic magnitude, say 25%, the adjustment coefficient and hence the risk depends very little on which premium principle is used. Thus, even if we have theoretically demonstrated that the commonly used safe rates principle is not optimal in the sense of risk minimization, our investigations do not produce any strong motivation to abandon it in favor of the optimal principle in Theorem 1.1 and Theorem 2.1. The difference between different premium principles becomes more marked as the loading increases, but even so, the safe rates principle is doing remarkably well. Both of these conclusions may, however, be of more theoretical interest than practical value because of the role of bonus in real life contracts which shifts the emphasis to other issues than risk minimization. See again Ramlau-Hansen (1991) and Norberg (1999). 2. From the values we have obtained for γ, it seems less risky to finance policies by level annual benefit premiums as opposed to a single premium payment at policy issue, and whole life insurance seems less risky than life annuities. These observations should be compared to the following conclusion in Gerber (1979, p. 68): “… The variance of L increases with the net premium reserve. Thus financing by a net single premium leads to a greater variance than financing by net annual premiums.” Thus, adjustment coefficient maximization and variance minimization seem to lead to the same conclusion (disregarding the choice of premium principle). 3. In Example 2.1 with exponentially distributed total surplus, the variance principle is close to optimal when adjustment coefficient maximization is used as the measure for risk minimization, but our numerical findings for the more realistic policies in Section 5 suggest a (small) difference in favor of the standard deviation principle. One implication of this is that the choice of premium principle for adjustment coefficient minimization depends crucially on the distribution of the risk so further theoretical as well as numerical studies would be interesting. 4. As mentioned in the Introduction, the literature on risk calculations in life insurance is scarce. Some relevant references are Papatriandafylou and Waters (1984) who use martingale models and Doob’s inequality to derive upper bounds of ψ(u), Martin-Löf (1986) who suggests a theory of life insurance where fluctuations in the benefit payments (dividends—to compensate for varying interest rates) are included in the models and Beard et al. (1984) who describe simulation as a method in risk theory in general and its use to assess the effect of various factors like the interest rate on the risk. However, none of these references are really close to the present paper.