چارچوب نامشخص شدت مرگ و میر: قیمت گذاری و توقف واحد مرتبط قراردادهای بیمه عمر

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.

Mortality is a major risk factor for life insurance companies and pension funds that needs to be modeled properly. In recent years, it has been widely accepted that mortality changes over time in an unpredictable way and stochastic models have been developed to adequately capture the systematic mortality risk. For stochastic models valuing of mortality-linked liabilities and determining the required market reserves, see for instance Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Dahl and Møller (2006), and Young (2008). Stochastic models with an emphasis on securitizing mortality risk by introducing survivor bonds as hedging instruments are discussed by e.g., Blake et al. (2006) and Cairns et al. (2006). Each mortality model is a possible description of the mortality risk. Melnikov and Romaniuk (2006) show that different mortality models perform differently in the risk management of a unit-linked pure endowment contract and warn us to be careful when choosing one mortality model against another. In this paper we provide a framework for assessing the mortality model risk embedded in unit-linked life insurance contracts arising from different specifications for the mortality intensity. Unit-linked life insurance contracts are popular and widely used on the insurance market.1 They provide either death benefit or maturity benefit or both. The benefits are linked to an underlying asset with or without certain guarantees so that the policyholders have the opportunity to participate in the financial market and (eventually) be protected from the downside development of the financial market. Many unit-linked life insurance contracts also embed options in them, e.g., the surrender option allowing the policyholders to terminate the contracts prematurely and the guaranteed annuity option giving the policyholders the right to convert a lump sum payment at the maturity into annuities at a predetermined rate. Depending on the payoff structures of the contracts, the effect of the mortality model risk may also be different. By investigating the effect of the mortality model risk we are able to know whether its importance is under or over-emphasized for different contract types. In our paper instead of inputting different mortality models into the same pricing and hedging problem and comparing their performances as Melnikov and Romaniuk (2006), we set up a more flexible framework saying that we do not know the exact process of the mortality intensity but are able to figure out its upper and lower bound under the statistical measure. Further, we restrict the set of equivalent martingale measures such that the same bounds apply to the mortality intensity under these measures. This setup allows us to study various contract types more efficiently and we call it the uncertain mortality intensity framework; see Avellaneda et al. (1995) for a related framework for pricing stock options when the volatility process is unknown but bounded. Within our framework we do not intend to find the fair value of a contract but its price bounds. The price bounds are solutions to the partial differential equations associated to a stochastic control problem. The upper price bound is found by choosing the worst-case mortality intensity at any time during the life time of the contract so that the contract value is maximized. Whereas the lower price bound is found by setting the mortality intensity to the best-case value in the sense that the contract value would be minimized. The effect of our approach is quite similar to that of the practice in traditional life insurance like pure endowment insurance and term insurance. An insurance company usually puts itself on the safe side by adjusting the premium by a loading factor defined as a percentage markup from the actuarially fair value of insurance. This is equivalent to assuming lower mortality intensity for pure endowment insurance and higher mortality intensity for term insurance. However, since our approach chooses the worst (or best) possible mortality intensity dynamically, we are able to deal with more complex contract structures where the safest mortality intensity at any time also depends on the price of the underlying asset. As a result, the higher the difference between the upper and the lower price bounds, the greater impact would the mortality model risk have on the contracts considered. In this way we are able to identify whether model risk is potentially deteriorating the fair evaluation of the contracts. Further we examine hedging strategies induced by the price bounds. The unsystematic mortality risk is diversified by pooling a large enough number of policyholders together as usually is the case. However, the systematic mortality risk, that is here the random fluctuations of the mortality intensity, can in general not be diversified away by using the pooling rationale. Instead of applying risk-minimizing or mean–variance hedging strategies to minimize either hedging costs or hedging error (see Dahl and Møller (2006) and Young (2008)) we suggest using hedging strategies induced by the upper and lower price bounds. By construction, these strategies produce a superhedge and subhedge, respectively, on average for an increasing number of policyholders. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters pricing and hedging is fairly robust with respect to misspecification of the mortality intensity, with at most a mispricing of 4% for single premium contracts and at most 2% for periodic premium payment. We conclude that model risk resulting from the uncertain mortality intensity is of minor importance. The structure of the paper is as follows. In Section 2 we describe both the financial market and the insurance market. In Section 3 we formalize the uncertain mortality intensity framework. Based on the model setup, we introduce in Section 4 the optimal control rule of the mortality intensity within its upper and lower bounds so that the price bounds are found. This enables us to build in mean superhedging strategies which are discussed in Section 5. Section 6 illustrates the theoretical results by providing a numerical analysis for different types of unit-linked life insurance contracts. Section 7 concludes.

We have investigated the influence of mortality model risk on unit-linked life insurance contracts. This investigation is undertaken within an uncertain mortality intensity framework where we assume reasonable bounds for the unknown mortality intensity. The magnitude of the mortality model risk can be easily identified by carrying out a stochastic control analysis and establishing upper and lower price bounds of unit-linked life insurance contracts; see Theorem 1. The hedging strategy induced by the upper (lower) price bound produces a superhedge (subhedge) under the statistical measure when pooling together an increasing number of similar contracts; see Corollary 1 of Theorem 2 and Remark 8. The unsystematic mortality risk is diversified away by the pooling rationale. The systematic mortality risk is addressed by dynamically assuming the worst (best) case for the stochastic mortality intensity within the given bounds. If the worst (best) case scenario does not occur then the hedging strategy generates a positive (negative) cash flow. We show that when the risk profiles of the death benefit and the survival benefit are not significantly different, the effect of the mortality model risk may not be very large indeed. The contract prices in our examples have little sensitivity with respect to changes in the mortality intensity. For the single premium version the overall contract price differences were well below 4%. In the periodic premium case the deviation from the fair price was in the same range, and was not exceeding a six month premium income. In this case, other risk sources such as interest rate risk and equity risk deserve more attention than mortality model risk. Our framework can be extended in many useful directions. The setup can be directly extended to include an American feature where the policyholder has the right to quit the contract for a pre-specified payoff, the surrender guarantee. Further, other risk factors such as interest rate risk and other facets of equity risk such as volatility risk can be included in the setup. The so extended framework can then be analyzed for model risk by similar methods as used here.