استراتژی های بهینه برای توقف پرتفوی قراردادهای بیمه عمر واحد مرتبط با تضمین حداقل مرگ

In this paper, we are interested in hedging strategies which allow the insurer to reduce the risk to their portfolio of unit-linked life insurance contracts with minimum death guarantee. Hedging strategies are developed in the Black and Scholes model and in the Merton jump–diffusion model. According to the new frameworks (IFRS, Solvency II and MCEV), risk premium is integrated into our valuations. We will study the optimality of hedging strategies by comparing risk indicators (Expected loss, volatility, VaR and CTE) in relation to transaction costs and costs generated by the re-hedging error. We will analyze the robustness of hedging strategies by stress-testing the effect of a sharp rise in future mortality rates and a severe depreciation in the price of the underlying asset.

The new frameworks (Accountant: IFRS/IAS, Prudential: Solvency II, and financial communication: Market Consistent Embedded Value) encourage insurance companies to adopt an economic approach when evaluating their liabilities (Thérond, 2007). On this subject, the concept of “Fair Value” is fundamental. The fair value of an asset or a liability is the amount for which two interested and informed parties would exchange this asset or this liability. Fair values are usually taken to mean arbitrage-free values, or values consistent with pricing in efficient markets. The arbitrage-free valuation of an item is one which makes it impossible to guarantee riskless profits by buying or selling the item. This leads to the concept that if two portfolios have identical cash flows, and the portfolios can be priced in an efficient market, then the two portfolios will have the same price. Otherwise, an investor could sell one portfolio, buy the other and make free money. The fair value is therefore the price that the market naturally assigns to any tradable asset. Risk-neutral valuation produces the fair value of any liability. As noted by Milliman Consultants and Actuaries (2005) the main reason for using risk-neutral or fair valuations is because they represent the objective market cost of purchasing a replicating portfolio in terms of the liability, thus ensuring that the company will have sufficient resources to meet the liability over all possible market movements. Risk-neutral valuation effectively translates the risky, market-dependent costs of the guarantee into a fixed cost item for the insurance company. Thus, using the logic of fair valuation, purchasing a replicating portfolio is essential in the evaluation of liabilities. Accordingly, in the case of unit-linked life insurance for example, Frantz et al. (2003) showed that fair valuation is only valid if the underlying hedging is actually applied.1 In such contracts, the return obtained by the policyholders on their savings is linked to some financial asset, and in this way it is the policyholder who supports the risk of the investment. The investment can be made on one asset or on a portfolio of assets, and various types of guarantees can be added to the pure unit-linked contract. In our study, we shall concentrate on the minimum death benefit guarantee. In this case, the insurer’s liability in the case of the death of the policyholder will be max(K,V)=V+[K−V]+max(K,V)=V+[K−V]+, where VV is the value of the unit-linked contract and KK is the guarantee. If V<KV<K then the insurer will pay the additional amount K−VK−V. It therefore stands that the risk related to these contracts is real. However, this risk is often underestimated by the insurance companies, which then expose themselves to massive losses connected to a market in strong decline. Frantz et al. (2003) analyzed delta hedging within the framework of Black and Scholes’ model (1973). The Black and Scholes model assumes that the return process is continuous, distributed according to a normal distribution, and that its volatility is constant during time. However, the empirical reports show that none of these assumptions are always true when applied to the markets, as shown by the works of Cont (2001). Moreover, the classic valuation of unit-linked contracts assumes a perfect mutualisation of the deaths in the insurance portfolio. It therefore follows logically that we can wonder about the effectiveness of the insurer setting up a hedging strategy in order to protect from abnormally high death rates in the portfolio in the future. Moreover, other hedging strategies exist. Hedging strategies which we can develop come primarily from the methods used for hedging derivatives. In practice, hedging a portfolio of derivatives is typically done by matching different sensitivities between the given portfolio and the hedging portfolio. As an alternative, a hedging portfolio can be chosen to minimise a measure of the hedge risk for a given time horizon. The object of this paper is to analyze the optimality of some hedging strategies being offered to the insurer to cover the risks related to unit-linked life insurance contracts with minimum death benefit guarantee. These contracts are subjected to two types of risks: financial risk and mortality risk. The financial risk is represented by the possibility of a poor evolution in the underlying asset, whereas the mortality risk results from the possibility of a strong fluctuation in the sample. In this last case, if the future mortality of the insured parties in the portfolio is stronger than foreseen, this may be due to the non-validity of the assumption of mutualisation of the deaths retained during the evaluation of the contract.

In our study, we analyzed the optimality of some of the available hedging strategies which allow the insurer to reduce the risk related to a portfolio of unit-linked life insurance contracts with minimum death guarantee. We noted that all of the hedging strategies developed generated costs which were on average higher than when hedging was not used, but that all of these strategies reduced the volatility of the future costs and the indicators of extreme risks (VaR and CTE). We were interested in 3 types of hedging strategies: the delta neutral strategies, which consist of matching the sensitivities of the hedging portfolio and the liabilities of the insurer; the strategies that minimise the variance of the hedging error; and the semi-static strategy of hedging using short-term options. In the Black–Scholes and Merton environments, we noticed that the optimal strategies were the delta neutral strategies and the variance-minimising dynamic strategy which provided the same results. (If we content ourselves with a portfolio constituted of only the underlying asset and the free-risk asset.) The static strategy reduced volatility, the VaR and the CTE, but in the BS model it can generate a maximum loss that is higher than when hedging was not used. This strategy is not particularly sensitive to an increase in transaction costs, to the periodicity of compensation for the policyholders, nor to a strong fall in the price of the underlying asset. On the other hand, in the case of an abnormally high death rate in the future, its costs and its volatility progressed in much the same way as when there was no hedging cover. Semi-static hedging is not very sensitive to an increase in transaction costs but it, along with no hedging and static hedging, is at the mercy of the risk of increased future mortality. Although the increase in the maturity of the short-term options reduces the risk for the portfolio, this strategy is not satisfactory. Indeed, semi-static hedging is based on a very strong assumption; the assumption that a continuum of options of maturity uu exists. For example, our semi-static hedging portfolio is composed of 420 options with a maturity of one year. The availability of enough short-term options on the market with the required maturity is not guaranteed. Moreover, the results provided by this strategy are no better than for the other hedging strategies. Certainly, the delta strategies are more expensive for the insurer, but they provide the best risk indicators. It is also clear that these strategies are sensitive to an increase in transaction costs, but in the case of an increase in future mortality rates, or a sharp fall in the price of the underlying asset, the impact on the risk indicators is less severe than for the other hedging strategies. Insofar as we are satisfied with a hedging portfolio made up only of the risky asset and the risk-free asset, it seems logical that the DFR strategy and the dynamic strategy by minimisation give the same results (see Gabriel and Sourlas, 2006). However, the framework of the Merton model is an incomplete market; the underlying asset is not enough to hedge the risk of the portfolio. The insurer is subjected to the risk of a jump. The introduction of a second instrument of cover allows us to counteract this insufficiency. Let us also note that the options in the semi-static hedging portfolio are exerted in their maturity and that the profit thus made is then reinvested in the acquisition of the underlying asset and the risk-free asset. An alternative to this strategy would be to reinvest this amount in the purchase of a new hedging portfolio of short-term options and to reproduce the operation. These aspects are not treated here. We will, however, return to them in a later study.