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The insurance product analysed here can be understood as an LTC insurance that complements permanent disability coverage. It was very popular in the 1960s and 1970s, mainly due to the disability coverage and to the weakness of public pensions at that time. A striking result is the age of policyholders at the inception of the insurance contract, which is only 28 years, and it increases with calendar time. An explanation is that this coverage was sold in tandem with a temporary disability policy, which could be used earlier in the life cycle by the policyholders. An explanation is that the policyholders first hedged permanent disability risk in a context where the public coverage was very weak. Usually buyers of LTC insurance are middle aged or are young retirees. In France, the age of LTC insurance purchasers decreased from 65 to 61 years on average in a few years. The application of a life cycle approach given later in the paper shows however that rather young adults may rationally opt for LTC coverage.
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A life cycle analysis of the consumer is the suitable framework to analyse the timing of LTC insurance purchase. In this section, we present an extension of Yaari's model. Yaari's paper is seminal for the life cycle analysis of consumption and savings with life duration risk. It complements the fixed horizon models, where randomness is considered at the income and asset return levels. The rationale for LTC insurance purchase can also be studied in the Merton-Samuelson framework. Gupta and Li follow this approach (the planning horizon is retirement), with a stochastic model on the health capital and also departures from the expected utility model. Our model starts from Yaari's specification and includes LTC insurance bought ex ante in a model where irreversible dependency is a possible transition between a good health state and death. Adverse selection is low for young buyers of LTC insurance (for which we might even have an “advantageous selection” effect), as discussed in the above section on the pros and cons of LTC insurance purchase. Our approach, which applies on LTC insurance demand statistical results that are derived from LTC policies should not create an important selection bias. Using the statistical estimations of the preceding section, we will give a numerical example of optimal insurance purchase and savings behaviour in this context.
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Other problems arise besides the insurance purchasing choice. The first is the optimal date of purchasing. Indeed, there is an irreversibility issue: the insurance contract often cannot be repaid before entry into the LTC state. Hence, there is a timing decision that must be made as in any real options issue. Second, there is an alternative solution to ex ante insurance, which consists in buying annuities once the LTC state is reached. Several products of this type are proposed, which are termed immediate annuities, reverse mortgages, etc. The wealth effect on the willingness to buy LTC insurance is not straightforward. On the one hand, the deterrence effect of repayment risk should decrease with income and wealth. Besides, bequest motives increase with wealth and motivate the hedging of LTC risk. On the other hand, the motivation to use immediate annuities as a substitute for ex ante insurance (i.e., to self-insure against occurrence risk) increases with wealth. For instance, home equity can be sold in return for a life annuity, should an LTC spell occur.
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