Optimal consumption, portfolio, and long‐term‐care health insurance in a dynamic framework

We study the optimal dynamic strategy of representative agents who can invest in the financial market and sign an insurance contract to optimise the utility of intertemporal consumption and face the risk of long‐term‐care (LTC) expenses. The time horizon of the agent coincides with the stochastic death time, and the health expenditure risk takes the form of a jump Poisson process. The agent may hedge against this health risk by signing an insurance contract, on which we assume there exists a mark‐up. We find a closed‐form solution for the optimal consumption, the optimal portfolio, and the optimal insurance hedge. We show that the decision to purchase LTC insurance is more complex than what emerges from most insurance models. The proportion of LTC expenditure insured decreases with age. Our model predicts substitution between private coverage and savings as a means to finance LTC expenditure. In response to a health shock requiring an increase in LTC expenditure, the individuals sell their assets to keep up the level of consumption (the so‐called “consumption smoothing” effect). Richer individuals dissave more than poorer ones. An increase in the interest rate has the same qualitative impact. The reduction in the mark‐up, either due to increasing competitiveness or through public subsidies, is likely to increase the welfare of well‐off/fit individuals, while an increase in the interest rate may reduce coverage in a very substantial way, an aspect that has been overlooked by the literature so far.