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 longterm‐
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 closedform
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.