De-risking long-term care insurance

Abstract

In this paper, we propose a de-risking strategy model for LTC insurers facing with longevity and disability risks, by constructing hedge positions with vanilla disability swaps and options. We rely on long-term care insurance in a multiple state framework. The optimal hedge level for each de-risking strategies is computed, respectively, by minimizing the total cost of the de-risking strategy under the Conditional Value-at-Risk (CVaR) constraint on the total unfunded liabilities and minimizing the CVaR under a total cost constraint. A numerical application is performed, and the results suggest that a de-risking strategy based on disability derivatives can be a viable solution to reduce the portfolio riskiness of LTC insurers.

Appendix

In the following, we show the results of the numerical application for females (see Table 10, 11, 12, 13, 14, 15 and Fig. 8).

Table 10 LTC insurer strategies without and with hedging

Table 11 Optimal disability option strategy under different $\delta ^\mathrm{DO}$ and $p^\mathrm{DO}$. Optimization problem in Eq. 4.1: min $E[\mathrm{TC}^\mathrm{DO}]$

Table 12 Optimal disability option strategy under different $\delta ^\mathrm{DO}$ and $p^\mathrm{DO}$. Optimization problem in Eq. 4.3: min $\mathrm{CVaR}_{99.5\%}[\mathrm{TUL}^\mathrm{DO}]$

Table 13 Optimal disability option strategy under different $\pi$ and $p^\mathrm{DS}$. Optimization problem in Eq. 4.1: min $E[\mathrm{TC}^\mathrm{DS}]$

Table 14 Optimal disability option strategy under different $\pi$ and $p^\mathrm{DS}$. Optimization problem in Eq. 4.3: min $\mathrm{CVaR}_{99.5\%}[\mathrm{TUL}^\mathrm{DS}]$

Table 15 Optimal hedging strategy under different $\psi _1^{j}=\psi _2^{j}$. Other optimization parameters: $p^{j}=0.1\%$$\forall j$, $\pi =4\%$ and $\delta ^\mathrm{DO}=4\%$

Fig. 8

Distribution of the total unfunded liabilities $\mathrm{TUL}^j$ without and with hedging $j=\mathrm{DO}, \mathrm{DS}$