Pooling Functional Disability and Mortality in Long-Term Care Insurance and Care Annuities: A Matrix Approach for Multi-State Pools

Doreen Kabuche, Michael Sherris, Andrés M. Villegas and Jonathan Ziveyi

 

Abstract: Mortality risk sharing pools such as pooled annuity funds and tontines provide an attractive and effective solution for managing longevity risk. They have been widely studied in the literature. However, such arrangements are not optimal for individuals in need of long-term care (LTC) insurance. Enhancing the design of pooled annuities and tontines factoring in LTC can aid in reducing the cost of LTC insurance. This paper presents a matrix-based approach for pooling mortality risk across heterogeneous individuals classified by functional disability states and chronic illness statuses. Based on multi-state models of functional disability and health statuses, we demonstrate how individuals with different health risks can share mortality risk in a pooled annuity design. A multi-state pool is formed by pooling annuitants vulnerable to longevity and LTC risks, determining the associated actuarially fair benefits based on individuals’ health states. We provide a general structure for setting up a pooled annuity product that can be applied even for complex multi-state models. An extensive analysis is also carried out to illustrate our approach with numerical examples using US Health and Retirement Study (HRS) data. From the numerical illustrations, there is an increasing trend in the expected annuity benefits with higher upsides for individuals in poor health than those in good health, especially when systematic trends and uncertainty are considered in pricing. Smaller pool sizes and higher mortality credits among ill and disabled individuals due to higher death probabilities are the two main factors for the increased benefits in dependency.

 

Keywords: long-term care insurance, pooled annuity, multi-state models, functional disability, health status.