A Unified Framework for Insurance Demand and Mortality Immunization

Abstract

This article explores an individual’s optimal insurance choice and an insurer’s optimal product mix consisting of whole life insurance and deferred life annuities in a market equilibrium framework. On the demand side, the insured decides an optimal insurance choice by maximizing lifetime expected utility. On the supply side, an insurer chooses an optimal product mix by minimizing the conditional value-at-risk (CVaR) in its lines of business. By varying the loading for each insurance product, we match demand and supply of these products to clear the market. Our results suggest that market equilibria may occur when life insurance loading is relatively high and annuity loading is relatively low. This calls for attention from insurance regulators and life insurers to review insurance/annuity underwriting and pricing.

Notes

1 According to the Health and Retirement Study (HRS) core 2020 data, though only a very small percentage of households chose to annuitize the balance in their pension plans, 60% of those who chose to annuitize owned life insurance.

2 The study of life insurance demand starts from theoretical work of Yaari (1965) and Richard (1975). Yaari (1965) incorporates life insurance demand in a consumption decision. In a continuous-time framework, Richard (1975) derives optimal consumption, life insurance, and investment choices. A recent work by Zhu (2007) investigates the same topic using a one-period model. There is a large body of empirical literature in this area too. Bernheim (1991) finds that elderly households demand life insurance to offset an excessive level of Social Security benefits in order to leave a bequest. Lin and Grace (2007) provide evidence that life insurance demand should be jointly determined as part of a household’s portfolio. Both Zietz (2003) and Outreville (2014) survey empirical literature and list a variety of characteristics in determining the demand of insurance, including age, income, employment, risk aversion, marital status, family size, and bequest motives. Inkmann and Michaelides (2012) find that term life insurance demand is positively correlated to bequest motives. They then show that a life‐cycle model of life insurance, saving, and portfolio choice can generate results consistent with this finding.

3 The literature on annuity demand mainly focuses on the low participation in the annuity market. Yaari (1965) views the purchase of a life insurance policy as the selling of an annuity and he shows that, under very restrictive conditions, i.e., no bequest motives and complete markets with actuarially fair premiums, an individual should fully annuitize her wealth at all times. Davidoff et al. (2005) extend Yaari’s work with additional risk factors and more general utility functions, and find it is still optimal to annuitize all retirees’ wealth for their retirement. Despite these strong theoretical results, the demand for life annuities remains low in annuity markets, leading to the “annuity puzzle.” Two most often cited explanations are adverse selection (e.g., Mitchell et al., 1999; Finkelstein and Poterba, 2004, 2014) and bequest motives (e.g., Friedman and Warshawsky, 1990; Brown, 2001; Ameriks et al., 2011; Inkmann et al., 2011; Lockwood, 2012; Pashchenko, 2013). Other reasons include the crowding effect from social security and private defined benefits plans (Bernheim, 1991; Brown et al., 2001; Dushi and Webb, 2004), out-of-pocket health shocks (Sinclair and Smetters, 2004), and irreversibility of annuity (Milevsky and Young, 2007), etc. In addition to these arguments based on rational choice theory, there are some behavioral related explanations, such as mortality salience (Salisbury and Nenkov, 2016), mental accounting (Thaler, 1999; Benartzi et al., 2011; Knoller, 2016), price-inelasticity (Chalmers and Reuter, 2012; Boyer et al., 2020), risk perceptions and knowledge of annuities (Boyer et al., 2020), and cumulative prospect theory (Kahneman et al., 1990; Benartzi and Thaler, 1995; Barberis et al., 2001; Knoller, 2016), among others.

4 Boyle et al. (2020) exploit a life-cycle model to investigate the demand of life insurance/annuities by considering habit formation. Zhang et al. (2021) apply hyperbolic discounting and luxury bequest to seek the optimal consumption, life insurance/annuity demand. Though these two studies consider both insurance products, their models view a short position of life insurance as a long position in annuity. Therefore, they do not consider simultaneous purchase of life insurance and annuities.

5 We choose the CBD model because of its popularity in mortality modeling literature. But we do not claim it is the best model or intend to compare it with other stochastic mortality models because this is not the focus of our article and many other articles have done so. Parameter estimation of the CBD model is fairly standard. We refer interested readers to Cairns et al. (2006) for details. There is also an R package “StMoMo” that can be easily implemented for estimation of various mortality models, including the CBD model.

6 Sometimes variable annuities offer other optional features, such as guaranteed minimum income benefits or long-term care benefits. However, investors need to pay extra charges to get these benefits.

7 See https://www.annuitywatchusa.com/understanding-annuities.

8 We assume this is the only time when the individual can purchase these two insurance products. Relaxing this assumption (i.e., allowing individuals to purchase insurance products at different points of time) greatly complicates the problem and increases computational burden.

9 Our model framework of demand and supply can be readily adapted to insurers’ profit maximization or other objective functions.

10 The same approach has been used in the actuarial science literature, such as Chen et al. (2013), Zhou et al. (2011, 2013, 2015), Zheng (2015), and Jevtić et al. (2022).

11 See https://www.ibisworld.com/industry-statistics/number-of-businesses/life-insurance-annuities-united-states.

12 Our premium calculation is consistent with the market data. Based on a quote from an A + rated carrier for a preferred plus male aged 45 years, a 20-year limited pay whole life insurance policy charges an annual premium of $0.02439–0.02698 per dollar of face value. The annual premium for a preferred plus male aged 55 years ranges from $0.05574 to $0.05979 per dollar of face value. See https://www.insuranceandestates.com/limited-pay-whole-life-insurance. Note that the fair premiums in Table 1 are based on population mortality forecasts from the CBD model. Even after taking into account the mortality experience of life insurance policyholders, the actuarially fair premium does not change significantly (see Section 4.2).

13 A quote from the Charles Schwab website indicates that a man aged 45 years needs to pay a lump-sum premium of $120,269 for 20-year deferred life annuities with a monthly annuity payment of $1,000. Assuming a 3% interest rate, this is equivalent to an annual premium of $0.6629 (for 20 years) per dollar of annuity payment. Comparing this with our calculated actuarially fair premium of $0.4855, the annuity loading is about 36.5%. Note that the fair premium in Table 1 is based on population mortality forecasts from the CBD model. Annuitants normally exhibit lower mortality rates than the general population due to the lack of underwriting. Taking that into account, our calculated actuarially fair premium for annuity for a 45-year-old man is $0.5560 (see Section 4.2). The quote from the Charles Schwab website actually adds a loading of 19.2%.

14 The average income for a 45-year-old was $74,123 in 2019. See https://dqydj.com/income-percentile-by-age-calculator. The average net worth with home for the age group 45–49 years was $761,560 in 2019. See https://dqydj.com/net-worth-by-age-calculator-united-states. Thus, the annual income to wealth ratio is about 9.73%. Considering that people aged 46–49 years usually have a higher net worth than people aged 45, we assume the income to wealth ratio for age 45 is 10%.

15 We collect the average and median income data in 2021 from the Current Population Survey and find that an individual’s average (or median) income increases before age 40 and plateaus after that. This is consistent with the finding of Rupert and Zanella (2015), who document that the wages do not decline before age 65 if working hours are taken into consideration. Since our baseline case considers an individual aged 45, it is reasonable to assume that labor income does not increase with age in the context of our model. Meanwhile, we have collected the national average wage index (https://www.ssa.gov/oact/cola/AWI.html) and the annual inflation rate (https://www.macrotrends.net/countries/USA/united-states/inflation-rate-cpi) from 1985 to 2019. We find that the national average wage index, adjusted for inflation, grows about 0.92% annually. We therefore assume that the labor income for age 45 grows over time at a constant rate of 1%.

16 The data are from https://data.oecd.org/pension/net-pension-replacement-rates.htm.

17 The data are available at https://dqydj.com/net-worth-by-age-calculator-united-states/.

18 We search the optimal proportions over a larger range [0, 10] with looser grids and find the optimal proportion never exceeds 3.6 times of the initial wealth.

19 Recall that we assume the initial wealth follows a Pareto distribution with parameters θ = 1.132 and the minimum wealth level W_m = $89,024. With such a Pareto distribution, the probability of wealth exceeding $1,500,000 is only 2.67%.

20 This is a corner solution. When the annuity loading is large enough to outweigh the advantage of natural hedging by offering both life insurance and annuity products, the insurer may be able to minimize the CVaR of their losses by selling annuities exclusively.

21 The equilibrium price might not exist, and if it exists, it might not be unique (see, e.g., Arrow and Hurwicz, 1958, 1960; Arrow, Block, and Hurwicz, 1959).

22 Substandard annuities are “medically underwritten, impaired risk, or age-rated annuities, in which greater payouts are given to those buyers with shorter-than-average life expectancies” (LIMAR International, 2006).

23 For annuities and life insurance, the surrender fee often starts at 10 percent if you cash in your investment in year 1. It goes down by 1 percent per year, so there is no surrender fees after year 10. Surrender charges can apply for time periods as little as 30 days or as much as 15 years on some annuity and insurance products. See https://www.investopedia.com/terms/s/surrendercharge.asp.

24 Policyholders surrender for various reasons. The most common reason is to get extra cash to cover large unexpected shocks in labor incomes or health-related costs. Thus, the timing of surrender actually depends on when these shocks occur in an individual’s life. In this article, we abstract from income shocks or health-related shocks to stay focus on mortality/longevity risk and individuals’ choices between consumptions, risky investments, and their joint purchase of life insurance and annuity. This is why we only allow surrender at retirement so that we can better understand how the retired with different wealth levels strike a balance between their consumptions after retirement and their bequest motive. Early surrender can cause significant surrender charges; thus, it is generally not desirable under our model setting. Allowing policy surrender at any time will complicate our optimization problem and significantly increase the computational burden.

25 We conduct additional analysis by assuming that the policyholder expects health shocks around the age of retirement, which we assume will increase his mortality rates by 20% for the first three years and 10% thereafter. Our results remain relatively unchanged. Basically, only those with low wealth levels choose to surrender life insurance, and they start with more insurance in total but less annuities in their insurance portfolio when surrender options are given to them. When such health shocks are unexpected, it shall not affect their initial insurance purchase decisions, but we predict that policyholders will surrender more annuity and less life insurance due to the shortened life expectancy.

26 The taxation of annuity is based on the exclusion ratio, which is a percentage that represents the portion of an annuity payment that is excluded from gross income and, therefore, not subject to ordinary income tax. The exclusion ratio is calculated by factoring in how much is paid into the annuity, how much it has earned, and how long payments will last, which generally is the life expectancy if it is a life-based income annuity.

27 Though life insurance may be purchased with qualified plan assets, the IRS imposes strict limitations, requiring that the life insurance protection be only “incidental” to the retirement benefits provided by the plan. 

28 There is no underwriting capacity constraint in a perfectly competitive market, because we focus on the market supply, which comes from lots of insurers.

Additional information

Funding

Wei Zhu acknowledges financial support from the Beijing Municipal Social Science Foundation (grant no. 17YJC039), the Fundamental Research Funds for the Central Universities in UIBE (grant no. ZD6-02), and the Program for Distinguished Young Talents at UIBE (Grant No. 20JQ05), and technical support from the Scientific Research Cloud Project in the School of Insurance and Economics at UIBE. Jin Gao acknowledges financial support from the Direct Grant (DR19B1/101108) at Lingnan University.