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Feature Articles

Optimal Portfolio Choice with Health-Contingent Income Products: The Value of Life Care Annuities

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Pages 278-302 | Published online: 15 Jul 2022
 

Abstract

We study optimal retirement portfolio choice and welfare when individuals can choose between stocks, bonds, and either a life-contingent (life) annuity or a health-contingent (life care) annuity. We develop a life-cycle model of annuitization, consumption, and investment decisions for a single retired individual who faces stochastic capital market returns, uncertain health status, differential mortality risks, and uncertain out-of-pocket health care expenditure, including at end of life. Using the calibrated model, we find that life care annuities are more attractive than life annuities and would enhance annuity demand by 12 percentage points. Life care annuities allow individuals to reduce precautionary savings and consume more throughout retirement, with individuals willing to pay a loading of up to 21% for access to the product.

ACKNOWLEDGMENTS

We are grateful to Elena Capatina, Hans-Martin von Gaudecker, Joachim Inkmann, Fedor Iskhakov, Jeffrey Tsai, Mark Warshawsky, Anthony Webb, and Peter Zweifel for their helpful comments. We are especially grateful to Jason Brown for generously sharing his code for estimating health transitions. We also thank the participants in seminars at UNSW Australia, the 19th International Congress on Insurance: Mathematics and Economics (Liverpool, 2015), the 23rd Annual Colloquium of Superannuation Researchers (Sydney, 2015), the World Risk and Insurance Economics Congress 2015 in Munich, APRU Ageing in the Asia Pacific Research Symposium 2015 in Sydney and the CEPAR Retirement Income Modelling Workshop (Sydney, 2015), 7th Australasian Actuarial Education and Research Symposium (Gold Coast, 2015), and Netspar International Workshop 2017.

CONFLICT OF INTEREST

The authors have no conflicts of interest to report.

Discussions on this article can be submitted until January 1, 2024. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Notes

1 Throughout this article, HCE refers to the out-of-pocket component of total health care expenditure unless specified otherwise.

2 Even in countries with social insurance, there is often substantial HCE (Muir Citation2017).

3 This is because liquidity constraints in their model are less likely to bind in early retirement. Individuals are predicted to annuitize more to take advantage of the mortality credit and save significant amounts out of their annuity income to build a buffer against HCE later in life.

4 We chose to abstract from modelling the existence of the public pension or defined benefit plans for two reasons. First, the aim of the article is to study the impact on optimal annuitization of having access to life care annuities. The crowding-out impact of pre-existing public pension or defined benefit (DB) plans on optimal annuitization has been discussed thoroughly in the literature (e.g., Bernheim Citation1991; Dushi and Webb Citation2004; Inkmann, Lopes, and Michaelides Citation2011). This crowding-out impact is likely to have similar impacts on the optimal annuitization rate with both standard life annuities and life care annuities. Second, abstracting from the existence of the public pension allows the results of the article to be interpreted in different institutional settings. One can just quantify the present value of the public pension, which differs by country, and deduct that from the optimal annuitization amount from the article to infer the additional private annuities to be purchased.

5 To enable the evaluation of the optimal allocation to health- and life-contingent annuities, the products should be provided under the same conditions; that is, single premium at retirement, no surrender value, and the discount rate for pricing.

6 Peijnenburg, Nijman, and Werker (Citation2016) argued that annuitization is often a one-off decision due to institutional constraints, the fact that retirees make financial decisions very infrequently rather than annually, and the decline of cognitive ability to make financial decisions at older ages. This is consistent with Turra and Mitchell (Citation2008) and Peijnenburg, Nijman, and Werker (Citation2017). However, Pang and Warshawsky (Citation2010), Horneff, Maurer, Mitchell, and Dus (Citation2008), and Horneff et al. (Citation2009) allowed annuities to be purchased at any time during retirement.

7 The wealth floor is equivalent to the consumption floor in Ameriks et al. (Citation2011), Capatina (Citation2015), and Peijnenburg, Nijman, and Werker (Citation2017), given that individuals cannot die with negative wealth with the specification of our bequest function in Section 2.4.

8 The model extends the one developed in Robinson (Citation1996), which is widely used in the academic literature as well as by insurance companies, regulators, and government agencies. Other actuarial models for health status and/or functional disabilities are also available (see, for example, Fong, Shao, and Sherris Citation2015; Li, Shao, and Sherris Citation2017; Sherris and Wei, Citation2021). We follow J. Brown and Warshawsky (Citation2013) for the sake of consistency in the definition of health states. In this article, we inherited the estimated monthly transition rates from J. Brown and Warshawsky (Citation2013) and computed the yearly transition probabilities. The health dynamics model not only allows recovery from disability and improvement of health (e.g., from risk category 5 to 1) but also allows different rates of recovery from disability based on the individual’s health history. However, J. Brown and Warshawsky (Citation2013) set some transition probabilities to zero either because of the transition is impossible (e.g., from a state with a history of stroke to a state without the history of stroke) or because the transition is extremely unlikely to happen (e.g., transition from severe disability with no chronic condition to no disability with a history of stroke). J. Brown and Warshawsky (Citation2013) reported 2-year transition probabilities to disabled status or death, based on initial health status, age, and gender.

9 Nielsen (2016) showed that self-reported health is a stronger predictor of mortality than objective health measures.

10 We use health status rather than age, because it has been shown to be a better determinant (see, for example, Carreras, Ibern, and Inoriza Citation2018; Howdon and Rice Citation2018).

11 Because national health insurance schemes (such as Medicare in the United States) pay for a fraction of total HCE for people at age 65 or above, this implies that national health insurance schemes are modeled implicitly rather than explicitly.

12 Modeling HCE using a Lognormal distribution would require either dropping observations with zero HCE or incorporating them as small positive values. The former approach overestimates HCE, whereas the latter approach biases the parameter estimates because the clustering of small values does not fit a Lognormal distribution well.

13 In particular, a Lognormal distribution has a fatter tail at very high percentiles (e.g., beyond the 99.5th percentile) compared with the empirical HCE distribution, resulting in unrealistically high simulated HCE. However, French and Jones (Citation2004) showed that the goodness of fit in the tail is important for life-cycle models, because extremely high HCE results in a low level of wealth, and sensitivity of utility to a wealth shock is larger at a low wealth level than at a high one. One way to deal with this issue is to calibrate the two parameters of the Lognormal distribution to the mean and a large percentile (French and Jones Citation2004). However, this approach can lead both to a poor fit in the lower part of the HCE distribution and to unrealistically high simulated HCE beyond the calibrated percentile, especially in the case of end-of-life HCE. An exponential distribution for the tail of nonzero HCE avoids these problems.

14 Note that individuals in the life-cycle model do not need to know Ht+1 to make decisions. Rather, they only need to know the mortality probability in health state Ht. With exogenous health transition and HCE, one can simulate Ht+1 and then the HCE in period t when solving the life-cycle model.

15 For individuals who are alive in the following year, the three models in have similar goodness of fit. Pooling surviving and deceased individuals results in differences in goodness of fit of the same type as shown in .

16 However, the mixture model shares the tendency of the fitted Lognormal to underestimate the standard deviation.

17 In J. Brown and Warshawsky (Citation2013) and Murtaugh, Spillman, and Warshawsky (Citation2001), the LCA has a 10-year guarantee period and two top-up income streams, one for health states 5–7 and the other for health states 8–10. For simplicity, this distinction is not made here.

18 We use different interpolation approaches because the consumption–wealth relationship is relatively linear, whereas the optimal allocation to the risky stock with respect to wealth is highly non-linear.

19 We are aware of the highly subjective discount factors found in experimental studies (e.g., Coller and Williams Citation1999; Andersen et al. Citation2008)

20 Pang and Warshawsky (Citation2010) used a 3.4% discount rate for life annuities, whereas J. Brown and Warshawsky (Citation2013) used a 6% discount rate for LCAs. To make the products comparable, we use rf to discount both.

21 We choose this value because the results in Section 3 focus on males. The optimal product design for females in health state 1 is L = 1. The difference in the optimal value of L is driven by the different observed HCE distribution between genders.

22 For the goodness of fit of the estimated model, in J. Brown and Warshawsky (Citation2013) compares fitted transition probabilities with observed ones, whereas their exhibits seven moments of mortality and disability projections. We do not repeat their work here.

23 As shown in , Lognormal or fitted Lognormal models start to significantly deviate from the data after the 90th percentile.

24 It should be noted that our HCE model is not necessarily increasing HCE risk for individuals. Rather, we calibrate the parameters to best match the data. In fact, the HCE risk is lower for survivors than in a model that does not capture the cost of dying.

25 Results for females are discussed in Section 4.

26 For example, Pang and Warshawsky (Citation2010), using the HCE model of De Nardi, French, and Jones (Citation2010), found that annuity demand is higher in the presence of uncertain HCE than in its absence. The reason is that the mortality credits provided by life annuities increase with age, helping individuals hedge against HCE that increases with age as well.

27 It is also because they anticipate their lower life expectancies.

28 In the absence of HCE, an LCA is ‘riskier’ than a standard life annuity because it leads to uncertain (health-dependent) income.

29 Results for individuals with some health issues (health state 4) are similar and are available upon request.

30 Also at advanced ages, an individual consumes, as a proportion of their wealth, less in the presence of HCE than without HCE. At advanced ages, due to lower wealth levels in the absence of HCE, the consumption in absolute value in the absence of HCE is lower than the case with HCE.

31 Note that it is also lower than $170,000 required in the case of stand life annuities in the absence of HCE.

32 See Appendix B for a discussion of the investment strategy during the life cycle.

33 This is better than holding liquid wealth, being impoverished by HCE, and relying on the subsidy for the remaining life.

34 It should be noted that we fixed b(1γ)—that is the ratio between marginal utility from bequest and marginal utility from consumption in the benchmark model—when we experimented with different values of relative risk aversion.

35 When individuals are alive in the next period, the mean HCE for females in health states 8–10 is almost twice the mean HCE in health states 5–7, whereas it is only 31% higher for males.

36 Because the value function is concave, the first-order conditions are sufficient for an optimum.

37 In fact, the proportion invested in the risky asset is constant across age for cases without HCE. The fluctuation for cases with HCE is due to simulation errors, given an extra source of uncertainty. Because we model HCE using a mixture distribution that incorporates end-of-life HCE, we are not able to (numerically) evaluate expectations on HCE using the quadrature method (which was applied in earlier studies; e.g., French and Jones, Citation2004). Instead, these expectations are computed by Monte Carlo integration (see Appendix A), which leads to larger simulation errors compared with the quadrature method.

Additional information

Funding

This research was supported by the Australian Research Council Centre of Excellence in Population Ageing Research (Project Nos. CE110001029 and CE170100005) and ARC Discovery Grant DP1093842.

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