Bequest-Embedded Annuities and Tontines

2.1 Payoff and Premium

In a bequest-embedded annuity contract, which is simply a combination of a life annuity and a life insurance contract, any policyholder continuously receives a deterministic annuity payment c(t) until death and a time-dependent deterministic death benefit A(t). Assuming the death benefits will be transferred to the heirs of deceased policyholder, these benefits can be considered as a bequest left to the policyholder’s heirs. Hence, we have decided to introduce the term “bequest-embedded annuity” to easily refer to this product below instead of mentioning its separate components “life annuity and life insurance”, which would make some explanations below unnecessarily lengthy. The payment stream of this bequest-embedded annuity can be written as

Note that c(t) describes the survival benefit of the annuity contract, and A ( T ϵ ) is the one-time death benefit which is paid directly to the heirs of the policyholder, upon the death time T ϵ . The actuarially fair premium for the bequest-embedded annuity can be obtained as

where m ϵ ( s ) : = E [ e s ϵ ] is the moment generating function of the random variable ε and r ∈ ℝ is the risk-free interest rate.

2.2 Optimal Bequest-Embedded Annuity

Consider a retiree endowed with an initial wealth v, which can be used to invest in one of the retirement products with death benefits. The retiree needs to decide for a specific bequest-embedded annuity, i.e. for a particular bundle of ( ( c ( t ) ) t ∈ [ 0 , T ϵ ] , A ( T ϵ ) ) . For this, we follow, for example, De Nardi et al. (2010), Lockwood (2012) and Bernhardt and Donnelly (2019) and assume that the retiree evaluates payments using the following discounted expected lifetime utility:

where ρ is the individual’s subjective discount rate. Recall that T ϵ is the time of death. The total utility of the retiree is given by the utility from the realized annuity over the retiree’s lifetime plus the utility of the bequest. We assume u and U are both increasing and concave utility functions. More specifically, we further assume that the retiree evaluates survival benefits through a power utility with a relative risk aversion coefficient γ ∈ ( 0 , ∞ ) , γ ≠ 1 , i.e.

and the bequest through a slightly modified power utility function

for all y > 0, where the constant b ≥ 0 measures the strength of the bequest motive relative to the desire for survival benefits. The power utility is abundantly used in both theoretical and empirical research because of its nice analytical tractability. Most importantly, the use of the power utility is also well-motivated economically, since the long-run behavior of the economy suggests that the long run risk aversion cannot strongly depend on wealth, see Campbell et al. (2002).

In the absence of bequest motives (b = 0), the optimal payoff of the annuity and the optimal Lagrangian multiplier have the following form (see Chen et al. 2019):

With a bequest motive (b > 0), the retiree solves the subsequent optimization problem at the retirement date t = 0:

In Theorem 2.1, we provide the optimal annuity payoff and death benefit for the case b > 0.

Let us make a few remarks about the optimal payoffs:

• Clearly, it holds λ A ≥ λ A NB , and the equality results when b is set to zero. This implies that c ∗ ( t ) ≤ c NB ∗ ( t ) , i.e. the bequest motive makes the policyholder pursue less survival benefits. It is a natural result given that in the absence of bequest motives all the initial wealth is annuitized and in the presence of bequests only a fraction.

• Under the assumption that the individual makes all her investment decisions at time t = 0, optimal bequest-embedded retirement products are superior to setting aside some fraction of wealth initially for bequest by the policyholder herself.^[2] This can be seen as follows: Assume that ϕ ∈ [ 0 , 1 ] is the fraction of the initial wealth set aside for the bequest and the amount ϕ v is invested in the risk-free asset at time 0. The invested amount plus interest ϕ v e r T ϵ is paid out to the heirs at the time of the policyholder’s death. The remainder ( 1 − ϕ ) v is invested in an optimal annuity (with no bequest). In other words, the death benefit function in this case is given by A ( t ) = ϕ v e r t and the annuity payoff is given by ( 1 − ϕ ) c N B ∗ ( t ) , where c N B ∗ ( t ) is the optimal annuity payoff with initial wealth v and no bequest. Note that this setting assumes a parameterized form for the death benefit and that the pair ( ( 1 − ϕ ) c N B ∗ ( t ) , A ( t ) ) is, for any ϕ (also the optimal one), one admissible solution to Problem (5) because E [ e − r T ϵ A ( T ϵ ) ] = ϕ v and thus V 0 ( ( 1 − ϕ ) c ∗ ( t ) , A ( t ) ) = v , where V ₀ denotes the initial value operator. In particular, A(t) is increasing, whereas the optimal A∗(t) can be increasing, decreasing or constant, depending on the relation between r and ρ.

• Note that the optimal payoff functions are proportional to the initial wealth v:

If ρ = r , the optimal retirement payout function c ^∗(t) is constant over time, and upon death, a constant death benefit A ∗ ( T ϵ ) will be paid out. If ρ > r , c ^∗(t) decreases in time, while the reverse holds for ρ < r , consistent with Yaari (1965). A later death leads to a smaller death benefit if the retiree’s subjective discount factor exceeds the risk free rate ( ρ > r ) . The reverse holds for ρ < r .

• Note that the optimal death benefit is a multiple of the optimal annuity payment:

where the constant b 1 / γ determines the magnitude of the death benefit relative to the annuity payment. In practice, one could thus think of the insurers asking the retirees how much of their regular annuity payments shall be paid out to the heirs as death benefit in case of death.

• As above, we use V ₀ to denote the initial value operator. The initial value of the death benefit in an annuity product is given by

and that of the retirement benefits is given by

• – λ A is increasing in b, which leads to a decrease of c ^∗(t) in b. Consequently, A ^∗(t) has to be increasing in b.

We now demonstrate some numerical examples. Throughout our numerical analyses, we assume that the survival curve p x t follows the well-known Gompertz law (see Gompertz 1825). To be precise, we assume that for any x and t ≥ 0 , the force of mortality is given by

which implies that the t-year survival probability of an x-year old is given by

where m > 0 denotes the modal age at death and β > 0 is the dispersion coefficient. Table 1 provides the base case parameters used in the majority of the subsequent numerical analyses.

Below, we provide some justifications for the above parameters:

• The pool size n is not relevant for the annuity and will be introduced in Section 3, where we introduce the retirement income tontine. For this parameter, we follow Qiao and Sherris (2013) who assume a similar pool size for their analysis of group self-annuitization schemes.

• The risk-free interest rate is chosen close to zero, based on the prevailing low interest rate environment in many developed countries. In Germany, for instance, the average risk-free interest rate in 2019 was equal to 1.1% (cf. Statista 2019).

• For the values of m and β, we follow Milevsky and Salisbury (2015).

• Concerning the longevity shock ε, we follow Chen et al. (2019) and assume that it follows a truncated normal distribution on the interval ( − ∞ , 1 ) . The parameters used for this distribution also stem from Chen et al. (2019).

• Following Bernhardt and Donnelly (2019) where a value ranging from one to seven has been used to describe the weighting parameter b, we have chosen a moderate parameter of three for our base case.

For the base case parameters provided in Table 1, we obtain an initial value of the death benefit (10) of 5.53. Consequently, the initial value of the annuity payments (11) is given by 94.47. In other words, in the setting and under the parameters we consider, an individual optimally spends 5.53% on a death benefit and 94.47% on retirement benefits. As we will see in later sections, by varying the parameters, particularly the weighting factor b of the bequest, a much higher amount of money can be used for bequests. Being concerned about leaving sufficient bequests to their heirs, the explicit incorporation of the death benefit can promote the incentive for the policyholders to purchase bequest-embedded annuities.

3.1 Payoff and Premium

While in an annuity, the insurer bears the longevity risk, in a tontine contract a pool of n ≥ 1 homogeneous policyholders shares the longevity risk.^[3] We assume the policyholders to be identical copies of each other. While the idiosyncratic mortality risk can initially be diversified by choosing a large enough pool size n, the aggregate risk cannot, as this type of risk affects the pool of policyholders as a whole. At older ages, the pool size will decrease and leave the remaining policyholders with both aggregate and idiosyncratic risk. Denoting by N ϵ ( t ) the number of policyholders alive at time t, each policyholder receives d ( t ) n / N ϵ ( t ) , where d(t) is a deterministic payout function specified at time 0. Following Milevsky and Salisbury (2015) and adding a time-dependent death benefit taking account of the bequest motive B(t), this yields the following continuous payment stream for each t > 0:

Similarly as for the annuity, we refer to this product as a “bequest-embedded tontine”. Note that this product simply consists of a tontine (as introduced in Milevsky and Salisbury 2015) and a life insurance. In contrast to the annuity payment (3), the tontine payment (12) depends substantially on the number of surviving policyholders N ϵ ( t ) . Note that, given the survival of the considered individual, the number of pool members follows a binomial distribution, that is, ( N ϵ ( t ) − 1 | ϵ , T ϵ > t ) ∼ Bin ( n − 1 , p x 1 − ϵ t ) from the insurer’s point of view. The actuarially fair premium of this contract can then be obtained as (cf. Chen et al. 2019):

If we assume that the death benefit B(t) is a deterministic function, the second term in (13) can be simplified similarly as for the annuity in (4). In total, this delivers the premium

3.2 Optimal Bequest-Embedded Tontine

First, note that the objective function can be written as

with all the parameters being identical to the annuity case. The term κ n , γ , ϵ ( p t x ) is derived in Chen et al. (2019) and given by

In the absence of bequest motives (b = 0), the optimal payoff of the tontine and the optimal Lagrangian multiplier have the following form (see also Chen et al. 2019):

If a bequest motive is present (b > 0), the policyholder with an initial endowment v solves the following optimization problem:

In Theorem 3.1, we solve Problem (15).

Let us make a few remarks about the optimal payoffs:

• Clearly, it holds λ O T ≥ λ O T NB , and the equality results when b = 0. This implies that d ∗ ( t ) ≤ d NB ∗ ( t ) , a natural result given that in the absence of bequest motives all the initial wealth is invested in the tontine and in the presence of bequest only a fraction.

• Note that the optimal payoff functions are proportional to the initial wealth v, because

• Similarly to the annuity, if ρ = r , a constant death benefit will be left, independent of when the retiree passes away, whereas ρ > r leads to a decreasing death benefit and ρ < r results in an increasing death benefit for later death.

• Note that the optimal death benefit is at each time t a multiple of the optimal tontine payoff:

where, at each time, the constant b 1 / γ ∫ - ∞ 1 1 - 1 - p x 1 - φ t n f ϵ φ d φ κ n , γ , ϵ p x t 1 / γ determines the magnitude of the death benefit relative to the tontine payoff at each time. In practice, one could thus think of the insurer asking the retirees how much of the tontine payoff shall be paid out the heirs as death benefit in case of death.

• Concerning the tontines with death benefits, the present value of the death benefit is given by

and that of the tontine payments is given by

• – λ O T is increasing in b, which has the consequence that d ^∗(t) diminishes in b. Consequently, B ^∗(t) has to be increasing in b.

Using the parameters in Table 1, we determine the initial values of the death benefit and the tontine payments. The initial value of the death benefit in the tontine (20) is 5.50, while the initial value of the tontine payments (21) is equal to 94.50. Rather similarly to the annuity, the policyholder optimally spends 5.5% on the death benefit and 94.5% on retirement benefits. Compared to the annuity, the retiree spends slightly less initial wealth on the death benefit. In the following section, we carry out a detailed comparison between the annuity and the tontine and determine, among other things, the main driving factors behind this observation.

3.3 Comparison to Bequest-Embedded Annuities

3.3.2 Safety Loadings

Note that, in an annuity, the insurer carries all the mortality risk, whereas in a tontine, the policyholders are left with (almost) all the mortality risk. Therefore, an annuity typically comes with a higher safety loading than a tontine. As pointed out in e.g. Stamos (2008), Hanewald et al. (2013), Milevsky and Salisbury (2015) and Chen et al. (2019, 2020b), tontines can deliver a higher lifetime utility level than annuities, if the safety loading for the annuities are sufficiently higher than for the tontines. In other words, a critical safety loading can be determined analytically. We examine this argument for bequest-embedded retirement products. We follow Stamos (2008), Hanewald et al. (2013) and Milevsky and Salisbury (2015) and assume that loadings are only charged for the annuity but not the tontine. Given the optimal payoffs of a bequest-embedded tontine, we can ask how the tontine payoffs (or the initial fair premium equal to v _OT ) shall change such that the tontine delivers the same expected lifetime utility as the optimal bequest-embedded annuity. This initial fair premium v _OT shall be larger than the initial fair premium for the bequest-embedded annuity, say v, as with the same initial wealth an optimal tontine leads to a lower lifetime utility level than an optimal annuity, and the lifetime utility increases in the initial wealth. The difference v O T / v − 1 gives the critical loading factor δ. Any loading for the annuity higher than this level makes the annuity worse than the tontines. In Milevsky and Salisbury (2015), a similar quantity is considered and called annuity indifference loading. Proposition 3.4 provides an analytic expression for this critical loading which makes the policyholder indifferent between the bequest-embedded tontine and the bequest-embedded annuity.

Proposition 3.4.

The critical annuity loading in the presence of bequest is given by

(22) δ = v O T v − 1  ,

where v is the initial price of the annuity and

(23) v O T = ( λ A v ( ∫ 0 ∞ e − r t ∫ − ∞ 1 ( 1 − ( 1 − p x 1 − φ t ) n ) d φ   ( e ( ρ − r ) t ∫ − ∞ 1 ( 1 − ( 1 − p x 1 − φ t ) n ) d φ κ n , γ , ϵ ( p x t ) ) − 1 γ d t + ∫ 0 ∞ e − r t ( e ( ρ − r ) t b ) − 1 γ μ x + t ∫ − ∞ 1 ( 1 − φ )   p x 1 − φ t   f ϵ ( φ )  d φ  d t ) − γ ) 1 1 − γ  .

Proof.

See Appendix A.4.□

Similarly, the loading in the absence of bequest is

δ NB = v O T NB v − 1  ,

where v O T NB is obtained from (23) by setting b = 0. Note that the Lagrangian multiplier λ A in (23) depends on b as well, and thus, is replaced by a Lagrangian multiplier λ A NB which is simply λ A for b = 0.

We provide the critical loadings in dependence of the bequest parameters b and the degree of relative risk aversion γ. In each plot, we compare the critical loading with bequests to that without bequests. In Figure 1, we start with the risk aversion. We observe that the annuity loading for the bequest-embedded tontine is for all risk aversions slightly lower than the loading without bequest, i.e. δ ≤ δ N B , with equality holding for b = 0. The bequest-embedded tontine is closer to the bequest-embedded annuity than the tontine without death benefit is to the annuity without death benefit. This is an interesting result, as it can provide a rationale to promote the tontine products, as embedding death benefits increases the similarity between tontines and annuities.

Figure 1:

Critical loading δ of the annuity in dependence of the risk aversion γ. The parameters are chosen as in Table 1.

Figure 2:

Critical safety loading of the annuity in dependence of the bequest factor b. The parameters are chosen as in Table 1.

In Figure 2, we present the critical loadings in dependence of the bequest factor b. We observe that the loading decreases if more weight is put on the utility of the death benefit. The reason for this is that the individual invests more wealth in the component in the two retirement products which is similar (the death benefit), if b increases. Thus, a higher b makes the tontine more similar to the annuity and hence, a lower loading can result. Note, however, that the changes in b are rather moderate, meaning that the critical safety loading is not affected too extremely by the bequest factor b. Finally and naturally, the loading with bequest is always dominated by the loading without bequest, i.e. δ ≤ δ N B .

4.1 Solvency Capital Requirement

We adopt the notations and references in Chen et al. (2019). Following Pitacco et al. (2009), the solvency capital requirement for our bequest-embedded retirement products is given by:

where BEL X ( t | shock ) is the value of the liabilities at time t under a longevity shock and BEL X ( t | ∗ ) is the value of the liabilities at time t under best-estimate assumptions for X ∈ { A , O T } . The amount of regulatory capital required by Solvency II is, in general, consistent with an assessment of the basic own funds via Value-at-Risk at a confidence level of 99.5% on a one-year time horizon, see Article 101 of Directive 2009/138/EC (2009). The standard approach for the longevity risk sub-module is to consider an instantaneous and permanent decrease in annual death probabilities by 20% (EIOPA 2014).

Note that, seen from time 0, solvency capital requirements at some future time t are actually random variables, as they depend on the evolution of mortality up to time t. We follow Börger (2010) and assume that mortality evolves according to best-estimate assumptions till time t in the computation of the SCR _X (t). More precisely, the probability to survive to time t is given by p x t . Following Chen et al. (2019), where the 99.5% quantile of the time-t liability value is expressed by the 99.5% quantile of the mortality shock, we thus obtain for the annuity:

where z _q is the q-quantile for any q ∈ ( 0 , 1 ) of the shock ε (to be more precise, of the truncated normal distribution given in Table 1). The best-estimate liabilities of the tontine are then given by

In the considered retirement products, death benefits are incorporated, which on some level dampens the longevity risk. According to EIOPA (2014), as long as a decrease in the mortality rate leads to an increase in the reserve required for the bequest-embedded retirement products,^[4] we can still use a longevity shock, namely permanent decreases in death probabilities, to compute the SCRs of these products. For instance, for the given parameters, this use is numerically confirmed in Figure 3 for the bequest-embedded annuity, where we determine the shocked best-estimate liability for the annuity under different deterministic shocks. We set the shock ε equal to the 90%- and 99.5%-quantile of the truncated normal distribution given in 1. We observe that a larger value of the shock, leading to higher survival probabilities, implies an increase in the solvency capital requirements.

Figure 3:

Shocked best-estimate liability of the bequest-embedded annuity and the regular annuity for different deterministic shocks. The remaining parameters are chosen as in Table 1.

Figure 4:

Solvency capital requirement of the bequest-embedded annuity (b = 3) and the regular annuity (b = 0). The remaining parameters are chosen as in Table 1.

In Figure 4, we compare the solvency capital requirements of the bequest-embedded annuity to a regular life annuity. We observe that the solvency capital requirements of the regular annuity dominates the solvency capital requirement with death benefit. This is due to the fact that the bequest-embedded annuity naturally hedges some longevity risk by the inclusion of the death benefit, which also lowers the retirement benefits paid out (as both products are purchased with the same initial wealth level).

For the tontine, a similar analysis as in Figure 3 shows that the mortality risk of the life insurance is, in fact, the dominant source of risk under the parameters in Table 1. More specifically, taking the above approach of using the longevity shock for the bequest-embedded tontine results in reserves which decrease in the longevity shock. Note that this observation is only due to the choice of the bequest factor b. By choosing b lower, it would be possible to achieve increasing reserves for the bequest-embedded tontine, making it a retirement product again. In fact, it is also possible to achieve decreasing reserves for the bequest-embedded annuity by choosing the bequest factor b large enough. We analyze this observation in more detail in the following subsection.

4.2 Risk Margin

Following EIOPA (2014), the risk margin is defined based on the solvency capital requirements in the following way:

where CoC is the cost-of-capital rate and X ∈ { A , O T } . The risk margin allows us to summarize the analyses carried out in the previous subsection using a single number. In Table 3, we provide the risk margin (25) in dependence of the bequest factor b for the bequest-embedded annuity and tontine, assuming a cost-of-capital rate of CoC = 6%, as suggested by EIOPA (2014). In the table, we incorporate the initial values of the death benefits (denoted by V ₀) as well. Note that we use different values of b for the annuity and the tontine, as the risk margin of the tontine becomes smaller than zero for choices of b close to zero. The reason for this is that the tontine leaves the policyholders with the mortality risk, whereas in an annuity, the insurer carries all the mortality risk. Therefore, in the annuity, there is more room regarding the range of the bequest factor b.

We observe that the risk margin decreases in the bequest factor b, an observation already clear from intuition. Hence, an insurer can reduce the risk margin if death benefits are included in the retirement products.

Table 3 suggests the existence of a critical bequest factor b ^∗ which perfectly diversifies the risks contained in a bequest-embedded retirement product, i.e. which sets RM _X  = 0. On some level, RM _X equal to 0 can be understood as the considered product containing neither mortality nor longevity risk, or more precisely, the two risks offsetting each other. In Table 4, we provide these critical bequest factors b∗ for the annuity and the tontine.

We make the following observations:

• For the annuity, we observe that the critical bequest factor is rapidly increasing in the risk aversion. The reason for this is equation (9): In order for the death and the retirement benefits to offset each other, under an increasing risk aversion, the bequest factor has to increase, because an increasing risk aversion lowers the death benefit compared to the retirement benefit under a constant factor b.

• For the tontine, the situation is more complex: Here, a higher risk aversion leads to a lower critical bequest factor b∗. The reason for this lies in equation (19): Note that b 1 / γ increases in γ for b ∈ ( 0 , 1 )  , which is the relevant range of values for the tontine as it does not contain much longevity risk. To compensate this increase, b∗ consequently becomes lower. Note that, due to the effects of the additional terms in (19), the decrease of b∗ in Table 4 for the tontine is less pronounced than the increase for the annuity and reverses at the risk aversion level γ = 10 .

Additionally and most obviously, we observe that the critical bequest factor of the tontine is extremely small compared to that of the annuity, as the annuity contains by far more longevity risk than a tontine for the insurer. This leaves the insurer a lot of freedom to design bequest-embedded annuities, as all the possible choices of bequest factors between 0 and the critical factor may be chosen by the policyholders and lead to some diversification effect, where the effect becomes more pronounced, the higher the bequest factor of the policyholder is. For the tontine, on the other hand, there is not much room left for the insurer, if we want to keep the bequest-embedded tontine a real retirement product. If the bequest factor becomes too large, the bequest-embedded tontine turns into a life insurance product. Of course, such a product could also be offered by insurers, but the product is no longer classified as a longevity risk carrying insurance product and the risk margin of the bequest-embedded tontine may, in fact, be larger than that of a regular tontine. As we want to focus on bequest-embedded retirement products in this article, we leave this analysis for future research.

Through the above analyses, we show that bequest-embedded products, with their natural hedge between the survival and death payments, reduce the solvency capital the insurers are required to set aside. In addition, without the inclusion of death benefits, the insurer plays purely an administrative role in a tontine product, while adding the death benefits makes the tontines real insurance products. These arguments together can incentivize the insurers to actively provide these innovative retirement products.