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Publicly Available Published by De Gruyter March 4, 2021

Bequest-Embedded Annuities and Tontines

An Chen and Manuel Rach

Abstract

In the present paper, we propose death benefits be directly embedded in retirement products to promote their attractiveness, where annuities and tontines are used as representative retirement products. As we assume that the death benefits will be passed on to the heirs of policyholders, these benefits can be considered as bequests. We find that both from the policyholders’ and insurers’ perspective, embedding bequests directly in the retirement products can increase the attractiveness of the retirement products and promote their sale. From the insurers’ perspective, without including death benefits, the insurers play purely an administrative role in a tontine product, while adding the bequest benefits makes the tontines real insurance products and incentivizes the insurers to actively provide these innovative retirement products. Further, adding the bequest benefits upon the death of the policyholder, some natural hedging is incorporated in the survival payments, which reduces the solvency capital the insurers are required to set aside.

JEL Codes: G22; J32

1 Introduction

Annuities provide a life-long payment stream to the policyholder and thus, yield efficient protection against longevity risk. Therefore, they are thought to be very desirable retirement products from a policyholder’s perspective, a well-known result due to the pioneering work of Yaari (1965). In practice, however, voluntary annuitization rates remain low, a phenomenon which is well-known as the annuity puzzle (see e.g. Friedmann and Warshawsky 1990; Yaari 1965). It is well-believed that one possible explanation for this phenomenon is the bequest motive – the desire to pass on financial wealth to heirs. For instance, Davidoff et al. (2005) find that bequest motives can explain that full annuitization is not optimal. Lockwood (2012) even finds that in the presence of bequest motives, no annuitization at all is preferable if premiums are not actuarially fair. While the existing literature deals with the bequest motive using life cycle models with bequests, in the present paper, we propose adding death benefits directly in retirement products. More specifically, the retirement product pays out not only upon the survival of the policyholders, but also on their death. As these death benefits will be bequeathed to the policyholder’s heirs, they can be considered as bequests. Many insurers already offer retirement products with death benefits.[1] However, little seems to be known about the optimal structure and benefits of such bundled products to policyholders and insurers. While the literature on optimal retirement planning frequently focuses exclusively on the perspective of the policyholders, we show that both policyholders and insurance companies can benefit from such bequest-embedded products.

The reason why bequest-embedded retirement products are attractive to insurers is simply because the products naturally hedge longevity risk by the inclusion of death benefits. Due to this hedging component, solvency capital requirements can be reduced drastically, given a large enough death benefit. This idea of natural hedging is not new and has been extensively analyzed in the literature, see e.g. Cox and Lin (2007), Bayraktar and Young (2007), Wetzel and Zwiesler (2008), Tsai et al. (2010), Wang et al. (2010), Gatzert and Wesker (2012), Zhu and Bauer (2014), Luciano et al. (2017) and Wong et al. (2017). Many of these references derive optimal hedging strategies from an insurer’s perspective. Our paper differs from these references by making a suggestion how new types of user-friendly, highly flexible products could be designed, by deriving the utility-maximizing retirement and death benefits and then showing that the resulting product design is beneficial for both policyholders and insurers.

As potential alternatives to conventional annuities to tackle the challenges faced by modern societies in today’s world, such as longevity risk, various products have been proposed in both the academia and the insurance industry, with the most prominent examples being tontines as well as pooled annuity funds and group self-annuitization schemes. These plans share the similarity that, in contrast to conventional annuities, a pool of policyholders carries the longevity risk. While the idiosyncratic risk can be diversified in a large enough pool, the individual policyholder is left with the systematic longevity risk (see e.g. Donnelly et al. 2013, 2014; Milevsky and Salisbury 2015; Piggott et al. 2005; Sabin 2010; Stamos 2008; Valdez et al. 2006). Milevsky and Salisbury (2015) show that, with an actuarially fair premium, constant relative risk aversion and assuming the absence of aggregate mortality risk, an optimal annuity delivers a higher lifetime utility level than an optimal tontine. This result is generalized to more general, stochastic mortality laws in Chen et al. (2020a). It is shown, for example, in Stamos (2008), Hanewald et al. (2013), Milevsky and Salisbury (2015) and Chen et al. (2019, 2020b) that tontines are preferred over annuities, when a comparably higher safety loading is charged for the annuities. While tontines were a popular retirement product in the seventeenth, eighteenth and nineteenth century, they do not play a major role in the retirement landscape of today’s world and are currently being resurrected (see also Li and Rothschild 2020). One of the reasons why tontines have not yet been extensively provided by insurers is that insurers play only an administrative role in tontine products.

In this article, we consider an explicit inclusion of death benefits in retirement products. As examples for the retirement products, we consider conventional annuities and the new alternative retirement products mentioned above (more specifically, Milevsky and Salisbury’s tontines, see Milevsky and Salisbury (2015)) and include direct death/bequest benefits in these products. These two products are representative, as they are two extreme products, when it comes to bearing longevity risk. While in an annuity, the longevity risk is fully borne by the insurance company, in a tontine contract, it is mainly borne by the policyholders. Our paper is related to the literature on optimal retirement planning, see e.g. Hanewald et al. (2013), Milevsky and Salisbury (2015), Bernhardt and Donnelly (2019), Dagpunar (2019) and Chen et al. (2019, 2020a, 2020b). The articles closest to ours among these are Hanewald et al. (2013), Bernhardt and Donnelly (2019) and Dagpunar (2019). Hanewald et al. (2013) study bequests with both life annuities and group self-annuitization schemes, allowing for phased withdrawals and longevity risk. The main differences of our article compared to Hanewald et al. (2013) are that we consider the tontine design proposed by Milevsky and Salisbury (2015), whereas Hanewald et al. (2013) consider the group self-annuitization scheme by Piggott et al. (2005), and that we analyze bequest-embedded retirement products from the insurer’s point of view, relying on solvency capital requirements. An additional difference is that our main goal in this article is to combine retirement products with life insurance products (paying out a death benefit at the time of death), whereas Hanewald et al. (2013) focus exclusively on retirement products and do not consider life insurance products which pay out a death benefit. In the recent article Bernhardt and Donnelly (2019), a bequest account is included in a modern tontine and an optimal consumption and asset allocation problem is analyzed. The work of Bernhardt and Donnelly (2019) is extended in Dagpunar (2019). The main differences of our article to Bernhardt and Donnelly (2019) and Dagpunar (2019) are that they actually ignore the mortality risk by considering an infinitely large portfolio and focus on the optimal asset allocation problem, while in our case, we incorporate death benefits in the retirement products and study their influences on insurer and policyholders. Additional differences lie, again, in the tontine design and in the fact that we compare annuities and tontines, whereas Bernhardt and Donnelly (2019) and Dagpunar (2019) focus exclusively on tontines.

In an expected utility framework, we determine the optimal survival and death benefits for the policyholder, which ensures her to leave a desirable proportion of money to her heirs. How much the policyholder chooses to leave for her heirs depends substantially on her risk aversion and on the weight she assigns to the bequest part. Being concerned about leaving sufficient bequests to heirs, it can create or promote the incentive for the policyholders to purchase these retirement products. Due to the increases in the life expectancy in last decades, low interest rates and tightening solvency regulations (e.g. Solvency II in EU), the insurers need to set aside rather high solvency capital for conventional products like annuities. Following the categorization provided in EIOPA (2014), we show that the bequest-embedded retirement products considered in this article can still be categorized as retirement products for certain parameter combinations. Due to this categorization, by adding the death benefits, we add some natural hedging to the survival payments, which reduces the solvency capital the insurers are required to set aside. A comparison between bundled and unbundled products reveals that bequest-embedded retirement products can lower the risk margins drastically compared to unbundled products. This supports the attractiveness of bequest-embedded retirement products not only from the insurer’s perspective, but also from the policyholder’s perspective, particularly if the insurer decides to shift the risk margin in part or fully to the policyholders, as reduced solvency capital requirements might allow insurers to offer products at lower prices. In this sense, our results are consistent with Cox and Lin (2007) and Bayraktar and Young (2007). Further, without including death benefits, the insurer plays purely an administrative role in a tontine product, while adding the death benefits makes the tontines real insurance products and incentivizes the insurers to actively provide these innovative retirement products.

A comparison of bequest-embedded annuities and tontines reveals that annuities yield a higher expected lifetime utility than tontines when death benefits are explicitly embedded in these products. However, naturally a lower survival benefit will be pursued by a retiree who has a bequest motive. For our retirement products with death benefits, we determine the critical safety loading for the annuity, beyond which the annuities are worse than the tontines, and find that this level is lower than the critical safety loading when a bequest motive is not present. The higher the policyholder weighs the bequest, the lower this critical safety loading gets. It indicates that adding death benefits makes the tontines closer to the annuity products, which might provide another rationale to promote the tontine products.

Extending our baseline model, we allow for treating bequests as a luxury good and derive the corresponding optimal retirement and death benefits under the generalized utility function. We find that, under this assumption, policyholders invest less wealth in death benefits compared to treating bequests as a regular good. As a second extension, we consider deferred annuities, which are frequently sold in practice. We derive the utility-maximizing retirement and death benefits for this product assuming that the payments to the policyholder will start at some retirement age in the future. We find that the optimal retirement and death benefits remain largely unchanged compared to bequest-embedded annuities without deferment periods. Furthermore, we show that such a deferred bequest-embedded annuity can be more beneficial to policyholders than the bequest-embedded annuity without deferment period.

The outline of the paper is as follows. In Section 2, we derive the optimal payout function, death benefit and optimal level of expected utility of the bequest-embedded annuity. In Section 3, similar derivations are carried out for the tontine and we compare the annuity to the tontine from the policyholder’s perspective. Section 4 provides an analysis of bequest-embedded retirement products from the insurer’s point of view. In Section 5, we compare bequest-embedded retirement products to unbundled products. In Section 6, we extend our baseline model by considering an alternative utility of bequests, treating bequests as luxury goods, and allowing for deferment periods in annuities. Section 7 concludes the article and is followed by the appendix, in which we collect some proofs.

2 Bequest-Embedded Annuity

The products considered in this paper contain exclusively mortality risks. There are two different kinds of mortality risk: idiosyncratic mortality risk stems from the lifetimes of people being unknown but still following a certain mortality law. Aggregate mortality risk stems from the fact that we cannot certainly determine the actual (“true”) mortality law. In the context of retirement products, this risk is also called longevity risk. For the x-year-old policyholder, the best-estimate t-year survival probability is denoted by p x t , which can be estimated from historical data.

To incorporate uncertainty in this mortality law, we follow e.g. Lin and Cox (2005) and apply a random shock ε to the survival curve such that the shocked survival curve is given by p x 1 ϵ t . This shock covers the aggregate mortality risk described above. We assume that ε is a continuous random variable with density f ϵ ( ) and support on ( , 1 ) . The special case where no shock is applied (and aggregate mortality risk is disregarded) is obtained by setting ϵ = 0 .

Consider a retiree at age x endowed with a wealth level v > 0 at her retirement date (in our framework at time t = 0). If we denote the time of death of the retiree by T ϵ 0 , we obtain the (conditional) survival probability at age x for time t 0 by

(1) ( T ϵ t | ϵ ) = p x 1 ϵ t = exp ( 0 t μ x + u d u ) 1 ϵ  ,

where { μ x + u } u 0 is a deterministic force of mortality process. Based on this, we can compute the conditional density function of T ϵ :

(2) f ( t ) | ϵ = ( T ϵ t | ϵ ) t = ( 1 ϵ ) μ x + t p x 1 ϵ t  .

Since nowadays retirees are highly encouraged to invest in private retirement plans, we assume that at time t = 0 the retiree spends her initial wealth on a retirement plan. To demonstrate how death benefits are explicitly embedded in a retirement product, we take conventional annuities and Milevsky and Salisbury’s tontines (see Milevsky and Salisbury 2015) as representative examples in the subsequent sections. They are two extreme products for longevity risk sharing between the policyholders and the insurer. While in an annuity, the longevity risk is fully borne by the insurance company, in a tontine contract, it is mainly borne by the policyholders.

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

(3) b A ( t ) : = 1 { T ϵ > t } c ( t ) + A ( t ) 1 { T ϵ [ t , t + d t ) }  .

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

(4) P 0 A = E [ 0 T ϵ e r t c ( t ) d t + e r T ϵ A ( T ϵ ) ] = 0 e r t E 1 T ϵ > t c t  d t + E e r T ϵ A T ϵ = 0 e r t E E 1 T ϵ > t | ϵ c t  d t + E E e r T ϵ A T ϵ | ϵ = 0 e r t E p x 1 ϵ t c t  d t + E 0 e r t A ( t )   f ( t ) | ε  d t = 0 e r t 1 p x 1 φ t f ϵ φ d φ c t  d t + 0 e r t A t E 1 ϵ μ x + t p x 1 ϵ t d t = 0 e r t p x t m ϵ ln p x t c t  d t + 0 e r t A t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ  d t  ,

where m ϵ ( s ) : = E [ e s ϵ ] is the moment generating function of the random variable ε and 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:

E [ 0 T ϵ e ρ t u ( c ( t ) ) d t + e ρ T ϵ U ( A ( T ϵ ) ) ] = 0 e ρ t p x t m ϵ ( ln p x t ) u ( c ( t ) ) d t + 0 e ρ t U ( A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t

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.

u ( y ) = y 1 γ 1 γ

and the bequest through a slightly modified power utility function

U ( y ) = b y 1 γ 1 γ  ,

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):

c NB ( t ) = ( e ( ρ r ) t λ A NB ) 1 / γ , λ A NB = ( 1 v ( 0 e r t p x t m ϵ ( ln p x t ) e r ρ γ t  d t ) ) γ .

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

(5) max c ( t ) , A ( t ) ( 0 e ρ t p x t m ϵ ( ln p x t ) u ( c ( t ) ) d t + 0 e ρ t U ( A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t ) s . t . v = P 0 A .

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

Theorem 2.1.

For the annuity, the optimal payoff functions are given by

(6) c ( t ) = ( e ( ρ r ) t λ A ) 1 / γ  ,

(7) A ( t ) = ( λ A e ( ρ r ) t b ) 1 / γ ,

where the optimal Lagrangian multiplier λ A is given by

λ A = ( 1 v ( 0 e r t p x t m ϵ ( ln p x t ) e r ρ γ t  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 .

The optimal level of expected discounted lifetime utility is then given by

(8) U A = λ A 1 γ v

Proof.

See Appendix A.1.

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:

c t = e r ρ γ t v ( 0 e r s p x s m ϵ ln p x s e r ρ γ s  d s + 0 e r s e ρ r s b 1 / γ μ x + s 1 1 φ p x 1 φ s f ϵ φ  d φ  d s ) 1 , A t = e ρ r t b 1 / γ v ( 0 e r s p x s m ϵ ln p x s e r ρ γ s  d s + 0 e r s e ρ r s b 1 / γ μ x + s 1 1 φ p x 1 φ s f ϵ φ  d φ  d s ) 1 .

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:

(9) A ( t ) = b 1 / γ c ( t )  ,

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

(10) V 0 ( { A ( t ) } t 0 ) = 0 e r t A ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t

and that of the retirement benefits is given by

(11) V 0 ( { c ( t ) } t 0 ) = v V 0 ( { A ( t ) } t 0 ) = 0 e r t c ( t ) 1 p x 1 φ t f ϵ ( φ )  d φ  d t  .

  • 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

μ x + t = 1 β e x + t m β  ,

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

p x t = e e x m β ( 1 e t β )  ,

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.

Table 1:

Base case parameter setup.

Initial wealth Pool size Risk aversion
v = 100 n = 1000 γ = 4
Risk-free rate Subjective discount rate Bequest
r = 0.01 ρ = r b = 3
Initial age Gompertz law Longevity shock
x = 65 m = 88.721, β = 10 ϵ N ( , 1 ) ( 0.0035 , 0.0814 2 )

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 Bequest-Embedded Tontine

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:

(12) b O T ( t ) = 1 { T ϵ > t } n d ( t ) N ϵ ( t ) + B ( t ) 1 { T ϵ ( t , t + d t ) }  .

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):

(13) P 0 O T = E [ 0 e r t 1 { T ϵ > t } n d ( t ) N ϵ ( t ) d t + e r T ϵ B ( T ϵ ) ] = 0 e r t 1 ( 1 ( 1 p x 1 φ t ) n ) d φ d ( t )  d t + E [ e r T ϵ B ( T ϵ ) ]  .

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

P 0 O T = 0 e r t 1 ( 1 ( 1 p x 1 φ t ) n ) d φ d ( t )  d t + 0 e r t B ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t  .

3.2 Optimal Bequest-Embedded Tontine

First, note that the objective function can be written as

E [ 0 T ϵ e ρ t u ( n d ( t ) N ϵ ( t ) ) + e ρ T ϵ U ( B ( T ϵ ) ) d t ] = 0 e ρ t κ n , γ , ϵ ( p x t ) u ( d ( t ) ) d t + 0 e ρ t U ( A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t  ,

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

(14) κ n , γ , ϵ ( p x t ) = E [ 1 { T ϵ > t } ( n N ϵ ( t ) ) 1 γ ] = k = 1 n ( n k ) ( k n ) γ 1 ( p x 1 φ t ) k ( 1 p x 1 φ t ) n k f ϵ ( φ ) d φ  .

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):

d NB ( t ) = ( e ( ρ r ) t λ O T NB 1 ( 1 ( 1 p x 1 φ t ) n ) d φ κ n , γ , ϵ ( p x t ) ) 1 / γ , λ O T NB = ( 1 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 ) ) γ  .

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

(15) max d ( t ) , B ( t ) ( 0 e ρ t κ n , γ , ϵ ( p x t ) u ( d ( t ) ) d t + 0 e ρ t U ( B ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t ) s . t . P 0 O T = v  .

In Theorem 3.1, we solve Problem (15).

Theorem 3.1.

For the tontine, the optimal payoff functions are given by

(16) d ( t ) = ( e ( ρ r ) t λ O T 1 ( 1 ( 1 p x 1 φ t ) n ) d φ κ n , γ , ϵ ( p x t ) ) 1 / γ  ,

(17) B ( t ) = ( λ O T e ( ρ r ) t b ) 1 / γ  ,

where the optimal Lagrangian multiplier λ O T is given by

λ O T = ( 1 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  .

The optimal level of expected discounted lifetime utility is then given by

(18) U O T = λ O T 1 γ v  .

Proof.

See Appendix A.2.

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

λ O T 1 γ = 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 ) 1  .

  • 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:

(19) B t = d t 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  ,

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

(20) V 0 ( { B ( t ) } t 0 ) = 0 e r t B ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t  ,

and that of the tontine payments is given by

(21) V 0 ( { d ( t ) } t 0 ) = v V 0 ( { B ( t ) } t 0 ) = 0 e r t 1 ( 1 ( 1 p x 1 φ t ) n ) d φ d ( t )  d t  .

  • 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

3.3.1 Actuarially Fair Pricing Framework

In the setting we consider, if an individual can invest her money in tontines and annuities, she will choose to invest all her wealth in annuities, given that these two products are priced actuarially fair. It is a well-known result and is due to the fact that an optimal annuity delivers a higher lifetime utility than an optimal tontine, if only actuarially fair premiums are charged. On the contrary, the individual would also choose to invest some of her wealth in tontines, if appropriate safety loadings to these products are taken into consideration (Chen et al. 2019, 2020b; Hanewald et al. 2013; Milevsky and Salisbury 2015; Stamos 2008). In the subsequent, we will discuss whether these arguments still hold true, if death benefits are incorporated in these products as modeled above.

Theorem 3.2.

The optimal level of expected utility of the annuity (8) is always at least as high as that of the tontine (18) , i.e. U A U O T .

Proof.

See Appendix A.3. □

Theorem 3.2 is a natural extension of the Milevsky and Salisbury (2015) result. Note that the setting of Milevsky and Salisbury (2015) is a special case of ours by setting b = 0, ϵ = 0 , ρ = r and redefining the tontine premium in a slightly modified way. As long as the retirees weigh the bequest in the identical way in their total utility functions (in our case particularly with the same weighting factor b) for the considered retirement products, adding death benefits does not change the preference order of the actuarially fairly priced annuities and tontines. Note that the optimal death benefits in Theorems 2.1 and 3.1 take an identical form, with the only difference lying in the Lagrangian multiplier. It consequently leads to different initial values of the death payments associated with the annuity and the tontine, which can already be seen from the numerical examples in the above two subsections.

Corollary 3.3 is a direct consequence of Theorem 3.2 and compares the optimal death benefits embedded in the annuity and the tontine.

Corollary 3.3.

If γ ( 0 , 1 ) , it holds A ( t ) B ( t ) and if γ > 1 , it holds A ( t ) B ( t ) for all t 0 and b 0 .

Proof.

By Theorem 3.2, it holds U A U O T , i.e.

λ A 1 γ λ O T 1 γ

for all γ. This implies that λ A λ O T for γ ( 0 , 1 ) and λ A λ O T for γ > 1 . From this, we obtain λ A 1 γ λ O T 1 γ for γ ( 0 , 1 ) and λ A 1 γ λ O T 1 γ for γ > 1 . From (7) and (17), we see directly that A ( t ) B ( t ) for γ ( 0 , 1 ) and A ( t ) B ( t ) for γ > 1 .

Corollary 3.3 tells us that a policyholder with a risk aversion between 0 and 1 invests more wealth in the death benefit in a tontine than in an annuity. On the other hand, a policyholder with a risk aversion above 1 invests more initial wealth in the death benefit in an annuity than in a tontine. This results because policyholders with γ > 1 optimally invest more initial wealth in the retirement benefits of the tontine (compared to the annuity) to be protected against extremely low retirement incomes, which leaves less wealth to be invested in the death benefit. Conversely, if γ < 1 , the opposite is true, because policyholders with a risk aversion below 1 can obtain low retirement benefits without drastic decreases in their utility.

Next, we compare the initial values of the death benefits. Table 2 provides the initial values of the death benefits derived above. We observe that the initial values differ slightly and that the initial value of the death benefit in the tontine is slightly lower than in the annuity. This is due to the fact that a relative risk aversion coefficient higher than 1 is used in this example, consistent with Corollary 3.3. As the pool size increases, the initial value of the death benefit in the tontine increases as well. λ A and λ O T become very close to each other as n becomes larger. Note that, if ϵ = 0 , a tontine with pool size n = is equal to an annuity in the framework we consider. Thus, as n , the initial value of the death benefit of the tontine converges to that of the annuity. In the stochastic mortality model we consider, this result is no longer valid. However, Table 2 suggests that the initial values of the death benefits of the tontine and the annuity still approximately coincide if the pool size tends to infinity.

Table 2:

Comparison of the initial values of the death benefits for annuity (10) and tontine (20) with different pool sizes. The parameters are taken from Table 1.

Annuity Tontine: n = 10 n = 100 n = 1000
Initial value 5.53 5.23 5.46 5.50

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 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.

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 The Perspective of the Insurer

In this section, we examine whether the insurer can benefit from explicitly embedding the death benefits in the retirement products. Insurance companies need to hold a sufficient buffer of risk capital to be protected from extreme situations. Adding the death benefit, the insurer adds some natural hedging to the retirement products, which consequently indicates that they need to bear less longevity risk. Hence, it is to expect that bequest-embedded retirement products can reduce the solvency capital requirement of an insurer. This type of natural hedging of risks is not new and has been studied extensively in the literature, see e.g. Cox and Lin (2007), Bayraktar and Young (2007), Wetzel and Zwiesler (2008), Tsai et al. (2010), Wang et al. (2010), Gatzert and Wesker (2012), Zhu and Bauer (2014), Luciano et al. (2017) and Wong et al. (2017).

Latest insurance solvency regulations require insurers to test their balance sheets against various stress-test scenarios. For instance, in the Canadian solvency regulation, a 10–20% decrease of mortality rates (depending on the type of annuity business) is assumed for a longevity shock. The U.S. regulation assumes a stress on mortality improvement between 16 and 40% (depending on the age). This results in lower mortality rates between 0.7 and 6%. In Solvency II which is implemented for insurance undertakings in EU, a longevity shock is defined as a decrease of annual death probabilities by 20%. In the following, we mainly focus on the capital requirement prescribed by Solvency II, from which the qualitative messages can be transferred to other regions.

Let us point out that the analysis in this section disregards many realistic frictions like expenses, profit margins and, for annuities, the potential presence of adverse selection. We focus on a rather simple illustration, as we want to work out the advantages of bequest-embedded retirement products under a solvency regulation. Adding these realistic expenses might change our results. For instance, if the same expenses are added for annuities and tontines, our comparison results do not change. If it is to expect that a higher expense loading will be charged for annuities than tontines, taking these realistic aspects into consideration will make annuities less preferable.

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:

(24) SCR X ( t ) = BEL X ( t | shock ) BEL X ( t | )  ,

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:

BEL A ( t | shock ) = p x t t e r ( s t ) p x + t 1 z 0.995 s - t c ( s )  d s + p x t t e r ( s t ) A ( s ) μ x + s ( 1 z 0.995 ) p x + t 1 z 0.995 s - t  d s , BEL A ( t | ) = p x t t e r ( s t ) p x + t s - t m ϵ ( ln p x + t s - t ) c ( s )  d s + p x t t e r ( s t ) A ( s ) μ x + s 1 ( 1 φ ) p x + t 1 φ s - t f ϵ ( φ )  d φ  d s  ,

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

BEL O T ( t | shock ) = p x t t e r ( s t ) ( 1 ( 1 p x + t 1 z 0.995 s - t ) n ) d ( s )  d s + p x t t e r ( s t ) B ( s ) μ x + s ( 1 z 0.995 ) p x + t 1 z 0.995 s - t  d s , BEL O T ( t | ) = p x t t e r ( s t ) 1 ( 1 ( 1 p x + t s - t ) n ) f ϵ ( φ ) d φ d ( s )  d s + p x t t e r ( s t ) B ( s ) μ x + s 1 ( 1 φ ) p x + t 1 φ s - t f ϵ ( φ )  d φ  d s  .

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 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.

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:

(25) RM X = CoC t = 0 e r ( t + 1 ) SCR X ( t )  ,

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.

Table 3:

Comparison of the risk margins (25) of the bequest-embedded annuity and tontine for different values of the bequest factor b along with the initial values of the death benefits. The parameters are taken from Table 1 with CoC = 6%.

Annuity RM V 0 RM V 0 Tontine
b = 0 7.6827 0 0.0188 0 b = 0
b = 3 7.1623 5.53 0.0061 0.73 b = 0.00075
b = 40 6.7359 10.06 0.0037 0.86 b = 0.0015
b = 1000 5.7998 20.01 0.0021 0.95 b = 0.00225
b = 10,000 4.7855 30.79 0.0009 1.03 b = 0.003

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.

Table 4:

Critical bequest factor b* which sets the risk margin of the bequest-embedded annuity and tontine equal to zero. The parameters are taken from Table 1 with CoC = 6%.

Risk aversion b* (Annuity) b* (Tontine)
γ = 1 / 2 10 0.406036
γ = 2 10,000 0.037909
γ = 4 99,998,899 0.003651
γ = 6 9.999847 ⋅ 1011 0.000892
γ = 8 9.999752 ⋅ 1015 0.000523
γ = 10 9.999727 ⋅ 1019 0.000814

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.

5 Why Bequest-Embedded Retirement Products?

The bequest-embedded retirement products presented in this paper are simply a combination of a life annuity (or a tontine) and a life insurance, but they are provided as bundled products. Hence, the question arises why one should consider a combination of the two products as a new product, when the two original products can also be purchased separately.

In the literature, the benefits of natural hedging are frequently expressed as a reduction in the price of insurance products, clearly an advantage for the policyholders and, potentially, also for the insurers. For example, Cox and Lin (2007) find that insurers which simultaneously sell annuities and life insurance products can offer annuities at lower prices than insurers acting in a single business line. Furthermore, Bayraktar and Young (2007) show that the price of a portfolio consisting of life insurance and endowment insurance policies is lower than the sum of prices of the separate portfolios of life and endowment insurance contracts. To further assess the situation of the insurer, various risk measures are considered in the literature on natural hedging, including but not limited to the conditional value at risk (Tsai et al. 2010), the ruin probability and mean loss (Gatzert and Wesker 2012), the value at risk (Wong et al. 2017; Zhu and Bauer 2014), the Greeks (Luciano et al. 2017) and a variance decomposition (Wetzel and Zwiesler 2008). Below, we will find that our results are, to some extent, consistent with Cox and Lin (2007) and Bayraktar and Young (2007).

In order to work out the benefits of the bundled bequest-embedded retirement products, we will again consider the risk margins according to Solvency II. This risk margin might be charged in part or fully by the insurer from the policyholders on top of the actuarially fair premium.[5] To compare the risk margins of bundled and unbundled products, we first describe the determination of the SCRs for the unbundled products below.

Assume that the policyholder purchases a (regular) life annuity and a (regular) life insurance. For a life annuity, the insurer determines the SCRs using the best-estimate liabilities taking only the retirement benefits into account:

BEL a ( t | shock ) = p x t t e r ( s t ) p x + t 1 z 0.995 s - t c ( s )  d s , BEL a ( t | ) = p x t t e r ( s t ) p x + t s - t m ϵ ( ln p x + t s - t ) c ( s )  d s

Now, if the policyholder purchases a separate life insurance, the insurer determines an additional risk margin for this product. The standard approach for the mortality risk sub-module is to consider an instantaneous and permanent increase in annual death probabilities by 15% (EIOPA 2014). Therefore, we determine the solvency capital requirements and the risk margin of a life insurance contract paying out the death benefit A ( T ϵ ) at the time of death analogously to (24) and (25), using the following best-estimate liabilities:

BEL D ( t | shock ) = p x t t e r ( s t ) A ( s ) μ x + s ( 1 z ˜ 0.005 ) p x + t 1 z ˜ 0.005 s - t  d s BEL D ( t | ) = p x t t e r ( s t ) A ( s ) μ x + s 1 ( 1 φ ) p x + t 1 φ s - t f ϵ ˜ ( φ )  d φ  d s  ,

where z ˜ 0.005 is the 0.005-quantile of a new shock ϵ ˜ which follows a truncated normal distribution on ( , 1 ) different from the longevity shock ε. We calibrate the parameters similarly to Chen et al. (2019) in such a way that the sum of squared errors between the Solvency II shock and the 0.005-quantile of the shocked survival probabilities and the sum of squared errors between the best-estimate survival probabilities and the survival probabilities of the stochastic mortality model are minimized. Mathematically, this means that we determine the mean and variance of ϵ ˜ in such a way that

t = 1 T ( p x t E [ p x 1 ϵ ˜ t ] ) 2 + ( p x SIIshock t p x 1 z ˜ 0.005 t ) 2

is minimized, where T = 55 . This calibration delivers the mean μ ϵ ˜ = 0.0019 and the standard deviation σ ϵ ˜ = 0.0595 . The 0.005-quantile of this distribution is z ˜ 0.005 = 0.155 .

Considering the base case parameters in Table 1, we obtain the following risk margins:

  • Bequest-embedded annuity: 7.1623,

  • Life annuity: 7.2578,

  • Life insurance: 0.0577,

where the risk margin of the life annuity itself is already larger than that of the bequest-embedded annuity. In total, the risk margin can thus be reduced by 1 7.1623 / ( 7.2578 + 0.0577 ) 2 % . In Table 5, we provide similar analyses as for the base case parameters for different bequest factors. We observe that the reduction increases in the bequest factor, i.e. the stronger the bequest motive of a policyholder is, the lower the risk margin becomes. The reason for this is that the risk margin of the bequest-embedded annuity decreases in b. While the risk margin of the regular annuity decreases in b as well, the risk margin of the life insurance increases, which leads to an increase in the percentage reduction of the risk margin.

Table 5:

Risk margins of the bequest-embedded annuity versus the risk margins of a life annuity and a life insurance along with the percentage reduction in dependence of the bequest factor. The parameters are taken from Table 1 with CoC = 6% and μ ϵ ˜ = 0.0019 , σ ϵ ˜ = 0.0595 .

Bequest factor Bequest-embedded annuity Annuity Life insurance Reduction
b = 0 7.6827 7.6827 0 0%
b = 3 7.1623 7.2578 0.0577 2%
b = 40 6.7359 6.9097 0.1050 4%
b = 1000 5.7998 6.1454 0.2088 9%
b = 10,000 4.7855 5.3172 0.3212 15%

For the tontine, similar analyses can be carried out using lower values of b. However, for the tontine only small reductions of the risk margin can be expected, which is simply due to the fact that the risk margin of a regular tontine is already extremely small compared to an annuity (see also Chen et al. 2019). As pointed out above, it is probable that embedding bequests turns a tontine to a life insurance product, potentially with a larger risk margin than a regular tontine. We leave such an analysis for future research as we want to focus on bequest-embedded retirement products in this article.

Note that Table 5 is more suitable to the situation of the policyholder than to the situation of the insurer. While it is to expect that the insurer will charge separate loadings for each product sold, SCRs are aggregated under the standard approach. In other words, the overall SCR of the life underwriting risk module is determined using a formula which takes into account the diversification of longevity and mortality risk. To be precise, the following formula shall be used (EIOPA 2014):

(26) SCR Life ( t ) = SCR longevity 2 ( t ) + SCR mortality 2 ( t ) + 2 ϱ SCR longevity ( t ) SCR mortality ( t )

for all t 0 , where ϱ is a the correlation coefficient between the mortality and longevity risk which is set equal to −0.25 (see EIOPA 2014). Table 6 provides the risk margins resulting from the bequest-embedded annuity and the unbundled products using (26).

Table 6:

Risk margins of the bequest-embedded annuity versus the risk margins of a life annuity and a life insurance determined using (26) along with the percentage reduction in dependence of the bequest factor. The parameters are taken from Table 1 with CoC = 6% and μ ϵ ˜ = 0.0019 , σ ϵ ˜ = 0.0595 .

Bequest factor Bequest-embedded annuity Unbundled products Reduction
b = 0 7.6827 7.6827 0%
b = 3 7.1623 7.2436 1%
b = 40 6.7359 6.8842 2%
b = 1000 5.7998 6.0965 5%
b = 10,000 4.7855 5.2462 9%

As one would expect, the reduction in the risk margin is less pronounced than in Table 5. Nevertheless, we can draw similar conclusions from Table 6 as from Table 5: The bequest-embedded annuity reduces the risk margin stronger, the more pronounced the bequest motive of the policyholder is.

We have seen in the above analyses that the risk margin of a bequest-embedded retirement product can be lowered substantially by the inclusion of a death benefit. As the risk margin may be charged in part or fully from the policyholders, we can see that the price reduction might, in fact, be two-fold: By buying a bequest-embedded retirement product the policyholder is not only entitled to a lower loading for the retirement product, but also does not pay a separate loading for the life insurance product.

For the insurer, bequest-embedded retirement products can serve as an attractive, diversified bundle of existing products which could be offered at a lower price than their single retirement and life insurance components. As the separate retirement and the life insurance products are already existent in (probably) all insurance companies in the world (except for the tontines, of course), they all could offer such a bundle easily and without larger efforts. Bequest motives are a frequently discussed explanation for the annuity puzzle and therefore, insurers could react to this by offering bequest-embedded retirement products, possibly increasing the demand for annuities by lowering their loadings and including a death benefit at the same time.

All in all, our results are largely consistent with the literature on natural hedging, particularly Cox and Lin (2007) and Bayraktar and Young (2007). However, our paper does not address the question of how an optimal diversification strategy for an insurer can be achieved (like e.g. Gatzert and Wesker 2012; Luciano et al. 2017; Tsai et al. 2010; Wang et al. 2010; Wetzel and Zwiesler 2008; Wong et al. 2017). Instead, the main goal of this article is to make a suggestion about how a new type of user-friendly, highly flexible product, which could be more attractive than the currently existing products for policyholders and insurers, could be designed. Additionally, our analyses show that bequest-embedded tontines are insurance products carrying a not insignificant portion of mortality risk, offering insurance companies an advantage against banks or other financial institutions unwilling to carry such risks (but which could still offer a regular tontine due to the low risk).

6 Some Further Discussions

6.1 Bequest as a Luxury Good

Here, we will solve our original problems, treating bequest as a luxury good: Following Ait-Sahalia et al. (2004), we can model the utility of the bequest as a luxury good as follows: U ( y ) = b ( y + η ) 1 γ 1 γ where η > 0 .

In Theorem 6.1, we provide the optimal annuity payoff and death benefit for the case b > 0. To simplify the notation, we rely on the typical actuarial notation

A x = 0 e r t μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t

for the actuarial present value of a continuous whole life insurance which pays out 1 at the time of death.

Theorem 6.1.

For the annuity, the optimal payoff functions are given by

c L ( t ) = ( e ( ρ r ) t λ A L ) 1 / γ  , A L ( t ) = ( λ A L e ( ρ r ) t b ) 1 / γ η  ,

where the optimal Lagrangian multiplier λ A L is given by

λ A L = ( 1 v + η A x ( 0 e r t p x t m ϵ ( ln p x t ) e r ρ γ t  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

The optimal level of expected discounted lifetime utility is then given by

U A = λ A L 1 γ ( v + η A x )  .

Proof.

See Appendix A.1.

Let us make some remarks about the optimal payoffs with bequests treated as a luxury good:

  • For all η > 0 , it holds λ A L < λ A . This implies that c L ( t ) > c ( t )  . As a consequence, it necessarily follows that A L ( t ) < A ( t )  . Considering bequest as a luxury good, the policyholder chooses to leave behind less wealth for bequest.

  • Note that the optimal death benefit A (t) might now become negative for large enough values η. In other words, the policyholder might leave behind some debts which shall be paid back by her heirs. This was a tradition in old China where debts could be paid back within three generations. In the following, let us examine with what values of η the policyholder leaves behind some real bequests. For this, we define

λ ˜ A L : = ( 0 e r t p x t m ϵ ( ln p x t ) e r ρ γ t  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 .

  • Then, η needs to fulfill the following inequality for all t 0 :

(27) ( v + η A x ) ( b e ( r ρ ) t λ ˜ A L ) 1 / γ η 0 η { v ( b e ( r ρ ) t λ ˜ A L ) 1 / γ A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ 1 , A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ < 1 v ( b e ( r ρ ) t λ ˜ A L ) 1 / γ A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ 1 , A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ > 1  .

  • Note that for all t such that A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ > 1 , the second inequality in (27) is fulfilled, since we choose η > 0 . If A x ( b e ( r ρ ) t λ ˜ A L ) 1 / γ = 1 , the death benefit is also positive. In other words, the maximum value for η is determined by the first inequality in (27).

In Theorem 6.2, we present the optimal retirement and death benefits in a tontine, treating bequests as a luxury good.

Theorem 6.2.

For the tontine, the optimal payoff functions are given by

d L ( t ) = ( e ( ρ r ) t λ O T L 1 ( 1 ( 1 p x 1 φ t ) n ) d φ κ n , γ , ϵ ( p x t ) ) 1 / γ  , B L ( t ) = ( λ O T L e ( ρ r ) t b ) 1 / γ η  ,

where the optimal Lagrangian multiplier λ O T L is given by

λ O T L = ( 1 v + η A x ( 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 .

The optimal level of expected discounted lifetime utility is then given by

U O T L = λ O T L 1 γ ( v + η A x )  .

Proof.

See Appendix A.2.

Let us make some remarks about the optimal payoffs with bequests treated as a luxury good:

  • Similarly to the annuity, for all η > 0 , it holds λ O T L < λ O T . This implies that d L ( t ) > d ( t )  . As a consequence, it necessarily follows that B L ( t ) < B ( t )  .

  • Different from the bequest-embedded annuity, note that the optimal death benefit B∗(t) might now become negative for large enough values η. Let us define

λ ˜ O T L : = ( 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

  • Then, similarly to the annuity, η needs to fulfill the following inequality for all t 0 :

(28) η { v ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ 1 , A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ < 1 v ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ 1 , A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ > 1 .

  • Note that for all t such that A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ > 1 , the second inequality in (28) is fulfilled, since we choose η > 0 . If A x ( b e ( r ρ ) t λ ˜ O T L ) 1 / γ = 1 , the death benefit is also positive. The maximum value for η is therefore again determined by the first inequality in (28).

6.2 Bequest-Embedded Deferred Annuity

In practice, annuities are frequently sold as deferred annuities, i.e. the policyholder purchases a pension annuity at age x, a pre-retirement age, but the contract stipulates that it does not start providing income until retirement age x + u > x. For instance, this product is bought by the policyholder at the age of 40, and pays out at the retirement age 65, i.e. x = 40 and u = 25. In this section, we study the bequest-embedded deferred annuity which pays out starting from the retirement age. Letting us assume that the contracting-issuing time is considered as 0, the contract starts to pay out at time u:

(29) b D A ( t ) : = 1 { T ϵ > t } c D A ( t ) + A D A ( t ) 1 { T ϵ [ t , t + d t ) } , t u  .

Note that c DA (t) describes again the survival benefit of the annuity contract, and A D 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 time-zero actuarially fair premium for the bequest-embedded deferred annuity can be obtained as

(30) P 0 D A = E [ u T ϵ e r t c D A ( t ) d t + e r T ϵ A D A ( T ϵ ) ] = u e r t p x t m ϵ ( ln p x t ) c D A ( t )  d t + u e r t A D A ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t  .

As we are focused on the evaluation of retirement products, we assume that the retiree evaluates payments using the following discounted expected lifetime utility in a deferred bequest-embedded annuity:

E [ u T ϵ e ρ t u ( c D A ( t ) ) d t + e ρ T ϵ U ( A D A ( T ϵ ) ) ] = u e ρ t p x t m ϵ ( ln p x t ) u ( c D A ( t ) ) d t + u e ρ t U ( A D A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t  ,

where u ( y ) = y 1 γ 1 γ and U ( y ) = b y 1 γ 1 γ for all y > 0, and the constant b 0 measures the strength of the bequest motive relative to the desire for survival benefits. The retiree solves the subsequent optimization problem at time t = 0:

max c D A ( t ) , A D A ( t ) ( u e ρ t p x t m ϵ ( ln p x t ) u ( c D A ( t ) ) d t + u e ρ t U ( A D A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t ) s . t . v D A = P 0 D A  .

where v DA is the initial wealth the policyholder owns to buy the deferred bequest-embedded annuity product. In Theorem 6.3, we provide the optimal deferred annuity payoff and death benefit for the case b > 0.

Theorem 6.3.

For the annuity, the optimal payoff functions are given by

(31) c D A ( t ) = ( e ( ρ r ) t λ D A ) 1 / γ  ,

(32) A D A ( t ) = ( λ D A e ( ρ r ) t b ) 1 / γ  ,

where the optimal Lagrangian multiplier λ D A is given by

λ D A = ( 1 v D A ( u e r t p x t m ϵ ( ln p x t ) e r ρ γ t  d t + u 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 .

The optimal level of expected discounted lifetime utility is then given by

(33) U D A = λ D A 1 γ v D A  .

Proof.

The proof can be carried out analogously to the proofs of Theorems 2.1 and 6.1 in Appendix A.1.

In order to set up a reasonable comparison between the immediate and deferred annuity product, and particularly work out the implications for the optimal bequest amount, we assume

(34) v D A = v e r u E p x u 1 ϵ .

Assuming that ϵ 0 allows us to carry out a theoretical comparison of the deferred annuity and the regular annuity considered in Section 2. We summarize the results in Proposition 6.4.

Proposition 6.4.

Let u 0 and assume that ϵ 0 , that x + u is the retirement age considered in Section 2 and that v DA is given as in (34) . Then, the following results hold:

  1. The optimal retirement benefits (6) and (31) are identical and the optimal death benefits (7) and (32) are identical.

  2. The optimal fractions of wealth invested in the retirement benefits are identical in the deferred and the regular annuity. Consequently, the optimal fractions of wealth invested in the death benefit are also identical.

  3. Assume that ρ 0 . Then, for γ > ( < ) 1 , it holds U DA >(<)U A .

Proof.

(i) The optimal payoffs mainly differ only in the Lagrangian multipliers λ A and λ D A and in their domains. For the Lagrangian multipliers, we observe the following relation:

λ D A = ( 1 v D A ( 0 e r ( t + u ) p x t + u e r ρ γ ( t + u )  d t + 0 e r ( t + u ) ( e ( ρ r ) ( t + u ) b ) 1 / γ μ x + t + u p x t + u  d t ) ) γ = ( e r u e r ρ γ u p x u v D A ( 0 e r t p x + u t e r ρ γ t  d t + 0 e r t ( e ( ρ r ) t b ) 1 / γ μ x + u + t p x + u t  d t ) ) γ = e ( r ρ ) u λ A .

  • Thus, the optimal payoffs in the deferred annuity are

c D A ( t ) = ( e ( ρ r ) ( t u ) λ A ) 1 / γ , A D A ( t ) = ( λ A e ( ρ r ) ( t u ) b ) 1 / γ  ,

  1. for all t u , equal to (6) and (7).

  2. The second point follows directly from the first one:

1 v D A 0 e r ( t + u ) p x t + u c D A ( t + u )  d t = e r u p x u v e r u p x u 0 e r t p x + u t ( e ( ρ r ) t λ A ) 1 / γ  d t = 1 v 0 e r t p x + u t c A ( t )  d t  .

  1. From (11) it follows directly that the optimal fractions of wealth invested in the death benefits are also identical.

  2. We have already shown that λ D A = e ( r ρ ) u λ A . Assume that γ > ( < ) 1 . Then, we get

U D A = 1 1 γ e ( r ρ ) e r u p x u v λ A = 1 1 γ e ρ u p x u 1 v λ A ( ) 1 1 γ v λ A = U A

Proposition 6.4 tells us that an individual allocates her wealth in the exact same way to retirement and death benefits, regardless of the deferment period. The third point results only because utility levels for a risk aversion level below 1 are positive, but negative for risk aversions above 1, similarly to Corollary 3.3. This result reflects that individuals with a higher risk aversion buy longevity insurance at earlier ages, whereas policyholders with a lower risk aversion obtain a higher utility when buying longevity insurance directly before retirement.

The results derived in Proposition 6.4 remain approximately valid under a non-degenerate shock. In the following, we will provide a small numerical illustration to emphasize this. We assume that x = 40 and u = 25. In addition, we consider the base case parameters in Table 1. Then, we obtain v DA  = 71.47 and an initial value of the death benefits of 3.95. The percentage of wealth invested in the death benefit is therefore given by 3.95 / 71.47 = 0.05530 5.53 %  . This percentage is (up to the fifth decimal) equal to the 5.53 % ( 0.05531 ) derived in Section 2. Furthermore, a simple calculation delivers

λ A = 0.0014245 0.0014256 = λ D A  .

Consequently, we obtain the following (constant) retirement and death benefits:

c D A ( t ) = 5.146 5.147 = c ( t ) A D A ( t ) = 6.773 6.774 = A ( t )  .

The extremely small difference is caused by the slightly higher systematic mortality risk which the policyholder faces in a deferred annuity. As suggested by Proposition 6.4, we obtain

λ D A 1 γ v D A = 0.0340  .

For the regular annuity, we get

λ A 1 γ v = 0.0475  .

7 Conclusion

We analyze the attractiveness resulting from bequest-embedded retirement plans, focusing on annuities and tontines. The tontine design we consider is the Milevsky and Salisbury (2015)-tontine. We suggest that the insurer embed a death benefit in the retirement products and determine the optimal death and retirement benefit from a policyholder’s perspective. These payments depend substantially on the risk aversion and how policyholders weigh the bequest in their lifetime utility. Policyholders who put a higher weight on the bequest motive can choose a lower survival benefit and a higher death benefit. We carry out an analysis of solvency capital requirements to examine the attractiveness of the bequest-embedded retirement products. We find that offering annuities and tontines with death benefits is beneficial for insurers, as the death benefits provide some natural hedge against the longevity risk contained in retirement products and, particularly, annuities. A comparison between bundled and unbundled products reveals that bequest-embedded retirement products lead to potentially drastically lower risk margins than unbundled products, supporting the attractiveness of bequest-embedded retirement products from the insurer’s and the policyholder’s perspective. Further, an analysis of an indifference safety loading which makes the policyholder indifferent between the tontine and the annuity reveals that the inclusion of death benefits can make tontines and annuities more similar to each other.

Extending our baseline model, we treat bequests as a luxury good and observe a decrease in the percentage of wealth invested in death benefits, compared to the baseline model. As a second extension, we consider deferred bequest-embedded annuities and find that the optimal payoff structures remain basically unchanged. Furthermore, we find that deferred bequest-embedded annuities can deliver a higher expected lifetime utility to some policyholders than the regular bequest-embedded annuity. To sum up, our analytical and numerical results suggest bequest-embedded retirement products can be promising both from the policyholders’ and from the insurers’ point of view. In spite of stylized settings, our models can provide some implications about how death benefits can be designed and incorporated in retirement products.

There are some questions which we want to openly acknowledge for future research. In this article, we limit ourselves to (bequest-embedded) annuities and tontines. Of course, the retirement landscape contains many more products than these two. There are other retirement plans taking into account investment risk such as variable annuities which might dominate both products considered in this article. Furthermore, portfolios of different retirement plans might be formed to offer more potential than the single products standing by themselves (as shown e.g. in Chen et al. (2020b) for annuities and tontines). Therefore, our article rather provides suggestions and illustrations of how bequests can easily be embedded in retirement plans to promote such bundled products rather than provide one optimal bequest-embedded retirement plan.


Corresponding author: Manuel Rach, Institute of Insurance Science, Ulm University, Helmholtzstr. 20, 89069 Ulm, Germany, E-mail:

Funding source: DFG

Award Identifier / Grant number: 418318744

Acknowledgements

An Chen and Manuel Rach acknowledge the financial support given by the DFG for the research project “Zielrente: die Lösung zur alternden Gesellschaft in Deutschland” (Grant number 418318744).

Proofs

A.1 Proof of Theorem 2.1 and 6.1

In the following proof, we assume that b > 0. The case b = 0 can be carried out analogously (see also Chen et al. 2019). We consider a general utility function of the form U ( y ) = ( y + η ) 1 γ 1 γ for the utility of bequests. To obtain the results in Theorem 2.1, simply set η = 0 . The Lagrangian function is given by

L = 0 e ρ t p x t m ϵ ( ln p x t ) u ( c ( t ) ) d t + b 0 e ρ t U ( A ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t + λ A ( v 0 e r t p x t m ϵ ( ln p x t ) c ( t )  d t + 0 e r t A ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t )  .

The first-order conditions are

L c ( t ) = e ρ t p x t m ϵ ( ln p x t ) u ( c ( t ) ) λ A e r t p x t m ϵ ( ln p x t ) = 0 c ( t ) = ( e ( ρ r ) t λ A ) 1 / γ , L A ( t ) = b U ( A ( t ) ) e ρ t μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ λ A e r t μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ = 0 A ( t ) = ( λ A e ( ρ r ) t b ) 1 / γ η  .

Plugging these terms into the budget constraint delivers an explicit formula for the Lagrangian multiplier:

v = 0 e - r t p x t m ϵ - ln p x t e ρ - r t λ A - 1 / γ  d t + 0 e - r t λ A 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 - η 0 e - r t μ x + t - 1 1 - φ t p x 1 - φ f ϵ φ  d φ  d t  .

This delivers the following formula for the Lagrangian multiplier:

λ A 1 γ = 1 v + η A x ( 0 e r t p x t m ϵ ( ln p x t ) e r ρ γ t  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 )

The expected discounted lifetime utility can then be determined as

E [ 0 T ϵ e ρ t u ( c ( t ) ) d t + b e ρ T ϵ U ( A ( T ϵ ) ) ] = 0 e ρ t p x t m ϵ ( ln p x t ) u ( ( e ( ρ r ) t λ A ) 1 γ ) d t + b 0 e ρ t U ( ( λ A e ( ρ r ) t b ) 1 / γ η ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t = λ A 1 1 γ 1 γ ( 0 e ρ t p x t m ϵ ( ln p x t ) ( e ( ρ r ) t ) 1 1 γ d t + b 0 e ρ t ( e ( ρ r ) t b ) 1 1 γ μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t ) = λ A 1 1 γ 1 γ ( 0 e r t p x t m ϵ ( ln p x t ) ( e ( ρ r ) 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 ) = λ A 1 1 γ 1 γ ( v + η A x ) λ A 1 γ = λ A 1 γ ( v + η A x )  .

A.2 Proof of Theorem 3.1 and Theorem 6.2

In the following proof, we assume that b > 0 . The case b = 0 can be carried out analogously (see also Chen et al. 2019). We consider a general utility function of the form U ( y ) = ( y + η ) 1 γ 1 γ for the bequest. To obtain the results in Theorem 3.1, simply set η = 0 . The Lagrangian function is given by

L = 0 e ρ t κ n , γ , ϵ ( p x t ) u ( d ( t ) ) d t + 0 e ρ t U ( B ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t + λ O T ( v 0 e r t 1 ( 1 ( 1 p x 1 φ t ) n ) d φ d ( t )  d t 0 e r t B ( t ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t )

The first-order conditions are

L d t = e ρ t κ n , γ , ϵ p x t u ' d t λ O T e r t 1 1 1 p x 1 φ n d φ = 0 d t = e ρ r t λ O T 1 1 1 p x 1 φ n d φ κ n , γ , ϵ p x t 1 / γ L B t = U ' B t e ρ t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ λ O T e r t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ = 0 B t = λ O T e ρ r t b 1 / γ η  .

Plugging these terms into the budget constraint delivers an explicit formula for the Lagrangian multiplier:

v = 0 e r t 1 ( 1 ( 1 p x 1 φ t ) n )  d φ ( e ( ρ r ) t λ O 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 ( λ O 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 η A x

This delivers the following formula for the Lagrangian multiplier:

λ O T = ( 1 v + η A x ( 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 .

The overall expected discounted lifetime utility is then given by

E [ 0 T ϵ e ρ t u ( n d ( t ) N ϵ ( t ) ) d t + b e ρ T ϵ U ( B ( T ϵ ) ) ] = 0 e ρ t κ n , γ , ϵ ( p x t ) u ( ( e ( ρ r ) t λ O 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 ρ t U ( ( λ O T e ( ρ r ) t b ) 1 / γ η ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t = λ O T 1 1 γ 1 γ ( 0 e ρ t κ n , γ , ϵ ( p x t ) ( e ( ρ r ) t 1 ( 1 ( 1 p x 1 φ t ) n ) d φ κ n , γ , ϵ ( p x t ) ) 1 1 γ  d t + b 0 e ρ t ( e ( ρ r ) t b ) 1 1 γ μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t ) = λ O T 1 1 γ 1 γ ( 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 ) = λ O T 1 1 γ 1 γ ( v + η A x ) λ O T 1 γ = λ O T 1 γ ( v + η A x )  .

A.3 Proof of Theorem 3.2

The proof is carried out in a similar way as in Chen et al. (2020a, 2020b). Consider the following optimization problem:

(35) max α [ 0,1 ] E [ 0 e ρ t 1 { T ϵ > t } u ( α c ( t ) + ( 1 α ) n N ϵ ( t ) d ( t ) )  d t + e ρ T ϵ U ( α A ( T ϵ ) + ( 1 α ) B ( T ϵ ) ) ]

where c (t) is the optimal annuity payoff, d (t) is the optimal tontine payoff, A (t) is the optimal death benefit in the annuity and B (t) is the optimal death benefit in the tontine. In particular, it holds v = P 0 A = P 0 O T , i.e. the individual maximizes her utility over the percentage of initial wealth invested in the bequest-embedded annuity, given access to both optimal bequest-embedded annuities and tontines. Note that this optimization problem differs from Problems (5) and (15), as, in (35), we maximize over the fraction of the initial wealth α. Therefore, we do not have to include a budget constraint in the problem, since the individual can only split her initial wealth v between the optimal bequest-embedded annuity with initial value P 0 A = v and the optimal bequest-embedded tontine with initial value P 0 O T = v . The objective function to Problem (35) is given by

F = 0 e ρ t E 1 T ϵ > t u α c t + 1 α n N ϵ t d t  d t + E e ρ T ϵ U α A T ϵ + 1 α B T ϵ = 0 e ρ t E p x 1 ϵ t E u α c t + 1 α n N ϵ t d t | T ϵ > t , ϵ  d t + 0 e ρ t U α A t + 1 α B t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ  d t = 0 e ρ t E p x 1 ϵ t k = 0 n 1 u α c t + 1 α n k + 1 d t n 1       k p x 1 ϵ t k 1 p x 1 ϵ t n 1 k  d t + 0 e ρ t U α A t + 1 α B t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ  d t = 0 e ρ t k = 0 n 1 u α c t + 1 α n k + 1 d t n 1       k 1 p x 1 φ t k + 1 1 p x 1 φ t n 1 k f ϵ φ  d φ  d t + 0 e ρ t U α A t + 1 α B t μ x + t 1 1 φ p x 1 φ t f ϵ φ  d φ  d t

We determine the first-order derivative to find a maximum of this function. The first-order condition with respect to α is

(36) α = 0 e ρ t k = 0 n 1 u ( α c ( t ) + ( 1 α ) n k + 1 d ( t ) ) ( c ( t ) n k + 1 d ( t ) ) n 1       k × 1 ( p x 1 φ t ) k + 1 ( 1 p x 1 φ t ) n 1 k f ϵ ( φ )  d φ  d t + 0 e ρ t U ( α A ( t ) + ( 1 α ) B ( t ) ) ( A ( t ) B ( t ) ) μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t = ! 0

Using (6) and (7), we can verify that α∗ = 1 fulfills the first-order condition (36). We can verify that α∗ = 1 is a maximum and that it is the only maximum of the objective function by taking a look at the second-order derivative:

2 α 2 = 0 e ρ t k = 0 n 1 u ( α c ( t ) + ( 1 α ) n k + 1 d ( t ) ) ( c ( t ) n k + 1 d ( t ) ) 2 n 1       k × 1 ( p x 1 φ t ) k + 1 ( 1 p x 1 φ t ) n 1 k f ϵ ( φ ) d φ  d t + 0 v e ρ t U ( α A ( t ) + ( 1 α ) B ( t ) ) ( A ( t ) B ( t ) ) 2 μ x + t 1 ( 1 φ ) p x 1 φ t f ϵ ( φ )  d φ  d t < 0  ,

since u ( α c ( t ) + ( 1 α ) n k + 1 d ( t ) ) < 0 and U ( α A ( t ) + ( 1 α ) B ( t ) ) < 0 for all α [ 0 , 1 ] . If the second-order derivative is strictly negative, the first-order derivative is strictly decreasing in α. Hence, α can only be equal to zero for exactly one value of α, which we have already found above (α∗ = 1). From this, we also see that the first-order derivative has to be greater than zero for all α < 1. Consequently, the expected utility in Problem (35) is increasing in α until it reaches its maximum at α = 1. Particularly, a 100% investment of initial wealth in the optimal bequest-embedded annuity delivers a higher expected lifetime utility than a 100% investment in the optimal bequest-embedded tontine.

A.4 Proof of Proposition 3.4

The optimal level of expected utility of the annuity is given by

U A = 1 1 γ λ A v

and that of the tontine is given by

U O T = 1 1 γ λ O T v O T  .

We want to find v O T v such that U A = U O T . We obtain

λ A v = v O T 1 γ ( 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

Solving this for v OT delivers (23).

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Received: 2020-09-23
Accepted: 2021-02-15
Published Online: 2021-03-04

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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