On the free boundary of an annuity purchase
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Abstract
It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies if the mortality force is a generic function of time and the subjective life expectancy of the investor differs from the objective one adopted by insurance companies to price annuities. In this paper, we address this problem by considering an individual who invests in a fund and has the option to convert the fund’s value into an annuity at any time. We formulate the problem as a real option and perform a detailed probabilistic study of the optimal stopping boundary. Due to the generic timedependence of the mortality force, our optimal stopping problem requires new solution methods to deal with nonmonotonic optimal boundaries.
Keywords
Annuities Mortality force Optimal stopping Free boundary problemsMathematics Subject Classification (2010)
91G80 62P05 60G40 35R35JEL Classification
J261 Introduction
In an ageing world, an accurate management of retirement wealth is crucial for financial wellbeing. It is important for working individuals to carefully consider the existing offer of financial and insurance products designed for retirement, beyond the state pension. This offer includes for example occupational pension funds and taxadvantaged retirement accounts (e.g. Individual Retirement Account (US)). Most of these products rely on annuities to turn retirement wealth into guaranteed lifetime retirement income. Life annuities provide a lifelong stream of guaranteed income in exchange for a (single or periodic) premium. The purchase of an annuity helps individuals to manage the longevity risk, i.e., the risk of outliving their financial wealth, but it is usually an irreversible transaction. In fact, most annuity contracts impose steep penalties for partial or complete cancellation by the policyholder, especially in the early years of the contract.
Timing an annuity purchase (socalled annuitisation) is a complex financial decision that depends on several risk factors as e.g. market risk, longevity risk, potential future need of liquid funds, and bequest motive. The study of this topic has motivated a whole research field since the seminal contribution of Yaari [19], who showed that individuals with no bequest motive should convert all their retirement wealth into annuities.
After Yaari, several authors have analysed the annuitisation decision under the socalled allornothing institutional arrangement, where a lifetime annuity is purchased in a single transaction (as opposed to gradual annuitisation). Initially, an individual’s wealth is invested in the financial market, and at the time of an annuity purchase, it is converted into a lifetime annuity. The central idea in this literature is to compare the value deriving from an immediate annuitisation with the value of deferring it while investing in the financial market. Therefore, a strict analogy holds with the problem of exercising an American option, and the annuitisation decision can be considered as the exercise of a real option.
Milevsky [12] proposed a model where an individual defers annuitisation for as long as the financial investment’s returns guarantee a consumption flow which is at least equal to the one provided by the annuity payments. In particular, [12] adopts a criterion based on controlling the probability of a consumption shortfall.
Other papers study the optimal annuitisation time in the context of utility maximisation, and formulate the problem as one of optimal stopping and control. The investor aims at maximising the expected utility of consumption (preretirement) and of annuity payments (postretirement).
Assuming a constant force of mortality and CRRA utility, Stabile [18] analytically solves a timehomogeneous optimal stopping problem. He proves that if the individual has the same degree of risk aversion before and after the annuitisation, then an annuity is purchased either immediately or never (the socalled nowornever policy). On the other hand, if the individual is more risk averse during the annuity payout phase, the annuity is purchased as soon as the wealth falls below a constant threshold (the optimal stopping boundary).
A constant force of mortality is also assumed in Gerrard et al. [7] and Liang et al. [11]. The model in [7] is analogous to the one studied in [18], but with quadratic utility functions, and the authors find a closedform solution: If \((X_{t})_{t\ge 0}\) represents the individual’s wealth process, then it is optimal to stop when \(X\) leaves a specific interval (hence the optimal stopping boundary is formed by the endpoints of such an interval). In [11], in contrast to the previous papers, the authors assume that the individual may continue to invest and consume after annuitisation. By using martingale methods, explicit solutions are provided in the case of CRRA utility functions. Contrarily to [7], the optimal annuitisation in [11] occurs when the wealth process enters a specific interval, whose endpoints form the optimal stopping boundary.
Assuming a timedependent force of mortality, Milevsky and Young [13] analyse both the allornothing market and the more general anythinganytime market, where gradual annuitisation strategies are allowed. For the allornothing market, they find that the optimal annuitisation time is deterministic as an artifact of CRRA utility. Thus, the annuitisation decision is independent from the individual’s wealth.
Our work is more closely related to work by Hainaut and Deelstra [9]. They consider an individual whose retirement wealth is invested in a financial fund which eventually must be converted into an annuity. The fund is modelled by a jumpdiffusion process and pays dividends at a constant rate. The mortality force is a timedependent, deterministic function and the individual aims at maximising the market value of future cashflows before and after annuitisation. According to the insurance practice, it is assumed that the individual can only purchase the annuity by a given maximal age. The authors in [9] cast the problem as an optimal stopping one and write a variational inequality for the value function. They then use the Wiener–Hopf factorisation and a time stepping method to solve the variational problem numerically. Hainaut and Deelstra argue that the decision to purchase the annuity should be triggered by either an upper or a lower, timedependent threshold in the timewealth plane. Numerical examples are provided in [9] where the annuitisation occurs when the value of the financial fund is high enough or, alternatively, low enough.
In this paper, we perform a detailed mathematical study of the optimal stopping problem associated to an annuitisation decision similar to that considered in [9]. In the interest of a rigorous analysis of the optimal stopping boundary, we simplify the dynamics of the financial fund by considering a geometric Brownian motion with no jumps. As in [9], we look at the maximisation of future expected cashflows for an individual who joins the fund and has the opportunity to purchase an annuity on a time horizon \([0,T]\). Time 0 is the time when the individual joins the fund and time \(T\) is the time by which the individual reaches the maximal age for an annuity purchase. The present value of future expected cashflows, evaluated at the optimum, gives us the socalled value function \(V\).
Notice that a closer inspection of the problem formulation in (2.4) below shows that at time \(T\), the fund is converted into an annuity (the same occurs in [9]). This means that the individual will eventually purchase the annuity at time \(T\), but she also has an option to buy it earlier. One could think of this feature as part of the fund’s contract specifications or as a commitment of the investor at time 0. It is, however, important to remark that the methods developed in this paper apply also to the case \(T=+\infty \), up to some minor changes (see also Remark 3.3 for further details).
One of the key features of the model presented here is the use of a rather general timedependent, deterministic mortality force. This is a realistic assumption commonly made in the actuarial profession. As in [13], we consider two different mortality forces: a subjective one, used by the individual to weigh the future cashflows (denoted \(\mu ^{S}\)), and an objective one, used by the insurance company to price the annuity (denoted \(\mu ^{O}\)). The interplay between these two different mortality forces contributes to some key qualitative aspects of the optimal annuitisation decision (see Sect. 5 for more details). Interestingly, the generic timedependent structure of the mortality force constitutes also the major technical challenge in the mathematical study of the problem.
On the one hand, standard optimal stopping results ensure that the time–wealth plane splits into a continuation region \(\mathcal{C}\), where the option to wait has strictly positive value, and a stopping region \(\mathcal{S}\), where the annuity should be immediately purchased. Denoting by \((X_{t})_{t\ge 0}\) the process that represents the fund’s value (or equally, the individual’s retirement wealth), an optimal stopping rule is given by stopping at the first time the twodimensional process \((t,X_{t})_{t\ge 0}\) enters the set \(\mathcal{S}\). Moreover, under some mild technical assumptions, we prove in Proposition 4.2 that these two sets are split by an optimal boundary (free boundary, in the language of PDEs) which only depends on time, i.e., \(t\mapsto b(t)\).
On the other hand, technical difficulties arise when trying to infer properties of the boundary \(b\). In fact, due to the generic timedependence of the mortality force, it is not possible to establish any monotonicity of the mapping \(t\mapsto b(t)\). It is well known in optimal stopping and free boundary theory that monotonicity of \(b\) is the key to a rigorous study of the regularity of the boundary (e.g. continuity) and of the value function (e.g. continuous differentiability). The interested reader may consult [15, Chaps. VII and VIII] for a collection of relevant examples, and the introduction in [5] for a deeper discussion.
We overcome this major technical hurdle by proving that the optimal boundary is in fact a locally Lipschitzcontinuous function of time. In order to achieve this goal, we rely only on probabilistic methods which are new and specifically designed to tackle our problem. This approach draws from similar ideas in [5], but we emphasise that our problem falls outside the class of problems addressed in that paper (see the discussion prior to Theorem 4.8 below).
Once Lipschitz regularity is proved, we then obtain also that the value function \(V\) is continuously differentiable in \(t\) and \(x\), at all points of the \((t,x)\)plane and in particular across the boundary of \(\mathcal{C}\). This is a stronger result than the more usual smoothfit condition, which states that \(z\mapsto V_{x}(t,z)\) is continuous across the optimal boundary. Finally, we find nonlinear integral equations that characterise uniquely the free boundary and the value function.
The analysis in this paper is completed by solving numerically the integral equation for some specific examples and studying their sensitivity to variations in the model’s parameters. It is important to remark that the optimal boundary turns out to be nonmonotonic in some of our examples, under natural assumptions on the parameters. This shows that the new approach developed in this paper is indeed necessary to study the annuitisation problem.
In summary, our contribution is at least twofold. On the one hand, we add to the literature concerning annuitisation problems in the allornothing framework by addressing models with timedependent mortality force. As we have discussed above and to the best of our knowledge, such models were only considered in [13] (which produces only deterministic optimal strategies) and in [9] (mostly in a numerical way). We provide a rigorous theoretical analysis of the optimal annuitisation strategy in terms of the optimal boundary \(b\). Our study also reveals behaviours not captured by [9] as e.g. lack of monotonicity of \(b\). The latter may reflect the change over time in the investor’s priorities, due to (deterministic) variations in the mortality force. On the other hand, it is rather remarkable that we started by considering an applied problem, with a somewhat canonical and seemingly innocuous formulation, but we soon realised that its rigorous analysis is far from trivial. Therefore we developed methods which are new in the probabilistic literature on optimal stopping and of independent interest.
Finally, in order to relate our work to the PDE literature in this area, it may be worth noticing that [6] (and later [2]) studies a free boundary problem motivated by optimal retirement. In that paper, an investor can decide to retire earlier than a given terminal time \(T\). Early retirement benefits are defined by a function \(\varPsi (t,s)\) of time \(t\) and the current salary \(s\). The problem is addressed exclusively with variational inequalities, and the free boundary depends on time since \(t\mapsto \varPsi (t,s)\) increases linearly. However, contrarily to our model, the mortality force in [6] and [2] is assumed constant.
The rest of the paper is organised as follows. In Sect. 2, we introduce the financial and actuarial assumptions and then the optimal annuitisation problem. In Sect. 3, we provide some continuity properties of the value function and useful probabilistic bounds on its gradient. In Sect. 4, we present sufficient conditions under which the shape of the continuation and stopping regions can be established, and we study the regularity of the optimal boundary. Moreover, we find nonlinear integral equations that characterise uniquely the free boundary and the value function. In Sect. 5, we present some numerical examples to illustrate the range of applicability of our assumptions. In Sect. 6, we provide some final remarks and extensions.
2 Problem formulation
In our model, we consider an individual (or investor) and an insurance company who are faced with two distinct sources of randomness: a financial market and the survival probability of the individual. We assume that the individual and the insurance company have different beliefs about the demographic risk, while they share the same views on the financial market. It is therefore convenient to construct initially two probability spaces: one that models the financial market and another that models the demographic component. The time horizon of the problem is fixed and denoted by \(T<+\infty \).
2.1 Financial and demographic models
For the demographic risk, we consider another probability space. Given a measurable space \((\varOmega ',\mathcal{F}')\), we let \(\mathbb{Q}^{S}\) and \(\mathbb{Q}^{O}\) denote two probability measures on \((\varOmega ', \mathcal{F}')\) and assume that \((\varOmega ',\mathcal{F}',\mathbb{Q}^{i})\), \(i=S,O\), are both complete. The measure \(\mathbb{Q}^{S}\) is associated with the subjective survival probability of the individual. In contrast, \(\mathbb{Q}^{O}\) refers to the objective survival probability used by the insurance company to price annuities, and it is public information.
Finally, we say that \(\mathcal{M}^{S}:=(\varOmega \times \varOmega ', \mathcal{F}\otimes \mathcal{F}',\mathbb{P}\times \mathbb{Q}^{S})\) is the probability space for the individual and \(\mathcal{M}^{O}:=(\varOmega \times \varOmega ',\mathcal{F}\otimes \mathcal{F}',\mathbb{P}\times \mathbb{Q}^{O})\) the probability space for the insurance company.
Remark 1
The functions \(\mu ^{S}\) and \(\mu ^{O}\) are given at the outset and are not updated during the optimisation. Updating in a nontrivial way would require the use of a stochastic dynamics for the mortality force which in general would lead to a more complex problem.
2.2 The optimisation problem
Remark 2
Thanks to a result in [1], we show in the Appendix that there is no loss of generality in using stopping times from \(\mathcal{T} _{t,T}\). That is, we obtain the same value in (2.4) as if we were using stopping times of the enlarged filtration \((\mathcal{G} _{t})_{t\ge 0}\), where \(\mathcal{G}_{t}=\mathcal{F}_{t}\vee \sigma ( \{\varGamma _{D}>s\},0\le s\le t)\).
We notice that the state process in our problem formulation (2.7) is a time–space Markov process \((Y_{s})_{s\in [0,Tt]}\) defined by \(Y_{0}=(t,x)\) and \(Y_{s}:=(t+s,X^{x}_{s})\), \(s \in (0,Tt]\).
2.3 The variational problem
3 Properties of the value function
In this section, we provide some continuity properties of the value function and useful probabilistic bounds on its gradient. In what follows, given a set \(A\subseteq [0,T]\times \mathbb{R}_{+}\), we sometimes write \(A\cap \{t< T\}:=A\cap ([0,T)\times \mathbb{R}_{+})\). Also, we make the next standing assumption in the rest of the paper.
Assumption 1
\(\mu ^{S}(\cdot )\) and \(\mu ^{O}(\cdot )\) are continuously differentiable on \([0,+\infty )\).
From (2.7), it is clear that \(V(t,x) \geq G(t,x)\) for all \((t,x)\in [0,T]\times \mathbb{R}_{+}\) so that \(w\) is nonnegative. Moreover, it is straightforward to check that \(w(t,x)\) is finite for all \((t,x)\in [0,T]\times \mathbb{R}_{+}\), thanks to wellknown properties of \(X\) and to Assumption 3.1.
Remark 2
Remark 3
The next proposition starts to analyse the regularity of \(w\) and provides a probabilistic characterisation for its gradient which is crucial for our subsequent analysis of the boundary of \(\mathcal{C}\).
Proposition 4
Proof
The proof is divided into several steps.
Step 3 (gradient bounds). Let \((t,x)\in [0,T)\times \mathbb{R} _{+}\) be a point of differentiability of \(w\). Dividing (3.12) and (3.13) by \(\varepsilon \) and letting \(\varepsilon \to 0\) gives (3.9), as claimed. Moreover, dividing (3.18) by \(\varepsilon \) and letting \(\varepsilon \to 0\), we obtain the lower bound in (3.10). Finally, dividing (3.20) by \(\varepsilon \) and letting \(\varepsilon \to 0\), we get the upper bound in (3.10). □
The next corollary follows by standard PDE arguments used normally in the optimal stopping literature; see e.g. [10, Theorem 2.7.7].
Corollary 5
The next technical lemma states some properties of \(w\) that will be useful to study the regularity of the boundary \(\partial \mathcal{C}\). Its proof is given in the Appendix.
Lemma 6
 (i)
\(x\mapsto w(t,x)\)is nonincreasing for all\(t\in [0,T]\).
 (ii)For any\(t\in [0,T]\), we have$$\begin{aligned} \lim _{x\to \infty }w(t,x)=0. \end{aligned}$$(3.25)
 (iii)
For all\(t_{1}< t_{2}\)in\([0,T]\), we have\(\mathcal{S}\cap ((t_{1},t_{2})\times \mathbb{R}_{+})\neq \emptyset \).
It is worth noticing that (iii) does not exclude that there may exist \(t\in (0,T)\) such that \(\mathcal{S}\cap (\{t\}\times \mathbb{R}_{+})= \emptyset \).
4 Properties of the optimal boundary
In this section, we provide sufficient conditions for the boundary \(\partial \mathcal{C}\) to be represented by a function \(b\) of time. We establish connectedness of the sets \(\mathcal{C}\) and \(\mathcal{S}\) with respect to the \(x\)variable and finally study Lipschitzcontinuity of \(t\mapsto b(t)\). It is worth emphasising that this study is mathematically challenging because of the lack of monotonicity of the map \(t\mapsto b(t)\) and falls outside the scope of the existing probabilistic literature on optimal stopping and free boundary problems. In Sect. 5, we show that the Gompertz–Makeham mortality law (a mainstream model in actuarial science) leads naturally to the set of assumptions that we make below.
Remark 1
(a) Note that if \(g(\cdot )\) and \(K\ell (\cdot )\) in (3.2) have the same sign on \([0,T]\) (i.e., \(\gamma (\,\cdot \,)\le 0\)), then either \(\mathcal{R}=\emptyset \) or \(\mathcal{R}=[0,T]\times \mathbb{R}_{+}\). In the former case, \(\mathcal{S}=[0,T]\times \mathbb{R}_{+}\), while in the latter we have \(\mathcal{C}=[0,T]\times \mathbb{R}_{+}\).
Motivated by (a) in the remark above, we later assume that \(\gamma ( \cdot )>0\) on \([0,T]\).
Now we exhibit sufficient conditions under which the probabilistic representation (3.9) easily provides the shape of the continuation and stopping regions.
Proposition 2
 (i)
If\(g(t)\ge 0\)for all\(t\in [0,T]\), then\(\mathcal{S}=\{(t,x)\in [0,T]\times \mathbb{R}_{+}: x\le b(t)\}\).
 (ii)
If\(g(t)\le 0\)for all\(t\in [0,T]\), then\(\mathcal{S}=\{(t,x)\in [0,T]\times \mathbb{R}_{+}: x\ge b(t)\}\).
Proof
For each \(t\in [0,T]\), we define \(\mathcal{S}_{t}:=\{x\in \mathbb{R} _{+}: (t,x)\in \mathcal{S}\}\) and its complement \(\mathcal{C}_{t}= \mathbb{R}_{+}\setminus \mathcal{S}_{t}\). These are the socalled \(t\)sections of the stopping and continuation set, respectively. Clearly, \(\mathcal{S}_{t}=[0,b(t)]\) under (i) and \(\mathcal{S}_{t}=[b(t),+ \infty )\) under (ii) of the above proposition.
In the rest of the paper, we make the next standing assumption. This will hold in all the results below without explicit mention. In Sects. 5 and 6, we discuss its range of applicability and some extensions.
Assumption 3
Remark 4
It is worth discussing a financial/demographic interpretation of our assumption. We start by noticing that the money’s worth \(f(\cdot )\) should change slowly as a function of time. Indeed, in models with a deterministic mortality force like ours, it is unreasonable to imagine that the individual or the insurer may have drastic changes in their views on survival probabilities (this could instead be the case in stochastic mortality models). Moreover, it is reasonable that an individual who is healthier than the average of the population at the outset will remain so over time, so that the map \(t\mapsto f(t)1\) should not change sign (this again would be less reasonable in stochastic mortality models). Finally, we observe that as \(t\to \infty \), one has \(f(t)\to 1\) because eventually the individual and the insurer should agree on the survival probabilities.
As explained, \(f'(\cdot )\) is small; hence the leading terms in \(g(\cdot )\) are those involving \(\beta \) and \(\theta \rho \) (see (3.3)). In particular, they highlight the interplay between the financial and demographic risk for the individual. On the one hand, \(\beta (t)=\alpha +\mu ^{S}(\eta +t)\) in (2.5) can be interpreted as the dividends arising from the financial investment adjusted by the demographic risk – notice that an individual with high subjective mortality force is less likely to annuitise and would rather enjoy the return on a financial investment; on the other hand, \(\theta \rho \) is closely related to the risk premium of the financial investment. In \(g(\cdot )\), these two terms are weighted exactly with \(1f(\cdot )\) and \(f(\cdot )\), respectively. Moreover, \(g(\cdot )\) in (3.4) represents the marginal gain, per unit of investment in the stock, arising from delaying the annuitisation by one time unit.
In light of all considerations so far (recall that \(f(\cdot )1\) does not change sign and \(f'\) is almost negligible), it is clear from the formula in (3.3) that if \(f(t)>1\) (\(f(t)<1\)) for all \(t\in (0,T)\), i.e., the individual finds the annuity underpriced (overpriced), and \(\theta <\rho \) (\(\theta >\rho \)), then \(g(\cdot )\) remains negative (positive) on \([0,T]\). This conclusion is financially clear because an underpriced annuity and the prospect of a negative risk premium would give the investor a negative marginal gain (per unit of stock) from delaying the annuity purchase (and vice versa).
Finally, if \(f(t)>1\) (\(f(t)<1\)) for all \(t\in (0,T)\) and \(\theta > \rho \) (\(\theta <\rho \)) then \(g(\cdot )\) may change its sign. However, if for example \(t\mapsto \beta (t)\) is monotonic (i.e., the mortality force is monotonic), then the change in sign is unlikely to occur more than once, due to the slow variation in time of the money’s worth \(f\). Motivated by this observation, we conduct numerical experiments in Sect. 5 using Gompertz–Makeham mortality force that suggest that, when a change in the sign of \(g(\cdot )\) occurs, this is likely to be over a time period longer than 20 years. It is therefore mainly for mathematical interest that we allow \(g(\cdot )\) to vanish at time \(T\). Indeed, we will also show in Sect. 6 that this extra flexibility enables us to extend our results to some cases when \(g(\cdot )\) changes its sign once on \((0,T)\).
Remark 5
The positivity of \(\gamma (t)\) in Assumption 4.3 rules out the cases when \(\mathcal{S}=[0,T]\times \mathbb{R}_{+}\) or \(\mathcal{C}=[0,T]\times \mathbb{R}_{+}\) (see point (a) of Remark 4.1).
Notice that Assumptions 3.1 and 4.3 imply \(\gamma \in C(0,T)\). It is not hard to verify that the following properties hold.
Lemma 6
The next simple lemma will be useful in what follows.
Lemma 7
If\(K\ell (t)\le 0\)for all\(t\in [0,T]\), then\(\lim _{x\to 0}w(t,x)=0\)for all\(t\in [0,T]\). Hence\([0,T]\times \{0\}\subseteq \mathcal{S}\).
Proof
Theorem 8
The optimal boundary\(b(\cdot )\)is locally Lipschitzcontinuous on\([0,T)\).
Proof
In this proof, most of the arguments are symmetric when we consider cases (i) and (ii) of Assumption 4.3.
To find the uniform bound on \(b'_{\delta }\), we divide the proof into steps.
Step 2 – Case (ii). Here \(\mathcal{S}_{t}=[b(t),+\infty )\). The analysis in this part is more involved than in the previous case. An argument similar to (4.11) gives \(\mathbb{E}[ \tau _{\delta }]\ge \widetilde{\mathbb{E}}[\tau _{\delta }]\), which unfortunately does not help with the estimate in (4.10). So we need to proceed in a different way.
Lipschitzcontinuity of the boundary has important consequences regarding the regularity of the value function \(w\), which we summarise below.
Proposition 9
The value function\(w\)is continuously differentiable on the set\([0,T) \times \mathbb{R}_{+}\). Moreover, for all\(\varepsilon >0\), the function\(w_{xx}\)is continuous on the closure of the set\(\mathcal{C}\cap \{t< T\varepsilon \}\).
Proof
Corollary 3.5 tells us that \(w_{t}\) and \(w_{x}\) are continuous at all points in the interior of \(\mathcal{C}\) and of \(\mathcal{S}\), and thus it remains to analyse the regularity of \(w\) across the boundary. In order to do that, it is crucial to observe that since \(t\mapsto b(t)\) is locally Lipschitz, the law of the iterated logarithm implies that \(\tau _{*}\) is indeed equal to the first time \(X\) goes strictly below the boundary, in case (i), or strictly above the boundary, in case (ii). In other words, the first entry time to \(\mathcal{S}\) is equal to the first entry time to its interior part. This is an important fact that can be used to prove that \((t,x)\mapsto \tau _{*}(t,x)\) is continuous ℙa.s., and is zero at all boundary points (see e.g. [4, Lemma 5.1 and Proposition 5.2]).
Fix a point \((t_{0},x_{0})\in \partial \mathcal{C}\cap \{t< T\}\) and take a sequence \((t_{n},x_{n})_{n\ge 1} \subseteq \mathcal{C}\) with \((t _{n},x_{n})\to (t_{0},x_{0})\) as \(n\to \infty \). Continuity of \((t,x)\mapsto \tau _{*}(t,x)\) and the discussion above imply that \(\tau _{*}(t_{n},x_{n})\to 0\) ℙa.s. as \(n\to \infty \). The latter and (3.9) and (3.10) give \(w_{x}(t_{n},x_{n})\to 0\) and \(w_{t}(t_{n},x_{n})\to 0\). Since \((t_{0},x_{0})\) and the sequence \((t_{n},x_{n})\) were arbitrary, we get \(w\in C^{1}([0,T)\times \mathbb{R}_{+})\).
Proposition 10
Proof
Remark 11
In general, the map \(s\mapsto e^{\int _{0}^{t+s}r(u)\,du}H(t+s,x)\) is not monotonic. As a consequence, it becomes extremely challenging (if possible at all) to determine whether \(b(\cdot )\) is monotonic or not. For numerical evidence of nonmonotonic boundaries, see Sect. 5.
4.1 Characterisation of the free boundary and the value function
In the next theorem, we find nonlinear integral equations that characterise uniquely the free boundary and the value function. Here we notice that all regularity properties of \(w\) obviously transfer to \(V\) of (2.7), due to (3.1) and (3.5). In particular, we have \(V\in C^{1}([0,T)\times \mathbb{R}_{+})\) and \(V_{xx}\in L^{ \infty }_{\mathrm{loc}}([0,T)\times \mathbb{R}_{+})\), with the only discontinuity of \(V_{xx}\) occurring across \(\partial \mathcal{C}\). It is important to remark that Corollary 3.5 and the remaining properties of \(w\) studied above imply that \(V\) indeed solves (2.8) in the almost everywhere sense (more precisely, at all points \((t,x)\notin \partial \mathcal{C}\)).
Theorem 12
Proof
Here we only show how to obtain (4.29). Then inserting \(x=b(t)\) in (4.29) and using \(V(t,b(t))=G(t,b(t))\), we get (4.30). The proof of uniqueness is standard in the modern optimal stopping literature and dates back to [14] (for more examples, see [15, Chaps. VII and VIII]). However, we provide the full argument in the Appendix for the interested reader.
5 Numerical findings
 (a)

\(f(t)\equiv f>0\) (see (2.6)) and \(\mu ^{S}(\cdot )= \mu (\cdot )\);
 (b)

\(\mu ^{O}(\cdot )=\mu (\cdot )\) and \(\mu ^{S}(\cdot )=(1+\bar{ \mu })\mu ^{O}(\cdot )\) with \(\bar{\mu } \in (1,+\infty )\).
In the first scenario, the money’s worth function (2.6) is constant. If the individual believes she is healthier than the average in the population, then \(\mu ^{S}(\,\cdot \,) < \mu ^{O}(\,\cdot \,)\) and therefore \(f>1\). Conversely, for an individual who is pessimistic about her health, \(\mu ^{S}(\,\cdot \,) > \mu ^{O}(\,\cdot \,)\) and therefore \(f<1\). It is important to notice that the function \(g\) in (3.3) is monotonic increasing (decreasing) if \(f\) is a constant smaller (greater) than 1.
The second scenario uses the socalled proportional hazard rate transformation, introduced in actuarial science by Wang ([20], see also [13]). If \(\bar{\mu }<0\) (resp. \(\bar{\mu }>0\)), the individual considers herself healthier (resp. less healthy) than the average. The limit case \(\bar{\mu }\to 1\) is not relevant in practice as it corresponds to an individual whose life expectancy is infinite. Similarly, the case \(\bar{\mu }\rightarrow +\infty \) is also irrelevant in practice as it corresponds to an individual who believes she is about to die. An important difference between scenarios (a) and (b) above is that in the latter, the money’s worth \(f\) varies over time. In particular, if \(\bar{\mu }<0\) (resp. \(\bar{\mu }>0\)), then \(f(t)>1\) (resp. \(f(t)< 1\)) for all \(t\in [0,T]\).
Unless otherwise specified, in what follows we take \(T=30\), \(\eta =50\), \(\theta =4.5\%\), \(\alpha =3.5\%\), \(\sigma =10\%\) and \(\widehat{\rho }=\rho =4\%\) (we take \(\widehat{\rho }=\rho \) just for simplicity).
Scenario (a). In Fig. 5.1, the function \(g\) in (3.3) is computed for different values of the constant \(f\). As noticed in [9], it is reasonable to expect that the value of \(f\) is close to 1. Notice that if \(f\) is high enough (\(f=1.2\)), \(g\) remains always negative even if \(\theta >\rho \). We observe that \(g\) varies slowly and in most cases it does not change sign, as required in Assumption 4.3. However, if \(g\) changes its sign (at most once, since \(g\) is monotonic), we can still apply our methods as described in more detail in Sect. 6.
6 Final remarks and extensions
As discussed in Remark 4.4, our main technical assumption (Assumption 4.3) is supported by numerical experiments on the Gompertz–Makeham mortality law. The latter is widely used in the actuarial profession; hence it is a natural choice from the modelling point of view. We also notice that Assumption 4.3 allows \(g(T)=0\). This condition is mainly of mathematical interest. Indeed, it enables extensions of our results to cover examples where \(g\) is monotonic on \([0,T]\) and changes sign once. The latter examples are observed in Figs. 5.1 and 5.3 (although the change of sign occurs only on rather long time horizons e.g. \(T>20\) years). On the other hand, it appears that the function \(\ell (\cdot )\) (see (3.3)) is positive in all our numerical experiments.
Here we explain how our results can cover extensions to the case of \(g(\cdot )\) changing sign once. We consider separately the cases \(K<0\) and \(K>0\). From now on, we assume that \(g\) is monotonic and there exists \(t_{0}\in (0,T)\) such that \(g(t_{0})=0\). We also assume that \(\ell (t)> 0\) for \(t\in [0,T]\) and recall ℛ and \(\gamma \) from (4.1) and (4.2).
Case \(K<0\) .
(1) (\(g(\cdot )\)decreasing). In this setting, we have \(\gamma (t)>0\) for \(t\in [0,t_{0})\) with \(\gamma (t)\uparrow +\infty \) as \(t\uparrow t_{0}\). Moreover, ℛ lies above the curve \(\gamma \) on \([0,t_{0})\). For \(t\in [t_{0},T]\), we have \(\mathcal{R}=\emptyset \) and therefore \(\mathcal{S}\cap \{t\ge t_{0} \}=[t_{0},T]\times \mathbb{R}_{+}\) (see Remark 4.1). This implies that \(t_{0}\) is an effective time horizon for our optimisation problem (2.7) since it is optimal to immediately stop for any later time. From a mathematical point of view, this means that we can equivalently study (2.7) with \(T\) replaced by \(t_{0}\). On the effective time interval \([0,t_{0}]\), part (i) of Assumption 4.3 holds and we can repeat the analysis carried out in Sects. 3 and 4.
(2) (\(g(\cdot )\)increasing). In this setting, we have \(\mathcal{R}=\emptyset \) for \(t\in [0,t_{0}]\) while on the interval \((t_{0},T]\), we have \(\gamma (\cdot )>0\) with \(\gamma (t)\uparrow + \infty \) as \(t\downarrow t_{0}\) and ℛ lies above the curve \(\gamma \). We can therefore study problem (2.7) on the restricted time interval \((t_{0},T]\) where our Assumption 4.3 holds. This gives \(\mathcal{S}_{t}=[0,b(t)]\) for \(t\in (t_{0},T]\) and all the results from the previous sections hold.
Moreover, we can show that \(\mathcal{S}_{t}=[0,b(t)]\) also for \(t\in [0,t_{0}]\) (with \(b(t)\) possibly infinite). For that we recall that \(w(t,\,\cdot \,)\) is convex for each \(t\in [0,T]\) (Proposition 3.4) and \(w(t,0+)=0\) (see Lemma 4.7). The latter two properties imply \(\mathcal{S}_{t}=[0,b(t)]\) for \(t\in [0,t_{0}]\) as claimed and \(w_{x}(t,\,\cdot \,)\ge 0\) for all \(t\in [0,t_{0}]\).
In summary, \(\mathcal{S}_{t}=[0,b(t)]\) for all \(t\in [0,T]\) and most of the analysis in Sect. 4 carries over to this setting. However, it should be noted that the methods used in Theorem 4.8 only allow to establish Lipschitzcontinuity of \(b\) on \([t_{0},T]\). A complete study of the boundary on \([0,t_{0}]\) requires new methods, and we leave it for future work.
Case \(K>0\) .
(1) (\(g(\cdot )\)decreasing). Here \(\mathcal{R}\cap \{t\le t _{0}\}=[0,t_{0}]\times \mathbb{R}_{+}\), so that it is optimal to delay the annuity purchase at least until \(t_{0}\), regardless of the dynamics of the fund’s value, because \(\mathcal{C}\cap \{t\le t_{0}\}=[0,t_{0}] \times \mathbb{R}_{+}\). On \((t_{0},T]\), we find that \(\gamma (\cdot )>0\) with \(\gamma (t)\uparrow +\infty \) as \(t\downarrow t_{0}\) and ℛ lies below the curve \(\gamma \). From a mathematical point of view, this means that we only need to study our problem (2.7) on the restricted time interval \((t_{0},T]\) where our Assumption 4.3 holds.
(2) (\(g(\cdot )\)increasing). Here \(\mathcal{R}\cap \{t\ge t _{0}\}=[t_{0},T]\times \mathbb{R}_{+}\) so that for \(t\ge t_{0}\), it is optimal to delay the annuity purchase until the maturity \(T\), regardless of the dynamics of the fund’s value, because \(\mathcal{C}\cap \{t \ge t_{0}\}=[t_{0},T]\times \mathbb{R}_{+}\). On \([0,t_{0})\), we find that \(\gamma (\cdot )>0\) with \(\gamma (t)\uparrow +\infty \) as \(t\uparrow t_{0}\) and ℛ lies below the curve \(\gamma \).
Noticing that \(H(t,x)\) and \(v(t_{0},x)\) are linear in \(x\), the asymptotic behaviour of \(v(t,x)/x\) as \(x\to \infty \) and convexity of \(v(t,\cdot )\) (see Proposition 3.4) suggest that for a fixed \(t\in [0,t_{0})\), we should have \(\mathcal{S}_{t}=[b_{1}(t),b _{2}(t)]\) with \(b_{2}(t)\) possibly infinite. This, however, leaves open several questions concerning the actual shape of \(\mathcal{S}\) and the regularity of its boundary. A complete answer to these questions requires the use of different methods, and we leave it for future work.
6.1 A comment on the regularity of the optimal boundary
One of the main mathematical challenges in this work was the lack of monotonicity for the optimal boundary, which we managed to overcome by showing that the boundary is indeed locally Lipschitz. While our methodology only relies on stochastic calculus, the idea of looking at the implicit function theorem and to provide bounds on \(b'_{\delta }\) in (4.6) somehow comes from PDEs (we refer to [5] for a more extensive review on the topic). In particular, we were inspired by [17] where a variational inequality associated to an optimal stopping problem^{3} is studied (see Eq. (1.3) therein). Interestingly, our assumptions are weaker than those contained in [17, Sect. 2, Eqs. (2.1)–(2.3)], as we explain in more detail in the next paragraph. In this sense, our probabilistic method extends results which were obtained in [17] with purely analytical tools. The parallel with our notation is better understood if we use the problem formulation (3.5), although using (2.7) is clearly equivalent.
Footnotes
Notes
Acknowledgements
We thank three anonymous referees, whose valuable comments helped improving the quality of our paper.
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