2.1. Benefits and GSA fund dynamics

We consider a homogeneous GSA pool consisting of n individuals aged x joining the fund at time 0, each of them providing an initial capital amount c. The total pool fund then amounts to $F_0=c\cdot n$ at time 0.

At anytime, the GSA fund value, $F_{t}$ , evolves according to the return on investment and the benefits paid out, the latter depending on the mortality experienced by the pool and realized investment returns. In what follows, we define the GSA fund dynamics in continuous time.

The number of surviving members at time t is $N_t$ and follows a pure death process mediated by a stochastic (or deterministic) transition intensity. More specifically, the transition rate from i to $i-1$ policyholders at time t is $i\mu_{x+t}$ where $\mu_{x+t}$ is the force of mortality for a life aged $x+t$ . Clearly, $N_0=n$ . If a death occurs at time t, then $dN_t=-1$ , otherwise $dN_t=0$ . Since $dN_t$ can be viewed as a Poisson decrement process, the probability of multiple deaths is of order o(dt) (Ross, Reference Ross2014) and can be neglected.

The GSA scheme pays living benefits to the surviving participants; we further incorporate the possibility of paying out death benefits upon member’s death in our framework. We assume that living benefits are continuously paidFootnote 1; consistently with the GSA rationale, they are not guaranteed, but their amount is assessed at the time of payment, so as to keep the actuarial balance in respect of the current GSA fund value $F_t$ . Let $\overline{a}_{x+t}$ denote the actuarial value at time t of a unitary annuity. Then, the total living benefit amount paid by the GSA scheme at time t is

(2.1) \begin{align} B_t=\frac{F_t}{\overline{a}_{x+t}},\end{align}

and

(2.2) \begin{align} L_t=\frac{B_t}{N_t}\end{align}

represents the amount cashed by each survivor at time t.

The actuarial value of the annuity $\overline{a}_{x+t}$ is computed taking deterministic assumptions about the discount rate and the force of mortality. Then $\overline{a}_{x+t} = e^{-\int_t^\infty (r + \mu_{x+s})\, ds}$ , where r is the (flat) short rate and $\mu_{x+s}$ is assumed to be deterministic. We point out that this definition of $\overline{a}_{x+t}$ implies that no future risk is accounted for in the living benefits assessed at time t; this is a natural choice within a GSA arrangement, where risks are retained by the surviving participants.

As suggested in many studies (see, e.g., Modigliani, Reference Modigliani1986; Brown, Reference Brown2009), individuals have bequest preferences, at least up to some (old) age. We then incorporate death benefits paid by the GSA scheme as members die. Upon one member’s death at time t, an amount equal to

(2.3) \begin{align} D_t=\beta\, \frac{F_t}{N_{t^-}}\end{align}

is paid to the member’s beneficiaries. Here, $\beta$ is a proportion of the fund value, and $N_{t^-}=N_t-dN_t$ is the number of survivors an instant before time t; then $\frac{F_t}{N_{t^-}}$ can be referred to as the notional individual’s share of the GSA fund at time t, prior to the deaths reported at that time. We note that, similar to living benefits, the amount of death benefits is not guaranteed either. It is also important to note that while death benefits meet bequest preferences, they reduce the possibility for the GSA arrangement in achieving a satisfactory pooling effect. For instance, if $\beta=1$ no mortality credits are left and the GSA scheme becomes a purely financial arrangement, where individual members fully retain their respective longevity risk. Conversely, if $\beta=0$ all the money is retained by the GSA fund upon a member’s death, and the individual longevity risk is pooled within the fund. While $\beta=0$ is acceptable, $\beta=1$ is not realistic in the logic of a GSA arrangement. In view of practical applications, we will access scenarios involving low values of $\beta$ in our numerical illustrations.

In this paper, we assume that the GSA fund is proportionally invested in equity and riskless cash. We denote with $S_t$ the equity price process, r the risk-free return of a cash account and $w_t$ the proportion of the fund invested into equity (while $(1-w_t)$ is the proportion invested in the cash account).

Under this setting, the GSA fund dynamics can be described as follows:

(2.4) \begin{align} dF_t = \left[w_t\frac{dS_t}{S_t} + (1-w_t)rdt\right]F_t - B_tdt + D_tdN_t,\end{align}

where $dS_{t}$ denotes the instantaneous change of the equity process over a time increment, dt. For a precise interpretation of Equation (2.4), and related expressions below, recall that $dN_t$ either takes a 0 or a negative value; then, $D_t\,dN_t$ denotes an outflow for the GSA fund.

2.2. The components of the GSA fund

As is well known, annuity benefits are funded by the initial capital (and this is trivial), investment returns and mortality credits. In a GSA arrangement, neither investment returns nor mortality credits are guaranteed. The latter, in particular, depend on the realized mortality rate and the death benefit paid out by the scheme. In order to better understand the dynamics of the living benefits, it is convenient to split the GSA fund into three components as follows:

1. 1. principal, that is, consumption of the fund (denoted as $F_t^1$ );

2. 2. interest ( $F_t^2$ );

3. 3. mortality credit ( $F_t^3$ ),

so that we have the fund value at time t as

(2.5) \begin{align} F_t=F_t^1+F_t^2+F_t^3 ,\end{align}

apart from time $t=0$ , when $F_0^1=F_0=n\cdot c$ , while $F_0^2=F_0^3=0$ .

A decomposition similar to (2.5) can be developed for benefits, living benefits in particular; for example (see Equations (2.1) and (2.2)), $L_t=L_t^1+L_t^2+L_t^3$ , where $L_t^i=\frac{F_t^i}{\overline{a}_{x+t}\,N_t}$ . Understanding the dynamics of components of the GSA fund thus provides information about the benefit dynamics.

Before describing the dynamics of each of these components, we note that all three contribute to the funding of the living and death benefits at time t. We assume, in particular, that they contribute proportionally to both. Then, from Equation (2.4) we derive that $\frac{B_t\,dt}{F_t}\cdot F_t^i$ is the living benefit covered by the fund component i ( $i=1,2,3$ ), while $\frac{D_t\,dN_t}{F_t}\cdot F_t^i$ is the death benefit covered by the same fund component.

We assume that the principal, $F^1_t$ , is simply the part of the fund value that can be attributed to the initial capital, without taking investment returns into account. Therefore, as already noted, at $t=0$ we have $F^1_0 = F_0$ . As mentioned above, the fund component $F_t^1$ contributes to the funding of the living and death benefits. Further, it must contribute to the mortality credits. The dynamics of $F_t^1$ can be defined as follows:

(2.6) \begin{align} dF_t^1 &= - \frac{B_t\,dt}{F_t}\cdot F_t^1+\frac{D_t\,dN_t}{F_t}\cdot F_t^1+(1-\beta)\cdot \frac{dN_t}{N_t}\cdot F_t^1 \nonumber \\ & = -\frac{F_t^1}{\overline{a}_{x+t}}\,dt+\frac{dN_t}{N_t}\,F_t^1.\end{align}

In order to interpret $(1-\beta)\cdot \frac{dN_t}{N_t}\cdot F_t^1$ , recall that $dN_t=-1$ if a death occurs, otherwise $dN_t=0$ . This means that $(1-\beta)\cdot \frac{dN_t}{N_t}\cdot F_t^1$ is either 0 or an outflow for $F_t^1$ . We can interpret it as the contribution of $F_t^1$ to the mortality credits.

The interest component, $F^2_t$ , is the cumulative amount of investment gains and losses incurred by the fund. This component changes by $(w_t \frac{dS_t}{S_t} + (1-w_t)r_t)dt)\cdot F_t$ in each time-interval dt. However, part of the benefits must be covered with the interest gained up to time t, so that the absolute amount of accumulated investment returns decreases as living and death benefits are paid out. The dynamics of $F_t^2$ can be described as follows:

(2.7) \begin{align} dF_t^2 &= \left(w_t \frac{dS_t}{S_t} + (1-w_t)r_t dt\right)F_t- \frac{B_t\,dt}{F_t}\cdot F_t^2+\frac{D_t\,dN_t}{F_t}\cdot F_t^2+(1-\beta)\cdot \frac{dN_t}{N_t}\cdot F_t^2 \nonumber \\& = \left(w_t \frac{dS_t}{S_t} + (1-w_t)r_t dt\right)F_t-\frac{F_t^2}{\overline{a}_{x+t}}\,dt+\frac{dN_t}{N_t}\,F_t^2.\end{align}

Similarly to the case for $F_{t}^{1}$ , $(1-\beta)\cdot \frac{dN_t}{N_t}\cdot F_t^2$ can be interpreted as the contribution of $F_t^2$ to the mortality credits.

Mortality credits, $F^3_t$ , accrue when an annuitant dies and correspond to the part of the fund (including interest) notionally belonging to the individual which is not paid back as a death benefit. As discussed above, when an annuitant dies at time t, an amount is transferred from the principal ( $F^1_t$ ) and interest components ( $F^2_t$ ) into mortality credits ( $F^3_t$ ). The dynamics of $F_t^3$ can be described as follows:

(2.8) \begin{align} dF_t^3 & = - \frac{B_t\,dt}{F_t}\cdot F_t^3+\frac{D_t\,dN_t}{F_t}\cdot F_t^3-(1-\beta)\cdot \frac{dN_t}{N_t}\cdot \left(F_t^1+F_t^2\right) \nonumber \\ & = -\frac{F_t^3}{\overline{a}_{x+t}}\,dt-\frac{dN_t}{N_t}\,F_t^3 +(1-\beta)\cdot \frac{dN_t}{N_t}\cdot \left(F_t^3-F_t^1-F_t^2\right).\end{align}

It can easily be checked that $dF_t^1+dF_t^2+dF_t^3=dF_t$ .

3.1. The equity model

We assume that the equity process, $S_{t}$ , evolves according to Heston (Reference Heston1993) stochastic volatility model

(3.1) \begin{align} dS_{t} &= \mu S_{t}dt + \rho\sqrt{v_{t}}S_{t}dW_{t}^1+\sqrt{1-\rho^2}\sqrt{v_{t}}S_{t}dW_{t}^2,\end{align}

(3.2) \begin{align} dv_{t} &= \kappa (\theta - v_{t})dt +\sigma_u\sqrt{v_t}dW_{t}^1, \end{align}

where $\mu$ is the instantaneous return of the equity processFootnote 2, $v_{t}$ is the instantaneous variance of $S_{t}$ which is a mean reverting process whose long-term average, speed of mean reversion and standard deviation are $\theta$ , $\kappa$ and $\sigma_u$ , respectively. As presented in Feller (Reference Feller1951), for Equation (3.2) to be a positive process, the condition $2\kappa\theta \geq \sigma_{u}^{2}$ has to be satisfied. Here, $dW_{t}^{1}$ and $dW_{t}^{2}$ are correlated Brownian motion increments for the equity and variance processes whose correlation is denoted as $\rho$ . An Euler scheme is adopted in implementing Equations (3.1) and (3.2). The fund allocates a proportion $w_t$ into equity and $(1-w_t)$ into a cash account which evolves deterministically according to $dC_t = r\,C_t$ . The parameters for the Heston model have been adapted from Andersen et al. (Reference Andersen, Benzoni and Lund2002) for all numerical illustrations presented in this paper.Footnote 3

3.2. The target volatility strategy

At any instant during the life of the fund and as benefits are paid, the weights of the equity and cash holdings are dynamically rebalanced so as to maintain a target volatility level of the fund. The target volatility strategy is a self-insurance strategy which facilitates more equity holdings during low volatility periods, and vice versa during high volatility periods. Constraining volatility of the fund around a given target enhances investment return targets to be attainable. We rebalance the weights of the equity by setting

\begin{align*} w_t' = \min{\bigg(\frac{TV}{\sqrt{\hat{\sigma}^2(t)}},1\bigg)},\end{align*}

where TV is the exogenously specified (annualized) target volatility and $\hat{\sigma}(t)$ is an estimated volatility level of equity returns between any two adjacent rebalancing points, see Morrison and Tadrowski (Reference Morrison and Tadrowski2013). In all the numerical illustrations which follow, we will implement an Euler scheme for the fund dynamics where the time domain is discretized into incremental time steps. We will jointly assume that the fund is rebalanced weekly and benefits (to surviving members or as death benefits) paid out immediately before rebalancing. That is, the proportional investment in equity and cash will be dynamically adjusted every week in line with changing market conditions, so as to maintain or achieve a target volatility level of the fundFootnote 4. Based on historical observations, the fund manager keeps track of the exponentially weighted moving average (EWMA) of the volatility such that

(3.3) \begin{align} \hat{\sigma}^2(t+\Delta t) = \lambda \cdot \hat{\sigma}^2(t) + \frac{(1-\lambda )}{\Delta t} \bigg( \ln \left( \frac{S_{t+\Delta t}}{S_t} \right) \bigg)^2,\end{align}

where $\Delta t = \frac{1}{52}$ which corresponds to weekly rebalancing (Engle, Reference Engle1982). The EWMA incorporates all prior observations, but with exponentially declining weights through time, whose rate of decay is detected by the parameter $\lambda$ . A higher value of $\lambda$ implies that the estimate volatility reacts slower to recent changes in the equity returns, thus putting more weight on past observations.

For all our numerical illustrations, we adopted the parameter set for the Heston stochastic volatility model presented in Andersen et al. (Reference Andersen, Benzoni and Lund2002) fitted using S&P 500 index returns. These parameters have been reproduced in Table 1 for completeness.

Table 1. Parameters of the Heston stochastic volatility model and the decay parameter for the EWMA. These parameters are for illustrative purposes and one can fit either the Heston model or adapt any equity modelling process to dataset of interest using standard techniques like that presented in Andersen et al. (Reference Andersen, Benzoni and Lund2002).

3.3. Illustrative performance of a target volatility strategy

In assessing the performance of the target volatility strategy, we compare the dynamic strategy with the static case which involves preassigning fixed weights to the equity-cash holding at initial time where the equity dynamics is governed by the Heston stochastic volatility process. The weights for the static case are chosen such that the initial fund volatility equates to the corresponding target volatility of the dynamic strategy. The static strategy works as follows: at $t=0$ , the fund manager estimates the long run volatility to be $\hat{\theta}$ and elects a constant proportion strategy in which $w_t = \frac{TV}{\sqrt{\hat{\theta}}}$ is allocated to equity and the remainder to cash. This strategy is static in the sense that the proportion $w_{t}$ is determined in advance and is not rebalanced. A detailed algorithm for implementing the static strategy is illustrated in Algorithm 1 of Appendix A.

In contrast, the target volatility strategy continuously updates $w_{t}$ based on observed returns. To be more precise, when the static weight allocation is $w^{static}$ , we will compare it to a dynamic strategy with target volatility $w^{static} \times \sqrt{ \theta }$ , where $\theta$ is the long run variance of the equity market. This facilitates a fair comparison between the ‘rewards’ of two strategies by ensuring that they have a similar level of ‘risk’. The dynamic target volatility strategy can be implemented by replicating the pseudo code outlined in Algorithms 1 and 2 of Appendix A. Table 2 shows the relationships between $w^{static}$ and the equivalent volatility based on our selected value of $\theta$ .

An illustrative example with sample paths for dynamic target volatility strategy and static Heston stochastic volatility case is depicted in Figure 1. The horizontal axis of all subplots of Figure 1 is in years. Sample paths in Figure 1(a) show that the dynamic rebalancing strategy offers improved returns relative to the static case in the long run. This is consistent with existing literature such as Morrison and Tadrowski (Reference Morrison and Tadrowski2013), Doan et al. (Reference Doan, Papageorgiou, Reeves and Sherris2018) and Li et al. (Reference Li, Labit Hardy, Sherris and Villegas2019) who note that target volatility strategies limit losses by disinvesting from equities during high volatility periods and vice versa. Figure 1(b) shows a typical sample on how the weights of the dynamic trading strategy change through time due to changing market conditions. As reflected in Figure 1(c), dynamic rebalancing of the equity-cash holdings ensures that the fund volatility is always constrained around a targeted level.

Table 2. Relationship between static equity allocation and ‘equivalent’ dynamic target volatility.

Figure 1(d) shows a simulated path of the volatility process in blue depicting a typical static case. Applying an exponential weighting used here for constraining volatility towards the target level yields the red plot. The red plot has less variability relative to the unconstrained case.

Figure 1. Illustrative simulation: Monthly target volatility rebalancing (‘dynamic’) is supposed to ensure that the model volatility remains close to the target. Rebalancing decisions are based on EWMA estimates of the volatility, since the true volatility is not observable in the market. The dynamic strategy offers some improvements in returns but is limited in actually constraining the volatility of the fund.

3.4. The mortality model

For all illustrations presented in this paper, we adopt the Gompertz–Makeham mortality law which expresses the force of mortality as

(3.4) \begin{align} \mu_{x} = a + e^{b_1 + b_2x},\end{align}

where a, $b_{1}$ and $b_{2}$ are constant parameters which are determined through calibrating to mortality data. In this paper, we calibrate the model to the US male mortality data for ages 50–110 for the cohort born in 1915 with the data extracted from the Human Mortality Database.Footnote 5 By fitting the Gompertz–Makeham mortality law using the steps presented in Appendix B, the resulting parameters are presented in Table 3. The corresponding mortality and survival curves are shown in Figure 2. From the left panel of this figure, we note that the fitted mortality rates track the realized mortality rate very well. As with the financial model, the Gompertz–Makeham law is adopted here for illustrative purposes. Any mortality model (either deterministic or stochastic) can be adapted to the approach presented in this paper.

Table 3. Gompertz–Makeham estimated parameters for US male mortality, ages 50–110.

Figure 2. Plots for the fitted force of mortality and survival function for the US male data for ages 50–110 for the cohort born in 1915.

4.1. Benchmark case with 70% equity allocation at initial time

This subsection presents the benchmark case with equity allocation of 70% at initial time corresponding to a target volatility of 12% per annum. Figure 3 shows trajectories of living benefits for varying death benefit payments. In each subplot, summary statistics of living benefits (namely, the median, the $10{\rm th}$ and the $90{\rm th}$ percentile) are plotted in the case of a static (red) and a dynamic (blue) target volatility strategy. All subplots reveal that the dynamic target volatility strategy consistently pays out higher living benefits compared to the static volatility strategy, which does not adjust the equity and cash composition due to changing market conditions. In particular, we point out that while the lower quantiles of living benefits do not differ that much in respect of the investment strategy (with the dynamic strategy usually providing higher benefit amounts), the median and the upper quantiles show much more favourable paths in the case of a dynamic strategy.

Figure 3. Individual living benefit payment ( $L_t$ ) quantiles for various death benefit proportions when the static equity weight = 70% and target volatility = 12%. Quantiles for the static case are in red and those for the dynamic case are in blue.

Figure 4. Quantiles for the principal component when the static equity weight = 70% and target volatility = 12%.

In respect of the death benefit (see Figure 3(a)–(d)), whose size is defined by the proportion $\beta$ , clearly higher death benefit payments result in reduced living benefit payments through time, as the fund value will be proportionally reduced due to lower mortality credits. This is reflected in Figure 3 where living benefits are decreasing as the benefit payments increase. When $\beta \geq 50\% $ , living benefit payments may fall below initial payments as revealed in Figure 3(c) and (d). Having adopted a static or a dynamic target volatility strategy does not seem to be significant in terms of the relative reduction of the living benefit amount when death benefits are included.

4.2. Analysis of different components of the GSA

The framework adopted in the paper makes it readily possible to analyse the fund dynamics at anytime in terms of the principal component, investment returns and mortality credits, as presented in Equations (2.6)–(2.8). Figures 4–7 present typical trajectories of these respective components through time. From Figure 4 we note that, on average, the principal repayments are consistent and exponentially decrease through time in a predictable fashion, and independent of the investment strategy. This is also apparent in Equation (2.6), as the stochastic differential equation for the principal component of the fund depends only on the mortality experience (that we are assuming to be deterministic; this is why summary statistics coincide in Figure 4). Minor differences visible in Figure 4 are due to discretization errors from Monte Carlo simulation.

Figure 5. Interest component quantiles for varying death benefit proportions when the static equity weight = 70% and target volatility = 12%.

Figure 6. Mortality credit quantiles for varying death benefit proportions when the static equity weight = 70% and target volatility = 12%.

Figure 7. Proportion of living benefit contributions from the principal, interest and mortality credits component for varying death benefits and investment strategies.

The interest component dominates during the initial stages of the pool with strategies for cases involving lower death benefit payments dominating those with higher as depicted in Figure 5(a)–(d). From these subplots, it is worth noting that regardless of how much is paid out as death benefits, interest accumulates aggressively until the $15{\rm th}$ year upon which the pool size starts to deteriorate and thus impacting more for cases within high death benefit payouts. Consistent with findings presented in Figure 3, the dynamic strategy returns dominate those obtained under a static strategy.

Mortality credits increase with increasing number of deaths as reflected in Figure 6(a)–(d). Again, the dynamic strategy yields higher mortality credits due to enhanced performance from the dynamic target volatility strategy. We note that as age increases, the variability in mortality credits returned also increases. This is reflective of a smaller pool size resulting in higher exposure to random fluctuations in mortality. As $\beta$ increases, we observe that the variability of mortality credits reduces. This is because the payment of death benefits is negatively correlated with survival benefits. Furthermore, these results provide comfort that mortality credits offer a useful qualitative measure of the variability in annuity payments that is attributable in part to the survival experience of the pool.

Figure 7 summarizes the contribution of each of these components to living benefits. During the initial phase of the pool, principal contributions make up the greater portion of benefit payouts with mortality credits gradually increasing as pool participants die through time. Investment returns exhibit a concave pattern, increasing to a climax before gradually decreasing as the pool size and principal depletes.

Figure 7 also reveals that interest accumulates at a faster rate on a target volatility strategy compared to associated static trading strategy cases. We note from Figure 7(a) that principal and interest equally contribute to living benefit payments much earlier compared to Figure 7(b) where interest matches principal components after the $10{\rm th}$ anniversary under the current setting. In addition to this, the interest component proportion attains a higher value under the dynamic case compared to the static case. We note that higher investment returns could help mitigating the risk of lower mortality credits arising in a scenario with unanticipated longevity improvements (which is a situation not included in our assessments).

4.3. Comparison of living benefits for varying equity compositions

Having presented the benchmark case with an initial strategy consisting of 70% equity and 30% cash investments, we now perform sensitivity analysis aimed at revealing the impact of various investment strategies on living and death benefits. In what follows, all analysis will be performed relative to the 70% equity strategy whose corresponding target volatility is 12% per annum. Table 4 presents living benefit quantiles for the benchmark case across various ages for the target volatility and static investment strategies. As noted from Figure 3, the target volatility strategy dominates the static case across all ages implying that a target volatility strategy enhances living benefits for pool participants.

Table 4. Individual living benefits for the base case with 70% equity allocation and $\beta=0$ .

Tables 5–7. present comparison results for living benefit quantiles across various ages and equity compositions relative to the results presented in Table 4. From these tables, we note that when the initial equity composition is 90% of the fund, the median and $90{\rm th}$ percentile are superior to those for the 70% initial equity strategy, while the $10{\rm th}$ percentile is lower than in the benchmark case. Note that corresponding to the 90% initial equity composition is an initial volatility of 16% for both the static and target volatility strategy, compared to the initial volatility of 12% for the benchmark case (see Table 2 for initial volatilities associated with initial equity compositions). When the strategy is less aggressive with more fund holdings invested in cash and less in equity, the living benefits decrease as reflected on columns corresponding to less equity holding in Tables 5–7.

Table 5. Relative individual living benefits at Age 75 for varying initial allocations and $\beta=0$ . In this table, quantile comparisons are performed relative to the benchmark case presented in Table 4.

Table 6. Relative individual living benefits at Age 80 for varying initial allocations and $\beta=0$ . In this table, quantile comparisons are performed relative to the benchmark case presented in Table 4.

Table 7. Relative individual living benefits at Age 85 for varying initial allocations and $\beta=0$ . In this table, quantile comparisons are performed relative to the benchmark case presented in Table 4.

As would be expected, across all ages presented in these tables, lower equity proportion implies lower expected value of living benefits and lower upper quantile. For the lower quantile, it turns out to be initially increasing, and then decreasing with respect to the reduction of the equity weight. The switch of the path occurs at different ages, depending on the equity weight. There is a trade-off between equity proportion and impact on the lower quantile, which suggest an optimal proportion of the equity.

Strategies with less equity holdings have low payout structures due to the limited performance of the underlying fund. In all cases (i.e., across all $w^{static}$ weights presented in Table 2), the expected value of the living benefits is higher in the dynamic setting, as well as the higher quantile, whereas the lower quantile is approximately the same in the dynamic and static case. When $w^{static}=0$ , that is when all the money is invested in cash, the living benefits tend to decrease, due to longevity cost not being adequately compensated by investment returns. In general, when there is less equity, there is less volatility in benefit amounts, but they are lower, even decreasing.

4.4. Conservative strategy from age 85

We now assess the behaviour of living benefits when the investment strategy becomes conservative with the entire fund invested in cash from age 85 and beyond. Figure 8 presents subplots for varying death benefit payments. Benefits for both target and static volatility strategies are increasing till age 85 and then flatten out beyond this age.

Figure 8. Living benefit payment quantiles for varying death benefit proportions when switching to all cash from age 85 when static strategy equity weight = 70% and target volatility = 12%.

A switch to cash yields a smooth and more stable living benefit payout pattern at the expense of potentially enhanced benefits from equity participation. Comparing Figure 3(a) with Figure 8(a), one notes that pool participants surviving to advanced ages tend to receive higher living benefits enhanced by equity investment returns from the unconstrained case. Limiting the fund to cash only investment strategy compromises its potential as reflected in Figure 8. This is also revealed in Figure 9 showing the investment return contributions to the fund for varying death benefits. Due to depleting fund value and low returns associated with cash investments, we note that all investment return contributions exponentially decrease beyond age 85.

Figure 9. Interest component quantiles for varying death benefit proportions when switching to all cash from age 85 when static strategy equity weight = 70% and target volatility = 12%.

Comparing Figures 6 and 10, one easily notes that the contribution of mortality credits to living benefits under both scenarios is equivalent as the two cases involve the same pool participants with similar mortality developments.

Figure 10. Mortality credit quantiles for varying death benefit proportions when switching to all cash from age 85 when static strategy equity weight = 70% and target volatility = 12%.

4.5. Comparison between different payout policies

To have a better perspective of the superiority of the target volatility strategy, we present Figure 11 showing comparisons of the contribution of investment returns to living benefits for the case where there are no death benefits. We assess trajectories of investment returns for the case where all funds are switched to cash from age 85 and the target volatility strategy for the whole duration.Footnote 6 Figure 11(a) and (b) is the same up to age 85, when in Figure 11(a) all the equity is disinvested. While returns are less dispersed after the switch, the investment return contribution significantly drops, due to forgoing the potential high returns of equities. This is a striking insight, particularly in respect of a more traditional life cycle investment approach, suggesting to disinvest equity as individuals age.

Figure 11. Comparison of investment returns between different payout policies during the tenure of the GSA plan when static strategy equity weight = 70% and target volatility = 12%.

4.6. Effects of mortality dynamics on living benefits

In this paper, we have assumed the Gompertz–Makeham mortality law which is a deterministic mortality model, implying a systematic way in which participants leave the GSA pool. In reality, mortality rates evolve in an unpredictable fashion with a great deal of research having been done on stochastic mortality modelling in both discrete and continuous time settings.Footnote 7 In as much our focus is on illustrating the effectiveness of the target volatility strategy in enhancing living benefit throughout the life of the fund, we can as well assess the impact of varying mortality rates on living benefit profiles. We accomplish this by shocking fitted parameters for the mortality process in Equation (3.4) so as to realize mortality rates which are $\pm$ 10% than those used in the preceding subsections.

Table 8 presents relative individual benefit comparisons between the benchmark case of Table 4 and the case where realized mortality is 10% lower across all ages. As reflected in Table 8, we note that irrespective of the investment strategy, mortality improvements result in lower living benefit payments across all ages. When realized mortality is higher than expected, we note higher living benefits across all ages irrespective of the investment strategy as reflected in Table 9 with all quantiles dominating the benchmark case.

Table 8. Relative individual living benefits for 70% initial equity allocation and $\beta=0$ when realized mortality is 10% lower across all ages. In this table, quantile comparisons are performed relative to the benchmark case presented in Table 4.

Table 9. Relative individual living benefits for 70% initial equity allocation and $\beta=0$ when realized mortality is 10% higher across all ages. In this table, quantile comparisons are performed relative to the benchmark case presented in Table 4.

For completeness, we present Appendix C which illustrates cases involving non-zero death benefit payouts for the 70% initial equity investment strategies with $\beta=20\%$ as presented in Tables C.1–C.2. In generating Tables C.2 and C.3, we have assumed corresponding quantiles in Table C.1 as reference and computed the relative differences to facilitate comparisons. Table C.2 corresponds to the case where realized mortality is 10% lower than expected with Table C.3 being the case with realized mortality being 10% higher than expected. As highlighted in Subsection 4.2, death benefit payouts result in lower living benefits across all ages.Footnote 8