2.1. Notation and model specification

This paper focuses on a defined benefit pension (TBP) plan with a discrete-time stochastic nature and decision-making process. The benefit payments to each retiring cohort are determined by an exogenous salary process whose source is random and which may be correlated with the financial market. The pension fund invests in a combination of a risk-free asset and multiple risky assets. The plan trustees aim to adjust the benefit payments to stay close to the target while avoiding excessive borrowing or leaving an excessive surplus for future cohorts. The study considers a discrete-time horizon from time 0 to T, divided into intervals of length one unit, $[k,k+1),$ where k ranges from 0 to $T-1$ .

The notations used in this paper are listed below, for time k,

• $(\Omega,\mathcal{F},\mathcal{P})$ is a complete probability space, where $\mathcal{F}\triangleq\{\mathcal{F}_{k}\, \textrm{for}\, k=0,1,..., T\}$ is the natural filtration generated by the processes for the securities in the economy;

• ${E_{k}}[{\cdot}]=E[{\cdot}\left|{\mathcal{F}_{k}}\right.]$ and ${Var_{k}}[{\cdot}]=Var[{\cdot}\left|{\mathcal{F}_{k}}\right.]$ represent, respectively, the expectation and variance operators under the condition of information set $\mathcal{F}_{k}$ .

• $x_{k}$ denotes the fund wealth;

• $A_k$ denotes the total number of active members in the pool, while $R_k$ is the total number of retiring members at k;

• $B_{k}$ denotes the total benefits distributed to retired members and $B_{k}^{*}$ denotes its corresponding target;

• $c_{k}$ denotes the fixed proportion of one’s wage that is contributed to the fund;

• $y_{k}$ denotes the average wage of the active members, and $p_{k}$ denotes the stochastic ratio of the average wage over the period $[k,k+1)$ , that is $y_{k+1}=p_{k} y_{k}$ ;

• $\mathcal {C}_{k}$ denotes the total contributions from the active members, then $\mathcal {C}_{k}=c_{k}y_{k}A_{k}$ ;

• $r_{k}$ denotes the deterministic gross rate of return of the risk-free asset, $e_{k}=(e_{k}^{1},...,e_{k}^{n})^{\prime}$ Footnote 5 denotes the vector of the stochastic gross rate of return from n risky assets, and we define

  \begin{equation*}\theta_{k}=e_{k}-r_{k}\textbf{{1}}, \quad {\eta_{k}}=\left({\begin{array}{c}{e_{k}}\\{p_{k}}\end{array}}\right);\end{equation*}

• $\mathcal{{S}_{+}}$ and $\mathcal{S}_{++}$ denote the set of positive semidefinite and positive definite matrices, respectively;

• For any symmetric matrices $\textrm{X}$ and $\textrm{Y}$ with the same order, we denote $\textrm{X}\ge\textrm{Y}$ if only if $\textrm{X}-\textrm{Y}\in\mathcal{S}_{+}$ ; and $\textrm{X}>\textrm{Y}$ if only if $\textrm{X}-\textrm{Y}\in\mathcal{S}_{++}$ . In particular, $\textrm{X}\ge0$ if only if $\textrm{X}\in\mathcal{S}_{+}$ ; and $\textrm{X}>0$ if only if $\textrm{X}\in\mathcal{S}_{++}$ ;

• $u_{k}=\left(u_{k}^{1},...u_{k}^{n}\right)^{\prime}$ denotes the vector for the amount of the fund’s wealth invested in the n risky assets. We point out that there is no exogenous injection or withdrawal of money throughout the running of our TBP fund: inflows result solely from members’ contributions, and outflows result from retirement withdrawals.

We make the following assumption in accordance with the positive nature of the average wage.

It is worth noting that we do not impose any particular parametric assumptions on $p_k$ . This reflects the stochastic nature of salary inflation, which has a direct impact on the fund’s wealth. The fund’s wealth process, denoted by $x_k$ , accumulates over discrete time intervals, pays benefits to retired members, and collects contributions from active members at the end of each period. Therefore, the dynamics can be modeled by:

(2.1) \begin{align}x_{k+1} & =(x_{k}-u^{\prime}_{k}\textbf{1})r_{k}+u^{\prime}_{k}e_{k}-B_{k+1}+\mathcal {C}_{k+1}\nonumber\\ & =x_{k}r_{k}+\theta^{\prime}_{k}u_{k}-B_{k+1}+c_{k+1}A_{k+1}y_{k}p_{k}.\end{align}

Following Equation (2.1), the fund trustee’s strategy for period $[k,k+1)$ is composed of two parts: the investment strategy $u_k$ implemented at time k, and the total benefit payment $B_{k+1}$ made to retired members at the end of the period, at time $k+1$ .

This model incorporates two sources of randomness along with time: the average wage growth rate (reflected by $p_k$ ) and the stochastic investment market returns (reflected by $e_k$ ). We make the following assumption.

Assumption 2. The covariance matrix

\begin{eqnarray*}Var_{k}[\eta_{k}]&=&cov(\eta_{k},\eta_{k})=E_{k}\left[{(\eta_{k}-E_{k}[\eta_{k}])(\eta_{k}-E_{k}[\eta_{k}]{)}^{\prime}}\right]\\&=&\left({\begin{array}{c@{\quad}c}{cov(e_{k},e_{k})}\quad & {cov(e_{k},p_{k})}\quad \\[5pt]{cov(e_{k},p_{k})}\quad & {cov(p_{k},p_{k})}\quad \end{array}}\right)>0,\end{eqnarray*}

for $k=0,1,\;\cdots,\;T-1$ .

Assumption 2 is a mild condition that assumes the rate of return from the risky asset $e_{k}$ and the rate of increase from the average salary $p_{k}$ are relatively independent in practice. Even in cases where they are dependent, the time-lag in the dependence structure results in a small value for $cov_{k}(e_{k},p_{k})$ .

2.2. The long-term objective of the TBP structure

This section discusses the long-term objectives of the trustee responsible for managing the TBP retirement fund and presents a mathematical formulation of these objectives as a stochastic optimal control problem.

First, unlike the traditional DB structure that guarantees benefits, the TBP structure establishes a benefit target $B_{k}^{*}$ at time 0. The fund trustee sets this target as a guide for future benefit payments. The actual benefit payment $B_k$ depends on the fund’s wealth level at the end of each period, which the trustee then declares and distributes. The trustee aims to minimize the squared distance between the benefit $B_k$ and its target $B_k^*$ , which usually remains deterministic and stable over time. The resulting benefit payment $B_k$ is also expected to be stable over time, providing an advantage of TBP schemes over traditional DC schemes. Additionally, from the members’ perspective, if the actual benefit is lower than the target benefit, the fund fails to meet their expectations, and this shortfall should be penalized.

Second, it is the trustee’s responsibility to maintain a balance in benefits between active and retired members. If retired members receive an excessive temporary benefit payment, it may come at the expense of younger generations. To safeguard the interests of younger generations, a target terminal wealth must be set for their retirement. This target also ensures the long-term sustainability of the fund. For instance, one can use $x_{0}\prod_{i=0}^{T-1}{r_{i}}$ as the target terminal wealth, which represents a conservative expected wealth accumulated from the initial wealth $x_0$ at time 0 to time T. The trustee is responsible for investing and distributing the fund’s wealth in an appropriate manner to ensure that the terminal wealth $x_T$ remains close to the target. For generality, we adopt $x^*_{0}\prod_{i=0}^{T-1}{r_{i}}$ as the target terminal wealth in this paper, where $x^*_0$ is a factor set at time 0.

Let the notation $\pi$ be the strategy consisting of $u_{k}$ and $b_{k+1}$ , and $\Theta$ be the transformed admissible set for this strategy. To reflect the target benefit, target wealth, and their relationships with the actual benefit payment and resulting terminal wealth, we adopt a mean-target objective function. When putting these elements together into the long-term objective function, $f_{k}(y,x)$ , at time k with wealth x and average wage y, we have

(2.2) \begin{equation}\left\{ \begin{array}{l}{f_{k}}(y,x)=\mathop{\min_{\pi\in\Theta_{k}}}E_{k}\left\{ {\sum\limits _{t=k}^{T-1}{\left[{{{\left({{B_{t+1}}-B_{t+1}^{*}}\right)}^{2}}-2{\lambda_{1}}\left({{B_{t+1}}-B_{t+1}^{*}}\right)}\right]{\rho^{t+1-k}}}}\right.\\[9pt]\quad \quad \quad \quad \left.{+{\lambda_{2}}{\left({x_{T}}-{x_{0}^*}\prod\limits _{i=0}^{T-1}{r_{i}}\right)^{2}}{\rho^{T-k}}}\right\} \quad \text{subject to }\quad y_{k+1}=p_k y_k\quad \text{and}\quad (2.1),\\[9pt]{f_{T}}(y,x)={\lambda_{2}}{\left(x-{x_{0}^*}\prod\limits _{i=0}^{T-1}{r_{i}}\right)^{2}},\end{array}\right.\end{equation}

where $\lambda_{1}\ge0$ and $\lambda_{2}>0$ are the penalty weights given to the deviation of the true value from the target, $\rho>0$ is a discount factor, and $x_0^*$ is a factor such that $x^*_{0}\prod\limits_{i=0}^{T-1}{r_{i}}$ reflects the target wealth at time T. The choice of $\lambda_1$ and $\lambda_2$ reflects the balance of risks between the benefit adequacy for the current retiring generation and the interest of future generations. It is important to note that $B_k$ is distributed for only one generation retiring at time k, while $x_T$ is considered for the overall future generations. Therefore, $\lambda_1$ and $\lambda_2$ adopt different magnitudes, which will be illustrated in Section 5.

It should be noted that each year, members retiring from our TBP scheme receive a lump-sum benefit and leave the fund. On the other hand, all active members are in their accumulation phase. The fund trustee makes the investment decision for all active members collectively, which has two implications. Firstly, the investment decision is uniform for all active members, including those retiring in the future. Secondly, this decision is made jointly with benefit distribution decisions, taking into account the already retired members.

The advantages of adopting a mean-target objective function are evident. As discussed in Section 1, this approach provides a clear indication of targets and allows for a comprehensive analysis of the distribution of benefits and remaining wealth. It also facilitates the weighting of parameters based on the risk-sharing mechanisms between generations, making the structure transparent and intuitive for interpreting a TBP plan. By examining the effects of model parameters, the study can provide valuable guidance for the day-to-day operations of a TBP fund. Additionally, it provides simplicity, allowing us to solve the problem analytically. The mean-target framework leads to a quadratic control structure, which can be solved analytically and backwardly through time via the dynamic programming approach. Although the penalty on both upside and downside risks is a by-product of the mean-target anatomy, this is precisely what we need in this IRS strategy. This objective is consistent with the mean-target” objective, as pointed out by Wang et al. (Reference Wang, Lu and Sanders2018):

Solving problem (2.2) with the presence of two control variables, $B_{k+1}$ and $u_{k}$ , is not straightforward in the multi-period case. To make the problem technically tractable, we need to transform the objective function as shown below. By following a completion-of-square procedure, we obtain

\begin{eqnarray*}&&E_{k}\left\{ {\sum\limits _{t=k}^{T-1}{\left[{{{\left({{B_{t+1}}-B_{t+1}^{*}}\right)}^{2}}-2{\lambda_{1}}\left({{B_{t+1}}-B_{t+1}^{*}}\right)}\right]{\rho^{t+1-k}}}+{\lambda_{2}}{\left({x_{T}}-{x_{0}^*}\prod\limits _{i=0}^{T-1}{r_{i}}\right)^{2}}{\rho^{T-k}}}\right\}\\[4pt]&=&E_{k}\left\{ {\sum\limits _{t=k}^{T-1}{\left[{{{\left({{B_{t+1}}-B_{t+1}^{*}-{\lambda_{1}}}\right)}^{2}}-\lambda_{1}^{2}}\right]{\rho^{t+1-k}}}+{\lambda_{2}}\left.{\left({x_{T}}-{x_{0}^*}\prod\limits _{i=0}^{T-1}{r_{i}}\right)^{2}}{\rho^{T-k}}\right)}\right\} \\[4pt]&=&E_{k}\left\{ {\sum\limits _{t=k}^{T-1}{{{\left({{B_{t+1}}-B_{t+1}^{*}-{\lambda_{1}}}\right)}^{2}}{\rho^{t+1-k}}}+{\lambda_{2}}\left.{\left({x_{T}}-{x_{0}^*}\prod\limits _{i=0}^{T-1}{r_{i}}\right)^{2}}{\rho^{T-k}}\right)}\right\}-\lambda_{1}^{2}\sum\limits _{t=k}^{T-1}{\rho^{t+1-k}}\end{eqnarray*}

The lengthy expression can be shortened by defining

\begin{equation*}\alpha_{k}=x_{k}-x^*_{0}\prod\limits _{i=0}^{k-1}{r_{i}},\end{equation*}

\begin{equation*}b_{k+1}^{\ast}=B_{k+1}^{\ast}+\lambda_{1},\end{equation*}

and

\begin{equation*}b_{k+1}=B_{k+1}-B_{k+1}^{\ast}-\lambda_{1}=B_{k+1}-b_{k+1}^{\ast}.\end{equation*}

In problem (2.2), $x^*_{0}\prod\limits_{i=0}^{T-1}{r_{i}}$ is the wealth target at the terminal time T; here, $x^*_{0}\prod\limits _{i=0}^{k-1}{r_{i}}$ can be taken as the wealth target at time k. Consequently, the difference between the wealth $x_k$ and its target at time k can be expressed as $\alpha_{k}=x_{k}-x^*_{0}\prod\limits _{i=0}^{k-1}{r_{i}}$ , which represents the excess of wealth at time k. Then based on (2.1), we have

\begin{eqnarray*} \alpha_{k+1} &=& x_{k+1}- x^*_{0}\prod\limits _{i=0}^{k}{r_{i}}\\ &=& x_{k}r_{k}+\theta^{\prime}_{k}u_{k}-B_{k+1}+c_{k+1}A_{k+1}y_{k}p_{k}- x^*_{0}\prod\limits _{i=0}^{k}{r_{i}}\\ &=& r_{k}\left(x_k-x^*_{0}\prod\limits _{i=0}^{k-1}{r_{i}}\right)+\theta^{\prime}_{k}u_{k}-B_{k+1}+c_{k+1}A_{k+1}y_{k}p_{k}\\ &=& r_{k}\alpha_{k}+\theta^{\prime}_{k}u_{k}-b_{k+1}-b_{k+1}^{\ast}+c_{k+1}A_{k+1}p_{k}y_{k}\end{eqnarray*}

The optimization problem (2.2) is equivalent to finding the optimal $b_{k+1}$ and $u_{k}$ for the following problem:

(2.3) \begin{equation}\left\{ {\begin{array}{l}V_{k}(y,\alpha)=\mathop{\min}\limits _{\pi\in\Theta_{k}}E_{k}\left\{ {\sum\limits _{t=k}^{T-1}{b_{k+1}^{2}\rho^{t+1-k}}+\lambda_{2}\alpha_{T}^{2}\rho^{T-k}}\right\}, \\[12pt]\text{subject to}\quad y_{k+1}=p_k y_k\text{ and}\\[8pt]\alpha_{k+1}\,=\alpha_{k}r_{k}+\theta^{\prime}_{k}u_{k}-b_{k+1}-b_{k+1}^{\ast}+c_{k+1}A_{k+1}p_{k}y_{k}\\[8pt]V_{T}(y,\alpha)=\lambda_{2}\alpha^{2}.\end{array}}\right.\end{equation}

Moreover, by comparing the objective functions of the optimization problems in Equations (2.2) and (2.3), we have $f_{k}(y,x)=V_{k}(y,\alpha)-\lambda_{1}^{2}\sum\limits_{t=k}^{T-1}{\rho^{t+1-k}}$ .

3.1. A further transformation into matrix form

To proceed with solving the problem in Equation (2.3), we first transform it into its matrix form. Let $\textrm{0}_{i\times j}$ denote the zero matrix with the dimensions $i\times j$ , and

(3.1) \begin{equation}\left\{ \begin{array}{l}{z_{k}}=\left({\begin{array}{c}{y_{k}}\\[4pt]{\alpha_{k}}\end{array}}\right),\quad {\pi_{k}}=\left({\begin{array}{c}{u_{k}}\\[4pt]{b_{k+1}}\end{array}}\right),\;{C_{k}}=\left({\begin{array}{c@{\quad}c}{p_{k}} & 0\\[4pt]{{c_{k+1}}{A_{k+1}}{p_{k}}} & {r_{k}}\end{array}}\right),\;\\[17pt]{D_{k}}=\left({\begin{array}{c@{\quad}c}{{\bf {0^{\prime}}}_{n\times1}} & 0\\[4pt]{{\theta^{\prime}}_{k}} & {-1}\end{array}}\right),{N_{k}}=\left({\begin{array}{c}0\\[4pt]{-b_{k+1}^{*}}\end{array}}\right),M=\left({\begin{array}{c@{\quad}c}0 & 0\\[4pt]0 & {\lambda_{2}}\end{array}}\right),L=\left({\begin{array}{c@{\quad}c}{{\bf {0}}_{n\times n}} & {{\bf {0}}_{n\times1}}\\[4pt]{{\bf {0^{\prime}}}_{n\times1}} & 1\end{array}}\right).\end{array}\right.\end{equation}

Then, problem (2.3) can be written in the form as:

(3.2) \begin{equation}V_{k}(z)=\mathop{\min}\limits _{\pi\in\Theta_k}E_{k}\left\{{\sum\limits_{t=k}^{T-1}{\rho^{t+1-k}{\pi}^{\prime}_{k}L\pi_{k}}+{z}^{\prime}_{T}Mz_{T}\rho^{T-k}}\right\},\text{subject to}\quad z_{k+1}=C_{k}z_{k}+D_{k}\pi_{k}+N_{k},\end{equation}

with a boundary condition $V_{T}(z)={z}^{\prime}Mz$ .

The optimization problem (3.2) is a discrete-time stochastic linear–quadratic (LQ) optimal control problem with a discount rate. The standard treatment of solving the stochastic LQ optimal control problem requires that the carrier matrices are positive definite matrices, that is, $L>0$ and $M>0$ in problem (3.2). However, both $L\in\mathcal{{S}_{+}}$ and $M\in\mathcal{{S}_{+}}$ in our model are irreversible (see (3.1)). This structure of $D_{k}$ , M, and L allows us to demonstrate the strictly positive definiteness and invertibility of some matrices critical for the existence of solutions (see Proposition 1 for more details). Thus, by combining our method with the classical method for solving the stochastic LQ optimal control problem, we can obtain the analytical solution of our model. The outline of the solving procedure is sketched in the next subsection.

3.2. The solution

Following the dynamic programming principle, we can derive the corresponding Bellman equation for Equation (3.2) as follows:

(3.3) \begin{equation}V_{k}(z)=\rho\mathop{\min}\limits _{\pi\in\Theta_k}E_{k}\left[{{\pi}^{\prime}_{k}L\pi_{k}+V_{k+1}(z_{k+1})}\right]=\rho\mathop{\min}\limits _{\pi\in\Theta_k}E_{k}\left[{{\pi}^{\prime}_{k}L\pi_{k}+V_{k+1}(C_{k}z+D_{k}\pi_{k}+N_{k})}\right].\end{equation}

To derive the expression for $V_{k}(z)$ , we construct a series of matrics $\Omega_{k}$ , $G_{k}$ , and $F_{k}$ for all $k=0,1,\;\cdots,\;T$ satisfying the following recurrence relation:

(3.4) \begin{equation}\left\{ {\begin{array}{l}\Omega_{k}=\rho\left({E_{k}\left[{{C}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]-E_{k}\left[{{C}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]}\right),\\[5pt]{G}^{\prime}_{k}=\rho\left({{N}^{\prime}_{k}\Omega_{k+1}E_{k}\left[{C_{k}}\right]+{G}^{\prime}_{k+1}E_{k}\left[{C_{k}}\right]-{N}^{\prime}_{k}\Omega_{k+1}E_{k}\left[{D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}}\right.\\[5pt]\quad \quad \left.{\times E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]-{G}^{\prime}_{k+1}E_{k}\left[{D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]}\right),\\[5pt]F_{k}=\rho\left({F_{k+1}+{N}^{\prime}_{k}\Omega_{k+1}N_{k}+2{G}^{\prime}_{k+1}N_{k}-{N}^{\prime}_{k}\Omega_{k+1}E_{k}\left[{D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}}\right.\\[5pt]\quad \quad \times E_{k}\left[{{D}^{\prime}_{k}}\right]\Omega_{k+1}N_{k}-{G}^{\prime}_{k+1}E_{k}\left[{D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}E_{k}\left[{{D}^{\prime}_{k}}\right]G_{k+1}\\[5pt]\quad \quad \left.{-2{G}^{\prime}_{k+1}E_{k}\left[{D_{k}}\right]\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}E_{k}\left[{{D}^{\prime}_{k}}\right]\Omega_{k+1}N_{k}}\right)\end{array}}\right.\end{equation}

with boundary conditions at time T:

(3.5) \begin{equation}\Omega_{T}=M,\quad G_{T}=\textrm{0}_{2\times1},\quad F_{T}=0,\end{equation}

where $\Omega_{k}$ is a symmetric matrix of order $2\times2$ , $G_{k}$ is a column vector of order 2 and $F_{k}$ is a scalar. These boundary conditions are derived by equating the boundary condition equation $V_{T}(z)={z}^{\prime}Mz$ in problem (3.2) with $V_{T}(z)={z}^{\prime}\Omega_{T}{z}+2G_{T}^{\prime} z+F_T$ . These conditions represent the scenario at time T when no investment strategy decisions need to be made. In this case, the value function is solely determined by the terminal conditions.

It is worth noting that the series $\Omega_{k}$ , $G_{k}$ , and $F_{k}$ are independent of the state variable $z_{k}$ , and their recursion formulas and boundary conditions do not depend on $z_{k}$ . Based on Equations (3.4)–(3.5), we can obtain the estimated numerical values of $\Omega_{k}$ , $G_{k}$ , and $F_{k}$ for all $k=0,1,\cdots,T-1$ using historical or stochastic simulated data. In Appendix A, we provide the formulas for calculating the expectations of those random matrices or vector multiplications, such as $E_{k}\left[{{C}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]$ , $E_{k}\left[{{C}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]$ and $E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]$ . These formulas can be computed using the market data $E_{k}[p_{k}]$ , $E[p_{k}^{2}]$ , $E_{k}[p_{k}{\theta}^{\prime}_{k}]$ , $E_{k}[\theta_{k}]$ and $E_{k}[\theta_{k}{\theta}^{\prime}_{k}]$ . The calculations in Section 5 can be simplified accordingly.

The following proposition shows that $L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]>0$ and hence $\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}$ exists, which guarantees that the definition of Equations (3.4)–(3.5) is meaningful.

The proof of Proposition 1 directly follows from Lemmas 1–3 in Appendix B. Lemmas 1 and 2 present a method for representing the returns from risky assets as a linear space. This allows for the investigation of their linear correlation, independence, and the representation of random variable groups. Additionally, necessary and sufficient conditions for the nonsingularity of the second-order moment matrix and the covariance matrix are provided. Lemma 3 provides the necessary and sufficient condition for the positive definiteness of the block matrix.

The proof for Proposition 1 is sketched as follows: we first simplify the expression of ${E_{k}}\left[{{{D^{\prime}}_{k}}{\Omega_{k+1}}{D_{k}}}\right]$ using the structure of $D_{k}$ and M (see (3.1)). Next, using mathematical induction and Lemmas 1–2, we show that $L+{E_{k}}\left[{{{D^{\prime}}_{k}}{\Omega_{k+1}}{D_{k}}}\right]>0$ for $k=1, 2, \cdots, T-1$ . We then decompose the matrix $\Omega_{k+1}$ into $J_{1}+J_{2}$ , where $J_{1}\in\mathcal{{S}_{+}}$ and $J_{2}$ has a special structure similar to M (see (3.1)). Using mathematical induction and the partition of positive semidefinite matrices (Lemma 3), we further prove $\Omega_{k}>0$ for $k=1, 2, \cdots, T-1$ . The complete proofs are presented in Appendix C.

We are now ready to state the main theorem of this paper.

Theorem 1. The solution to Bellman Equation (3.3), namely, the value function of problem (3.2) is given by:

(3.6) \begin{equation}V_{k}(z)={z}^{\prime}\Omega_{k}z+2{G}^{\prime}_{k}z+F_{k},\end{equation}

and the corresponding optimal strategy is given by:

(3.7) \begin{equation}\pi_{k}=-\left({L+E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}D_{k}}\right]}\right)^{-1}\left({E_{k}\left[{{D}^{\prime}_{k}\Omega_{k+1}C_{k}}\right]z+E_{k}\left[{{D}^{\prime}_{k}}\right]\Omega_{k+1}N_{k}+E_{k}\left[{{D}^{\prime}_{k}}\right]G_{k+1}}\right),\end{equation}

where $\Omega_{k}$ , $G_{k}$ , and $F_{k}$ are determined by (3.4) and (3.5).

The proof of Theorem 1 is based on the classical stochastic LQ optimal control theory and the concept of mathematical induction. Specifically, when $k=T-1$ , we apply the method of differentiation (i.e., the first-order condition) to Bellman Equation (3.3) to obtain the optimal solution and the optimal value of the objective function, which is a quadratic function. This establishes the validity of the theorem for $k=T-1$ . Assuming the theorem holds for $k+1$ , we demonstrate, using first-order conditions and Bellman Equation (3.3), that it also holds for k (the methodology is akin to that of the $T-1$ case). By virtue of the principle of mathematical induction, we thus establish the result. The details of this proof are presented in Appendix D.

With reference to Theorem 1 and bearing in mind that $\alpha_{k}=x_{k}-x_{0}^*\prod\limits _{i=0}^{k-1}{r_{i}}$ and $f_{k}(y,x)=V_{k}(y,\alpha)-\lambda_{1}^{2}\sum\limits_{t=k}^{T-1}{\rho^{t+1-k}}$ , we have the following results for the original problem (2.2).

Theorem 2. Let $x=x_{k},\;y=y_{k}$ for $k=0,1,\;\cdots,\;T-1$ , the optimal value for problem (2.2) is

\begin{equation*}f_{k}(x,y)=\left({y,\;x-x_{0}^*\prod\limits_{i=0}^{k-1}{r_{i}}}\right)\Omega_{k}\left({\begin{array}{l}y \\{x-x_{0}^*\prod\limits _{i=0}^{k-1}{r_{i}}} \end{array}}\right)+2{G}^{\prime}_{k}\left({\begin{array}{l}y \\{x-x_{0}^*\prod\limits _{i=0}^{k-1}{r_{i}}}\quad \end{array}}\right)+F_{k}-\lambda_{1}^{2}\sum\limits _{t=k}^{T-1}{\rho^{t+1-k}},\end{equation*}

and the corresponding optimal strategy is given by:

\begin{align*}{\pi_{k}} & =\left({\begin{array}{c}{u_{k}}\\[4pt]{b_{k+1}}\end{array}}\right)=-{\left({L+{E_{k}}\left[{{{D^{\prime}}_{k}}{\Omega_{k+1}}{D_{k}}}\right]}\right)^{-1}}\\[7pt] & \quad \times\left({{E_{k}}\left[{{{D^{\prime}}_{k}}{\Omega_{k+1}}{C_{k}}}\right]\left({\begin{array}{c}y\\[4pt]{x-{x_{0}^*}\prod\limits _{i=0}^{k-1}{r_{i}}}\end{array}}\right)+{E_{k}}\left[{{D^{\prime}}_{k}}\right]{\Omega_{k+1}}{N_{k}}+{E_{k}}\left[{{D^{\prime}}_{k}}\right]{G_{k+1}}}\right).\end{align*}

Theorem 2 demonstrates that the optimal value function of the optimal problem (2.2) is quadratic in nature and depends on both the present average wage level $y_{k}$ and the current excess wealth level (relative to risk-free investment) ${x_{k}}-{x_{0}^*}\prod\limits_{i=0}^{k-1}{r_{i}}$ . The optimal strategy $\pi_k$ takes the form of a linear feedback control, that is, a linear function of the present average wage level $y_{k}$ and the current excess wealth level ${x_{k}}-{x_{0}^*}\prod\limits_{i=0}^{k-1}{r_{i}}$ . Therefore, when determining the investment strategy $u_k$ and benefit payment strategy $B_{k+1}=b_{k+1}+B^*_{k+1}+\lambda_1$ (since $b_{k+1}=B_{k+1}-B^*_{k+1}-\lambda_1$ ), the fund trustee must take into account both the current levels of wage and wealth.

The economic implications are not immediately apparent due to the stochastic nature and matrix form of the solution. However, we can gain insights by considering a special case where $k=T-1$ , $n=1$ (only one risky asset), and the coefficient matrices are deterministic. In this case, the optimal strategy can be simplified as follows:

\begin{align*}{\pi_{T-1}} =\left({\begin{array}{c}{u_{T-1}}\\[5pt]{b_{T}}\end{array}}\right)=\frac{1}{\theta_{T-1}}\left({\begin{array}{c}b_{T}^*-c_{T}A_{T}p_{T-1}y-r_{T-1}\left({x-{x_{0}^*}\prod\limits _{i=0}^{T-2}{r_{i}}}\right)\\[5pt]0\end{array}}\right).\end{align*}

In the time period $[T-1, T]$ , since this scenario is deterministic, the benefit distribution is given by $B_{T}=B_{T}^*+\lambda_1$ (derived from $b_{T}=0$ ). This means that we distribute the benefit according to the planned target $B_{T}^*$ , with the consideration of the weight $\lambda_1$ . Regarding the investment strategy, a higher target benefit $B_{T}^*$ or a higher weight $\lambda_1$ (then a higher $b_{T}^*=B_{T}^*+\lambda_1$ ), as well as a higher wealth target ${x_{0}^*}\prod\limits_ {i=0}^{T-2}{r_{i}}$ , will result in a more aggressive investment in the risky asset. The implications of this relationship will be further explored in Section 5.

5.1. Data structure and statistical estimation for the financial market and earnings

The equity market holds a prominent position among Australia’s investment markets. According to recent data, Australian listed shares account for over 22% of superannuation funds, while international stocks make up 25% of such funds.Footnote 7 Australian equities are known for offering higher dividends compared to other countries, which can be attributed to specific tax treatments as discussed in Bergmann et al. (Reference Bergmann2016). Therefore, when measuring the returns on Australian equities, it is essential to account for dividend payments. In this paper, equity returns refer to the total shareholder return (TSR), which is the sum of capital gains and dividends. To obtain the time series for TSR, we rely on a newly compiled dataset on the equity market that was published by the Reserve Bank of Australia (RBA) in August 2019. The dataset provides quarterly data from different types of companies, including the financial sector (especially banks), resources sector (mainly miners), and others (excluding financials and resources). We extract the time series from 1965 to 2018, totaling 54 years, to model the three risky assets (financial, resources, and others).

Regarding the risk-free rate, we use the deposit interest rate paid by commercial or similar banks for demand, time, or savings deposits. The International Monetary Fund collects and documents this rate, which is published in the International Financial Statistics. For earning data, we rely on the Average Weekly Earnings report published by the ABS. We extract annualized data from 1965 to 2018 to use in this study.

To obtain the conditional expectations and covariance matrices, we utilize an autoregressive vector structure of the time series data. The parameters for this model are estimated using the Bayesian method. There is an extensive body of literature on Bayesian vector autoregression and its associated estimation and forecasting techniques in macroeconomics. Detailed information on the Bayesian vector autoregressive model can be found in Kadiyala and Karlsson (Reference Kadiyala and Karlsson1997), while the appropriate choice of prior is discussed in Chan et al. (Reference Chan, Jacobi and Zhu2019). To avoid introducing additional mathematical notation in this section, we provide a brief description of the estimation procedure in Appendix G.

In this section, our focus is on the in-sample forecasts of the state variables. Using the Bayesian Markov chain Monte Carlo approach, we can conveniently generate in-sample forecasts for these variables conditional on the posterior draws. Therefore, we can easily compute the conditional mean and covariance, as used in Theorem 1.

5.2. Parameter settings

This subsection outlines the parameter values used in the model.

The initial wealth of TBP. It should be noted that retiring members are already accounted for in the TBP pension at the fund’s setup, as an initial fund is necessary in the TBP scheme to meet upcoming payment obligations. In line with Cui et al. (Reference Cui, De Jong and Ponds2011), this paper sets the initial fund value as the product of $f_0$ and the target benefit at the end of the first period $B^{*}_1$ , where $f_0$ can be adjusted to observe the impact of initial wealth. Specifically, $x_0=f_0 B^{*}_1$ , and the base value of $f_0$ is set to 1. Notably, compared to Cui et al. (Reference Cui, De Jong and Ponds2011), who use the fund liability (including the benefit for all the generations) as the initial wealth, our approach is conservative, as $B_{1}^{*}$ represents the benefit for only one generation. The optimal setting for initial wealth is beyond the scope of this paper, but we analyze the impacts of varying initial wealth in the next subsection. As noted by Gollier (Reference Gollier2008), this initial fund can be accumulated from existing individual accounts or raised through a privatization program.

The target benefit and target wealth. To simplify the presentation, we define the target benefit as a function of the target replacement rate denoted by $R_{tar}$ . Based on the demographic structure of OLG, the target benefit $B^{*}_k$ is computed as $R_{tar}\times 14 y^{f}_{k}$ . Additionally, we use ${x_{0}}\prod\limits _{i=0}^{T-1}{r_{i}}$ as the base value for the wealth target and adjust it by multiplying with $(W_{tar})^{T}$ to study the effects of different target wealth values. We set the base values as $R_{tar}=0.8$ and $W_{tar}=1.05$ .

The weights $\lambda_1$ and $\lambda_2$ . By choosing a higher value of $\lambda_{1}$ , the analyst places more emphasis on the well-being of members retiring before time T. To balance the welfare interests across generations, the parameter $\lambda_{2}$ represents the weight assigned to the fund surplus after deducting benefits. The goal of the trustee is to provide generous benefits to retired generations and accumulate large surpluses for active members. Thus, $\lambda_{1}$ and $\lambda_{2}$ can be viewed as the weights given to the interests of retiring and future generations, respectively. Since the benefit for a single retiring generation is much smaller than the total wealth of all active generations, the magnitude of $\lambda_2$ is typically much larger than $\lambda_1$ . We set the base values as $\lambda_1=1$ and $\lambda_2=10$ .

Other parameters. We set the discount factor to $\rho=0.95$ . The contribution rate remains constant at 10% throughout the period from 1965 to 2018. The number of active members is fixed at $A_k=40$ , while each generation has $R_k=1$ retired member staying with the fund, for $k=1,2,...,T$ .

The notations and base parameter values are summarized below.

• Time span is from 1965 to 2018, that is, $T=54$ .

• A member enters the TBP at the age of 25 years, retires at the age of 65 years, and leaves the fund with a lump-sum benefit payment. The survival time is 14 years until the limit age of 80 years.

• The number of active members $A_k=40$ . The number of retired members in the fund $R_k=1$ .

• Final salary for members retiring at time k is $y_k^f$ .

• Target benefit $B_k^*=R_{tar}\times 14 y_k^f$ where $R_{tar}=0.8$ .

• Target wealth $(W_{tar})^{T}\times {x_{0}}\prod\limits _{i=0}^{T-1}{r_{i}}$ where $W_{tar}=1.05$ .

• Initial wealth $x_0=f_0 B_1^*$ where $f_0=1$ .

• $\lambda_1=1$ and $\lambda_2=10$ .

• Discount factor $\rho=0.95$ .

• Contribution rate 10%.

• Three risky assets: Financial, Resources, and Others.

5.3. The features of TBP fund

TBP members are primarily concerned with the amount and stability of benefit payments distributed from the fund. On the other hand, the fund trustee focuses more on the investment strategy, the progression of the wealth process, and the corresponding funding ratio. In this subsection, we analyze how the model parameters in problem (2.2) affect these areas of interest.

The Effects of weights $\lambda_{1}$ and $\lambda_{2}$ .

The formulation of (2.2) suggests that parameter $\lambda_{1}$ controls the distribution of $B_{k}-B_{k}^{*}$ over time. This is confirmed by Figure 1, where we adopt three sets of $\lambda_{1}$ and $\lambda_{2}$ . Figure 1(a) shows the benefit payments $B_k$ plotted against time from 1966 to 2018, where the difference between the plots is not visible due to the large magnitude of the benefit payment. The excess benefit, $B_{k}-B_{k}^{*}$ , is mainly determined by the value of $\lambda_{1}$ , and the role of $\lambda_2$ is minimal, as shown in Figure 1(b). In other words, the benefit payment $B_k$ is primarily determined by $B_k^*$ and $\lambda_1$ , with minimal impact from $\lambda_2$ , which is further verified by Figure 2, where additional sets of $\lambda_1$ and $\lambda_2$ are plotted.

Figure 1.

The effects of $\lambda_{1}$ and $\lambda_{2}$ on the benefit payments $B_k$ for $k=1966, 1967,..., 2018$ . (a) The value of $B_k$ . (b) gives the deviation of $B_k$ from the target benefit $B_k^*$ .(c) The value of benefit payment in terms of replacement ratio.

Figure 2.

The joint effects of $\lambda_{1}\in[1, 10,000]$ and $\lambda_{2}\in[1, 10,000]$ on the benefit payments $B_k$ for 1974 and 2014. (a) $B_{1974}$ . (b) $B_{2014}$ . (c) $B_{1974}-B^*_{1974}$ . (d) $B_{2014}-B^*_{2014}$ .

Another notable observation is that the resulting benefit payment $B_k$ approaches $B_k^*$ as k approaches T. This trend is evident in Figure 1(c), where we plot the replacement rate against time. The replacement rate converges to 0.8, which can be attributed to the discount factor $\rho$ in (2.2). As we approach T under the mean-target framework, the level of stochastic randomness decreases, providing more certainty for the optimal control variable to attain the target.

Turning to investment strategies, Figure 3 shows how the $\lambda$ values affect the amount invested in the resources sector. Note that short-selling is allowed in the model formulation, and negative investment amounts around 1980 and 1990 can be attributed to two well-known economic recessions in Australia’s history, one caused by the 1973 oil crisis and the other by the early 1990s global recession that followed Black Monday in October 1987. It is reasonable for fund trustees to short-sell stock before prices drop and repurchase it at a lower price. Unlike the benefit payment, $\lambda_2$ plays a leading role in the investment behavior. When the investment amount is positive, the green line is the highest among the three scenarios, but when the amount is negative, it is the lowest. This means that the higher the emphasis on $\lambda_2$ , that is, the greater the focus on the wealth target, the more aggressive the investment strategies, as observed in Figure 3. Figure 4 provides an overview of the investment allocation to three risky assets, where we see that the investment allocation to the resources sector has declined since the peak in 2012/2013, consistent with the practice observations from RBA.Footnote 8

Figure 3.

The effects of $\lambda_{1}$ and $\lambda_{2}$ on the amount invested in the resources sector.

Figure 4.

The allocations to three risky assets along with time with fixed $\lambda_1=1$ and $\lambda_2=10$ . (a) The investment amounts to three risky assets and the value of wealth process. (b) The percentage of wealth invested in the three risky assets.

The effects of the benefit target and final wealth target.

From the perspective of the fund trustee, how much flexibility is available in establishing the fund’s target benefits and wealth? This subsection aims to address this question by examining the impact of benefit targets and wealth targets on benefit payments and investment strategies. Setting appropriate benefit targets ( $B^{*}_k$ ) and wealth targets is crucial to achieving a balance between the benefits of retiring members and those of active members. However, when the fund is significantly in surplus and the initial wealth can support all generations financially, there may be less conflict between a high $B^{*}_k$ and a high wealth target. In such cases, higher $B^{*}_k$ would result in greater benefits for retiring members. Nevertheless, in other circumstances, the establishment of $B^{*}_k$ and the wealth target could lead to conflicting objectives.

In Figure 5, we present the benefit payments for three target replacement rates $R_{tar}$ : 0.8, 0.85, and 0.9, using base values of $\lambda_{1}=1$ and $\lambda_{2}=10$ . As per our previous findings, we anticipate the difference between $B_k$ and $B^{*}_k$ to be approximately $\lambda_{1}$ initially, gradually converging to 0 by 2018, as shown in Figure 5(b). The investment strategies proposed in this paper enable the achievement of the target replacement rate, as demonstrated in Figure 5(c), indicating adequate funding. Figure 6 displays the benefit payments for varying wealth targets $W_{tar}=$ 1, 1.1, and 1.2, with a fixed benefit target of $R_{tar}=0.8$ . In cases where the wealth target is unrealistically high, such as $W_{tar}=1.2$ , the trustee must decrease benefits for members retiring between 0 and T. As time progresses toward T, the deviation between $B_k$ and $B^{*}_k$ in Figure 6(b) ultimately becomes negative, suggesting that achieving the target benefit may be unsustainable without jeopardizing the benefits of the current generation.

Figure 5.

The effects of the benefit target $B_k^*$ on the benefit payments $B_k$ in terms of target replacement rate $R_{tar}$ . (a) The value of $B_k$ . (b) The deviation of $B_k$ from the target benefit $B_k^*$ . (c) The value of benefit payment in terms of replacement ratio.

Figure 6.

The effects of the wealth target on the benefit payments $B_k$ . (a) The value of $B_k$ . (b) The deviation of $B_k$ from the target benefit $B_k^*$ . (c) The value of benefit payment in terms of replacement ratio.

Figure 7.

The effects of the benefit (left) and wealth (right) targets on the amount invested in the resources sector.

With regard to investment strategies, Figure 7 illustrates the amount invested in the resources sector for varying benefit targets (left) and wealth targets (right). Notably, both higher targets result in more significant investments in the long position (positive) for risky assets, and more selling in the short position (negative). The short positions observed around 1980 and 1990 could be attributed to economic recessions, consistent with the findings in Figures 3 and 4.

Figure 8 presents the joint impacts of benefit and wealth targets, showing the trends in benefit and investment strategies for 1974 and 2014. Figures 8(a) and (b) indicate that, with fixed values of $\lambda_1$ and $\lambda_2$ , the benefit distribution is mainly influenced by the target benefit, regardless of the wealth target. On the other hand, regarding investment strategies, Figures 8(c) and (d) suggest that the wealth target plays a more significant role and leads to a more aggressive strategy to achieve a higher wealth target.

Figure 8.

The effects of the benefit and wealth targets in 1974 and 2014. (a) The impacts on benefit payment in 1974. (b) The benefit payment in 2014. (c) The investment amount in resources sector in 1974. (d) The investment amount in resources sector in 2014.

Is this structure sustainable over the long term?

Figure 9 illustrates the progression of the TBP wealth process with varying wealth targets (see (a)), initial wealth $x_0=f_0 B^*_1$ (see (b)), and benefit targets (see (c)). Among these factors, the effect of initial wealth is most noticeable in Figure 9(b), as higher values of $x_{0}$ correspond to greater wealth for the fund.

Figure 9.

The effects of the target wealth, initial wealth $x_0$ , and target benefit on the wealth process $x_k$ along with time.

Figure 9(a) depicts the impact of target wealth on the wealth process. As indicated in the previous analysis, an extremely ambitious target poses a challenge to the investment strategy across generations. However, setting an unreasonably low target is also inadequate. The red curve in Figure 9(a) represents a low target wealth at time T. With the results derived in this paper, the curve initially rises and then falls as the expiry date approaches. A low target wealth at T acts like a brake on wealth accumulation, and as time approaches T, the need to reach the low target wealth becomes more pressing. The investment strategy must react by forcing wealth decumulation, which can potentially result in negative values close to the expiry date.

Figure 9(c) illustrates the impact of target benefit on the wealth process. A low target replacement rate, such as 30%, which is significantly below the market average of 41% in Australia, can result in an unsatisfactory wealth process, as indicated by the blue curve. With each member’s retirement, the low target benefit undermines the wealth accumulation momentum in the early stages. As a result, the blue curve exhibits a declining trend, and even negative values, over the first several years, spanning about 20 years. As the deadline approaches, the pressure of the target benefit necessitates more aggressive strategies, resulting in an increasing trend as seen from 1995.

In summary, adjusting the model parameters carefully is crucial to establish a robust optimal strategy and to ensure the sustainability of the fund in the long run.

The funding ratio process.

The funding ratio expresses the ratio between a pension fund’s available assets and liabilities, reflecting its current financial position. In practice, fund managers often target the funding ratio at one. However, the empirical analysis of this paper, without considering rebalancing, implies a funding ratio that is purely determined by the market. The liability, defined as the total benefit payments for $A_k$ active members at $k=1,2,...,54$ , can be expressed as $A_k B_k$ , where $B_k$ denotes the lump-sum benefit to the retiring member at time k (with only one member). The asset is simply defined as the wealth $x_{k}$ , leading to the funding ratio process $x_k/(A_k B_k)$ . Figure 10 illustrates the impact of initial wealth on the funding ratio process. As we explained in Section 5.2, when $f_0=1$ , the initial wealth $x_0=f_0 B^*_1=B^*_1$ represents the target benefit payment for only one retiree at each time k. Considering $A_k=40$ active members in the fund, this base value for $x_0$ is far from adequate to provide an adequate funding ratio. This explains the low values in the early stages of Figure 10. We then observe a small spike in growth until 1972. The subsequent relatively flat period from 1975 to 1985 corresponds to Australia’s economic recession. With the gradual improvement in the economy, the ratio climbs to the end of the planning horizon. Therefore, carefully managing the initial wealth is essential for maintaining an adequate funding ratio and ensuring the sustainability of the fund over the long term.

Figure 10.

The effects of initial wealth $x_0$ on the funding ratio process along with time.

Figure 11.

The wealth process of the DC account in 14 entry time scenarios.

5.4. The features of DC and comparison with TBP

In this subsection, we investigate the features of our optimal investment model for an individual DC account facing the same financial market as in Section 5.3. We assume that a member of the DC fund earns the average salary published by the ABS and contributes 10% of their salary to the fund for 40 years before retiring in 2005, 2006,…, 2018. We consider 14 scenarios in total. Unlike the TBP scenario, where the fund as a whole requires initial wealth, we assume that members of the DC fund have no savings at the beginning of their contribution period, that is, $\bar{{x}}_{0}=0$ . The 40 years of contributing and investing in the financial market lead to an accumulated value that will financially support their retirement. We adopt the investment strategy derived in Section 4 that corresponds to a target 100% replacement rate.

Table 1.

Static investment strategies.

The volatile wealth process.

Figure 11 illustrates the wealth processes for the 14 DC account entry time scenarios. Compared to the wealth processes in the TBP, the DC wealth processes are highly volatile, particularly in the first several years. Since the initial wealth is 0, the wealth processes are primarily driven by returns from the risky assets, which can fluctuate considerably. However, after 10–25 years of accumulation, the wealth processes smooth out and maintain an increasing trend over time. We also observe that late entry into the labor market can lead to higher earnings and salary inflation. However, due to the high volatility, it is still possible to retire with a low balance after 40 years of contribution. Furthermore, later generations require more financial support due to the effect of consumer inflation, which is the relative risk of DC pension schemes.

Additionally, it appears from Figure 11 that the entry time has a significant impact on achieving a desirable account balance at retirement. Those who start accumulating in 1978 have a much higher balance than those who start in other years. This higher balance could be attributed to the mining boom in Australia during the late 1970s and early 1980s, driven primarily by the energy market, particularly steaming coal, oil, and gas. Members who began accumulating around this period adopted an aggressive investment strategy and benefited from the booming financial market, especially in the resource sector, which left them with a substantial account balance by the late 1980s and the opportunity to switch dynamically to a more conservative direction.

Comparison with the static investment strategy.

The pension industry still favors static investment allocations due to their straightforward approach and strong historical performances. For instance, the Australian government’s MySuper initiative offers default products that are devoid of unnecessary features and charges. If employees do not choose their superannuation fund, they are automatically enrolled in the default MySuper product. The fund’s investment options have the same asset classes, but with varying weightings that match different risk appetites. We have utilized their weightings and titles from Table 1 for our static investment strategies.

The wealth processes resulting from the optimal strategy obtained in this study and four static strategies (cash, conservative, balanced, and growth) are compared in Figure 12. The DC account balance at retirement is shown on the y-axis for 14 scenarios, ranging from 2005 to 2018. Blue lines depict static investment cases, while red lines depict dynamic cases. As expected, the “growth” strategy has the highest balance among the four static options due to its high proportion of risky assets. However, a concerning observation for this strategy is its declining trend after 2015, which corresponds to workers who joined the DC fund in 1975. This trend could be due to the oil price shock that occurred during that period.

Figure 12.

A comparison of the resulting wealth processes between the optimal investment strategy (with different target replacement rates) obtained in this paper and the four static investment strategies in Table 1.

While the dynamic cases exhibit slightly more volatility than the static cases, they often result in higher balances. We computed the account balance for four target levels, with higher targets indicating more aggressive dynamic investment strategies and consequently, greater accumulated account balances. To surpass static investment strategies, the critical adjustment required is to modify the target, which stimulates a steeper wealth accumulation trajectory.

DC versus TBP. Figure 13 compares the retirement benefits from a TBP plan and a DC account using optimal investment and benefit payment strategies. As expected, the TBP trustees provide a more stable benefit over time than a DC account. However, it is not guaranteed that retirees will receive a higher benefit from a TBP plan compared to a DC account. The two solid lines in the graph (red for DC and blue for TBP, with the same replacement rate target of 80%) demonstrate that a DC account can potentially yield a greater benefit amount than a TBP. To achieve a higher benefit payment, the TBP trustees must adjust either the $\lambda_1$ value or the target benefit, which aligns with our findings in Figures 1 and 2.

Figure 13.

A comparison between the retirement benefit from a TBP (with different settings on $\lambda_1$ , $\lambda_2$ , and the target replacement rate) and a DC account.