2.1 Economic and financial framework

Over the last 40 years, various frameworks have been proposed to model economic and financial variables relevant to actuaries. These frameworks – called economic scenario generators or ESGs – are comprehensive models that allow actuaries and risk managers to grasp the long-term uncertainty underlying financial market values and economic variables. The main end-users of these frameworks are pension, life insurance, and banking practitioners, which use them for various purposes: financial planning, asset and liability management, investment strategy, and regulatory compliance, among others (see, e.g., Pedersen et al., Reference Pedersen, Campbell, Christiansen, Cox, Finn, Griffin, Hooker, Lightwood, Sonlin and Suchar2016, for more details).

Wilkie proposed the first cascade model in a pioneering 1986 paper. His framework, which relies on the Box–Jenkins approach, is based on four connected models: an inflation model, a long interest rate model, a dividend yield model, and a stock index return model. In a follow-up article, Wilkie (Reference Wilkie1995) generalized his model by allowing for an earnings index, short-term interest rates, and property prices.

Modern technological and methodological advances have paved the way for more sophisticated frameworks. Specifically, in 2005, Ahlgrim, D’Arcy, and Gorvett proposed a model similar to Wilkie’s but allowed for regime-switching dynamics for stock index returns to capture bull and bear markets. More recently, Bégin (Reference Bégin2021) introduced a new cascade-type ESG based on the monetary policy, which is modeled via observable regime-switching dynamics. The model also considers the changing nature of the variance via generalized autoregressive conditional heteroskedasticity (GARCH) models. In the present article, we rely on a modified version of Bégin’s (Reference Bégin2021) model. Specifically:

1. – We use a homoskedastic autoregressive model for price inflation, wage inflation, and the dividend yield with (monetary) regime-dependent long-run levels, as supported by the conclusions of Bégin (Reference Bégin2021, Reference Bégin2023).

2. – We rely on sophisticated models for the short rate, the stock index and alternative asset returns, as well as the investment grade corporate bond yields. Specifically, we use GARCH models to capture heteroskedasticity and a regime-switching component – based on the monetary policy – to capture changes in the level, as in Bégin (Reference Bégin2021).

3. – The risk-free term structure is constructed based on observable factors: the level (proxied by the short rate), the slope, and the curvature of the term structure.Footnote 10 The slope and the curvature are modeled by a two-dimensional autoregressive model.

4. – Investment grade bond yields are used to obtain bond portfolio returns. The bond returns are proxied using a portfolio of ten zero-coupon bonds with maturities increasing between one and ten years.Footnote 11

We use monthly Canadian economic and financial data to estimate the ESG parameters. Our sample period extends from 1998 to 2020. The parameters are obtained by Markov chain Monte Carlo methods similar to those used in Bégin (Reference Bégin2021). Section SM.A of the Supplementary Material gives additional details about the ESG, the data used, and some of the estimation results.

The ESG yields monthly economic and financial variables that are aggregated to annual time intervals, matching the time step used in the pension plan operation. Section SM.A of the Supplementary Material also explains how the monthly quantities are used to obtain annual price and wage inflation, short and long rates, and the returns of various representative asset classes such as long-term risk-free government bonds (RF), investment grade bonds (IG), stock index (S), and alternative assets proxied by private equity (P).Footnote 12

2.2 Mortality and longevity modeling

Merging pension plans changes the characteristics of the plan’s membership, and this may have unexpected consequences. Generally speaking, adding more members to a plan should have a positive effect in terms of diversification when future lifetimes are independent and identically distributed. The diversification effect is less clear when populations with heterogeneous mortality and longevity are combined (Barrieu et al., Reference Barrieu, Bensusan, El Karoui, Hillairet, Loisel, Ravanelli and Salhi2012). For example, if pension mergers occur between groups of members from different industries and geographic regions, new members’ life expectancies may not match those of members already in the plan, and this may affect the overall benefit of the merger. Moreover, the future evolution of mortality improvements might be different for diverse groups of individuals. To study this effect, we use a stochastic mortality model for the Canadian population that accounts for mortality improvements in subpopulations that are not necessarily identically and independently distributed, producing heterogeneous membership in the merged plans.

A common approach when dealing with multipopulation modeling is to consider different age, period, and cohort effects for each subpopulation (e.g., Li & Lee, Reference Li and Lee2005; Cairns et al., Reference Cairns, Blake, Dowd, Coughlan and Khalaf-Allah2011; Dowd et al., Reference Dowd, Cairns, Blake, Coughlan and Khalaf-Allah2011; Villegas & Haberman, Reference Villegas and Haberman2014, among others). This strategy works relatively well when the number of subpopulations considered is small. When the number of subpopulations under study is large, however, this is impractical. To reduce the number of parameters – age, period, and cohort effects – we aim our attention at subpopulation features instead of focusing on each subpopulation, and we stratify some of our parameters according to these features as done in Bégin et al. (Reference Bégin, Sanders and Xu2023b). In the present study, we rely on three different features: sex, region, and socioeconomic status.

1. 1. We first denote the set of sexes by $\mathcal{I}_1 \equiv \{\text{Male},\text{Female}\}$ .

2. 2. Second, we divide the ten Canadian provinces into five main regions that share similar features: Atlantic provinces (Newfoundland and Labrador, New Brunswick, Nova Scotia, Prince Edward Island), Quebec, Ontario, Prairies (Alberta, Manitoba, Saskatchewan), and British Columbia; the latter regions are denoted by $\mathcal{I}_2$ .Footnote 13

3. 3. Third, socioeconomic groups are constructed based on individuals’ pension amounts at retirement under second-pillar earnings-related programs (i.e., Canada Pension Plan and Quebec Pension Plan), which is related to their salary at retirement. The set of socioeconomic groups is denoted by $\mathcal{I}_3 \equiv \{1,2,\ldots,11,\text{Others}\}$ , where low numbers refer to low socioeconomic statuses and high numbers to high statuses.Footnote 14

Based on the insights of Wen et al. (Reference Wen, Kleinow and Cairns2020) and Bégin et al. (Reference Bégin, Sanders and Xu2023b), we consider two period factors and no cohorts effects. Similar to Cairns et al. (Reference Cairns, Kallestrup-Lamb, Rosenskjold, Blake and Dowd2019), we begin our modeling by using a multipopulation gravity model of the two-factor CBD-X type (Hunt & Blake, Reference Hunt and Blake2020; Dowd et al. Reference Dowd, Cairns and Blake2020):

(1) \begin{align} & m^{(i_1,i_2,i_3)}(x,t)\nonumber\\[5pt] &\quad = \frac{D^{(i_1,i_2,i_3)}(x,t)}{E^{(i_1,i_2,i_3)}(x,t)} \notag \\[5pt] &\quad = \exp\!\left ( \beta ^{(i_1)}_0(x) + \kappa ^{(i_1,i_2,i_3)}_1 (t) + \kappa ^{(i_1,i_2,i_3)}_2(t) \left ( x - \overline{x} \right )\right ) \notag \\[5pt] &\quad = \exp\!\left ( \beta ^{(i_1)}_0(x) + \left (\kappa ^{(i_1)}_1 (t) + \kappa ^{(i_2)}_1 (t) + \kappa ^{(i_3)}_1 (t)\right ) + \left ( \kappa ^{(i_1)}_2(t) + \kappa ^{(i_2)}_2(t)+ \kappa ^{(i_3)}_2 (t) \right ) \left ( x - \overline{x} \right )\right ), \end{align}

where $m^{(i_1,i_2,i_3)}(x,t)$ is the time- $t$ crude death rate for age $x$ , sex $i_1 \in \mathcal{I}_1$ , region $i_2 \in \mathcal{I}_2$ , and socioeconomic group $i_3 \in \mathcal{I}_3$ . Moreover, $\bar{x}$ denotes the average age. We assume that the baseline age effect is sex-specific but does not depend on other features. This choice allows our model to stay parsimonious by reducing the number of age effects. The first period effects $\kappa _1^{(i_1)}(t)$ , $\kappa _1^{(i_2)}(t)$ , and $\kappa _1^{(i_3)}(t)$ pick up the time changes in the mortality level, and the second period effects $\kappa _2^{(i_1)}(t)$ , $\kappa _2^{(i_2)}(t)$ , and $\kappa _2^{(i_3)}(t)$ capture changes in the slope of the log-mortality curve. Together, these two period effects capture the future evolution of death rates and potential for future longevity trends.

Following Dowd et al. (Reference Dowd, Cairns, Blake, Coughlan and Khalaf-Allah2011) and Cairns et al. (Reference Cairns, Kallestrup-Lamb, Rosenskjold, Blake and Dowd2019), we model the first stratum period effects via a common long-term trend with a short-term gravity effect. The second and third dimension’s period effects are modeled through dynamics similar to those of the first stratum. More details on the dynamics of the period effects are available in Section SM.B of the Supplementary Material.

This model is estimated using a Kalman filter-based Markov chain Monte Carlo method. It uses Canadian data that span from 1980 to 2015 and ages from 65 to 109 years.Footnote 15 Details about the datasets are also included in Section SM.B of the Supplementary Material.

From this mortality model, we generate mortality scenarios that are used to obtain death probabilities, annuity prices, and death counts. As is commonly done in actuarial science, we denote the time- $t$ one-year death probability of an individual age $x$ by $q_{x,t}$ , which is obtained by calculating $1-e^{\,-m(x,t)}$ based on the most current information in the model. Throughout the paper, we remove the superscripts $i_1$ , $i_2$ , and $i_3$ denoting the individual features for convenience; the reader should remember that these death probabilities are member-specific and depend on their sex, region, and socioeconomic status.

The time- $t$ annuity-due and deferred annuity prices for individuals age $x$ are obtained by taking an expectation over possible mortality scenarios (and conditional to current conditions). Notice that future death probabilities and annuity prices are random and path-dependent as period effects change through time.Footnote 16

2.3 Model for economies of scale and the cost of pension funds

Pension plans can be expensive to run. Typically, costs are separated into two different categories: administrative and investment costs. The former cost generally depends on the number of members, the service provided by the plan, the plan complexity, and other controls, whereas the latter cost is tied to the amount of assets under management.

In the present study, we rely on the administrative cost model estimated by Bikker et al. (Reference Bikker, Steenbeek and Torracchi2012). We focus on the parameters estimated based on Canadian pension plans for consistency. The administrative cost model is given by:

(2) \begin{align} \log \!\left ( \text{Administrative Cost}_t \right ) = &\, \alpha + \beta \, \log \!\left ( \text{Members}_t \right ) + \gamma \, \text{Service}_t + \theta \, \text{Complexity}_t \notag \\ &\, + \sum _{k} \zeta _k \, \text{Controls}_{k,t} \, + \, \sum _{s=1}^t q_s, \end{align}

where $\text{Members}_t$ is the number of plan members, $\text{Service}_t$ is a standardized service quality score, and $\text{Complexity}_t$ is a standardized complexity score.Footnote 17 The controls include the share of retired members, the share of deferred members, and multiple dummy variables related to the number of plans being offered, the pension fund type, and the type of employees in the plan. The authors also control for the effects of pension fund types.Footnote 18 To Bikker et al.’s (Reference Bikker, Steenbeek and Torracchi2012) model, we add price inflation to account for the increase in the prices of goods and services, where $q_t$ is the time- $t$ annual inflation rate. Note that the parameters in Table 8 of Bikker et al. (Reference Bikker, Steenbeek and Torracchi2012) are estimated based on costs in euros expressed in the 2005 price level; we therefore adjust these costs for inflation and convert them to Canadian dollars.Footnote 19 We report the adjusted parameters in Table 1.

Table 1. Administrative cost of Canadian pension fund

Note: This table reports the parameters estimated by Bikker et al. (Reference Bikker, Steenbeek and Torracchi2012). The parameters were adjusted for inflation and converted to Canadian dollars. The exchange rate is set to the 2005 average rate as given by the Canada Revenue Agency, that is, 1.5090. The total increase in the inflation index between 2005 and 2021 was 28.4%. The parameter standard errors are reported in parentheses.

Fig. 1 illustrates the per member annual administrative cost implied by the parameters of Table 1; we assume that all scores and control dummies are zero and that the share of retired members is set to 40% for simplicity’s sake in this figure. This cost decreases as the number of members increases, capturing economies of scale in the pension plan’s administrative fees.

Figure 1 Per member annual administrative cost of Canadian pension fund.

Note: This figure reports the per member administrative cost as a function of the plan membership using the parameters of Table 1. We assume that all scores and control dummies are set to zero and that the share of retired members is set to 40%.

The investment cost assumption is simpler: we set the cost to 0.5% of assets under management, consistent with the 2010–2020 average cost across the 38 member countries of the Organization for Economic Cooperation and Development (see OECD, 2022, for more details):

(3) \begin{equation} \text{Investment Cost}_t = 0.005 \times A_{t}, \end{equation}

where $A_{t}$ is the value of the plan’s assets at time $t$ . In the recent literature, there is limited evidence for economies of scale in investment costs (see, e.g., Bikker & Meringa, Reference Bikker and Meringa2022).

3.1 Assets, liabilities, and normal cost calculation

We start by assuming that we know the time- $t$ total assets and show how to obtain the contributions, benefits, and normal costs at time $t$ as well as the total assets and total liabilities at time $t+1$ . We assume that the contributions, benefits, administrative cost, and investment cost are paid at the beginning of the year for simplicity’s sake. Fig. 2 shows the timing of the various cash flows of interest.

Figure 2 Assets, contributions, benefits, costs, and liabilities timing.

Note: This figure shows the timing of the assets, contributions, benefits, costs, and liabilities.

Let the time- $t$ (continuously compounded) valuation rate used to compute the liabilities be denoted by $v_t$ (see Section 3.2 for more details on this rate). Let us further assume that $\mathcal{L}_t$ represents the set of members (actives and retirees) at time $t$ ; that is, $k \in \mathcal{L}_t$ if and only if the $k^{\text{th}}$ member is alive at time $t$ . The age of the $k^{\text{th}}$ life is denoted by $x_k + t$ at time $t$ . We divide the membership into two subsets: $\mathcal{A}_t$ and $\mathcal{R}_t$ are subsets of $\mathcal{L}_t$ representing the active members and retired members at time $t$ , respectively

As commonly done for DB plans, we use a formula to determine the benefits. Specifically, throughout this study, we assume that the $k^{\text{th}}$ member’s annual benefit at retirement is calculated as 1.5% of their final salary, multiplied by the number of years of service.Footnote 20

Their estimated pension based on past service is given by:

(4) \begin{equation} B_{k,t} = \left \{ \begin{array}{l@{\quad}l} 0.015 \times \text{Projected Final Salary}_{k,t} \times \text{Years of Service}_{k,t} & \text{ if } x_k + t \lt R \\[5pt] 0.015 \times \text{Final Salary}_{k} \times \text{Years of Service}_{k,t} & \text{ if } x_k + t \geq R \end{array} \right ., \end{equation}

where $R$ is the retirement age and $\text{Final Salary}_{k}$ is the salary of member $k$ in their final year of employment. Salary is projected until retirement based on the (continuously compounded) unconditional wage inflation average $\overline{w}$ and the (continuously compounded) merit-based salary increases $m$ set to 0.5% in this study:

\begin{equation*} \text{Projected Final Salary}_{k,t} = \text{Salary}_{k,t} \times e^{m (R-x_k-t)} \times e^{ \overline{w} (R-x_k-t)}, \end{equation*}

where $\text{Salary}_{k,t}$ is the time- $t$ salary of member $k$ .

The time- $t$ liability associated with this specific member is thus given by:

\begin{equation*} L_{k,t} = \left \{ \begin{array}{l@{\quad}l} B_{k,t} \, \times \,{}_{R-x_k-t\,|}\ddot{a}_{x_k+t,t} & \text{ if } x_k + t \lt R \\[5pt] B_{k,t} \, \times \, \ddot{a}_{x_k+t,t} & \text{ if } x_k + t \geq R \end{array} \right ., \end{equation*}

where ${}_{n\,|}\ddot{a}_{x_k+t,t}$ is a time- $t$ $n$ -year deferred annuity using rate $v_t$ for discounting and $\ddot{a}_{x_k+t,t}$ is a time- $t$ annuity-due using rate $v_t$ for discounting.Footnote 21 Again, notice that the superscripts $i_1$ , $i_2$ , and $i_3$ are removed from the annuity factors for convenience’s sake.

The total current liability is then the sum of each member’s liability; that is,

(5) \begin{equation} L_t = \sum _{k \in \mathcal{L}_t} L_{k,t} . \end{equation}

Each member’s normal cost is based on their (one-year-ahead) projected liability at time $t+1$ , which is a function of the information available at time $t$ . Specifically, the projected benefit is given by:

\begin{equation*} \tilde{B}_{k,t+1} = 0.015 \times \text{Projected Final Salary}_{k,t} \times \left ( \text{Years of Service}_{k,t} + 1\right ) . \end{equation*}

The $k^{\text{th}}$ member’s (one-year-ahead) projected liability isFootnote 22

\begin{equation*} \tilde{L}_{k,t+1} = \left \{ \begin{array}{l@{\quad}l} \tilde{B}_{k,t+1} \, \times \,{}_{R-x_k-t-1\,|}\ddot{a}_{x_k+t+1,t} & \text{ if } x_k + t \lt R \\[5pt] \tilde{B}_{k,t+1} \, \times \, \ddot{a}_{x_k+t+1,t} & \text{ if } x_k + t \geq R \end{array} \right . . \end{equation*}

The normal cost is thus given by the difference between the (discounted) projected liability at time $t+1$ and that at time $t$ :

\begin{equation*} \text{Normal Cost}_{\,k,t} = \left \{ \begin{array}{l@{\quad}l} \tilde{L}_{k,t+1} \, e^{-v_t} \, p_{x_k+t,t} -{L}_{k,t} & \text{ if } k \in \mathcal{A}_t \\[5pt] 0 & \text{ if } k \in \,\mathcal{R}_t \end{array} \right ., \end{equation*}

where $p_{x_k+t,t} = 1 - q_{x_k+t,t}$ is the (random) one-year survival probability.Footnote 23 The total contribution at time $t$ is given by the sum of the normal costs across the active members, the various administrative and investment costs, and special payments that smooth the surpluses and deficits arising over time:

(6) \begin{equation} C_{t} = \sum _{k\in \mathcal{A}_t} \text{Normal Cost}_{\,k,t} + \text{Administrative Cost}_{t} + \text{Investment Cost}_{t} + \kappa \!\left (L_{t} - A_{t} \right ), \end{equation}

where $\kappa$ is the smoothing factor (set to 20% here) used to determine the special payments.Footnote 24

The total benefit paid for year $t$ is obtained by summing up all benefits paid to each retired member, that is,

(7) \begin{equation} B_t = \sum _{k\in \mathcal{R}_t} B_{k,t}, \end{equation}

where $B_{k,t}$ is given in Equation (4). From the time- $t$ total contributions and benefits from Equations (6) and (7), respectively, we can obtain the value of the assets at time $t+1$ recursively:

(8) \begin{align} A_{t+1} =&\, \left ( A_{t} - B_{t} + C_{t} \overbrace{\vphantom{\Big (} - \text{Administrative Cost}_{t} - \text{Investment Cost}_{t}}^{-\,\text{Costs}_{t}} \right ) \notag \\[5pt] &\, \, \times \underbrace{\left ( \omega ^{\text{(RF)}}_{t} e^{R^{(\text{RF})}_{t+1}} + \omega ^{\text{(IG)}}_{t} e^{R^{(\text{IG})}_{t+1}} + \omega ^{\text{(S)}}_{t} e^{R^{(\text{S})}_{t+1}} + \omega ^{\text{(P)}}_{t} e^{R^{(\text{P})}_{t+1}} \right )}_{e^{R_{t+1}^{\text{(PF)}}}} \,, \end{align}

where $\omega ^{(a)}_{t}$ is the portfolio allocation to asset $a$ decided based on information at time $t$ , $R^{(a)}_{t+1}$ is the return on asset $a$ at time $t+1$ , and $a\in \{\text{RF},\text{IG},\text{S},\text{P}\}$ . Moreover, $R_{t+1}^{\text{(PF)}}$ is the (continuously compounded) total asset return at time $t+1$ .

Finally, from both the total assets and liabilities at time $t+1$ , we can obtain the funded ratio at time $t+1$ as:

(9) \begin{equation} F_{t+1} = \frac{A_{t+1}}{L_{t+1}}. \end{equation}

3.2 Valuation rate

The valuation rate $v_t$ , introduced at the beginning of Section 3.1, is determined in a building block fashion reminiscent of the methodology proposed by the Canadian Institute of Actuaries in their 2015 educational note (see CIA, 2015, for more details). In this study, we rely on the current long rate and unconditional expected future investment returns for the various asset classes and then combine these estimates to obtain a valuation rate that is aligned with the plan’s investment policy and the current economic environment.

We start from the current long rate $l_{t}$ as our baseline and add to it the weighted average of long-term risk premiums – defined as the unconditional expected future investment returns minus the current long rate – based on the asset mix selected at time $t-1$ . Specifically, the time- $t$ valuation rate is given by:

(10) \begin{equation} v_t = l_{t} \quad + \sum _{a\in \{\text{RF},\text{IG},\text{S},\text{P}\}} \omega ^{(a)}_{t-1} \left ( \overline{R}^{(a)} - l_{t} \right ), \end{equation}

where the unconditional expected return on asset $a$ is denoted by $\overline{R}^{(a)}$ . This simplifies to $v_t = \sum _{a\in \{\text{RF},\text{IG},\text{S},\text{P}\}} \omega ^{(a)}_{t-1} \overline{R}^{(a)}$ in our context – the valuation rate is the long-run investment return based on the asset mix at time $t-1$ .

3.3 Asset allocation

The asset dynamics of Equation (8) depend on the asset allocation

\begin{equation*}\boldsymbol {\omega }_{t} = \left [ \,\, \omega ^{\text {(RF)}}_{t} \quad \omega ^{\text {(IG)}}_{t} \quad \omega ^{\text {(S)}}_{t} \quad \omega ^{\text {(P)}}_{t} \,\, \right ]^{\top },\end{equation*}

which has to be selected by the plan. To mimic the process used by the plan to decide on an optimal asset allocation strategy, we use a utility-based approach, as presented in Warren (Reference Warren2019). This is not the first use of utility functions for forming portfolios in the literature (see, e.g. Adler & Kritzman, Reference Adler and Kritzman2007; Blake et al., Reference Blake, Wright and Zhang2013; Levy, Reference Levy2016; Estrada & Kritzman, Reference Estrada and Kritzman2018).

In our context, utility functions are convenient as they provide a mechanism to evaluate all potential outcomes by giving each of them a score; these scores can then be averaged to obtain an expected utility, which is maximized to obtain the optimal allocation. This approach is comprehensive as it captures all information about the range of potential outcomes (Adler & Kritzman, Reference Adler and Kritzman2007). A related advantage is that we do not have to use a specific distribution, as the method is flexible enough to accommodate any type of distribution.Footnote 25

Choosing the right utility function is difficult and subjective; yet, some have been used in the pension context thus far, and we build on this past research. Specifically, in this study, we use a reference-dependent utility function to evaluate the outcomes – the future funded ratio $F$ – relative to some reference level – a fully funded plan with a funded ratio of 1:

(11) \begin{equation} u_{A}(F) = \left \{ \begin{array}{l@{\quad}l} w_L \!\left ( F^{c_L} - 1\right ) & \text{if } F \lt 1 \\[5pt] w_G \!\left ( F^{c_G} - 1\right ) & \text{if } F \geq 1 \end{array} \right ., \end{equation}

where $c_L$ and $c_G$ are the curvature parameters on losses and gains, respectively, and $w_L$ and $w_G$ are the weighting parameters on the losses and gains, respectively.Footnote 26 In this article, we use parameters similar to those proposed in Warren (Reference Warren2019): $c_L = 0.88$ , $c_G = 0.44$ , $w_L = 9$ , and $w_G = 1$ . The horizon of the future funded ratio $h$ is set to three years as in Warren (Reference Warren2019).Footnote 27

Figure 3 Reference-dependent utility as a function of the funded ratio.

Note: This figure shows the reference-dependent utility function used in this study. The parameters are inspired from Blake et al. (Reference Blake, Wright and Zhang2013), and this methodology was recently used by Warren (Reference Warren2019).

Fig. 3 reports the shape of the reference-dependent utility. This function is kinked around the reference funded ratio of 1. It reflects the loss aversion of the plan, that is, the penalty applied to deficits is much greater than the reward applied to surpluses, all other things being equal. The function also exhibits curvature, especially as the funded ratio approaches 0.

We find the optimal allocation by maximizing the expected reference-dependent utility:

(12) \begin{equation} \underset{\boldsymbol{\omega }_{t}}{\text{argmax}} \, \, \mathbb{E}_{t} \!\left [ u_A (F_{t+h}) \right ], \end{equation}

where $\mathbb{E}_{t}$ represents the expected value operator based on the information available at time $t$ . This optimization problem is solved numerically using the Nelder–Mead method (fminsearch in Matlab) and by simulating a sample of future funded ratios based on possible realizations of the ESG and the mortality model, thus leading to nested simulations. Specifically, for a given allocation, the expected utility is obtained by first generating values of the future funded ratio at time $t+h$ based on current economic and mortality conditions at time $t$ (see Section 3.1 for more details).Footnote 28 Then, we evaluate the utility score for each scenario, and we average these values across the scenarios to obtain the expected utility of Equation (12). The optimization routine changes the allocation until it finds the one that maximizes the expected reference-dependent utility given above.

To be consistent with the three-year horizon used in the optimization, we update the optimal asset allocation every three years in our setup (i.e., at time 0, 3, 6, and so forth).

3.4 Pension fund dynamics and plan operation

We start our plan operation dynamics by assuming a given membership structure. Based on it and an exogenous estimate of the initial funded ratio $F_0$ , we get the total liabilities at time 0, $L_0$ , and the total assets at time 0, $A_0$ . Our economic and mortality projections start in 2021, meaning that we assume that 2020 is time 0, and we project the variables of interest for $\tau =50$ years. Then, for each economic and mortality scenario and for each year, we follow the steps of Algorithm 1 to project the plan operation.

Algorithm 1. Plan operation dynamics

4.1 Funding adequacy via economic capital

The notion of economic capital has been extensively used for the purpose of risk quantification in the financial and insurance sectors. More recently, a few studies considered its application in the pension context (see, e.g., Porteous et al. Reference Porteous, Tapadar and Yang2012; Yang & Tapadar, Reference Yang and Tapadar2015; Tapadar et al., Reference Tapadar, Andrews, Bonnar, Curtis, Oberoi and Pittea2019; Andrews et al., Reference Andrews, Bonnar, Curtis, Oberoi, Pittea and Tapadar2021). Similar to these studies, we adopt the following definition of economic capital.Footnote 29

This definition is very general, and some subjective choices must be made to operationalize the measure. First, we select a very long horizon of 50 years (same as $\tau$ above) to ensure that short-term fluctuations do not impact our end results. Second, we focus on a going-concern valuation, which assumes that the pension plan continues into the future indefinitely. This strategy is different than that used by Porteous et al. (Reference Porteous, Tapadar and Yang2012), Yang & Tapadar (Reference Yang and Tapadar2015), Tapadar et al. (Reference Tapadar, Andrews, Bonnar, Curtis, Oberoi and Pittea2019), and Andrews et al. (Reference Andrews, Bonnar, Curtis, Oberoi, Pittea and Tapadar2021); the latter authors use a solvency basis and assume that no new members join the plan.

Based on the quantities of Section 3, we define the time- $t$ net cash outflow of the pension plan (excluding investment returns):

\begin{equation*} X_t = B_{t} - C_{t} + \text{Administrative Cost}_{t} + \text{Investment Cost}_{t}, \end{equation*}

which accounts for the benefits, the contributions, and the various costs. The time- $t$ profit is then defined as:

\begin{equation*} P_t = L_{t-1} \exp\!\left ( R_t^{\text{(PF)}}\right ) - X_t - L_t, \quad \text{for } t \in \{1,2,3,\ldots,\tau \}, \end{equation*}

and the initial profit is given by $P_0 = A_0 - X_0 - L_0$ .Footnote 31 The present value of the future profits over the time horizon from 0 to $\tau$ is thus given by:

(13) \begin{align} V_0^{(\tau )} = &\, \sum _{t=0}^{\tau } P_t \exp\!\left ( - \sum _{s=1}^{t} R_s^{\text{(PF)}}\right ) \notag \\[5pt] = &\, A_0 - \sum _{t=0}^{\tau } X_t \exp\!\left ( - \sum _{s=1}^{t} R_s^{\text{(PF)}}\right ) - L_{\tau } \exp\!\left ( - \sum _{s=1}^{\tau } R_s^{\text{(PF)}}\right ). \end{align}

One of our main aims is to compare pre- and post-merger versions of $V_0^{(\tau )}$ ; yet, these values would depend on the initial asset values of the respective plans. To remove any impacts related to the initial asset values, we express $V_0^{(\tau )}$ as a percentage of $A_0$ and denote it by $\check{V}_0^{(\tau )}$ . This standardization is helpful as it allows for consistent and comparable values across different plans.

The actual economic capital risk measures can be found in terms of the distribution of $\check{V}_0^{(\tau )}$ . As commonly done in the actuarial literature, we rely on the value-at-risk (VaR) and the expected shortfall (ES). The former measure is defined as:

(14) \begin{equation} \mathbb{P}_0 \!\left ( \check{V}_0^{(\tau )} \leq \text{VaR}_p^{\text{Plan}} \right ) = p, \end{equation}

for a given probability $p$ , and it represents the percentage of initial assets required at time 0 for the plan to meet all future obligations with probability $1-p$ . The latter measure is defined as:

(15) \begin{equation} \text{ES}_p^{\text{Plan}} = \mathbb{E}_0 \!\left [ \check{V}_0^{(\tau )} \, \middle | \, \check{V}_0^{(\tau )} \leq \text{VaR}_p^{\text{Plan}} \right ] \end{equation}

for a given probability level $p$ , and it represents the average of all losses (in percentage) that are greater than or equal to the value of $\text{VaR}_p^{\text{Plan}}$ (or the gains that are lesser than or equal to $\text{VaR}_p^{\text{Plan}}$ ).

4.2 Member’s welfare and consumption utility

Comparing pre- and post-merger consumption streams is relevant for determining whether merging is beneficial from the members’ perspective. In economics and other related fields, it is common to compare utility (or preferences) instead of dollar amounts as this represents better how members assess their welfare. The utility framework is thus used to mimic this assessment process and compares different options – pre- and post-merger satisfaction.

Plan members’ utility functions are assumed to be to isoelastic (power utility) in this study, such that

\begin{equation*} u_M(\chi ) = \frac{\chi ^{1-\eta }}{1-\eta }, \end{equation*}

where $\chi$ is the consumption and $\eta$ is a constant that is positive for risk-averse agents (set to 5 in this study).

From the quantities defined in Section 3, we can determine the actual consumption stream under each economic and mortality scenario. Specifically, a retired member is receiving their benefits as per Equation (4); an active member, on the other hand, is receiving their salary, minus their share of the contributions, which is computed as half of the contribution rate times the member’s salary.Footnote 32 The contribution rate, in turn, is calculated by taking the total contribution $C_t$ and dividing it by the total payroll. In summary, the time- $t$ consumption for the $k^{\text{th}}$ member is determined by the following equation:

\begin{equation*} \chi _{k,t} = \left \{ \begin{array}{l@{\quad}l} \text{Salary}_{k,t} - \frac{1}{2} \left ( \frac{C_t}{\sum _{l\in \mathcal{A}_t} \text{Salary}_{l,t} }\right ) \times \text{Salary}_{k,t} & \text{if } x_k + t \lt R \\[5pt] B_{k,t} & \text{if } x_k + t \geq R \\[5pt] \end{array} \right . . \end{equation*}

Because members receive streams of consumption – and not just lump sums – we need to aggregate their consumption over their future lifetimes. In the present study, we use the expected discounted utility model to combine the consumption payments:

(16) \begin{equation} \mathbb{E}_0 \!\left [ \sum _{t = 0}^{\infty } e^{-\delta t} \, u_M \big ( \! \underbrace{ \chi _{k,t} \, e^{-\sum _{s=1}^{t} q_s} \!}_{\text{Real consumption}} \big ) \,{}_tp_{x_k} \right ], \end{equation}

where $\delta$ is the (continuously compounded) rate associated with the member’s subjective discount factor (set to 2% in this study).Footnote 33 We consider consumption in real terms to make it comparable from one year to the next; this can be easily done in our framework by using each scenario’s realization of price inflation. We also consider mortality by multiplying the utility function by the $t$ -year survival probability for member $k$ , ${}_tp_{x_k}$ ; note that this probability is also random and depends on the realized mortality scenario.

The expected discounted utility is by itself not informative unless it is transformed into a meaningful quantity. For this reason, many authors rely on the certainty equivalent consumption (CEC) to compare consumption streams. In our case, the $k^{\text{th}}$ member’s CEC is defined as the solution to the following equation:

(17) \begin{equation} \mathbb{E}_0 \!\left [ \sum _{t = 0}^{\infty } e^{-\delta t} \, u_M \big ( \text{CEC}_{k} \big ) \,{}_tp_{x_k} \right ] = \mathbb{E}_0 \!\left [ \sum _{t = 0}^{\infty } e^{-\delta t} \, u_M \big ( \chi _{k,t} \, e^{-\sum _{s=1}^{t} q_s} \big ) \,{}_tp_{x_k} \right ]. \end{equation}

The CEC can be interpreted as the guaranteed consumption in real terms that a member receives in exchange for their random consumption stream once risk and preferences are accounted for.

4.3 Employer risk and contributions

The employer is an important stakeholder in the context of pension plan mergers. Comparing pre- and post-merger employer contributions (and their risk) allows us to quantify the financial benefits derived from the merger.

To assess contribution risk from the employer’s perspective, we first aggregate the employer’s contributions by taking their present value over time horizon $\tau$ :

\begin{equation*} \zeta _0^{(\tau )} = - \sum _{t=0}^{\tau -1} \frac{C_t}{2} \exp\!\left ( - \sum _{s=1}^t R_s^{\text{(PF)}} \right ), \end{equation*}

where, again, we assume that half of the contributions are paid by the members and the other half by the employer; note the negative sign reflects that contributions are losses from the employer’s perspective. Then, because comparing between pre- and post-merger plans is difficult as the total dollar contribution depends on the size of the respective plans, we express the present value of the contributions as a percentage of initial liabilities, $L_0$ . This standardization allows us to compare the various plans under study; we denote the standardized present value of the contributions by $\check{\zeta }_0^{(\tau )}$ .

Finally, to understand the risk, we again rely on VaR and ES. In the context of employer’s contributions, the first measure is defined as:

(18) \begin{equation} \mathbb{P}_0 \!\left ( \check{\zeta }_0^{(\tau )} \leq \text{VaR}_p^{\text{Employer}} \right ) = p, \end{equation}

for a given probability $p$ . This measure loosely represents the percentage of initial liabilities required at time 0 to pay for all future employer contributions with probability $1-p$ . The second measure is defined as:

(19) \begin{equation} \text{ES}_p^{\text{Employer}} = \mathbb{E}_0 \!\left [ \check{\zeta }_0^{(\tau )} \, \middle | \, \check{\zeta }_0^{(\tau )} \leq \text{VaR}_p^{\text{Employer}} \right ] \end{equation}

for a given probability level $p$ . Similar to $\text{ES}_p^{\text{Plan}}$ , it represents the average of all contributions (in percentage terms) that are greater than or equal to the value of $-\text{VaR}_p^{\text{Employer}}$ .

5.1 Comparison between the small, large, and merged plans

We begin this comparison exercise with a caveat about the size of the differences: one should not expect massive deviations between the small and merged plans. A merger has the potential to improve solvency and welfare, but the size of these improvements will most likely be modest in the grand scheme of things. Note, however, that even the smallest improvements in percentage terms can be worthwhile as it involves substantial gains in dollar terms.

Before turning to the various solvency and welfare measures of Section 4, we first look at different (long-run) outputs that are relevant to understanding the plans under study. Specifically, we compare the merged plan’s behavior to that of the unmerged small and large plans. Unmerged here assumes that the small and large plans continue to operate without merging; these are denoted by “Small plan” and “Large plan” below.

Table 3 reports statistics about the long-run behavior of the small (Panel A), large (Panel B), and merged plans (Panel C); specifically, we report statistics on the distribution of the portfolio weights, the total asset return, the funded ratio, the valuation rate, and the contribution rate, all at the end of the horizon of $\tau = 50$ years.

Table 3. Long-run statistics of asset allocation as well as asset- and liability-related quantities

Note: This table reports the long-run average and standard deviation (std. dev. in the table) as well as the $10^{\text{th}}$ , $25^{\text{th}}$ , $50^{\text{th}}$ , $75^{\text{th}}$ , and $90^{\text{th}}$ percentiles of the weights invested in each asset class, the total asset return, the funded ratio, the valuation rate, and the contribution rate. Long-run quantities are taken at a horizon of 50 years, at which time the random quantities have converged to their long-run distributions. This is done for the small plan (Panel A), the large plan (Panel B), and the merged plan (Panel C). Note that the allocations sum to one in all scenarios.

The asset allocation of the small plan is generally different than that of the merged plan. In addition to rounding allocations to the nearest 5%, the small plan cannot invest in the alternative asset. This implies very different weights, on average. Even though both the small and the merged plans invest about 18% of their assets in long-term government bonds, the allocations are different for the other, riskier assets. On average, the small plan invests about 41% of its assets in investment grade bonds and 41% in the stock index. For the merged plan, it is 35% in investment grade bonds, 31% in the stock index, and 16% in the alternative asset. Access to additional asset classes allows the merged plan to obtain more benefits from diversification, which ultimately leads to an average asset return that is 30 bps higher than that of the small plan. The asset allocation of the large plan, on the other hand, is virtually identical to that of the merged plan – having 1,000 additional members does not impact the plan’s allocation strategy. This leads to also having the same average asset return of 6.8% in the merged plan as in the large plan.

The different asset allocation strategy also impacts the valuation rate used to calculate the liabilities. In the case of the small plan, the average valuation rate is about 6.4%; it is 6.8% for the large and merged plans. This difference of 40 bps makes the per-capita liabilities of the merged plan smaller, which explains the higher average funded ratio at the end of the horizon to some extent. Indeed, the average funded ratio increases from 1.066 in the small plan to 1.081 in the merged plan. The contribution rate also drops, from 12.7% to 10.2%, on average. However, these changes in the funded ratio and the contribution rate are only partially explained by the change in the valuation rate. For instance, plan size differences create gaps between the various plans’ administrative and investment costs and could affect the long-run dynamics of the plans’ funded status and contribution rate. Indeed, administrative and investment costs are 1.3% lower in the merged plan than in the small plan, on average. Moreover, having more (similar) members reduces the risk associated with idiosyncratic mortality, which ultimately increases the funded ratio and decreases the contribution rate.

Figure 4 Distribution of standardized present value of the future profits.

Note: This figure reports the distribution of the standardized present value of the future profits as given by Equation (13) for the small plans in the top panel and for the merged plan in the bottom panel. The first, fifth, and tenth quantiles of both distributions are reported with dotted, dashed, and solid lines, respectively.

To assess the merger from the plan’s, the members’, and the employer’s perspective, we use the measures explained in Section 4. The main goal of this comparison is to assess the improvement (or deterioration) associated with the merger for the plan via economic capital risk measures, for members via CECs, and for employers via contribution risk measures.

Fig. 4 shows histograms of the standardized present value of the future profits for the small (top panel), large (middle panel), and merged plans (bottom panel). These distributions are the primary building blocks in obtaining the economic capital risk measures (i.e., from the plan’s perspective). Note that the left tail – associated with negative profits – is fatter in the case of the small plans. The tenth percentile of the distribution (equivalent to $\text{VaR}_{0.10}^{\text{Plan}}$ ) is improved by about 12% (jumping from $-0.085$ to $-0.075$ ). Similar improvements are observed for other probability levels $p$ as well as for ES measures (see first rows denoted by “Base” in Table 4).

Table 4. Economic capital risk measures and improvements for the small plan

Note: This table reports economic capital risk measures for the small plan and the post-merger improvements (in percentage) in parentheses. The value-at-risk and the expected shortfall are computed using Equations (14) and (15), respectively, for probability levels of 1%, 5%, and 10%. In addition to the base case, the risk measures are also computed for all relevant robustness cases: a small plan initial funded ratio of 0.8 and 1.2, a less mature small plan for which members and retirees are younger than 71 years old, a small plan with no retirees at inception (only active members), a small plan for which members are coming from British Columbia (i.e., lower mortality rates), a small plan for which members are coming from Nova Scotia (i.e., higher mortality rates), a small plan with a higher starting salary of $80,000, a small plan with a lower starting salary of $20,000, a small plan with 90% of members that are female, a small plan with 90% of members that are male, a small plan for which members retire at 70 years old, a larger small plan with 5,000 members, a smaller small plan with 500 members, and a small plan that does not need to round its asset allocation.

The large plan does not seem to be impacted much by the merger when the counterpart is small in comparison: most percentiles in Fig. 4 are the same for both the large and the merged plan. Indeed, improvements in economic capital risk measures (and all the other metrics considered in this study as a matter of fact) are virtually nil for the large plan; see Section SM.C of the Supplementary Material for more details. We note that, although a single merger between a large and small plan may bring negligible benefits to the large plan, multiple successive mergers produce material benefits. This may justify some large plans’ aggressive agenda for expansion through repeated mergers.Footnote 39 For the rest of this section, we focus on the impacts of mergers from the perspective of small plans.

The first rows of Table 5 report relevant CEC measures for male and female members aged 25, 35, 45, and 55 years for the base case.Footnote 40 Overall, the CECs are noticeably larger for the merged plan, meaning that merging is beneficial from the members’ perspective; for instance, 25-year-old males (females) get an additional $265 ($252) per annum in terms of CEC if the plans merge. Differences in the contribution rates can explain the discrepancy between the small and merged plan CECs – lower contribution rates translate into better average consumption profiles.

Table 5. Certainty equivalent consumptions and improvements for the small plan

Note: This table reports certainty equivalent consumptions (CECs) for the small plan and their post-merger improvements (in dollar) in parentheses. The CECs are computed using Equation (17) for males and females aged 25, 35, 45, and 55 years old. In addition to the base case, the CECs are also computed for all the robustness cases mentioned in the caption of Table 4.

Lower contribution rates also impact the employer’s measures. The first rows of Table 6 document the decreases in the present value of the standardized employer’s contributions. The median of the distribution (i.e., $\text{VaR}_{0.50}^{\text{Employer}}$ ) is 24% lower for the merged plan when compared to the small plan. We obtain similar decreases for other probability levels $p$ and for the employer’s ES measures.

Table 6. Employer contribution risk measures and improvements for the small plan

Note: This table reports employer contribution risk measures for the small plan and their post-merger improvements in percentage (in parentheses). The value-at-risk and the expected shortfall are computed using Equations (18) and (19), respectively, for probability levels of 5%, 10%, and 50%. In addition to the base case, the employer contribution risk measures are also computed for all the robustness cases mentioned in the caption of Table 4.

The lower contributions, which affect both members’ and the employer’s welfare, are explained by various factors in our setting. As noted earlier, the small plan’s suboptimal investment strategy due to rounding and its lack of access to alternative investments makes its asset returns lower, on average, which also decreases the valuation rate. The reduced asset returns then need to be compensated by higher contributions from the members and the employer. Second, investment and administrative costs are higher for small plans, on average; this additional cost requires extra contributions from the employer and the members. Finally, having more members in the merged plan creates less uncertainty in the plan’s benefits due to better mortality pooling; the standard deviation of annual benefit payments is 1% lower for the merged plan than that observed in the small plans, on average.

All in all, merging plans is beneficial for both small plan’s members and employers, which is ultimately reflected in the plan’s perspective via lower economic capital measures in Table 4.

5.2 Robustness tests

The results above depend on the assumptions selected at the beginning of this section. To verify the robustness of our conclusions, we consider some modifications that create heterogeneity in the merged plan:

1. – We first consider cases where the small plan is underfunded (with funded ratios of 0.8) or overfunded (with funded ratios of 1.2) before the merger.

2. – We assess the impact of the small plan having younger members and populations that have not reached stationarity at time 0. We look at a case where the small plan has no retirees at time 0 (i.e., “No retirees at inception” throughout the article) and another case where the small plan has retirees but no members older than 71 years of age (i.e., “Less mature plan” hereafter).Footnote 41

3. – We change the demographic profile of the plan, thus impacting the mortality and longevity of our pool; that is, the province of residence (British Columbia or Nova Scotia instead of Ontario), the starting salary at time 0 as a proxy for the socioeconomic group ($80,000 in the case of “High starting salary” and $20,000 in the case of “Low starting salary”), and the sex (90% of members being female in “More female” and 90% of members being male in “More male”).

4. – We investigate the impact of members retiring later in life – at 70 years old.

5. – We modify the size of the small plan from 1,000 members to 500 (i.e., “Smaller plan” hereafter) and to 5,000 members (i.e., ‘Larger plan” henceforth). In the former, the small plan represents about 1% of the large plan’s membership; in the latter, it is about 10%.

6. – We consider a small plan that does not round its asset allocation to verify whether this assumption has a material impact on our results.

Table 4 reports the economic capital risk measures – taken from the plan’s perspective – for the base case of Section 5.1 and each robustness test. Tables 5 and 6 do the same from the members’ and the employer’s perspectives by reporting CECs and risk measures based on the present value of standardized contributions, respectively. Overall, the conclusion that the merger of a small and a large plan is beneficial remains robust to the aforementioned changes; the rest of this section is devoted to explaining these results.

5.3 Merger of three small-sized plans as a robustness test

As an additional robustness test of our results, we apply our framework and metrics to a second case study – a merger involving three small-sized Ontario universities of about 2,000 members each, inspired by UPP’s experience. This case is distinct from the previous one as the plans merging are more similar in nature. The features of this second case study are summarized in Table SM.7 of the Supplementary Material. Detailed assumptions and results are presented in Section SM.D of the Supplementary Material. Overall, the merger of three small-sized plans is beneficial for all parties involved – the plan, the members, and the employers – under all circumstances considered.

6.1 Asset allocation

Smaller plans tend to be more conservative than larger plans with their asset allocation. To capture this possibility, we consider different reference-dependent utility parameter values in Equation (11); specifically, we change the small plan’s loss curvature parameter $c_L$ and the loss weighting parameter $w_L$ to assess the impact of a more risk-averse allocator. We consider three cases: (1) a high loss curvature parameter of $c_L = 2.2$ , (2) a high loss weighting parameter of $w_L = 18$ , and (3) a combination of the two (i.e., high curvature and weighting parameters).

Changing these parameters significantly impacts the long-run asset allocation of the small plan. For instance, the allocation to the stock index is between 32% and 36% in the three modifications considered in this section; it was about 41% in the base case. These lower allocations to the stock index are compensated by larger allocations to the long-term government bond portfolio (increasing from 17% in the base case to 19–21%) and the investment grade bond portfolio (increasing from 41% in the base case to 44–46%).

Having a small plan allocate more of its assets to conservative asset classes enhances the merger’s overall benefits, aligning favorably with the interests of the plan, its members, and the employer. These enhancements typically surpass the outcomes achieved in the base scenario outlined in Section 5. For the three cases considered and on average, the improvements in CECs attributable to the merger are nearly double when compared to the base case, whereas the employer contribution risk measures are elevated by 57% in comparison to the baseline case. Merging allows the small plan members and employer to reduce the average contribution level – a by-product of having higher asset returns, on average, in the merged plan – which improves their risk–reward profiles. From the plan’s perspective, we report a similar behavior to that of the base case of Section 5, meaning that it is still beneficial for the small plan to merge.

6.2 Administrative and investment costs

The small plan’s higher administrative and investment costs could also impact the plan’s stakeholders in nontrivial ways. Accordingly, we consider additional tests in which we raise these costs. Specifically, we investigate administrative costs that are doubled, tripled, and quadrupled, as well as investment costs that are doubled, tripled, and quadrupled with respect to the base case.

Overall, increasing the small plan’s investment or administrative cost makes the merger significantly more beneficial from the members’ and the employer’s perspectives. For instance, doubling the administrative cost increases the impact of the merger on CECs by about 20% when compared to the base case, on average. Doubling the investment cost – from 0.5% to 1.0% of the small plan’s assets – also increases the improvements in the CECs by about 55%, on average.

The employer risk measures exhibit noteworthy enhancements in comparison to the base case scenario outlined in Section 5. On average, the improvement factors surge by 17% when the administrative cost is doubled. Similarly, a substantial increase is observed when the investment cost is increased, surpassing the improvements obtained under the base case.

From the plan’s perspective, the improvements are neither better nor worse when compared to the base case of Section 5. This is expected as the administrative and investment costs are absorbed by plan members and the employer. Indeed, the stream of profits do not change as a function of these expenses.

To conclude this section, considering more risk-averse small plans or higher costs tends to make the merger more beneficial for members and the employer.