2.1. Overlapping generations

We adopt a multi-period overlapping generation model in the continuous-time setting. We assume homogeneous individuals in the sense that their employment entry age is fixed at A and their retirement age is fixed at R, with a maximum attainable age of ω. For the numerical experiments in the later sections, we set A = 25, R = 65, and ω = 115.

Denote _t p _x as the probability that an individual aged x will survive for at least t years, and n _x(t) as the number of lives aged x at time t. We assume the population is stationary with 100 new employees every year entering at the age of 25 and retiring at the age of 65, and assume no pre-exit other than death.

We ignore the longevity risk, and assume the mortality risk is fully diversifiable, which means that the actual number of death is exactly the same as its expectation (i.e., n _x(t) = 100 × _x−25p ₂₅). Mortality rates are calculated based on the Makeham Law so the force of mortality at age x is defined as m _x = A _m + B _m × (c _m)^x. Parameters are fitted to the Canadian Pensioners' Mortality Table 2014 (CPM2014, male), such that A _m = 0.0015, B _m = 1.6605 × 10⁻⁶, and c _m = 1.1339.

2.2. Economic assumption

Since the pioneer work of Merton (Reference Merton1969), the continuous-time model plays a key role in financial modeling. However, Merton's model is not suitable for the applications that require long-term modeling. Regime switching models, as initially proposed by Hamilton (Reference Hamilton1989), include a finite-state continuous-time Markov chain that represents the uncertainty of the market conditions, and thus grant a more accurate representation of the long-term financial market (see Hardy, Reference Hardy2001). In this paper, we follow the model that has been studied in Sotomayor and Cadenillas (Reference Sotomayor and Cadenillas2009), in which the regimes are modeled by an observable continuous-time Markov chain, and all coefficients of the market are regime-dependent.

Denote $( {\Omega , \;{\cal F}, \;{\rm {\opf P}}} )$ as a complete probability space with a right-continuous filtration ${\cal F} = \{ {{\cal F}_t} \} _{t \ge 0}$ generated by a two-dimensional standard Brownian Motion $( {Z( t ) , \;\bar{Z}( t ) } )$ and a continuous-time stationary Markov chain ɛ(t) taking values in a finite state space ${\cal M} = \{ {1, \;2, \;\ldots , \;m} \}$. We further assume that ɛ(t) is independent of Z(t) and $\bar{Z}( t )$. The Markov chain ɛ(t) has a generator Q = (q _ij)_m×m (transition rate matrix) and stationary transition probabilities p _ij(t) = ℙ(ɛ(t) = j|ɛ(0) = i) ≥ 0, for t ≥ 0, and i, j = 1, 2,.., m. Moreover, we stipulate that $\bar{Z}( t )$ correlates with Z(t) with a correlation coefficient ρ _ɛ(t) at regime ɛ(t).

The pension fund trustee allocates across two representative assets, one risk-free and one risky. The value evolution of the risk-free asset S ₀(t) and the risky asset S ₁(t) is given by

$$\eqalign{& dS_0( t ) = r_{\varepsilon ( t ) }S_0( t ) dt, \;\cr & dS_1( t ) = \mu _{\varepsilon ( t ) }S_1( t ) dt + \sigma _{\varepsilon ( t ) }S_1( t ) dZ( t ) , \;} $$

with the initial prices S ₀(0) = 1 and S ₁(0) = s > 0, and the initial state $\varepsilon ( 0 ) = \varepsilon _0\in {\cal M}$, where r _ɛ(t), μ _ɛ(t), and σ _ɛ(t) are ɛ(t)-dependent risk-free rate, equity return, and volatility, respectively. Note that each state of the Markov chain ɛ(t) represents a certain market regime, and the condition μ > r, which is often proposed in a single regime model, is no longer required.

In addition to the stochastic modeling of the equity price, we also model the uncertainty of the income process. Here we denote L(t) as the inflation index and its dynamics is given by

$$\matrix{ {dL( t ) = \alpha _{\varepsilon ( t ) }L( t ) dt + \eta _{\varepsilon ( t ) }L( t ) d\bar{Z}( t ) , \;} \hfill & \qquad \hfill & {} \hfill \cr } $$

where α _ɛ(t) and η _ɛ(t) are the regime-dependent instantaneous growth rate and the volatility of the inflation index, respectively.

We assume the uncertainty in the salary growth is fully captured by the inflation index L(t). The entry salary has been unitized to be the same as the inflation index, and we further stipulate a salary growth rate of γ ₁ (e.g., 50bps) on top of the inflation index L(t) for the active employees to represent the promotional increases. On the other hand, the retirement benefit is subjected to a depreciation rate of γ ₂ (e.g., 100bps) to reflect the inflation adjustment (often referred to as the partial inflation indexation).

For our numerical experiments later, we adopt a two-regime model for the risky asset with its parameters calibrated to the monthly data of S&P 500 from January 2000 to October 2020, which is sufficient to capture the two most shocking events, the 2008–09 financial crisis and the recent COVID-19 pandemic. The two regimes represent the low-volatility regime (regime 1, regular market) and the high-volatility regime (regime 2, recession). Although the existing literature has investigated on modeling with more regimes (e.g., a three-regime model; Cerboni Baiardi et al., Reference Cerboni Baiardi, Costabile, De Giovanni, Lamantia, Leccadito, Massabó, Menzietti, Pirra, Russo and Staino2020), a two-regime model remains popular in practice, and including additional regimes adds only marginal improvement to the overall likelihood (Hardy, Reference Hardy2001). The parameters are given in Table 1.

Table 1. The annualized parameters for a two-regime model

μ _ɛ and σ _ɛ are the equity return and the volatility of the risky asset, p _ɛɛ and q _ɛɛ are the transition probability and the transition rate between regimes, where ɛ = 1 represents the regular market and ɛ = 2 represents the recession.

In addition, we take α ₁ = α ₂ = α = 0.03, η ₁ = η ₂ = η = 0.01, and ρ = 0.1 for the salary process, and assume r ₁ = 0.04 and r ₂ = 0.02 for the risk-free asset. Note that the parameters are chosen based on the related literature (e.g., Sotomayor and Cadenillas, Reference Sotomayor and Cadenillas2009; Yuen and Yang, Reference Yuen and Yang2009; Cui et al., Reference Cui, De Jong and Ponds2011; Wang et al., Reference Wang, Lu and Sanders2018, etc.). Since the values are rather arbitrarily set, additional sensitivity tests are performed in the later sections.

Admittedly, it is impossible to actually detect the instantaneous moments of the switches between the market regimes. However, different indices in practice can be used as the signals for the changes in the underlying regimes. For example, Esteve et al. (Reference Esteve, Navarro-Ibanez and Prats2020) demonstrate that VIX is a good indicator for any structural changes in the market, and the National Financial Conditions Index (NFCI) is introduced to reflect the ‘tightness’ of the overall financial conditions. Cyclical risk sharing design can rely on these observable external financial stress indices to monitor the switches between different regimes.

2.3. Plan provisions

The TB pension plan can be either funded by the active members, the sponsor, or both, via contributing a fixed percentage of the active workers' salary c continuously into the fund until they are either retired or deceased. The retirement benefit, which will be adjusted according to the financial conditions, is expressed as a percentage b(t) of their pre-retirement income (inflation adjusted). Therefore, b(t) can be viewed as the instantaneous replacement ratio at time t. In a TB plan, the actual benefit b(t) may fluctuate around a target level $\bar{b}$. Since we are considering a homogeneous population without the mortality risk, the aggregate contribution C _ɛ(t)(t) and the aggregate benefit B(t) at time t can be expressed as:

(1)$$C_{\varepsilon ( t) }( t) = \mathop {\underbrace{{\mathop \int \nolimits_A^R }}_{{}}}\limits_{{\rm Active\ Workers}} c_{\varepsilon ( t) } \times \mathop {\underbrace{{L( t) \times e^{( x-A) \gamma _1}}}_{{}}}\limits_{{\rm Current\ Salary}} \times \mathop {\underbrace{{n_x( t) }}_{{}}}\limits_{\# {\rm \ of\ Age\ x}} dx = c_{\varepsilon ( t) } \times L( t) \times {\cal A}( t) , \;$$

(2)$$\matrix{ {B( t) } \hfill & { = \mathop {\underbrace{{\mathop \int \nolimits_R^\omega }}_{{}}}\limits_{{\rm Retirees}} b( t) \times \mathop {\underbrace{{L( t) \times e^{( R-A) \gamma _1} \times e^{-( x-R) \gamma _2}}}_{{}}}\limits_{{\rm Salary\ Adjustment}} \times n_x( t) dx = b( t) \times L( t) \times {\cal R}( t) , \;} \hfill \cr\cr {{\cal A}( t) } \hfill & { = \int_A^R {e^{( x-A) \gamma _1}} \times n_x( t) dx, \;} \hfill \cr\cr {{\cal R}( t) } \hfill & { = \int_R^\omega {e^{( R-A) \gamma _1}} \times e^{-( x-R) \gamma _2} \times n_x( t) dx, \;} \hfill \cr } $$

where $e^{( {x-A} ) \gamma _1}$ is the promotion adjustment to the salary for the age x, and since the mortality risk is ignored, the salary immediately before the retirement at time t is simply $L( t ) e^{( {R-A} ) \gamma _1}$. When we ignore the salary growth rate and the retirement benefit adjustment rates such that γ ₁ = γ ₂ = 0, then ${\cal A}( t )$ and ${\cal R}( t )$ will represent the population size of the active workers and the retirees at time t, respectively. Furthermore, if the population becomes stationary, i.e., n _x(t) = n _x, then we have ${\cal A}( t ) = {\cal A}$ and ${\cal R}( t ) = {\cal R}$.

In addition to the contribution and the retirement benefit, we assume that a death benefit is payable immediately as the member deceases. Here we denote D _x(t) as the death benefit for an aged x worker at time t, and assume it is simply a multiple δ of the member's current annualized salary if s/he is an active worker, or the target retirement benefit if s/he is a retiree. Specifically,

$$D_x( t ) = \left\{{\matrix{ {\delta \times L( t ) \times e^{( {R-A} ) \gamma_1}, \;\quad \quad \quad \quad\quad \!\!} & {{\rm \quad\quad \!\! for\;active\;employees}, \;} \cr {\delta \times \bar{b} \times L( t ) \times e^{( {R-A} ) \gamma_1-( {x-R} ) \gamma_2}, \; } & {{\rm \!\!\!\!\!\!\!\!\!\!\!\!\!\! for\;retirees}.} \cr } } \right.$$

Therefore, the aggregate death benefit payment made by the sponsor at time t is:

$$\matrix{ {D( t ) } \hfill & { = \mathop \int \nolimits _A^\omega n_x( t ) \times m_x( t ) \times D_x( t ) dx\qquad } \hfill & {} \hfill & \qquad \hfill \cr {} \hfill & { = \delta \times L( t ) \times {\cal D}_A( t ) + \delta \times \bar{b} \times L( t ) \times {\cal D}_R( t ) \qquad } \hfill & {} \hfill & \qquad \hfill \cr {} \hfill & { = L( t ) \times {\cal D}( t ) , \;\qquad } \hfill & {} \hfill & \qquad \hfill \cr } $$

where m _x(t) is the force of mortality for workers aged x + t, ${\cal D}_A( t ) = \mathop \smallint \nolimits_A^R n_x( t ) \times m_x( t ) \times e^{( {x-A} ) \gamma _1}dx$, ${\cal D}_R( t ) = \mathop \int \nolimits _R^\omega n_x( t ) \times m_x( t ) \times e^{( {R-A} ) \gamma _1-( {x-R} ) \gamma _2}dx$, and ${\cal D}( t ) = \delta \times {\cal D}_A( t ) + \delta \times \bar{b} \times {\cal D}_R( t )$. Again, if γ ₁ = γ ₂ = 0, then ${\cal D}_A( t )$ and ${\cal D}_R( t )$ represent the number of death for the active workers and the retirees at time t, respectively.

2.4. Pension assets

Let π _ɛ(t) be the portfolio weight invested in the risky asset when the market is in regime ɛ(t), then the pension asset X(t) has the following SDE:

(3)$$\matrix{ {dX( t) } \hfill & { = \pi _{\varepsilon ( t) }X( t) \displaystyle{{dS_1( t) } \over {S_1( t) }} + ( 1-\pi _{\varepsilon ( t) }) X( t) \displaystyle{{dS_0( t) } \over {S_0( t) }} + ( C_{\varepsilon ( t) }( t) -B( t) -D( t) ) dt} \hfill \cr {} \hfill & { = [ X( t) ( r_{\varepsilon ( t) } + \pi _{\varepsilon ( t) }( \mu _{\varepsilon ( t) }-r_{\varepsilon ( t) }) ) + ( C_{\varepsilon ( t) }( t) -B( t) -D( t) ) ] dt + \sigma _{\varepsilon ( t) }\pi _{\varepsilon ( t) }X( t) dZ( t) , \;} \hfill \cr } $$

where X (0) = x ₀. Here we assume the trustee of the pension fund adopts the constant weighting portfolio. In contrast to the common assumption in the optimal control literature where a dynamic asset allocation is often used, the pension portfolio in practice is often closer to a constant weighting strategy where the asset allocation is closely tied to a target mix.Footnote ⁴ The traditional approach for the portfolio mix is 60/40 in Anglo-American countries, and 70/30 in the UK (Franzen, Reference Franzen2010). In addition, it has also been demonstrated by, for example, Graf (Reference Graf2017) and Forsyth and Vetzal (Reference Forsyth and Vetzal2019), that it is not easy to out-perform the constant weight portfolio. We do include the equity share as an additional control variable in Section 4 and illustrate the limitations of the market friction-less model.

2.5. Target benefit and pension liability

Since most of the TB plans are transitioned from the existing DB plans, the benefit accrual formulae, as well as the liability valuation methods, are often kept the same. The TB level under a TB plan is often set as the promised benefit level in the original DB plans. Therefore, we assume the target replacement rate $\bar{b}$ is externally givenFootnote ⁵ such that the TB is $\bar{b} \times L( t )$ for a newly retired person at time t and is $\bar{b} \times L( t ) \times e^{( {R-A} ) \gamma _1-( {x-R} ) \times \gamma _2}$ for a retiree aged x. The aggregate amount of target retirement benefit can be simplified as:

$$\matrix{ {\bar{B}( t ) = \bar{b} \times L( t ) \times {\cal R}( t ) .} \hfill & \qquad \hfill & {} \hfill \cr } $$

Based on the TB level, we adopt the traditional unit credit (TUC) approach for the liability valuation.Footnote ⁶ When early leaves are not allowed (except in case of death), the liability can be expressed as:

(4)$$\matrix{ {{\rm Liability}( t) } \hfill & { = \int_A^\omega {n_x} \times \mathop {\underbrace{{\displaystyle{{\min ( x, \;R) -A} \over {R-A}}}}_{{}}}\limits_{{\rm Accrued\ Benefit}} \times \mathop {\underbrace{{\int_{\max ( x, R) }^\omega {v^{u-x}} \times _{u-x}p_x \times \bar{b} \times L( t) du}}_{{}}}\limits_{{\rm Present\ Value\ of\ Future\ Retirement\ Benefit}} dx } \hfill \cr {} \hfill & {\qquad + \int_A^\omega {n_x} \times \mathop {\underbrace{{\int_x^\omega {_{u-x} } p_x \times m_u \times v^{u-x} \times D_u( t) du}}_{{}}}\limits_{{\rm Present\ Value\ of\ Death\ Benefit}} dx, \;} \hfill \cr } $$

where v = (1 + r _d)⁻¹ is the discount factor with r _d being the discount rate. Although the choice of the discount rate should depend on both the current interest rate and the expected return of the pension funds, in practice, changes in the discount rate are rather slow to avoid volatile liability values between the adjacent years. Therefore, in this paper, the discount rate is chosen to be a regime-independent constant, and the liability value will also be regime-independent.

2.6. Problem formulation

One of the main purposes of introducing risk sharing structure in a pension plan is to postpone the pension deficits of the unfortunate cohorts, or to spread excess profits from the windfall cohorts. Therefore, a successful TB design should be able to limit the variability of benefits across all generations. Specifically, the objective is to minimize the variation between the actual and the TB level. In this paper, we choose a quadratic function as the measure to gauge the income stability.Footnote ⁷ In light of the above, the income stability at time t is measured as

$$\matrix{ {{( {{\rm Aggregate\;Benefit}-{\rm Aggregate\;Target\;Benefit}} ) }^2 = {( {b( t ) \times L( t ) \times {\cal R}( t ) -\bar{b} \times L( t ) \times {\cal R}( t ) } ) }^2.} \hfill & \qquad \hfill & {} \hfill \cr } $$

Notice that changing the TB level is equivalent to adding a penalty term. For example, in Wang et al. (Reference Wang, Lu and Sanders2018), the authors add a linear penalty term $\varrho \times [ {\bar{b} \times L( t ) \times {\cal R}( t ) -b( t ) \times L( t ) \times {\cal R}( t ) } ]$, which is equivalent to changing the target replacement rate to $\bar{b} + \varrho /2$. For most DB plans, the promised benefit level will be kept the same regardless of the market regimes. Therefore, the TB is selected to be regime-independent in this paper.

From both the numerical and the practical aspects, the projection period is often a finite number (e.g., T is chosen between 20 and 100 years). Thus, some restrictions must be made at the terminal time T to avoid excessive deficits being transferred to the future generations. Similar to the income stability, here we impose a quadratic terminal condition at the end of the projection period to penalize the instability of the terminal asset value. Specifically, we minimize the quadratic difference between the actual and the target asset levels:

$$\matrix{ {{\rm Penalty\;Weight} \times {( {{\rm Terminal\;Asset\;Level}-{\rm Target\;Asset\;Level}} ) }^2 = \lambda _{\varepsilon ( T ) }\cdot {( {X( T ) -X_{\varepsilon ( T ) }^\ast ( T ) } ) }^2, \;} \hfill & \qquad \hfill & {} \hfill \cr } $$

where λ _ɛ(T) is the penalty weight, and $X_{\varepsilon ( T ) }^\ast ( T )$ is the target asset level at time T. The target asset level is decided at the plan inception, and here we define $X_{\varepsilon ( T ) }^\ast ( T ) = X_{\varepsilon ( T ) }^\ast{\times} L( T )$ where $X_{\varepsilon ( T ) }^\ast$ is the target asset level expressed in the real terms, and L(T) is the inflation adjustment. Note that λ and X* are chosen to be regime-dependent to reflect the changes in the risk appetites and the regulatory requirements under different market conditions.

Given the objective function above, we solve the following optimization problem:

(5)$$\matrix{ {\mathop {\inf }\limits_{b\in \Pi } {\rm {\opf E}}_{t, x, l, \varepsilon }\{ \int_t^T {e^{{-}r_0s}} ( b( s) L( s) {\cal R}( s) } \hfill {-\bar{b}L( s) {\cal R}( s) ) ^2ds + \lambda _{\varepsilon ( T) }e^{{-}r_0T}{( X( T) -X_{\varepsilon ( T) }^\ast{\times} L( T) ) }^2\} , \;} \hfill \cr } $$

where Π denotes the set of all admissible strategies with its mathematical definition given in the online appendix (Definition Appendix A.1).

Notice that all measures of the income stability are multiplied by an exponential discounting function $e^{{-}r_0t}$, where r ₀ > 0 is referred to as the time-preference rate which is not necessarily the same as the risk-free rate r _ɛ(t). The choice of r ₀ and λ reflects a balance between the current and the future generations. More specifically, a large value of r ₀ or a small value of λ essentially means that the sponsor has no interest in delivering secured retirement benefits to the future generations.

3.1. Transparent risk allocation rules

To construct a stable risk sharing structure based on (6), some restrictions must be applied to the parameters. Often in an optimal control problem, these constraints are applied to the economic parameters and are too strict to be realistic. However, in our case, constraints can be applied to the parameters that are subjectively chosen by the sponsor, making a stable design practically feasible.

Remark 2. A regime-stable design such that

(7)$$Actual \, Benefit_{\varepsilon ( t) }( t) = Target \, Benefit_{\varepsilon ( t) }( t) + \beta _{\varepsilon ( t) }\left({\displaystyle{{Asset( t) -\psi_{\varepsilon ( t) } \times Liability( t) } \over {{\cal R}( t) }}} \right), \;$$

can be constructed if the pension sponsor is restricted to the choices of $\pi _\epsilon$, $\beta _\epsilon$, and $X_\epsilon ^\ast$ (see Equations (A.12) and (A.13)). Furthermore, we have β _ε(t) = λ _ε(t) and $\psi _{\epsilon ( t ) } \times {\rm Liabilit}{\rm y}_{\epsilon ( t ) }( t ) = X_{\epsilon ( t ) }^\ast{\times} L( t )$.

The simplicity of this regime-stable TB structure enhances the transparency of the risk sharing mechanism. The plan consists of only two sets of linear risk allocation rules with only regime-dependent parameters. Practically, once the market regime is determined through some publicly available indices such as NFCI, the members should easily understand how their actual benefit is being calculated. It is important to strengthen again that the constraints are applied to subjective chosen parameters π, β, and X*, and all economic parameters are externally given.

Recall that λ _ɛ measures the discontinuity risk, and is defined as the penalty weight for the difference between the actual and the target terminal asset levels. Although this definition is qualitatively intuitive, it fails to provide any quantitative justification for specific choices of its values. For example, λ = 10% is a common choice in the literature, but the question is what would be the practical meaning by saying that the stability of future liability (X(t) − X* × L(T))² is weighted only 10% compared to the stability of the retirees' income $( {b( t ) \times L( t ) -\bar{b} \times L( t ) } ) ^2$. We are essentially comparing apples and oranges in one equation.

Remark 2 indicates that λ _ɛ now represents the proportion of deficit/surplus shared to the retirees. For example, λ = 10% simply represents that 10% of the deficit/surplus will be distributed. The new interpretation also helps restrict the range of λ that should be used for the sensitivity tests. For example, λ > 100% corresponds to distribute more deficit than the pension plan has, which immediately loses its practical meaning. We would emphasize that based on our numerical analysis later, even for a non-transparent plan as Equation (6), the value of β _ε(t)(t) is still close to the value of λ _ε(t), and therefore the natural boundary for β should apply to λ as well.

Furthermore, the terminal condition $X_{\epsilon ( t ) }^\ast$ which represents the target asset level (in real terms) is now equivalent to the benchmark level that decides the risk allocation. Once the equity share is chosen, it also decides, implicitly, the optimal asset transfer to the future generation. Therefore, to construct a regime-stable risk sharing design, the sponsor is essentially facing the trade-off between the flexible asset allocation and the amount of intergenerational deficit transfer. In fact, the regime-stable structure is the solution of an infinite time problem, where ψ and X* will not be directly involved in the objective function.

The following remark states that the regime-stable structure (7) is indeed a solution to an infinite-time problem, where ψ and X* represent the income stability of all future generations that are taken into consideration.

Remark 3. The regime-stable structure (7) is the solution of the following infinite time problem:

(8)$$\mathop {\inf }\limits_{b\in \Pi } {\rm {\opf E}}\left\{{\left[{\mathop {\underbrace{{\int_0^T {e^{{-}r_0 \times s}} \times {( B( s) -\bar{B}( s) ) }^2ds}}_{{}}}\limits_{{\rm Generations\ getting\ benefit\ before\ T}} } \right] + {\vector E}\left[{e^{{-}r_0 \times T} \times \mathop {\underbrace{{\int_T^\infty {e^{{-}r_0 \times ( s-T) }} \times {( B( s) -\bar{B}( s) ) }^2ds}}_{{}}}\limits_{\lambda_{{\rm \epsilon }( T) } \times {( X( T) -X_{{\rm \epsilon }( T) }^\ast{\times} L( T) ) }^2} } \right]} \right\}.$$

The regime-stable TB structure implicitly balances the income stability between the current and the future generations. The weight given to the stability of a benefit payment at time t is the time-preference factor $e^{{-}r_0 \times t}$. A higher discount rate simply represents that the pension design sacrifices the income security of the future generations for the benefits of the nearby generations.

Clearly, the parameters $\lambda _\epsilon$ and $X_\epsilon ^\ast$ are not externally given, and implicitly represent the cumulative income uncertainty of all future generations.

We end this section by presenting an additional remark that relates our problem to Cui et al. (Reference Cui, De Jong and Ponds2011). By constraining $\psi _\epsilon$ to be fixed constants that are externally given, the problem becomes determining the optimal equity share and the optimal risk distribution percentage $\beta _\epsilon$.

Remark 4. If ψ ₁ and ψ ₂ are pre-determined, then the regime-stable structure (7) is indeed the solution of the following problem:

$$\matrix{ {\mathop {\inf }\limits_{{\boldsymbol \beta }, {\boldsymbol \pi }} {\rm {\opf E}}\left\{{\mathop \int \nolimits_t^T e^{{-}r_0s}{( {b( s ) L( s ) {\cal R}-\bar{b}L( s ) {\cal R}} ) }^2ds + \beta_{\varepsilon ( T ) }e^{{-}r_0T}{( {X( T ) -\psi_{\varepsilon ( T ) }{\rm Liabilit}{\rm y}_{\varepsilon ( T ) }( T ) } ) }^2} \right\}.} \hfill & \qquad \hfill & {} \hfill \cr } $$

The problem is similar to Cui et al. (Reference Cui, De Jong and Ponds2011) except that they work on a single regime model and aim to maximize an individual's lifetime utility.

3.2. Cyclicity

Based on the benchmark parameters, here we illustrate the cyclicity of the optimal risk allocation rules. We first examine each of the risk sharing parameters separately, and then discuss their cumulative effects.

3.2.1 Counter-cyclical risk sharing of λ

Figure 1 displays the relationship between the level of risk sharing λ and the equity share π for different time-preference rates r ₀ in two market regimes. In this figure, the curves in blue correspond to the regular market case and the curves in red to the market recession. Notice that λ ranges from 0 to 1 in this figure. As discussed in the previous section, λ represents the level (percentage) of risk sharing in the regime-stable structure. In all scenarios, λ lies between 0% and 10%, which means a deficit will be smoothly distributed for more than 10 years (${\rm deficit\;recovery\;period}\approx {1 \over \beta }$).

Figure 1. The relationship between the optimal surplus/deficit sharing (λ) and the equity share (π), for different time-preference rates r ₀.

We first observe that the higher the proportion of risky investment is, the higher risk sharing will be required. This is not surprising as the risk sharing mechanism is meant to offset the risks embedded in the pension assets. It is also clear that the risk sharing level λ ₁ in the regular market (blue) is always higher than λ ₂ in the recession (red). For the same amount of deficit, the pension fund actually distributes less during a recession, which indicates a counter-cyclical risk sharing policy with respect to the sustainability of the plan. In this case, current generations' income stability outweighs the sustainability of the fund. During the recession, to lessen the income volatility, risk sharing must be lowered to safeguard the retirees' benefits.

Another highlight drawn from Figure 1 is that the risk sharing level λ has a negative relationship with the time-preference rate of r ₀. This is because the lower the time-preference rate is, the higher weight is given to the stability of terminal asset level and thus the asset is required to be closer to the target asset level, which can be achieved through a higher level of risk sharing.

Under the circumstances that the time-preference rate r ₀ is large (which favors the nearby generations) and the investment strategy is conservative (i.e., small π), the TB plan degenerates to the traditional DB plan (where λ = 0) in which risk-sharing simply becomes risk-transferring to the future generations.

3.2.2 Pro-cyclical risk sharing threshold ($\psi _\epsilon$)

Before we proceed with the cyclicity of the risk allocation implied by the parameter ψ, we stress that the liability valuation in our paper is not regime-dependent. Despite the fact that a cyclical policy is often adopted for the traditional DB plans, the regulation always recommends liability valuation assumptions that are robust with respect to the changes in the market regimes (see, e.g., Yermo and Severinson, Reference Yermo and Severinson2010).Footnote ⁹ In this spirit, we assume that the liability valuation is independent of the underlying regime. Recall that $X_\varepsilon ^\ast$ is ψ _ɛ × Liability (in real terms) as shown in Remark 2, and thus we may compare ψ _ɛ through the relationship $X_2^\ast{/}X_1^\ast = \psi _2/\psi _1$. We further assume that ψ ₁ = 1, i.e., TB shares the surplus/deficit when the pension asset deviates from the liability, to better compare risk sharing thresholds in different market regimes.

Figure 2 presents the relationship between the risk sharing threshold ψ ₂ during the market recession and the equity share when ψ ₁ = 1 under different values of time-preference rates r ₀. As can be seen in this figure, ψ ₂ is larger than ψ ₁ by a factor between 1.02 and 1.14. The higher values of ψ ₂ mean that during the recession, the benefits will be reduced to a greater degree than the regular time. Hence, contradicting with λ _ɛ, the values of ψ _ɛ indicate a pro-cyclical policy which means that during the recession, more surplus will be held by the fund and more deficit will be distributed.

Figure 2. The risk sharing threshold ψ ₂ during recession (ψ ₁ = 1), for different equity share π and different time-preference rates r ₀.

Notice that the discount rate r _d only indirectly affects ψ through the liability values. However, the ratio ${{\psi _2} \over {\psi _1}}$ and the values of β ₁ and β ₂ will be irrelevant to the choice of r _d.

3.2.3 Overall cyclicity

The analysis above has shown that the optimal risk sharing parameters λ _ɛ and ψ _ɛ have counter- and pro-cyclical features, respectively. Therefore, the overall cyclicity of the deficit sharing remains unclear.

Heuristically, the pro-cyclical regulation in a DB plan requires an increase in the pension contribution during the economic downturn and allows the contribution holidays if both the market and the pension fund perform well, and the counter-cyclical policy refers to the opposite. For a TB plan, we refer the pro-cyclical deficit sharing in the sense that more deficit will be distributed during a market crash. In other words, for the same level of deficit, Actual Benefit₁(t) > Actual Benefit₂(t). Counter-cyclical, on the other hand, corresponds to Actual Benefit₁(t) < Actual Benefit₂(t). Since λ is counter-cyclical for the deficit sharing but ψ is pro-cyclical, the TB design will exhibit both pro- and counter-cyclicity features. Due to the linearity of the optimal risk allocation rules, pro- and counter-cyclical policy will be applied depending on the funding ratio. Mathematically, the pro-cyclical policy will be applied if the funding ratio falls into the set:

$$\matrix{ {\left\{{\displaystyle{{Asset} \over {Liability}}\vert {} \lambda_1\left({\displaystyle{{Asset} \over {Liability}}-\psi_1} \right)> \lambda_2\left({\displaystyle{{Asset} \over {Liability}}-\psi_2} \right)} \right\}.} \hfill & \qquad \hfill & {} \hfill \cr } $$

It is easy to see that we may find a benchmark funding level f* such that pro-cyclical deficit sharing will apply to any level higher than f*.

Figure 3 illustrates the value of f*. Clearly, counter-cyclicity only applies to the extremely under-funded pension plans, which surprisingly aligns with the market practice for the DB plans. However, we should emphasize that the motivation for the counter-cyclical TB design is to relieve the risk sharing obligations borne by the retirees. On the contrary, counter-cyclicity in the DB regulation is to relieve the stress of the DB sponsors who are facing extreme financial difficulties.

Figure 3. The benchmark funding level f* for different equity share π, for different time-preference rates r ₀.

One thing we would like to highlight from Figures 2 and 3 is the convergence of different curves when the equity weight increases. As the percentage of equity investment is larger than 80%, the choice of the time-preference rate has an insignificant impact on the risk sharing structure. Contradicting to the standard 60/40 portfolio for a DB plan, the sponsor may choose more aggressive investment strategies for a TB plan if the time-preference rate is hard to decide.

3.3. Comparison to DB plan

Until now we have not yet compared a TB plan with a DB plan. In our model, since the traditional DB plan is only a special case of a TB plan when β _ɛ(t) = 0, it is undoubted that a DB plan cannot over-perform a TB plan when both income security and fund sustainability are considered jointly. Since no risk sharing exists in the DB plan, it would be clear that DB is preferable to TB for short-term income security at the expense of sacrificing long-term sustainability. The question is on the magnitude of the trade-off.

The left panel of Figure 4 displays the 95% fan-chart of actual benefit payments during the regular market (light) and the recession (dark), and the right panel displays the 95% fan-chart of funding level for both TB (grey) and DB (green) plans. We use the base case parameters with r ₀ = 0.05. We also use an aggressive strategy of 80/20 for the regular market, and 60/40 for the recession. The initial pension asset is assumed to be fully funded in the regular market regime (i.e., $X_0 = X_1^\ast$).

Figure 4. The 95% fan-chart for the income security and the sustainability of the pension plans. The left panel displays the income security of a TB plan during the regular market (light/green) and the recession (dark/grey). The right panel displays the funding levels of both DB (light/green) and TB (dark/grey) plans.

Here we emphasize that if the default risk is considered, then the income variation does exist in a DB plan. Once a DB plan is terminated, the promised retirement income would no longer be deliverable, and a large income reduction is expected. TB plans essentially break down those large reductions into a stream of smaller shares. The default risk (Asset < 0) for a DB plan which initially is fully funded is 14% over a 20-year horizon which is significantly large, and this figure jumps to 47% if it is 70% funded initially. On the other hand, these numbers are only 0.57% and 0.43% for a TB plan. Despite the fact that the TB looks volatile for the retirees at the first sight, risk sharing effectively avoids the situation where the current generation will be left empty-handed when they retire. Cui et al. (Reference Cui, De Jong and Ponds2011) illustrates that TB plan may be the solution in the sense that the new cohort joining an underfunded risk sharing scheme may not be significantly worse-off in the welfare terms; here we reinstate this argument, but in terms of the income security.

Notice also that although we did not constrain on b _t > 0, the possibility of getting a negative benefit is negligible in our benchmark study. We also want to emphasize that when the ‘optimal’ benefit is negative, it simply reflects the fact that the asset becomes way below the liability value. In such a scenario, any pension plan will become insolvent, and the risk sharing structure is no exception. However, as shown in the right panel, the sustainability of the pension fund has been greatly improved when risk sharing is involved.

3.4. Time- and regime-independent structure

The transparent plan consists of two sets of linear risk sharing rules, one for each market regime. Therefore, it is natural to consider a universal risk sharing rule that is regime-independent. Technically, it only requires more restrictions on the subjectively chosen parameters as shown in the following remark.

Remark 5. Given the penalty λ, if the investment strategies under different regimes satisfy the following system of quadratic functions:

$$\matrix{ {\pi _1^2 \sigma _1^2 + \pi _1 \times 2( {\mu_1-r_1} ) + 2( {r_1-r_0} ) -\lambda = 0, \;} \cr\hfill & \qquad \hfill & {} \hfill \cr {\pi _2^2 \sigma _2^2 + \pi _2 \times 2( {\mu_2-r_2} ) + 2( {r_2-r_0} ) -\lambda = 0, \;} \hfill & \qquad \hfill & {} \hfill \cr } $$

then the risk sharing parameter λ _ɛ(t) = λ will be both regime- and time-independent. In addition, to obtain a time-invariant ψ, the terminal target asset level X* must satisfy:

$$\matrix{ {X^\ast{ = } \displaystyle{{c_1{\cal A}-{\cal D}-\bar{b}{\cal R}} \over {r_1 + \pi _1( {\mu_1-r_1} ) -2r_0 + \alpha _1 + \pi _1\sigma _1\eta _1\rho _1-\lambda }}.} \hfill & {} \hfill & {} \hfill \cr } $$

Notice immediately that π ₁≠π ₂, and π _ɛ is a function of λ + 2r ₀.

Given different values of λ + 2r ₀, we observe the corresponding investment strategies in Figure 5. For example, by setting λ = 0.05 and r ₀ = 0.05, the asset allocation used in the regular market is close to the rule of thumb 60/40 strategy (indicated by the blue circle in the figure). However, we need to short the equity during the recession (see the corresponding red line).

Figure 5. The equity exposure π given different time-preferences (r ₀) and surplus deficit sharing (λ).

Indeed, there is no set of optimal strategies that hold positive equity in both market regimes. In most countries, pension regulators set strict rules on the asset mix where the equity weight is often required to be positive. In addition, in response to the financial turmoil, many regulators impose short-selling bans that further limit the sponsors' ability in short-selling. Apparently, a stable TB design that is independent of both time and regime leads to practically infeasible investment strategies.

4.1. Risk-free rate

Most economic parameters are direct calibrated from the historical data; however, the choice of the risk-free rate r _ɛ is rather arbitrary. Here we perform its sensitivity test to examine its impacts on the values of β and ψ.

In Figure 6, we present the impact of the risk-free rate r _ɛ on the risk sharing parameters β _ɛ and ψ _ɛ as well as the liability (or the risk sharing benchmark) $X_\varepsilon ^\ast$. More specifically, we fix the risk-free rate during the market recession (r ₂ = 2%) and explore the impact of the risk-free rate in the regular market r ₁. In this figure, the blue curve in each panel corresponds to the base case scenario.

Figure 6. The impact of the risk-free rate during the regular market r ₁, given the time-preference rate r ₀ = 0.05 and the risk-free rate during the recession r ₂ = 2%. The figure displays the risk sharing parameters during the regular market β ₁ (left top), during the recession β ₂ (right top), the liability value (left bottom), and the risk sharing threshold ψ ₂ (right bottom). Each variable is plotted against different equity exposure π.

Overall, a similar pattern on the risk sharing parameter β has been observed for different choices of r _ɛ. For the effect on the risk sharing benchmark $X_\varepsilon ^\ast$, we have noticed that the relationship between r ₁ and r ₀ is a crucial factor. The benchmark level $X_\varepsilon ^\ast$ is rather stable when r ₁ ≥ r ₀, but becomes sensitive when r ₁ < r ₀. Nonetheless, when the portfolio leans toward fully investing in the equity (i.e., when the risk-free asset will not be involved), the risk-free rate no longer has any impact on the TB design.

While Figure 6 is based on an expansionary fiscal policies where r ₁ > r ₂, we run additional tests for the contractionary policies during the recession where r ₁ < r ₂ in Figure 7. Conclusions obtained from the previous observations remain valid.

Figure 7. The impact of the risk-free rate in recession r ₂, given the time-preference rate r ₀ = 0.05 and the risk-free rate during the regular market r ₁ = 0.04. The figure displays the risk sharing parameters during the regular market β ₁ (left top), during the recession β ₂ (right top), the liability value (left bottom), and the risk sharing threshold ψ ₂ (right bottom). Each variable is plotted against different equity exposure π.

4.2. Salary assumption

In the base case scenario, we omit the effect of the promotional salary growth γ ₁ and the partial inflation indexation γ ₂. In fact, they impact the optimal TB design only through the terms ${\cal A}$, ${\cal R}$, and ${\cal D}$. Immediately from Equation (A.12), we notice that β _ɛ is independent of the choice of γ _ɛ.

Figure 8 displays the impact of salary growth γ ₁ on the overall risk sharing weight as well as the terminal target asset $X_1^\ast$.

Figure 8. The impact of the salary growth γ on liability value (real term), given different equity weights (x-axis), and time-preference rate r ₀ = 0.05.

Here we stress that although it is expected that the liability value can be significantly different under different salary growth assumptions, the ratio between $X_2^\ast{/}X_1^\ast$ (which is ψ ₂ if ψ ₁ = 1) can be shown to have the following expression based on Equation (A.13):

$$\displaystyle{{X_2^\ast } \over {X_1^\ast }} = \displaystyle{{\zeta _1 \times [ c_2{\cal A}( t) -{\cal D}( t) -\bar{b}{\cal R}( t) ] -\displaystyle{{q_{21}\lambda _1} \over {\lambda _2}} \times [ c_1{\cal A}( t) -{\cal D}( t) -\bar{b}{\cal R}( t) ] } \over {\zeta _2 \times [ c_1{\cal A}( t) -{\cal D}( t) -\bar{b}{\cal R}( t) ] -\displaystyle{{q_{12}\lambda _2} \over {\lambda _1}} \times [ c_2{\cal A}( t) -{\cal D}( t) -\bar{b}{\cal R}( t) ] }}.$$

It is clear that when c ₁ = c ₂ (i.e., contribution rate is independent of the market regimes), γ ₁ and γ ₂ have no impact on ψ ₂. Therefore, the risk sharing mechanism is stable for different salary growth assumptions.

Similar to the effect of γ _ɛ, parameters such as the contribution level c _ɛ and the target replacement rate $\bar{b}$ affect the optimal plan design only through ${\cal A}$, ${\cal D}$, and ${\cal R}$, and therefore are irrelevant to the optimal values of ψ ₂ and β _ɛ.

4.3. Effect of contribution policy

DB regulations in many OECD countries allow contribution relief during the recession to avoid putting further pressures on a company's profitability. Therefore, we include the scenarios when c ₁ ⇔ c ₂ in Figure 9 to illustrate the effect of different contribution patterns.

Figure 9. The effect of contribution during the recession (c ₂) on the risk sharing threshold during the recession ψ ₂, given the contribution during the regular regime (c ₁) remains the same, the time-preference rate r ₀ = 0.05 and the risk sharing threshold during the regular market ψ ₁ = 1.

When a counter-cyclical contribution policy is adopted (i.e., decreasing contribution during a recession), it increases the pro-cyclicality in the deficit sharing. Although we have chosen a large range of c ₂ from 5% to 30%, the risk sharing threshold is relatively stable in the sense that ψ ₂ is always higher than ψ ₁ and lies between 1.05 and 1.18.

4.4. Single-regime model

To demonstrate the necessity of introducing a regime-switching model, we compare our result with that of a single regime model (similar to Wang et al., Reference Wang, Lu and Sanders2018). With the same objective function but all parameters being regime-independent, it is not difficult to establish the following remark:

Remark 6. The optimal benefit structure is

(9)$$\matrix{ {Actual\;Benefit( t ) } \hfill & { = \bar{b} \times L( t ) + \beta ( t ) \times \left({\displaystyle{{X( t ) -\psi ( t ) \times L( t ) } \over {{\cal R}( t ) }}} \right), \;} \hfill & {} \hfill \cr } $$

where β(t) and ψ(t) can be derived from Equation (A.6). Similarly, a time-invariant strategy can be obtained by constraining the value of π and λ:

(10)$$\matrix{ {\pi = \displaystyle{{\sqrt {{( {\mu -r} ) }^2-2( {r-r_0} ) \sigma ^2 + \lambda \sigma ^2} -( {\mu -r} ) } \over {\sigma ^2}}, \;} \hfill & {} \hfill & {} \hfill \cr } $$

and we have β(t) = λ.

The left panel in Figure 10 demonstrates the cyclical design using a single-regime model. Specifically, the TB design is derived from Equation (9) using the parameters from the base case (where the market parameters differ in different regimes). We observe a negative risk sharing parameter (β ₂) during the recession regardless of the asset allocation. This design completely loses its practical meaning since it increases the retirees' benefits for an under-funded plan and reduces the benefits for an over-funded plan. The right panel in Figure 10 further illustrates that this counter-intuitive behavior remains an issue for a wider range of reasonable market parameters.

Figure 10. Time-invariant strategy in a single regime model. The left panel displays the optimal risk sharing λ using the benchmark parameters, and the right panel displays the universal risk sharing under different parameter sets. Both panels are plotted against different equity weights π.

4.5. Equity share as control variable

In this paper, we assume that the pension fund trustees adopt constant weighting strategies for investment in each regime. However, for most stochastic control problems in pension literature, e.g., Wang et al. (Reference Wang, Lu and Sanders2018), equity weight is often treated as a control variable. Therefore, it would be also interesting to consider the dynamic investment strategies. In this subsection, we take both the benefit payment and the portfolio weight as the control variables, and minimize the same objective function. Mathematically, we have

(11)$$\mathop {\inf }\limits_{b, \pi } {\rm {\opf E}}_{t, x, l, \varepsilon }\left\{{\int_t^T {e^{{-}r_0s}} {( {b( s) L( s) {\cal R}( s) -\bar{b}L( s) {\cal R}( s) } ) }^2ds + \lambda_\varepsilon e^{{-}r_0T}{( X( T) -X_\varepsilon^\ast{\times} L( T) ) }^2} \right\}.$$

The problem can be solved similarly to Problem (5) via dynamic programming. The resulting optimal risk allocation rule preserves linearity as Equation (6) with an explicit form given in the online appendix.

With the dynamic asset allocation, the actual benefit can be perfectly maintained at the target level ($\bar{b} \times L( t )$), which seems that a TB design is not necessary. However, the stable benefit payments are guaranteed with a cost in the sense that a highly volatile portfolio will be observed. Figure 11 displays the 95% fan-chart for the optimal equity share.

Figure 11. The optimal equity share π during the regular market, when including the equity share as a control variable.

The optimal strategy requires the weight in equity to fluctuate between −100,000% and 100,000%. Thus, the optimal equity share obtained from Problem (11) is highly impractical. This also suggests that a risk sharing design is necessary for the market risk mitigation as long as the trustee does not have absolute control over its portfolio mix.