2.1 Stochastic model for state variables and risky assets

We let [0, T ₀] denote the (finite) time span of the economy, where uncertainty is described through a standard probability space $\lpar \rmOmega \comma {\cal A}\comma {\bb P}\rpar$. In what follows, T ₀ can be thought of as the date of the last pension payment by a pension fund (after which the pension fund will be terminated), or as the duration of pension liabilities. It is to be distinguished from the investment horizon, which can be some arbitrary date denoted by T⩽T ₀. We assume that financial markets are frictionless.

Regarding the liability side, inflation risk and interest rate risk appear as the two most relevant risk factors. This is because pension benefits are typically inflation-indexed, and the typically long duration of liability payments make their current value highly sensitive to changes in interest rates. In what follows, we model the nominal short-term interest rate as an Ornstein–Uhlenbeck process (Vasicek, Reference Vasicek1977) and the price index as a Geometric Brownian motion

$$\eqalign{ \tab {\rm d}r_{t} \equals a\lpar b \minus r_{t} \rpar \,{\rm d}t \plus \sigma _{r} \,{\rm d}z_{t}^{r} \comma \cr \tab {{{\rm d}\rmPhi _{t} } \over {\rmPhi _{t} }} \equals \phiv \,{\rm d}t \plus \sigma _{\rmPhi } \,{\rm d}z_{t}^{\rmPhi } \comma \cr}$$

where z^r and z ^Φ follow standard correlated Wiener processes under ℙ. ϕ represents the instantaneous expected inflation rate, which we assume to be constant for simplicity.Footnote ⁵

On the asset side, we assume that the menu of asset classes includes a unit zero-coupon bond with payoff 1 at maturity τ₁ and a price B(t, τ₁) at time t. So as to stay within a complete market environment, we also assume that the pension fund can trade in an inflation-indexed zero-coupon bond of maturity τ₂, i.e., a bond with payoff given by $\rmPhi _{\tau _{\setnum{2}} }$. The price at time t of the inflation-linked bond is denoted by I(t, τ₂), which will be made explicit in proposition 1 below. Moreover, we assume that the pension fund can trade in a stock index whose price S_t evolves as

$${\rm d}S_{t} \equals S_{t} \lsqb \lpar r_{t} \plus \sigma _{S} \lambda _{S} \rpar \,{\rm d}t \plus \sigma _{S} \,{\rm d}z_{t}^{S} \rsqb \comma$$

where λ_S is a constant Sharpe ratio. In addition to these assets, the pension fund can also invest in a cash account, whose value is the continuously compounded interest rate.

The dynamics of these state variables can be rewritten in vector form as

(1)$$\openup3\eqalign{ {\rm d}r_{t} \equals \tab a\lpar b \minus r_{t} \rpar \,{\rm d}t \plus {\bimsigma} \prime_{\hskip-2ptr} \,{\rm d}{\bi z}_{t} \comma \cr {\rm d}\rmPhi _{t} \equals \tab \rmPhi _{t} \lsqb \varphi \,{\rm d}t \plus {\bimsigma} \prime_{\hskip-2pt\rmPhi } \,{\rm d}{\bi z}_{t} \rsqb \comma \cr {\rm d}S_{t} \equals \tab S_{t} \lsqb \lpar r_{t} \plus {\sigma} _{S} \lambda _{S} \rpar \,{\rm d}t \plus {\bimsigma} \prime_{\hskip-3pt S} \,{\rm d}{\bi z}_{t} \rsqb \comma \cr}$$

where z is a three-dimensional Wiener process. Throughout the paper we assume that the information available to the investor at time t is ℱ_t, the augmented sigma-field generated by z up to time t.

Assuming a complete financial market, there exist prices of interest and inflation risks, λ_r and λ_Φ, hence there exists a unique price of risk vector λ, given by

$${\bimlambda} \equals {\bimsigma} \lpar {\bimsigma} \prime \; {\bimsigma} \rpar ^{ \minus \setnum{1}} \left( {\matrix{ {{\sigma} _{r} {\lambda} _{r} } \cr {{\sigma} _{\rmPhi } {\lambda} _{\rmPhi } } \cr {{\sigma} _{S} {\lambda} _{S} } \cr} } \right)\comma$$

with σ being the volatility matrix of traded risks, that is

$${\bimsigma} \equals \lpar \matrix{ {{\bimsigma} _{r} } \tab {{\bimsigma} _{\rmPhi } } \tab {{\bimsigma} _{S} } \cr} \rpar .$$

The vector λ defines a unique equivalent martingale measure by

(2)$${{{\rm d}{\bb Q}} \over {{\rm d}{\bb P}}} \equals {\rm exp}\left[ { \minus {\bimlambda} \prime {\bi z}_{T_{\setnum{0}} } \minus \textstyle{1 \over 2}\vert \vert {\bimlambda} \vert \vert ^{\setnum{2}} T_{\setnum{0}} } \right]\comma$$

and a unique pricing kernel process M through

(3)$$M_{t} \equals {\rm exp}\left( { \!\minus\! \int_{\setnum{0}}^{t} r_{s} \,{\rm d}s} \right){\bb E}_{t} \left[ {{{{\rm d}{\bb Q}} \over {{\rm d}{\bb P}}}} \right].$$

We are now able to write the prices of the nominal and indexed zero-coupon bonds that are available for trading.

Proposition 1

The prices of the nominal and of the real zero-coupon bonds of respective maturities τ₁and τ₂are given by

$$B\lpar t\comma \tau _{\setnum{1}} \rpar \equals {\rm e}^{\alpha \lpar \tau _{\setnum{1}} \minus t\rpar r_{t} \plus \beta _{\setnum{1}} \lpar \tau _{\setnum{1}} \minus t\rpar } \quad {\rm and}\quad I\lpar t\comma \tau _{\setnum{2}} \rpar \equals \rmPhi _{t} {\rm e}^{\alpha \lpar \tau _{\setnum{2}} \minus t\rpar r_{t} \plus \beta _{\setnum{2}} \lpar \tau _{\setnum{2}} \minus t\rpar }$$

where

$$\openup3\eqalign{ \alpha \lpar s\rpar \tab \equals \minus {{1 \minus {\rm e}^{ \minus as} } \over a}\comma \quad \tilde{b} \equals b \minus {{{\bimsigma} \prime_{r} {\kern 1pt} {\bimlambda} } \over a}\comma \quad \tilde{\varphi } \equals \varphi \minus {\bimsigma} \prime_{\hskip-3pt\rmPhi } {\bimlambda} \comma \cr \beta _{\setnum{1}} \lpar s\rpar \tab \equals \minus \tilde{b}s \plus \tilde{b}{{1 \minus {\rm e}^{ \minus as} } \over a} \plus {{\vert \vert {\bimsigma} _{r} \vert \vert ^{\setnum{2}} } \over {2a^{\setnum{2}} }}\left[ {s \minus 2{{1 \minus {\rm e}^{ \minus as} } \over a} \plus {{1 \minus {\rm e}^{ \minus \setnum{2}as} } \over {2a}}} \right]\comma \cr \beta _{\setnum{2}} \lpar s\rpar \tab \equals \left( {\tilde{\varphi } \minus {{\vert \vert {\bimsigma} _{\rmPhi } \vert \vert ^{\setnum{2}} } \over 2} \minus \tilde{b}} \right)s \plus \tilde{b}{{1 \minus {\rm e}^{ \minus as} } \over a} \plus {1 \over 2}\int_{\setnum{0}}^{s} \,\mathop {\left\Vert {{{1 \minus {\rm e}^{ \minus au} } \over a}{\bimsigma} _{r} \minus {\bimsigma} _{\rmPhi } } \right\Vert}\nolimits^{\setnum{2}} \,{\rm d}u. \cr}$$

In particular, the volatility vectors of B(·, τ₁) and I(·, τ₁) are given by

$${\bimsigma} _{B} \lpar t\comma \tau _{\setnum{1}} \rpar \equals \alpha \lpar \tau _{\setnum{1}} \minus t\rpar {\bimsigma} _{r} \quad {\rm and}\quad {\bimsigma} _{I} \lpar t\comma \tau _{\setnum{2}} \rpar \equals \alpha \lpar \tau _{\setnum{2}} \minus t\rpar {\bimsigma} _{r} \plus {\bimsigma} _{\rmPhi }$$

2.2 Net wealth process

We consider a pension fund managing financial assets and paying a stream of pension payments. For simplicity, we do not model the stream of contributions from the sponsor company, and instead assume that it can be summarized by an initial endowment A ₀ to the pension fund. This initial wealth can be invested in the stock, the nominal and the indexed bonds, and the cash account.

We denote with ω_t the vector of weights describing the portfolio at time t, and with σ_t the volatility matrix of the stock and the two zero-coupon bonds at time t, defined as

$${\bimsigma} _{t} \equals \lpar \matrix{ {{\bimsigma} _{S} } \tab {{\bimsigma} _{B} \lpar t\comma \tau _{\setnum{1}} \rpar } \tab {{\bimsigma} _{I} \lpar t\comma \tau _{\setnum{2}} \rpar } \cr} \rpar .$$

The value of the financial portfolio, A, evolves as

(4)$${\rm d}A_{t} \equals A_{t} \lsqb r_{t} \plus {\bimomega} \prime_{\hskip -2pt t} {\bimsigma} \prime_{\hskip -2pt t} {\bimlambda} \rsqb \,{\rm d}t \plus A_{t} {\bimomega} \prime_{\hskip -2pt t} {\kern 1pt} {\bimsigma} \prime_{\hskip -2pt t} \,{\rm d}{\bi z}_{t} \minus {\rm d}V_{t} \comma$$

where dV_t is the payment to pensioners between dates t and t + dt. This representation can accommodate continuous payments as well as lump-sum payments at dates t ₁, …, t_n. In this latter case, dV_t should be formally written as ${\rm d}V_{t} \equals \sum _{i \equals \setnum{1}}^{n} l_{t_{i} } \;{\rm d}H_{t}^{t_{i} }$, where $H^{t_{i} }$ is an Heaviside function, $H_{t}^{t_{i} } \equals \mathbbm{1}_{\lcub t\geqslant t_{i} \rcub }$. In what follows, we shall sometimes consider a specific case of the discrete payment model, where a single payment takes place, at time T ₀, a situation we shall refer to as the zero-coupon case. The generic case will be the generic situation, where the continuous or discrete nature of the payments is not specified.

Since the financial market is complete, the stream of future payments can be valued as the dividend flow of a financial asset. Hence, for T ₁<T ₂, the quantity

(5)$$L_{t}^{T_{\setnum{1}} \comma T_{\setnum{2}} } \equals {\bb E}_{t}^{{\bb Q}} \left[ {\int_{\left] {T_{\setnum{1}} \comma T_{\setnum{2}} } \right]} {\rm e}^{ \minus \int _{t}^{s} r_{u} \,{\rm d}u} \,{\rm d}V_{s} } \right]\comma \quad t \leqslant T_{\setnum{1}} \lt T_{\setnum{2}} \comma$$

is the price that an agent would have to pay at time t to receive the payment stream dV from date T ₁ excluded to date T ₂ included. We will also let: $L_{t} \equals L_{t}^{t\comma T_{\setnum{0}} }$ denote the total liability value, i.e., the discounted value of all future liability payments, at date t. With these notations, the budget constraint (see proposition 2.2 in Cox and Huang, Reference Cox and Huang1991) becomes

(6)$$A_{t} \equals {\bb E}_{t} \left[ {{{M_{s} } \over {M_{t} }}A_{s} } \right] \plus L_{t}^{t\comma s} \comma \quad t \lt s.$$

Throughout the paper we take l_t=n_tΦ_t, where n is a non-negative deterministic function of time representing the size of the population to which benefits will be provided for. Since the pair (r, Φ) is Markov under ℚ, $L_{t}^{T_{\setnum{1}} \comma T_{\setnum{2}} }$ is a function ${\cal L}^{T_{\setnum{1}} \comma T_{\setnum{2}} }$ of (t, r_t, Φ_t). It's lemma then gives the volatility vector of $L^{T_{\setnum{1}} \comma T_{\setnum{2}} }$

(7)$${\bimsigma} _{L\comma t}^{T_{\setnum{1}} \comma T_{\setnum{2}} } \equals {{{\cal L}_{r}^{T_{\setnum{1}} \comma T_{\setnum{2}} } {\bimsigma} _{r} \plus {\cal L}_{\rmPhi }^{T_{\setnum{1}} \comma T_{\setnum{2}} } \rmPhi _{t} {\bimsigma} _{\rmPhi } } \over {L_{t}^{T_{\setnum{1}} \comma T_{\setnum{2}} } }}.$$

We will set ${\bimsigma} _{L\comma t} \equals {\bimsigma} _{L\comma t}^{t\comma T_{\setnum{0}} }$.

In the zero-coupon case, it is assumed that the pension fund makes a single payment, at time T ₀. We thus have $L_{t}^{T\comma T_{\setnum{0}} } \equals n_{T_{\setnum{0}} } I\lpar t\comma T_{\setnum{0}} \rpar$ for any T∊[t, T ₀[ and the volatility vector of L is σ_{L ,t}=σ_I(t, T ₀). In particular, the volatility vector is deterministic in the zero-coupon case (while it is stochastic when a stream of continuous or discrete liability payments are considered). This property allows for explicit pricing of the terminal optimal net wealth, and the associated dynamic asset allocation strategy, as will be made clear below.

2.3 Objectives and optimal asset allocation decisions

The objective of this paper is to adopt the perspective of the pension fund manager, an agent who acts on behalf of the shareholders and workers of the company.Footnote ⁶ Preferences of the manager are expressed here on the terminal funding ratio F _T, where

(8)$$F_{t} \equiv {{A_{t} } \over {L_{t} }}\comma \quad t \lt T_{\setnum{0}} .$$

Several comments are in order. First, we assume that the pension fund is concerned with the funding ratio rather than the asset per se. This amounts to using the liability value process (L_t)_{t ⩾0}, as opposed to the bank account, as a numeraire, an approach that has already been used for example by Van Binsbergen and Brandt (Reference Van Binsbergen and Brandt2009). Second, we choose to focus on the funding ratio because it is the variable of interest for managers of DB pension plans and regulators of pension plans.Footnote ⁷ Third, we assume that the pension fund has utility from the terminal funding ratio, not from the intermediate ones. This reduced-form objective can be interpreted in two ways. A first interpretation is that the pension fund is closed, i.e., no new participants join the pension plan between 0 and the liability horizon T ₀. All liability payments that take place between 0 and the investment horizon date T are already taken into account in the dynamics of the net asset A, so it is not necessary to consider utility from intermediate funding ratios. A second interpretation is that the pension fund is in a stationary state, has liabilities with a constant maturity T ₀ and does not make any payments between dates 0 and T. Since there are no intermediate liability payments, intermediate funding ratios do not enter the utility function. Whichever perspective is taken, the choice of the horizon T is somewhat arbitrary, so we will let this parameter vary in our simulations.

The reduced-form program assumed for the pension fund manager is thus

(9)$$\mathop {\rm max}\limits_{{\bimomega} } {\bb E}\lsqb U\lpar F_{\rm T} \rpar \rsqb .$$

Unless otherwise stated, we assume that the pension fund manager has CRRA preferences, with relative risk aversion γ, i.e., we take U=u whereFootnote ⁸

$$u\lpar x\rpar \equals \left\{ {\openup3\matrix{ {\displaystyle{{x^{\setnum{1} \minus \gamma } } \over {1 \minus \gamma }}\comma } \hfill \tab {{\rm for}\;x \gt 0\comma } \hfill \cr { \minus \infty \comma } \hfill \tab {{\rm for}\;x \leqslant 0.} \hfill \cr} } \right.$$

To obtain the solution to (9), we use the martingale approach in complete markets developed by Cox and Huang (Reference Cox and Huang1989). The following proposition presents the expression for the optimal policy in the zero-coupon case.Footnote ⁹ Details of derivation are relegated to appendices.

Proposition 2

The optimal payoff and wealth process in (9) in the generic case are

(10)$$A \ast_{\hskip-5pt\rm T}^{ u} \equals {{A_{\setnum{0}} \minus L_{\setnum{0}}^{\setnum{0}\comma T} } \over {{\bb E}\lsqb \mathop {\lpar M_{\rm T} L_{\rm T} \rpar }\nolimits^{\setnum{1} \minus \lpar\setnum{1}\sol \gamma\rpar } \rsqb }}M_{\rm T}^{ \minus \setnum{1}\sol \gamma } L_{\rm T}^{\setnum{1} \minus\lpar \setnum{1}\sol \gamma\rpar } \comma$$

$$A\ast _{\hskip-5pt t}^{ u} \equals L_{t}^{t\comma T} \plus {{A_{\setnum{0}} \minus L_{\setnum{0}}^{\setnum{0}\comma T} } \over {{\bb E}\lsqb \mathop {\lpar M_{\rm T} L_{\rm T} \rpar }\nolimits^{\setnum{1} \minus \lpar\setnum{1}\sol \gamma\rpar } \rsqb }}M_{t}^{ \minus \setnum{1}\sol \gamma } \mathop {\lpar L_{t}^{T\comma T_{\setnum{0}} } \rpar }\nolimits^{\setnum{1} \minus \lpar\setnum{1}\sol \gamma\rpar } g\lpar t\comma r_{t} \comma \rmPhi _{t} \rpar \comma$$

where

$$g\lpar t\comma r_{t} \comma \rmPhi _{t} \rpar \equals {\bb E}_{t} \left[ {\mathop {\left( {{{M_{\rm T} L_{\rm T}^{T\comma T_{\setnum{0}} } } \over {M_{\rm T} L_{\rm T}^{T\comma T_{\setnum{0}} } }}} \right)}\nolimits^{\setnum{1} \minus \lpar\setnum{1}\sol \gamma\rpar } } \right].$$

• • In the zero-coupon case, the optimal strategy reads

  $${\bimomega} \ast _{\hskip-5ptt}^{ u} \equals {{{\bf 1}\prime{\bimsigma} _{t}^{ \minus \setnum{1}} {\bimlambda} } \over \gamma }{\omega} _{t}^{{\rm PSP}} \plus {\bf 1}\prime{\bimsigma} _{t}^{ \minus \setnum{1}} {\bimsigma} _{I} \lpar t\comma T_{\setnum{0}} \rpar \left( {1 \minus {1 \over \gamma }} \right){\bimomega} _{t}^{{\rm LMP}} \lpar T\comma T_{\setnum{0}} \rpar \comma$$
  where
  $${\bimomega} _{t}^{{\rm PSP}} \equiv {{{\bimsigma} _{t}^{ \minus \setnum{1}} {\bimlambda} } \over {{\bf 1}\prime{\bimsigma} _{t}^{ \minus \setnum{1}} {\bimlambda} }}\comma \quad {\bimomega} _{t}^{{\rm LMP}} \lpar T\comma T_{\setnum{0}} \rpar \equals {{{\bimsigma} _{t}^{ \minus \setnum{1}} {\bimsigma} _{I} \lpar t\comma T_{\setnum{0}} \rpar } \over {{\bf 1}{\prime } {\bimsigma} _{t}^{ \minus \setnum{1}} {\bimsigma} _{I} \lpar t\comma T_{\setnum{0}} \rpar }}\comma \quad {\bf 1} \equals \lpar \matrix{ 1 \tab 1 \tab 1 \cr} \rpar {\prime } .$$

3.1 Portfolio optimization with minimum funding ratio constraint

We complement the investor's objective by introducing the following explicit funding ratio constraint:

(11)$$\mathop {\rm max}\limits_{{\bimomega} } {\bb E}\lsqb U\lpar F_{\rm T} \rpar \rsqb \comma \quad {\rm s}.{\rm t}.\;A_{\rm T} \geqslant kL_{\rm T} .$$

Imposing such an explicit lower bound intuitively means that the pension fund has infinitely low utility from funding ratios below k. In fact, it follows from the results in Bouchard et al., (Reference Bouchard, Touzi and Zeghal2004) that (11) is a special case of (9) for a non-smooth utility function U:

(12)$$\mathop {\rm max}\limits_{{\bimomega} } {\bb E}\lsqb \tilde{U}^{\hskip1pt k} \lpar F_{\rm T} \rpar \rsqb \comma$$

where Ũ^k is a non-smooth utility function defined as follows:

(13)$$\tilde{U}^{\hskip1pt k} \left( x \right) \equals \left\{ {\openup3\matrix{ {u\lpar x\rpar \comma } \hfill \tab {{\rm if}\;x\geqslant k\comma } \hfill \cr { \minus \infty \comma } \hfill \tab {{\rm if}\;x \lt k.} \hfill \cr} } \right.$$

It seems from program (11) that only the terminal funding ratio is constrained to be larger than k. In fact, taking expectations on both sides of the inequality A _T⩾kL _T and using the budget constraint (6), we obtain that the constraint ${\bb P}\lpar F_{\rm T} \geqslant k\rpar \equals 1$ is equivalent to

$$A_{t} \minus L_{t}^{t\comma T} \geqslant kL_{t}^{T\comma T_{\setnum{0}} } \comma \quad {\rm for}\;{\rm all}\;t\;{\rm in}\;\lsqb 0\comma T\rsqb \;{\rm with}\;{\rm probability}\;1.$$

Hence, the long-term funding ratio constraint is equivalent to a series of short-term constraints. It can easily be seen that for k < 1, these short-term constraints imply that F_t⩾k with probability 1 at all dates t. In particular, it is necessary that $A_{\setnum{0}} \geqslant L_{\setnum{0}}^{\setnum{0}\comma T} \plus kL_{\rm T}^{T\comma T_{\setnum{0}} }$ holds at time 0, for the maximization problem to be feasible. It should also be noted that the complete market assumption is critical here, since the presence of a non-hedgeable source of risk would make it impossible for the terminal constraint to hold almost surely.

The following proposition provides the solution to the optimization process in the presence of funding ratio constraints.

3.2 Introducing maximum funding ratio constraints

As noted in the introduction, pension funds have no interest in building up exceedingly large surpluses. Overfunding can be formally prohibited by pension law, or may be simply discouraged by tax law or complexity of surplus sharing rules. The purpose of this subsection is to introduce a maximum funding constraint along with a minimum one. The value of the maximum can be interpreted as the funding level beyond which the utility function of the pension fund is constant. The optimization programme is thus given by (9) subject to the additional constraints F _T⩾k and F _T⩽k′:

(18)$$\mathop {\rm max}\limits_{{\bimomega} } {\bb E}\lsqb U\lpar F_{\rm T} \rpar \rsqb \comma \quad {\rm s}.{\rm t}.\;k\leqslant F_{\rm T} \leqslant k&#x0027;.$$

In order to have the constraint k⩽F _T⩽k′ satisfied almost surely, the initial asset A ₀ should lie between $L_{\setnum{0}}^{\setnum{0}\comma T} \plus kL_{\setnum{0}}^{T\comma T_{\setnum{0}} }$ and $L_{\setnum{0}}^{\setnum{0}\comma T} \plus k&#x0027;L_{\setnum{0}}^{T\comma T_{\setnum{0}} }$. The following proposition presents the solution to these optimization programmes in the presence of minimum and maximum funding ratio constraints (or targets).

Proof

See appendix A.5. □

The optimal terminal net wealth is given by the payoff of a static portfolio strategy that consists of investing kL ₀ in the LMP, with the remainder, A ₀−kL ₀, being invested in a bull spread strategy extended to an exchange option context. This strategy consists of a long position in an exchange option and a short position in an exchange option on the same underlying payoff and a higher strike price. The former option gives its owner the right to exchange a portfolio of terminal value kL _T for a portfolio of terminal value ξ′A _T*^u, while the latter option gives its owner the right to exchange a portfolio of terminal value k′L _T for the same portfolio of terminal value ξ′A _T*^u.

A noteworthy property of the strategy of proposition 5 is that it allows for the respect of the maximum funding constraint at all times, provided the maximum funding level k′ > 1 and the initial wealth satisfies the conditions stated in the proposition. The proof of this statement is straightforward: given that the terminal wealth satisfies $A \ast_{\hskip-5pt \rm T}^{ k\comma k\prime } \leqslant k\prime L_{\rm T}^{T\comma T_{\setnum{0}} }$ with probability 1, we obtain, taking the present values of both sides of the inequality

$$A_{t} \minus L_{t}^{t\comma T} \leqslant k\prime L_{t}^{T\comma T_{\setnum{0}} } \comma$$

which implies that A_t⩽k′L_t if k′ > 1. Similarly, the minimum funding constraint is respected at all dates if k < 1: the proof is exactly the same as for the strategy with a lower bound only (see Section 3.1).

Finally, comparing propositions 3 and 5, it can be seen that the imposition of an explicit upper bound involves a short position in an exchange option. This short position allows one to reduce the cost of downside protection by giving up some access to the upside potential beyond the funding ratio threshold k′, as shown in the following proposition.

4.1 Unconstrained strategy

With no loss of generality, we assume that the investment opportunity set includes a single stock index, regarded as an efficient combination of individual stocks, in addition to a zero-coupon bond and an inflation-indexed bond with maturity corresponding to the duration of pension payments. Our base case parameters are taken from Munk et al., (Reference Munk, Sørensen and Vinther2004), who also model the nominal interest rate as an Ornstein–Uhlenbeck process and the price index as a Geometric Brownian motion.Footnote ¹¹Table 2 summarizes our base case set of parameter values.

Table 2. Base case parameters

This table contains parameter values for interest rate, price index and stock return processes. These parameter values, as well as the price for interest rate risk, are borrowed from Munk et al., (Reference Munk, Sørensen and Vinther2004), while the equity risk premium parameter is taken from Brennan and Xia (Reference Brennan and Xia2002) and the inflation risk premium is set to zero.

In all cases, we have assumed that the pension fund was initially fully funded, i.e., A ₀=L ₀, and have estimated the distribution of the final funding ratio using 5,000 points. In Table 3, we provide information regarding the distribution of the funding ratio when no constraint is introduced. As expected, we find that the dispersion increases with T and decreases with γ. Indeed, a lower risk-aversion parameter implies a higher investment in the PSP, and hence a higher performance potential coupled with a higher funding risk. On the other hand, for a given risk-aversion parameter value, we find that the range of funding ratio values increases with the time-horizon T as more time is allowed for uncertainty to play a role. Even for γ = 10, we find that the minimum funding ratio obtained is lower than 90% for a one-year horizon, and lower than 70% for a 20-year horizon. This provides justification for the introduction of funding ratio constraints aiming at imposing a left-truncation of the final distribution.

Table 3. Distribution of the final funding ratio when utility is from terminal funding ratio and no lower bound is imposed

This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the volatility. Also reported are the shortfall probability, the expected shortfall and the conditional mean of the funding ratio given it lies above k = 0.9, or between 0.9 and k′ = 1.1. Parameters are fixed at their base case values (see Table 2). The pension fund is initially fully funded, and liability payments are given in Table 1.

4.2 Minimum funding ratio constraints

In Table 4, we test for the introduction of explicit funding ratio constraints, with a minimum set at k = 90%. In this case, we find that the minimum funding ratio is indeed limited to k, a value that is reached with a relatively high probability, suggesting that the margin for error is fully utilized with these strategies. In fact, the dispersion of the funding ratio distribution is narrower on both sides when the pension fund follows the strategy with explicit constraints than when it chooses the unconstrained counterpart. For instance, in the case T = 20 years and γ = 5, the maximum funding ratio is 3.1 in the explicit constraint case, while it reaches 4.01 in the unconstrained case. That downside protection comes at the cost of a more limited access to the upside potential can also be seen from the fact that the expected terminal funding ratio conditional upon being larger than the constraint k, ${\bb E}\lsqb F_{\rm T} \vert F_{\rm T} \geqslant k\rsqb$, is always strictly lower in the constrained case than in the unconstrained case. For example, when T = 20 years and γ = 5, the conditional mean reaches 1.16 in the constrained case, while it is 1.51 in the unconstrained case. On the other hand, the average deficit is also significantly higher in the latter case (13%, as opposed to 8% in the constrained case).

Table 4. Distribution of the final funding ratio when utility is from terminal funding ratio and a lower bound is imposed

This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the volatility. Also reported are the shortfall probability, the expected shortfall and the conditional mean of the funding ratio given that it lies between k = 0.9 and k′ = 1.1. Parameters are fixed at their base case values (see Table 2). The lower bound k is set to 0.9. The pension fund is initially fully funded, and liability payments are given in Table 1.

4.3 Minimum and maximum funding ratio constraints

In Table 5, we introduce an additional upper bound constraint, with a maximum funding ratio value set at k′ = 110%. Giving up access to the upside potential above 110% allows one to decrease the cost of downside protection, as can be seen from the fact that the average of terminal funding ratio values conditional upon being in the range between 90% and 110% are higher when the upper bound is imposed than when it is not introduced. In fact, focusing again on T = 20 years and γ = 5, we have that the conditional expected funding ratio ${\bb E}\lsqb F \ast_{\hskip-5pt\rm T}^{ k\comma k\prime } \vert k\leqslant F \ast_{\hskip-5pt\rm T}^{k\comma k\prime } \leqslant k\prime \rsqb \equals 1.04$ when both constraints are imposed, while it merely reached 0.95 when only the lower constraint was imposed. In the unconstrained case, we had that ${\bb E}\lsqb F \ast_{\hskip-5pt\rm T}^{k\comma k\prime } \vert k\leqslant F \ast_{\hskip-5pt\rm T}^{k\comma k\prime } \leqslant k\prime \rsqb \equals 1.01 \lt 1.04$. Hence, the addition of the short option position allows for an improvement of the mean funding ratio over the range of values between 0.9 and 1.1, not only with respect to the case with minimum funding requirement only, but also with respect to the unconstrained case.

Table 5. Distribution of the final funding ratio when utility is from terminal funding ratio and lower and upper bounds are imposed (k′ = 1.1)

This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the volatility. Also reported are the shortfall probability, the expected shortfall and the conditional mean of the funding ratio given that it lies between k = 0.9 and k′ = 1.1. Parameters are fixed at their base case values (see Table 2). The pension fund is initially fully funded, and liability payments are given in Table 1. The lower bound k is set to 0.9 and the upper bound k′ to 1.1.

4.4 Dynamic properties of the funding ratio

As noted in the theoretical section of this paper, the optimal strategies presented in propositions 3 and 5 allow for the respect of minimum funding constraints at all times if the minimum funding level k < 1, k′ > 1, and the initial funding ratio is compatible with the respect of these constraints (see conditions in the propositions).

Nevertheless, given the regulatory focus on intermediate funding ratios, it is useful to consider the evolution of this variable over time. In order to keep the analysis as simple as possible, we assume that there is a single liability payment at the terminal date T ₀, which corresponds to the zero-coupon case. We take T ₀ to be the duration of the payments in Table 1 (see equation (19)), numerically equal to 11.32 years, and we set the horizon to 10 years.Footnote ¹² Then it follows from propositions 2, 3 and 5 that the unconstrained funding ratio and the two constrained wealth funding ratios are given by

$$\eqalign{\hfill F \ast_{\hskip-5pt t}^{ u} \equals \tab {{A_{\setnum{0}} } \over {L_{\setnum{0}}^{\setnum{1} \minus \setnum{1}\sol \gamma } }}L_{t}^{ \minus \setnum{1}\sol \gamma } M_{t}^{ \minus \setnum{1}\sol \gamma } \exp\left[ {\int_{\setnum{0}}^{t} \vert \vert {\bimsigma} _{I} \lpar s\comma T_{\setnum{0}} \rpar \minus {\bimlambda} \vert \vert ^{\setnum{2}}\,{\rm d} s} \right]\comma \hfill\cr\hfill F \ast_{\hskip-5pt t}^{ k} \equals \tab {{\xi A\ast _{t}^{ u} } \over {L_{t} }}{\cal N}\lpar d_{\setnum{1}\comma t} \rpar \plus k{\cal N}\lpar \minus d_{\setnum{2}\comma t} \rpar \comma\hfill \cr\hfill F \ast_{\hskip-5pt t}^{k\comma k\prime } \equals \tab {{\xi \prime A \ast _{\hskip-5pt t}^{u} } \over {L_{t} }}{\cal N}\lpar d_{\setnum{1}\comma t} \rpar \minus {{\xi \prime A \ast_{\hskip-5pt t}^{ u} } \over {L_{t} }}{\cal N}\lpar d\prime _{\setnum{1}\comma t} \rpar \plus k{\cal N}\lpar \minus d_{\setnum{2}\comma t} \rpar \plus k\prime{\cal N} \lpar d\hskip1pt\prime _{\hskip-3pt\setnum{2}\comma t} \rpar \comma\hfill \cr}$$

where ξ, ξ′, d_{i ,t} and d′_{i ,t} are defined in propositions 3 and 5.

Using these expressions, we simulate the intermediate funding ratios at dates t = 1, 3, 5, 7 and 9 years for a strategy with horizon T = 10 years. Table 6 reports the volatility, mean, minimum and maximum after t years. It appears that the increase in volatility over time is larger for the unconstrained strategy than for the strategies that impose at least one bound: when γ = 2, volatility ranges from 0.18 after 1 year to 0.78 after 9 years with no constraint, and from 0.06 to 0.41 with a lower bound only. This effect is straightforward: by cutting either the left tail or both tails of the distribution, these strategies reduce the volatility. Similarly, the mean, the minimum and the maximum vary less over time for the two constrained strategies. The unconstrained strategy is found to respect neither the minimum funding constraint nor the maximum one, whatever the horizon. The violations of these constraints are, however, more limited in size at short horizons and for high levels of risk aversion.

Table 6. Distribution of intermediate funding ratios

(a) Optimal strategy with no lower bound

(b) Optimal strategy with lower bound

(c) Optimal strategy with lower and upper bounds

This table reports descriptive statistics on the intermediate optimal funding ratio in three situations: no lower bound is imposed, a lower bound k = 0.9 is imposed, and a lower bound and an upper bound k′ = 1.1 are simultaneously imposed. The pension fund is initially fully funded, and liabilities are represented by a single payment of $\rmPhi _{T_{\setnum{0}} }$ at date T ₀ = 11.32 years, and the horizon of the strategy is T = 10 years. Other parameters are fixed at their base case values (see Table 2).

5.1 Introducing dynamic incompleteness

The optimal terminal net wealth obtained in proposition 3 can only be generated through dynamic trading in continuous-time. In practice, continuous trading is impossible because of transaction costs or liquidity constraints. It is therefore relevant to test whether the strategy described in the proposition satisfies the minimum funding constraints when it is implemented in discrete time. We focus on the zero-coupon case, because it leads to a fully analytical expression for the optimal strategy in continuous time. Hence, there is a single liability payment at the terminal date T ₀ = 11.32 years.

We assume that trading takes place at dates 0<t ₁< ⋅⋅⋅<t_n<T, where the time step Δt=t_{i +1}−t_i can be either one week, one month, two months or one year. The portfolio is rebalanced at each date t_i, and is buy-and-hold over the period [t_i, t_{i +1}]. The weights allocated to the stock index, the nominal bond and the indexed bond at date t_i are given by the vector ${\bimomega} \ast^{ k}_{t_{i} }$, where ω*^k is the portfolio rule that has been shown to be optimal in continuous time (see equation (15)). The weight allocated to cash is one minus the sum of the weights allocated to the other assets. This strategy generates a certain terminal wealth, and panel (a) of Table 7 summarizes the distribution of the terminal funding ratio. It also reports descriptive statistics on the distribution of the terminal funding ratio when the strategy is implemented in continuous time. The continuous-time distributions have been obtained through direct simulation of the terminal payoff (14). All the results in this table have been obtained assuming an investment horizon of T = 10 years, and a lower bound k = 0.9.

Table 7. Summary of robustness checks

(a) Discrete-time trading

(b) Jump risk in stock index

(c) Unspanned liability risk

In panel (a), we study the impact of the rebalancing frequency. In panel (b), we introduce downside jumps in the price process of the stock index. In panel (c), liabilities include a specific risk component orthogonal to all traded risks. Liabilities are paid at date T ₀ = 11.32 years, the investment horizon is T = 10 years, and the lower bound k is set to 0.9. Other parameters are fixed at their base case values (see Table 2).

We find that the introduction of dynamic incompleteness leads to a positive probability of violating the minimum funding constraint. Indeed, the minimum value for the funding ratio happens to be lower than the minimum funding value k when trading is yearly or bimonthly. On the other hand, these violations are very limited in probability, since the minimum funding level is respected in at least 97.5% of the scenarios, except for the somewhat extreme assumption of annual rebalancing. For a reasonable trading frequency of one month, we find that the constraint is violated with positive probability only in the case γ = 10, but is respected at the 2.5% level. Overall, these results show that discrete trading does not lead to worrying deteriorations of the performance of the continuous-time strategy, as far as the objective of respecting a minimum funding level is concerned.

5.2 Introducing jump risk on the asset side

As a second robustness check, we now test the behavior of the strategy in the presence of jumps in the dynamics of stock returns. Jumps make the market incomplete, since they cannot be hedged. Formally, we follow Merton (Reference Merton1976) in assuming that the stock price evolves as a mixture of a pure diffusion process and a Poisson process

(20)$${\rm d}S_{t} \equals S_{t} \lsqb r_{t} \plus {\sigma} _{S} {\lambda} _{S} \minus \iota x\rsqb \,{\rm d}t \plus S_{t} {\bimsigma}{\prime}_{\hskip-3ptS} \,{\rm d}{\bi z}_{t} \plus S_{t \minus } x\,{\rm d}N_{t} .$$

In this equation, N denotes a Poisson process with constant intensity under ℙ denoted by ι, a process that we assume to be independent of z. x represents the jump size. This quantity is negative, but ensuring that the price remains positive requires x > − 1. The price process is right continuous, but no longer continuous, hence the S_{t −} in the above equation.

The presence of jump risk introduces a form of dynamic incompleteness. As pension cash-flows are not impacted by the presence of jumps in the stock price, the no-arbitrage value of liabilities equals the same L ₀ as in the complete market case. On the other hand, the optimal strategies that were derived with or without funding constraints in the absence of jumps (see propositions 2, 3 and 5) are now suboptimal. We do not attempt here to compute optimal strategies in the presence of jumps.Footnote ¹³ Instead, we use the same weights as in the complete market case, and we apply them to the two zero-coupon bonds and the stock with jump risk. These weights are given by (15). They involve, in particular, the constant ξ which is solution to the equation

(21)$$\xi A_{\setnum{0}} {\cal N}\lpar d_{\setnum{1}\comma \setnum{0}} \rpar \minus kL_{\setnum{0}} {\cal N}\lpar d_{\setnum{2}\comma \setnum{0}} \rpar \equals A_{\setnum{0}} \minus kL_{\setnum{0}} \comma$$

where d _1,0 and d _2,0 are given by equations (16) and (17). It should be noted that because of market incompleteness, the left side is no longer the no-arbitrage price of the payoff [ξA _T*^u−kL _T]⁺; therefore (21) is taken here as a definition, rather than a property, of ξ. We then implement strategy (15) with monthly rebalancings, which corresponds to a commonly chosen frequency in practice.

Panel (b) of Table 7 summarizes the distribution of the terminal funding ratio. We find that jumps of exceedingly large size or frequency are needed for the strategy to exhibit severe and frequent violations of the minimum funding ratio requirements. For example, assuming that a jump takes place on average every 5 years, with a −20% intensity, we find for γ = 2 that the minimum funding ratio is 75%, with the 2.5% bottom percentile already very close to the minimum funding requirement, with the value 87%. If a jump of the same size were to occur on average every year, the strategy would not perform as well: for γ = 2, the probability of not satisfying the minimum funding requirement would be more than 50%. However, a −20% jump on average every year appears to be an extreme assumption.Footnote ¹⁴ For γ = 2, in each but this particular case, the probability of not breaching the floor is greater than 75%. For greater values of the relative risk aversion, the terminal net wealth is less impacted by jump risk since stocks receive a lower allocation. For instance, when γ = 10, the constraint is satisfied with a probability of 2.5% in all cases, except if a −20% jump occurs on average every year.

5.3 Introducing specific liability risk

So far we have assumed that pension payments were only subject to interest rate and inflation risks. Since nominal and indexed bonds are assumed to be traded on the financial market, liability risk is therefore entirely spanned by existing securities. In practice, however, the presence of specific liability risk related, in particular, to actuarial uncertainty induces a particular form of market incompleteness. We do not attempt here to present a realistic model for the dynamics of the work force demography, but instead to introduce a stylized model that will allow us to test the robustness of the dynamic strategies in the presence of actuarial risk that is not spanned by traded securities. Formally, we assume that the amount to be paid at time T ₀ is equal to $\varepsilon _{T_{\setnum{0}} } \rmPhi _{T_{\setnum{0}} }$, where

$$\varepsilon _{T_{\setnum{0}} } \equals {\rm exp}\lsqb \minus {\sigma} _{\varepsilon }^{\setnum{2}} T_{\setnum{0}} \plus {\sigma} _{\varepsilon } \tilde{z}_{T_{\setnum{0}} } \rsqb .$$

In this equation, is a ℙ-Brownian motion independent of z that represents actuarial risk. This risk is therefore assumed to be independent of financial risk, which is arguably a reasonable approximation of reality. As the traded assets only span interest rate, inflation and stock price risks, the market is no longer complete. Technically speaking, there exist infinitely many equivalent martingale measures (see Harrison and Kreps, Reference Harrison and Kreps1979). To each martingale measure corresponds a pricing kernel through equation (3), and one price process for the liability payment $L_{T_{\setnum{0}} }$. As shown by He and Pearson (Reference He and Pearson1991), the solution to the utility maximization problem (9) involves the ‘minimax’ pricing kernel. In this section, we do not solve for minimax pricing kernel, and we fix the martingale measure exogenously by taking the ‘minimal’ martingale measure (see Föllmer and Schweizer, Reference Föllmer and Schweizer1990): this measure, which assigns a zero price to non-traded risks, leads to the same price process for the liability as in the complete market setting:Footnote ¹⁵

$$L_{t} \equals {\bb E}_{t} \left[ {{{M_{\rm T} } \over {M_{t} }}\varepsilon _{T_{\setnum{0}} } \rmPhi _{T_{\setnum{0}} } } \right] \equals \varepsilon _{t} I\lpar t\comma T_{\setnum{0}} \rpar .$$

We then solve for ξ in equation (21), and we implement the dynamic strategy (15) on a monthly basis, using the minimal entropy price L_t for the liability wherever needed. Panel (c) in Table 7 reports summary statistics on the distribution of the terminal funding ratio for different values of σ_∊, which is the parameter driving the uncertainty on $\varepsilon _{T_{\setnum{0}} }$ through the equality:

$${\bb V}\lsqb \varepsilon _{T_{\setnum{0}} } \rsqb \equals {\rm exp} \lsqb {\sigma} _{\varepsilon }^{\setnum{2}} T_{\setnum{0}} \rsqb \minus 1.$$

Since there is a specific risk factor in the liability payment, the payoff L _T is not replicable, so it is impossible to ensure that the terminal funding ratio is greater than k with probability 1. The results confirm that the introduction of unspanned liability risk causes a deterioration of the performance of the risk-controlled strategy. For reasonable parameter values leading to 5% standard deviation in $\varepsilon _{T_{\setnum{0}} }$, we find that minimum funding requirement is respected with a 2.5% probability when γ = 10, and close to being respected at the 2.5% confidence level for γ = 2 and γ = 5. Overall, it appears that the benefits of the risk-controlled strategies are relatively robust with respect to the introduction of various forms of market incompleteness.