2.1 Structure of the financial market

The components of the model are as follows:

• • The (commodity) price level Π follows a diffusion process:

  (1)$$\displaystyle{{d\Pi _t} \over {\Pi _t}} = \pi _tdt + \xi _SdW_{S, t} + \xi _rdW_{r, t} + \xi _\pi dW_{\pi , t}, \;$$

where Π₀ = 1 and π _t is the instantaneous expected rate of inflation and follows an Ornstein–Uhlenbeck process:

(2)$$d\pi _t = \kappa _\pi ( {\bar{\pi }-\pi_t} ) dt + \sigma _\pi dW_{\pi , t}.$$

The Ornstein–Uhlenbeck process is mean-reverting, which means that in the long run, the process tends to drift towards its long-term mean $\bar{\pi }$. The intensity of this mean-reverting tendency is scaled by the parameter $\kappa _\pi$. ξ _S, ξ _r, and $\xi _\pi$ represent the constant loadings on the stochastic innovations. All the Ws are standard Brownian motions under a probability measure ℙ. dW _t = [dW _S,t, dW _r,t, dW _π,t]^′. The Brownian motions $dW_S, \;\;dW_r, \;\;{\rm and}\;dW_\pi$ are assumed to be correlated with correlation coefficient ρ as follows:

$$\rho = \left({\matrix{ 1 & {\rho_{Sr}} & {\rho_{S\pi }} \cr {\rho_{Sr}} & 1 & {\rho_{r\pi }} \cr {\rho_{S\pi }} & {\rho_{r\pi }} & 1 \cr } } \right).$$

Consequently,

(3)$$\displaystyle{{\Pi _s} \over {\Pi _t}} = \exp \left\{{\int\limits_t^s {\left({\pi_u-\displaystyle{1 \over 2}{\xi }^{\prime}\rho \xi } \right)du + \int\limits_t^s {{\xi }^{\prime}dW_u} } } \right\}\quad {\rm where}\;\xi = [ {\xi_S, \;\xi_r, \;\xi_\pi } ] ^{\prime}.$$

• • The instantaneous real riskless interest rate, r _t, also follows the Ornstein–Uhlenbeck process:

  (4)$$dr_t = \kappa _r( {\bar{r}-r_t} ) dt + \sigma _rdW_{r, t}$$

where $\bar{r}$ is the long-term mean level of the real interest rates.

• • The real pricing kernel of the economy which determines the expected returns on all securities, M _t is given by:

  (5)$$\displaystyle{{dM_t} \over {M_t}} = {-}r_tdt + {\omega }^{\prime}dW_t, \;$$
  where M ₀ = 1 and $\omega = [ {\omega_S, \;\;\omega_r, \;\;\omega_\pi } ] ^{\prime}$ represents the constant loadings on the stochastic innovations in the economy. The pricing kernel relative can be written as follows:
  (6)$$\displaystyle{{M_s} \over {M_t}} = \exp \left\{{\int\limits_t^s {\left({-r_u-\displaystyle{1 \over 2}{\omega }^{\prime}\rho \omega } \right)du + \int\limits_t^s {{\omega }^{\prime}dW_u} } } \right\}.$$

If there are at least four securities whose instantaneous variance–covariance matrix has rank three, the state variables can be spanned. Moreover, the variance–covariance matrix of real returns on cash, stock, and two finite maturity bonds with different maturities has rank three.

• • The nominal price at time t of a bond which matures at time T with payoff of one nominal unit, satisfies:

  (7)$$P( {t, \;T} ) = \exp \{ {A( {t, \;T} ) -B( {t, \;T} ) r_t-D( {t, \;T} ) \pi_t} \} , \;$$
  where $A( {t, \; T} ) , \; B( {t, \; T} )$, and $D( {t, \;T} )$ are time-dependent constants, in particular $B( {t, \; T} ) = \kappa _r^{{-}1} ( {1-e^{\kappa_r( {t-T} ) }} )$, $D( {t, \;T} ) = \kappa _\pi ^{{-}1} ( {1-e^{\kappa_\pi ( {t-T} ) }} )$, and the expression for $A( {t, \;T} )$ is given in Appendix A. As a result,
  (8)$$\eqalign{\displaystyle{{dP( {t, \;T} ) } \over {P( {t, \;T} ) }} &= ( R_t-B( {t, \;T} ) \sigma _r\lambda _r-D( {t, \;T} ) \sigma _\pi \lambda _\pi ) dt \cr & \quad -B( {t, \;T} ) \sigma _rdW_{r, t}-D( {t, \;T} ) \sigma _\pi dW_{\pi , t}, \;} $$
  where λ _r and $\lambda _\pi$ are the constant unit risk premiums associated with the innovation dW _r,t and dW _π,t, respectively. R _t is the instantaneous nominal risk-free interest rate at time t. $R_t = r_t + \pi _t-\xi _S\lambda _S-\xi _r\lambda _r-\xi _\pi \lambda _\pi$, where λ _S is the constant unit risk premium associated with the innovation dW _S,t. The derivation of equation (8) and R _t are included in Appendix A. Moreover, λ = ρξ − ρω. The component ${-}B( {t, \;\;T} ) \sigma _r\lambda _r-D( {t, \;\;T} ) \sigma _\pi \lambda _\pi$ represents its nominal risk premium. In addition, the real price at time t of an indexed bond which matures at time T with a payoff of one real unit, is given as follows:
  (9)$$p^\ast ( {t, \;T} ) = \exp \{ {A^\ast ( {t, \;T} ) -B( {t, \;T} ) r_t} \} , \;$$
  where A*(t, T) is a time-dependent constant and its expression is given in Appendix A. Similarly,
  (10)$$\eqalign{\displaystyle{{dp^\ast ( {t, \;T} ) } \over {\,p^\ast ( {t, \;T} ) }} & = ( r_t-B( {t, \;T} ) \sigma _r( {-\omega_s\rho_{Sr}-\omega_r-\omega_\pi \rho_{r\pi }} ) ) dt \cr & \quad -B( {t, \;T} ) \sigma _rdW_{r, t}.} $$

Its nominal return is P*(t, T) = Π_tp*(t, T):

(11)$$\eqalign{\displaystyle{{dP^\ast ( {t, \;T} ) } \over {P^\ast ( {t, \;T} ) }} = & ( {r_t + \pi_t-B( {t, \;T} ) \sigma_r\lambda_r} ) dt + \xi _SdW_{S, t} \cr & \quad + ( {\xi_r-B( {t, \;T} ) \sigma_r} ) dW_{r, t} + \xi _\pi dW_{\pi , t}.} $$

• • The nominal stock price S _t follows a geometric Brownian motion:

  (12)$$\displaystyle{{dS_t} \over {S_t}} = ( {R_t + \sigma_S\lambda_S} ) dt + \sigma _SdW_{S, t}.$$

• • Premiums are payable in one of two forms: either as an initial single premium or regular premium, that is, continuously into the plan member's individual account at a rate of salary. The contribution rates and the rates of consumption before retirement are determined exogenously, and there are no non-pension savings in the model. For the purpose of clarity, we assume the plan member contributes an initial single premium throughout the paper, the value of which is equal to the market price for the future premiums, denoted by X ₀. This assumption will not affect the expected utility of the wealth at retirement. X ₀ is our initial capital. In addition, we assume early withdrawal is not possible.

• • The fund at retirement, denoted by X _T, is used to purchase a life annuity paying one real unit annually in arrears so long as the annuitant is alive. The price for the annuity is denoted by Π_Ta_T. a _T is the price for the life annuity at time T in real terms, and Π_T is the inflation rate at time T. The value of the ratio X _T/(Π_Ta _T) is the amount of the yearly payment in real terms the annuitant receives. This life annuity will be treated in detail in the next subsection.

2.2 Benchmark

To model the pension assets at retirement T, we assume the plan member may live τ more years after reaching the retirement age. In particular, we consider an individual who starts working at age 25, retires at age 65, and passes away at age 85. The current time is at age 25. Then the investment horizon is 40 years, that is, T = 40 and τ = 20. The life annuity pays one real unit or Π_T+i (i ∈ [0, τ]) nominal unit at the end of each year for τ years. Early withdrawal before the retirement age is not allowed. We use τ indexed bonds to generate these payments. To buy these indexed bonds at retirement T, the required amount of asset in real terms at time T is:

(13)$$\eqalign{a_T = & \,\, p^\ast ( {T, \;T + 1} ) + p^\ast ( {T, \;T + 2} ) + \cdots + p^\ast ( {T, \;T + \tau } ) \cr = & \,\, e^{A^\ast ( {T, T + 1} ) -B( {T, T + 1} ) r_T} + e^{A^\ast ( {T, T + 2} ) -B( {T, T + 2} ) r_T} + \cdots + e^{A^\ast ( {T, T + \tau } ) -B( {T, {\rm T} + \tau } ) r_T}.} $$

At time 0, the following equations hold:

(14)$$\eqalign{{\rm {\opf E}}[ a_T] = & \sum\limits_{i = 1}^\tau {e^{A^\ast( T, T + i) -B( T, T + i) {\rm {\opf E}}[ r_T] + {1\over 2} B{( T, T + i) }^2{\rm {\opf V}}{\rm ar}[ r_T] }} , \;\cr {\rm {\opf E}}[ a_T^2 ] = & \sum\limits_{i = 1}^\tau {\sum\limits_{\,j = 1}^\tau {e^{( A^\ast( T + i) + A^\ast( T, T + j) ) -( B( T, T + i) + B( T, T + j) ) {\rm {\opf E}}[ r_T] + {1\over 2} {( B( T, T + i) + B( T, T + j) ) }^2{\rm {\opf V}}{\rm ar}[ r_T] }} } , \;\cr {\rm {\opf V}}{\rm ar}[ a_T] = & {\rm {\opf E}}[ a_T^2 ] -( {\rm {\opf E}}[ a_T] ) ^2, \;} $$

where ${\rm {\opf E}}[ {r_T} ] = \bar{r} + e^{-\kappa _rT}( {r_0-\bar{r}} )$, ${\rm {\opf V}}{\rm ar}[ {r_T} ] = ( {\sigma_r^2 /(2\kappa_r)} ) ( {1-e^{{-}2\kappa_rT}} )$, and r _T is normally distributed with ${\rm {\opf E}}[ {r_T} ]$ and ${\rm {\opf V}}{\rm ar}[ {r_T} ]$. This functional form of a _T does not permit integration in closed-form. Hence, we use the principle of the Fenton–Wilkinson approximation method (Fenton, Reference Fenton1960) to generate a log-normal variable L _T. It is important to note that this method is a log-normal approximation for sums of log-normal variables based on the first and the second moment-matching. It gives the following:

(15)$$\eqalign{L_T = & e^{m-nr_T}{\rm \;( }m{\rm , \;}\;n\;{\rm are \; constant) }, \;\cr {\rm {\opf E}}[ {a_T} ] = & {\rm {\opf E}}[ {L_T} ] = e^{m-n{\rm {\opf E}}[ {r_T} ] + {1\over 2} n^2{\rm {\opf V}}{\rm ar}[ {r_T} ] }, \;\cr{\rm {\opf V}}{\rm ar}[ {a_T} ] = & {\rm {\opf V}}{\rm ar}[ {L_T} ]=( {\rm {\opf E}}[{L_T}] ) ^2( {e^{n^2{\rm {\opf V}}{\rm ar}[ {r_T} ] }-1} ) . } $$

Solving for m and n gives

(16)$$n = \sqrt {\displaystyle{{\ln ( {{\rm {\opf V}}{\rm ar}[ {a_T} ] /{( {{\rm {\opf E}}[ {a_T} ] } ) }^2 + 1} ) } \over {{\rm {\opf V}}{\rm ar}[ {r_T} ] }}} , \;\quad m = \ln ( {{\rm {\opf E}}[ {a_T} ] } ) + n{\rm {\opf E}}[ {r_T} ] -\displaystyle{1 \over 2}n^2{\rm {\opf V}}{\rm ar}[ {r_T} ] .$$

L _T is our benchmark in real terms. Its value approximates that of the required amount of asset at time T in real terms, that is, a _T. This approximation method is satisfactory in our case. We will illustrate it with numerical simulations.

2.2.1 Simulation study for a _T and L _T

In this subsection, we provide numerical simulations to show how well the value of a _T can be approximated by that of L _T. Table 1 shows model parameters for the simulations. In particular, the starting value of the real interest rate r ₀ is 1%, and its long-term mean $\bar{r}$ is 2%. The intensity of its mean-reverting tendency parameter κ _r has the value of 0.1 or 0.6. The Brownian motion increments underlying the real interest rate level are negatively correlated with that underlying the stock price level and the instantaneous expected rate of inflation.

Table 1. Overview of model parameters

We run 100,000 simulations and draw the histogram of a _T and L _T to study their distributions. We also make plots and tables to study the gap between the estimated value of a _T and L _T. The results for κ _r = 0.6 and κ _r = 0.1 are presented in Figure 1 and Table 2.

Figure 1. Asset a _T and L _T.

Table 2. Estimated values of a _T and L _T

Panels a and b in Figure 1 show that the distribution of the benchmark L _T closely matches that of the required asset at time T, that is, a _T. When κ _r is 0.6, panels c and e reveal that the estimated value of L _T and a _T are close to each other, and Table 2 shows that there is almost no gap between the values of a _T and L _T when the real interest rate at time T is in the range from −10% to 15%. On the other hand, in case κ _r is 0.1, panels d and f suggest that L _T underestimates a _T when the real interest rate at time T becomes very high or very low. Table 2 shows additionally that when the real interest rate at time T is in the range of −10% and 15%, the gap is below 0.5 in absolute value. This applies to around 95% of the cases.

We denote the ratio X _T/(Π _TL _T) by C _T, that is, C _T: = X _T/(Π_TL _T), and define C _T as the replacement ratio at retirement time T. We relate X ₀ to the benchmark L _T in the following way such that $X_0=\phi{\rm {\opf E}}[M_T L_T]$. ϕ is the initial funding level. When the value of ϕ is between 0 and 1, the starting position is underfunded.

4.1 CRRA risk preference with benchmark

Theorem 1. In case of CRRA risk preference, the optimal terminal real wealth is:

(26)$$\displaystyle{{X_t^\ast } \over {\Pi _t}} = X_0M_t^{-{ 1\over \gamma } } \displaystyle{{F( {t, \;T, \;1-{1\over \gamma} } \;) } \over {F( {0, \;T, \;1-{1\over \gamma} }\; ) }}, \;$$

where

(27)$$F( t, \;T, \;\gamma ) = {\rm {\opf E}}_t\left[{{\left({\displaystyle{{M_TL_T} \over {M_t}}} \right)}^\gamma } \right] = e^{\gamma c_1( t, T) + {1\over 2}\gamma ^2c_2( t, T) }, \;$$

(28)$$\eqalign{c_1( t, \;T) = & -\displaystyle{1 \over 2}\omega {\rm ^{\prime}}\rho \omega ( T-t) + m-( {n-ne^{\kappa_r( t-T) } + ( T-t) -B( t, \;T) } ) \bar{r} \cr & \quad -( {B( t, \;T) + ne^{\kappa_r( t-T) }} ) r_t, \;\cr c_2( t, \;T) = & \sigma _r^2 \eta _2( t, \;T) + \omega {\rm ^{\prime}}\rho \omega ( T-t) + ( n\sigma _r) ^2\eta _4( t, \;T) \cr & \quad -2\sigma _r( \omega _S\rho _{Sr} + \omega _r + \omega _\pi \rho _{\pi r}) ( \eta _1( t, \;T) + n\eta _3( t, \;T) ) + 2n\sigma _r^2 \eta _5( t, \;T) , \;} $$

and

(29)$$\eqalign{\eta _1( t, \;T) = & \displaystyle{1 \over {\kappa _r}}( T-t) -\displaystyle{1 \over {\kappa _r}}B( t, \;T) , \;\cr \eta _2( t, \;T) = & \displaystyle{1 \over {2\kappa _r^3 }}\{ 2\kappa _r[ ( T-t) -B( t, \;T) ] -\kappa _r^2 B^2( t, \;T) \} , \;\cr \eta _3( t, \;T) = & B( t, \;T) , \;\quad \eta _4( t, \;T) = B( t, \;T) -\displaystyle{{\kappa _r} \over 2}B^2( t, \;T) , \;\cr \eta _5( t, \;T) = & \displaystyle{1 \over 2}B^2( t, \;T) .} $$

Proof. See Appendix B. □

Assume that at time t ($0{\kern 1pt} {\kern 1pt} {\rm \leqslant }{\kern 1pt} {\kern 1pt} t{\kern 1pt} {\kern 1pt} {\rm \leqslant }{\kern 1pt} {\kern 1pt} T$) the plan member invests an optimal proportion of wealth in stock, a bond with maturity T ₁, and a bond with maturity T ₂. The remaining wealth is invested in cash. The vector of optimal proportional wealth allocation to the stock and two bonds is denoted by $\theta _t^\ast{ = } ( {\theta_{S, t}^\ast , \;\;\theta_{1, t}^\ast , \;\;\theta_{2, t}^\ast } ) ^{\prime}$.

Theorem 2. In case of CRRA risk preference, the vector of optimal proportional wealth allocations $\theta _t^\ast$ can be derived as follows:

(30)$$\theta _t^\ast{ = } \displaystyle{1 \over \gamma }\Omega ^{{-}1}\Lambda + \left({1-\displaystyle{1 \over \gamma }} \right)\Omega ^{{-}1}\sigma \rho ( {\xi_S, \;\xi_r-( B( {t, \;T} ) + ne^{\kappa_r( {t-T} ) }) \sigma_r, \;\xi_\pi } ) ^{\prime}, \;$$

where Ω = σρσ ^′, Λ = σλ.

If the portfolio consists of the stock, a nominal bond with maturity T ₁, a nominal bond with maturity T ₂ and cash:

$$\sigma = \left({\matrix{ {\sigma_S} & 0 & 0 \cr 0 & {-B( {0, \;T_1} ) \sigma_r} & {-D( {0, \;T_1} ) \sigma_\pi } \cr 0 & {-B( {0, \;T_2} ) \sigma_r} & {-D( {0, \;T_2} ) \sigma_\pi } \cr } } \right); \;$$

if the portfolio consists of the stock, a nominal bond with maturity T ₁, an indexed bond with maturity T ₂ and cash:

$$\sigma = \left({\matrix{ {\sigma_S} & 0 & 0 \cr 0 & {-B( {0, \;T_1} ) \sigma_r} & {-D( {0, \;T_1} ) \sigma_\pi } \cr {\xi_S} & {\xi_r-B( {0, \;T_2} ) \sigma_r} & {\xi_\pi } \cr } } \right).$$

Proof. See Appendix C. □

Equation (30) expresses the optimal portfolio as the sum of two portfolios. The first portfolio Ω⁻¹Λ is the nominal mean-variance tangency portfolio. The amount is inversely related to γ. The second portfolio with weight 1 − 1/γ has the largest correlation with the market price for the benchmark in nominal terms. The proportion invested in stock is given by the expression ξ _S/σ _S − ω _S/(σ _Sγ). As in Brennan and Xia (Reference Brennan and Xia2002), ω _s takes negative value, then the proportion invested in stock decreases when the value of γ increases.

The optimal terminal wealth without benchmark is a special case of that with benchmark, which can be achieved by equating m and n to 0. The optimal proportional wealth allocation under CRRA risk preference without a benchmark is shown in Brennan and Xia (Reference Brennan and Xia2002) as follows:

(31)$$\theta _t^\ast{ = } \displaystyle{1 \over \gamma }\Omega ^{{-}1}\Lambda + \left({1-\displaystyle{1 \over \gamma }} \right)\Omega ^{{-}1}\sigma \rho ( {\xi_S, \;\xi_r-B( {t, \;T} ) ) \sigma_r, \;\xi_\pi } ) ^{\prime}.$$

The difference between allocation equations (31) and (30) is the term $ne^{\kappa _r( {t-T} ) }$, which is originated from the element c ₁(t, T) of the benchmark L _T.

4.2 SAHARA risk preference with benchmark and lower bound

To assure that the final replacement ratio does not fall below a predetermined floor, a lower bound can be imposed on the level of the replacement ratio as long as the market price for the lower bound is smaller than the initial wealth. Denote the lower bound by KL _T. K is a constant and its value is between 0 and ϕ. In general, the objective function can be stated as follows:

(32)$$\matrix{ {\mathop {\max }\limits_{X_T} } \hfill & {{\rm {\opf E}}\left[{U\left({\displaystyle{{X_T} \over {\Pi_TL_T}}} \right)} \right], \;} \hfill \cr {{\rm s}.{\rm t}.{\rm \;}} \hfill & {{\rm {\opf E}}\left[{M_T\displaystyle{{X_T} \over {\Pi_T}}} \right] = X_0, \;} \hfill \cr {} \hfill & {\qquad \;{\kern 1pt} \displaystyle{{X_T} \over {\Pi _T}}{\kern 1pt} {\kern 1pt} {\rm \geqslant }{\kern 1pt} {\kern 1pt} KL_T.} \hfill \cr } $$

The paper by Grossman and Zhou (Reference Grossman and Zhou1996) showed that the optimal real terminal wealth $X_T^{{\ast}{\ast} } /\Pi _T$ can be expressed as:

(33)$$\displaystyle{{X_T^{{\ast}{\ast} } } \over {\Pi _T}} = \max \{ {I( {vM_TL_T} ) L_T, \;KL_T} \} = [ {I( {vM_TL_T} ) L_T-KL_T} ] ^ + { + } KL_T, \;$$

where I(vM _TL _T)L _T is the expression for the optimal real terminal wealth when no lower bound is imposed on the real terminal wealth, though the value of v in equation (33) needs to be recalculated. Denote the optimal real wealth in case of no lower bound by $X_T^\ast{/}\Pi _T$.

Lemma 3. In case of SAHARA risk preference with no lower bound on the terminal real wealth, the optimal terminal real wealth is

(34)$$\displaystyle{{X_T^\ast } \over {\Pi _T}} = \displaystyle{1 \over 2}( ( {vM_TL_T} ) ^{-( 1/\alpha ) }-\beta ^2( {vM_TL_T} ) ^{1/\alpha }) L_T + w_0L_T, \;$$

where

(35)$$v = \displaystyle{{\beta ^{-\alpha }e^{-\alpha\; {\rm arcsinh( }( X_0/F( {0, T, 1} ) -w_0) /( \beta e^{ c_2( {0, T} ) /(2\alpha ^2) })}} \over {e^{c_1( {0, T} ) + c_2( {0, T} ) }}}.$$

Proof. A detailed proof is presented in Appendix D. □

As mentioned in Section 3, the level of wealth is always positive under CRRA risk preference, which is not the case under SAHARA risk preference. Hence, the optimal real wealth at retirement derived in Theorem 1 is always positive, while that derived in Lemma 3 can be negative.

Theorem 4. In case of SAHARA risk preference with a lower bound KL _T (0 < K < ϕ) on the terminal real wealth:

1. (i) The optimal real wealth at time t ( $0{\kern 1pt} {\kern 1pt} {\rm \leqslant }{\kern 1pt} {\kern 1pt} t{\kern 1pt} {\kern 1pt} {\rm \leqslant }{\kern 1pt} {\kern 1pt} T$) is

   (36)$$\eqalign{\displaystyle{{X_t^{{\ast}{\ast} } } \over {\Pi _t}} & = L_t e^{{c_2( t, T) /( 2\alpha ^2)}}\displaystyle{1 \over 2}( vM_te^{c_1( t, T) + c_2( t, T) })^{-{1\over \alpha }}\Phi\left(d(t, \;T, \;1-\displaystyle{1 \over \alpha }) \right)\cr& \quad - L_t e^{{c_2( t, T) /( 2\alpha ^2)}}\displaystyle{1 \over 2}\beta^2( vM_te^{c_1( t, T) + c_2( t, T) })^{1\over \alpha }\Phi\left(d(t, \;T, \;1+\displaystyle{1 \over \alpha }) \right) \cr & \quad + ( w_0-K) L_t\Phi ( d( t, \;T, \;1) ) + KL_t,} \; $$
   where Φ is the cumulative standard normal distribution function, and
   (37)$$\eqalign{L_t & = F( {t, \;T, \;1} ) = e^{c_1( {t, T} ) + {1\over2} c_2( {t, T} ) }, \;\cr d( {t, \;T, \;\tilde{\alpha }} ) & = \displaystyle{{-\ln \left({{\left({K-w_0 + \sqrt {{( {K-w_0} ) }^2 + \beta^2} } \right)}^\alpha vM_t} \right)-c_1( {t, \;T} ) -\tilde{\alpha }c_2( {t, \;T} ) } \over {\sqrt {c_2( {t, \;T} ) } }}.} $$

v is solved by imposing the condition: $X_0^{{\ast}{\ast} } = X_0$.

1. (ii) The vector of optimal proportional wealth allocations is

   (38)$$\theta _t^{{\ast}{\ast} } = \displaystyle{1 \over {\alpha _t}}\Omega ^{{-}1}\Lambda + \left({1-\displaystyle{1 \over {\alpha_t}}} \right)\Omega ^{{-}1}\sigma \rho ( {\xi_S, \;\xi_r-( B( {t, \;T} ) + ne^{\kappa_r( {t-T} ) }) \sigma_r, \;\xi_\pi } ) ^{\prime}, \;$$

where $\alpha _t = \alpha {C_t^{\ast\ast} } /\sqrt {{( {\beta e^{c_2( {t, T} )/(2\alpha^2) }} ) }^2\Phi ( {d( {t, \;T, \;1-{1\over\alpha} } ) }) \Phi ( {d( {t, \;T, \;1+{1\over\alpha} } ) })+ {( C_t^{{\ast}{\ast} } -( {w_0-K} ) \Phi ( {d( {t, \;T, \;1} ) } ) -K) }^2 }$ and $C_t^{{\ast}{\ast} } = X_t^{{\ast}{\ast} } /(\Pi _tL_t)$.

Proof. See Appendix E. □

The probabilities and the lower bound parameter K in the expression for α _t force the generated terminal wealth not to fall below the specified lower bound. The function c ₂(t, T) is a time-dependent positive constant and its value decreases with time. Hence, the term $\beta e^{c_2( {t, T} )/(2\alpha ^2) }$ decreases with time and becomes β at time T. It is interesting to point out that given the parameters and time t, the value of $C_t^{{\ast}{\ast} }$ and α _t are determined by vM _t. Once the value of $C_t^{{\ast}{\ast} }$ is known, and so is the value of α _t.

5.1 CRRA risk preference with vs. without benchmark

In this subsection, we present the probability density distribution of the optimal replacement ratio at retirement time T under CRRA risk preference with or without the benchmark. We consider the following four cases:

$$\eqalign{& \kappa _r = 0.6, \;\;\phi = 60\% ; \;\quad \kappa _r = 0.6, \;\;\phi = 80\% ; \;\cr & \kappa _r = 0.1, \;\;\phi = 60\% ; \;\quad \kappa _r = 0.1, \;\;\phi = 80\% .} $$

Figure 3 displays the corresponding outcomes in panels a, b, c, and d, respectively. The solid line indicates the distribution generated with the benchmark and the dashed line that without the benchmark.

Figure 3. Distribution of the optimal replacement ratio at retirement under CRRA risk preference.

Panels a and b show that under the CRRA risk preference with γ being 5, when κ _r is 0.6, the benchmark merely affects the probability density distribution of the optimal replacement ratio at retirement, whenever the initial funding level is 60% or 80%. Panels c and d show that when κ _r decreases to 0.1, the benchmark has a clear effect on the probability density distribution: the left tail shifts to the right and the right tail to the left, such that the distribution is more centred around the threshold. Table 4 provides the tail probabilities of the optimal replacement ratio at retirement time T when κ _r is 0.1, that is, ${\rm P}( {C_T^\ast {\kern 1pt} {\kern 1pt} {\rm \geqslant }{\kern 1pt} {\kern 1pt} \chi } )$, for different values of χ.

Table 4. Tail probabilities under CRRA risk preference

With the benchmark, when the initial funding level is 60%, the probability of the optimal replacement ratio at retirement being not below 90% improves by around 3%, and that being not below 80% around 6%. When the initial funding level is increased to 80%, these probabilities increase to around 9%. In addition to that, the probability of the optimal replacement ratio at retirement being not below 100% improves by around 8%, which is not the case when the initial funding level is 60%. On the other hand, optimization with the benchmark results in a smaller probability of the optimal replacement ratio at retirement being not below 200%, whenever the initial funding level is 60% or 80%. These results imply that CRRA utility function defined over the ratio of wealth to a benchmark could affect the probability density distribution of the optimal replacement ratio at retirement in such a way that it is more centred around the benchmark, though this effect is sensitive to the value of κ _r, at least for γ is 5.

5.2 SAHARA risk preference vs. CRRA risk preference

In this subsection, we present the investment strategies under CRRA risk preference with the benchmark, and that under SAHARA risk preference with the benchmark as well as a lower bound on the replacement ratio at retirement time T. We also present the generated distributions of the optimal replacement ratio at retirement time T under these risk preferences. The lower bound parameter K takes one of the two values: either 10% or 50%. For SAHARA risk preference, we start with α = 5, β = 1, then decrease α to 2 and β to 0.1, to ensure the average of plausible values of α _t is around 5. The parameter κ _r is 0.6 from now on.

5.2.1 Investment strategies

Under SAHARA risk preference, the term α _t plays an important role in determining investment strategies. Figure 4 plots the distribution of α _t as a function of the replacement ratio in two different points of the investment period. The distributions displayed in panels a and c are generated with a lower bound of 10%, that is, K = 10%, and that in panels b and d with a lower bound of 50%, that is, K = 50%. The outcomes for α being 5 and β being 1 are coloured in green, and that for α being 2 and β being 0.1 are coloured in purple.

Figure 4. Distribution of α _t as a function of replacement ratio.

Figure 4 shows that, when the replacement ratio falls below the threshold, the value of α _t decreases, which indicates the plan member's tendency of ‘risk-taking for resurrection’; when the replacement ratio approaches the lower bound, the value of α _t increases, which implies that the plan member becomes more risk averse at these positions and is less willing to invest in risky assets. The value of α _t goes to infinity when the replacement ratio is on the lower bound, which prevents the optimal replacement ratio from falling below the lower bound. From panels c and d we see that when the value of β decreases, a peak is formed around the threshold, which indicates the plan member's tendency of ‘locking-in the desired target’. Furthermore, α _t is time-dependent, as seen in equation (38). This time effect is obvious around the turning points, that is, around the threshold and around the lower bound, such that the value of α _t is lower around the lower bound and higher around the threshold when the time horizon gets short, particularly in panels c and d where α is 2 and β is 0.1. This implies that when the time approaches the retirement, plan members tend to be more risk averse around the reference level, while accept more risk if the level of the obtained wealth is far from the reference level, though not on the lower bound. When the level of the lower bound is increased from 10% to 50%, we observe on panels b and d that particularly the distribution of α _t to the left of the threshold shifts up. Compared to CRRA risk preference, SAHARA risk preference is more flexible in adapting the level of risk aversion to the level of the replacement ratio.

Figure 5 plots the optimal proportional wealth allocation to the stock, a 5-year nominal bond, a 5-year indexed bond, and cash as a function of the investment horizon under the CRRA risk preference with γ being 5. Figure 6 plots the optimal proportional wealth allocation to these assets at time t = 10 as a function of replacement ratio under the SAHARA risk preferences with a lower bound of 10%, and Figure 7 plots that with a lower bound of 50%. Tables 5 and 6 report the optimal proportional wealth allocation for the following replacement ratios: 50%, 100%, and 150%, when the lower bound is imposed at 10% and 50%, respectively.

Figure 5. Optimal proportional wealth allocations under CRRA risk preference.

Figure 6. Proportional wealth allocation under SAHARA risk preference with K = 10%.

Figure 7. Proportional wealth allocation under SAHARA risk preference with K = 50%.

Table 5. Optimal proportional wealth allocation with K = 10%

Table 6. Optimal proportional wealth allocation with K = 50%

Under CRRA risk preference, the optimal proportional allocation to assets does not vary with the level of the replacement ratio. Additionally, the proportion invested in the stock also does not vary with the investment horizon. The allocation pattern indicates that the plan member takes a long position in the stock and the bonds, while a short position in cash. This is consistent with one of the findings in Brennan and Xia (Reference Brennan and Xia2002).Footnote ¹ Regarding the short position in cash, Ma (Reference Ma2011) pointed out that the cash account is not a safe asset in real terms and the returns on cash are lower than the returns on bonds and stocks, and that investors with a high level of risk aversion would keep a short position in cash to hedge against the inflation risk. A 5-year indexed bond turns out to be the most dominant asset in the pension portfolio. When time t gets close to the retirement, the plan member under the CRRA risk preference decreases the weight on the bonds and shorts less in cash, as shown in Figure 5.

Figure 6 highlights how the proportional wealth allocation varies with the level of the replacement ratio under the SAHARA risk preferences. When α is 5 and β is 1, this difference is obvious when the replacement ratio is below 100%. Table 5 shows that when the replacement ratio falls from 100% to 50%, the proportional wealth allocated to cash decreases significantly from −0.95 to −1.70, while that allocated to the stock, the 5-year nominal bond and the 5-year indexed bond increase slightly from 0.26 to 0.48, from 0.44 to 0.80, and from 1.25 to 1.42, respectively. This applies to a greater extent the smaller the value is of the replacement ratio, until the replacement ratio approaches the lower bound. In other words, in comparison with CRRA risk preference, a plan member under SAHARA risk preference realizes a more aggressive investment policy to overcome an underfunded position. When the replacement ratio rises from the threshold to 150%, the proportional wealth allocation to the stock and bonds all decrease slightly. When α and β decrease to 2 and 0.1 respectively, the plan member shorts much less at and around the threshold. When the replacement ratio rises above the threshold, the plan member gradually increases the proportion invested in the stock and the bonds. This occurs to a lesser extent than in the case where the replacement ratio falls below the threshold.

When the lower bound is set at 50%, Figure 6 and Table 6 show that particularly the optimal proportional wealth allocation corresponding to the level of wealth which is between the previous lower bound and the threshold will be affected, such that plan members tend to be more risk averse, short less in cash, and invest less in the bonds and the stock at these positions.

5.2.2 Distribution of retirement outcomes

For comparison, we consider subsequently the following four initial funding levels: ϕ = 60%, 80%, 100%, and 120%. Figure 8 plots the corresponding probability density distribution of the optimal replacement ratio at retirement time T in panels a, b, c, and d, respectively, under the CRRA risk preference with the benchmark, as well as under the SAHARA risk preferences with the benchmark and a lower bound of 10% on the replacement ratio at retirement time T. Table 7 provides the tail probabilities of the optimal replacement ratio at retirement time T when ϕ = 60%, 80%, and Table 8 provides that when ϕ = 100%, 120%.

Figure 8. Distribution of the optimal replacement ratio at retirement.

Table 7. Tail probabilities when K = 10%, ϕ = 60%, 80%

Table 8. Tail probabilities when K = 10%, ϕ = 100%, 120%

Figure 8 shows that under the SAHARA risk preference with α being 2 and β being 0.1, a peak is formed around the threshold. When ϕ is 60% or 80%, Table 7 shows that compared to CRRA risk preference, the probability of the optimal replacement ratio at retirement time T being not below 100% decreases greatly while that not below 90% or 80% increases. The lower the initial funding level, the greater the extent. This results in a probability distribution more centred around the threshold. In case ϕ is 100% or 120%, Table 8 shows that this risk preference leads to slightly higher tail probabilities compared to the other two types of risk preference.

Under the SAHARA risk preference with α being 5 and β being 1, Table 7 shows that compared to CRRA risk preference, the probability of the optimal replacement ratio at retirement time T being not below 100% or 90% or 80% increases. The lower the initial funding level, the more obvious the improvement. When the initial funding level is 60%, this increase is substantial and its value is around 20%, 15%, and 10%, respectively. In other words, the probability of overcoming underfunding could be improved by being less risk averse when the replacement ratio falls below the threshold. Tables 7 and 8 show additionally that this advantage gradually disappears when the initial funding level is increased from 60% to 120%.

To examine the effect of lower bound on the probability distribution of the optimal replacement ratio at retirement, we increase the level of the lower bound from 10% to 50% for the SAHARA risk preferences. We only consider the case where the initial funding level is 60%. The results are displayed in Figure 9. The solid line indicates the distribution when K = 10%, and the dashed line when K = 50%.

Figure 9. Impact of lower bound on distribution of the optimal replacement ratio at retirement.

Figure 9 shows that a higher level of the lower bound shifts the mode of the distribution to the left. It also leads to a higher probability of the optimal replacement ratio at retirement ending up on the lower bound. In case the level of the lower bound is equal to the initial funding level, that is, K = ϕ, plan members will invest in the lower bound for the entire investment period.