Robust time-consistent strategy for the defined contribution pension plan with a minimum guarantee under ambiguity

Abstract

This paper investigates the robust time-consistent investment strategy for the defined contribution pension plan with a minimum guarantee in a stochastic interest rate and stochastic inflation environment under the mean-variance criterion. To protect the purchasing power of pension members after retirement, the amount of the minimum guarantee is assumed to be adjusted with the stochastic interest rate and inflation level in the model. The fund manager is considered as ambiguity averse, and her objective is to prevent herself from the worst-case scenario and ensure that the fund wealth exceeds the minimum guarantee at retirement. The ambiguity-averse manager is allowed to invest in a financial market consisting of four assets: a risk-free asset, a nominal zero-coupon bond, an inflation-indexed bond and a stock. The nominal zero-coupon bond and the inflation-indexed bond are designed to hedge against interest rate risk and inflation risk. Following the robust optimal control theory, both the robust time-consistent strategy and the equilibrium value function are derived by an introduction of an auxiliary problem. Some degenerate cases are also provided. Finally, a numerical simulation demonstrates the results obtained.

Appendix A. Proof of Theorem 3.6

According to Eqs. (38) and (37), the HJB equation can be written as

$$\begin{aligned} \begin{aligned} \underset{\hat{\pi }\in \Pi }{\mathop {\sup }}\,\underset{{\mathbb {Q}}\in Q}{\mathop {\inf }}\,&\left\{ {{V}_{t}}+y\left[ {{{\hat{\pi }}}^{\top }}\Gamma \left( \Lambda +\varphi \right) +r \right] {{V}_{y}}+p\left( k+\sigma _{P}^{\top }\varphi \right) {{V}_{p}}+\left( a-br+\sigma _{R}^{\top }\varphi \right) {{V}_{r}} \right. \\&+\frac{1}{2}{{y}^{2}}{{{\hat{\pi }}}^{\top }}\Sigma \hat{\pi }\left( {{V}_{yy}}-\gamma g_{y}^{2} \right) +\frac{1}{2}{{p}^{2}}{{\left\| {{\sigma }_{P}} \right\| }^{2}}\left( {{V}_{pp}}-\gamma g_{p}^{2} \right) +\frac{1}{2}{{\left\| {{\sigma }_{R}} \right\| }^{2}}\left( {{V}_{rr}}-\gamma g_{r}^{2} \right) \\&+yp{{{\hat{\pi }}}^{\top }}\Gamma {{\sigma }_{P}}\left( {{V}_{yp}}-\gamma {{g}_{y}}{{g}_{p}} \right) +y{{{\hat{\pi }}}^{\top }}\Gamma {{\sigma }_{R}}\left( {{V}_{yr}}-\gamma {{g}_{y}}{{g}_{r}} \right) \\&\text { }+p\sigma _{P}^{\top }{{\sigma }_{R}}\left( {{V}_{pr}}-\gamma {{g}_{p}}{{g}_{r}} \right) +\frac{1}{2\xi }\left. {{\left\| \varphi \right\| }^{2}} \right\} =0. \end{aligned} \end{aligned}$$

(A1)

Based on the first-order optimality condition, the worst-case drift function ${{\varphi }^{*}}(t)$ takes the form as

$$\begin{aligned} {{\varphi }^{*}}(t)=-\xi \left( y{{\Gamma }^{\top }}\hat{\pi }{{V}_{y}}+p{{\sigma }_{P}}{{V}_{p}}+{{\sigma }_{R}}{{V}_{r}} \right) . \end{aligned}$$

(A2)

Plugging Eq. (A2) into Eq. (A1), we have

$$\begin{aligned} \begin{aligned} \underset{\hat{\pi }\in \Pi }{\mathop {\sup }}\,&\left\{ {{V}_{t}} +y( {{{\hat{\pi }}}^{\top }}\Gamma \Lambda +r){{V}_{y}}+pk{{V}_{p}}+\left( a-br \right) {{V}_{r}}+\frac{1}{2}{{{\hat{\pi }}}^{\top }}\Sigma \hat{\pi }{{y}^{2}}{{\chi }_{yy}} \right. \\&+\frac{1}{2}{{p}^{2}}{{\left\| {{\sigma }_{P}} \right\| }^{2}}{{\chi }_{pp}}+\frac{1}{2}{{\left\| {{\sigma }_{R}} \right\| }^{2}}{{\chi }_{rr}}+{{{\hat{\pi }}}^{\top }}\Gamma {{\sigma }_{P}}yp{{\chi }_{yp}}+{{{\hat{\pi }}}^{\top }}\Gamma {{\sigma }_{R}}y{{\chi }_{yr}} \\&\left. +\sigma _{P}^{\top }{{\sigma }_{R}}p{{\chi }_{pr}} \right\} =0, \\ \end{aligned} \end{aligned}$$

(A3)

where

$$\begin{aligned}&{{\chi }_{yy}}={{V}_{yy}}-\xi V_{y}^{2}-\gamma g_{y}^{2}, \quad {{\chi }_{yp}}={{V}_{yp}}-\xi {{V}_{y}}{{V}_{p}}-\gamma {{g}_{y}}{{g}_{p}}, \quad {{\chi }_{yr}}={{V}_{yr}}-\xi {{V}_{y}}{{V}_{r}}-\gamma {{g}_{y}}{{g}_{r}}, \\&{{\chi }_{pp}}={{V}_{pp}}-\xi V_{p}^{2}-\gamma g_{p}^{2}, \quad {{\chi }_{rr}}={{V}_{rr}}-\xi V_{r}^{2}-\gamma g_{r}^{2}, \quad {{\chi }_{pr}}={{V}_{pr}}-\xi {{V}_{p}}{{V}_{r}}-\gamma {{g}_{p}}{{g}_{r}}. \end{aligned}$$

In the same manner, we could obtain the optimal control ${{\hat{\pi }}^{*}}(t)$ by using the first-order condition for $\hat{\pi }(t)$ in Eq. (A3) as follows:

$$\begin{aligned} {{\hat{\pi }}^{*}}(t)=-\frac{1}{y{{\chi }_{yy}}}{{\Sigma }^{-1}}\Gamma \left( \Lambda {{V}_{y}}+p{{\chi }_{yp}}{{\sigma }_{P}}+{{\chi }_{yr}}{{\sigma }_{R}} \right) . \end{aligned}$$

(A4)

Substituting Eq. (A4) into Eq. (A3) implies

$$\begin{aligned} \begin{aligned} \underset{\hat{\pi }\in \Pi }{\mathop {\sup }}\,&\left\{ {{V}_{t}}+yr{{V}_{y}}+pk{{V}_{p}}+\left( a-br \right) {{V}_{r}}-{{\Lambda }^{\top }}\left( \frac{V_{y}^{2}}{2{{\chi }_{yy}}}\Lambda +\frac{p{{V}_{y}}{{\chi }_{yp}}}{{{\chi }_{yy}}}{{\sigma }_{P}}+\frac{{{V}_{y}}{{\chi }_{yr}}}{{{\chi }_{yy}}}{{\sigma }_{R}} \right) \right. \\&\text { }+\frac{1}{2}{{p}^{2}}{{\left\| {{\sigma }_{P}} \right\| }^{2}}\left( {{\chi }_{pp}}-\frac{\chi _{yp}^{2}}{{{\chi }_{yy}}} \right) +\frac{1}{2}{{\left\| {{\sigma }_{R}} \right\| }^{2}}\left( {{\chi }_{rr}}-\frac{\chi _{yr}^{2}}{{{\chi }_{yy}}} \right) \text { } \\&\text { }\left. +p\sigma _{P}^{\top }{{\sigma }_{R}}\left( {{\chi }_{pr}}-\frac{{{\chi }_{yp}}{{\chi }_{yr}}}{{{\chi }_{yy}}} \right) \right\} =0. \\ \end{aligned} \end{aligned}$$

(A5)

Next, the objective is to find the specific form of the value function, which is conjectured to be of the following form:

$$\begin{aligned} V\left( t,y,p,r \right) ={{n}_{1}}\left( t,r \right) \frac{y}{p}+{{n}_{2}}\left( t,r \right) , \end{aligned}$$

(A6)

$$\begin{aligned} g\left( t,y,p,r \right) ={{n}_{1}}\left( t,r \right) \frac{y}{p}+{{\bar{n}}_{2}}\left( t,r \right) , \end{aligned}$$

(A7)

with boundary conditions ${{n}_{1}}\left( T,r \right) =1$ and ${{n}_{2}}\left( T,r \right) ={{\bar{n}}_{2}}\left( T,r \right) =0$.

Remark 4

Based on the evidence gathered, we have sufficient reasons to suspect that the coefficient of the term $\frac{y}{p}$ is identical in forms of two coupling functions $V\left( t,y,p,r \right)$ and $g\left( t,y,p,r \right)$. We have also assumed the situation where they are different and derived the solutions, the results confirmed that they are exactly identical. To simplify the derivation, we propose the conjecture for the value functions with the forms of Eqs. (A6) and (A7).

Substituting the derivatives of Eqs. (A6) and (A7) into Eq. (A5) and differentiating on $\hat{\pi }(t)$, we get

$$\begin{aligned} {{{\hat{\pi }}}^{*}}(t)={{\left( {{\Gamma }^{\top }} \right) }^{-1}}&\left[ {{\sigma }_{P}} +\frac{p}{y{{n}_{1}}\left( \gamma +\xi \right) }\left( \Lambda -{{\sigma }_{P}} \right) +\frac{p{{n}_{1r}}}{yn_{1}^{2}\left( \gamma +\xi \right) }{{\sigma }_{R}} \right. \nonumber \\&\left. \text { }-\left( \frac{{{n}_{1r}}}{{{n}_{1}}}+\frac{p\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) }{y{{n}_{1}}\left( \gamma +\xi \right) } \right) {{\sigma }_{R}} \right] . \end{aligned}$$

(A8)

Then, inserting Eq. (A8) into Eqs. (37) and (38), and separating terms with respect to $\frac{y}{p}$, we can derive that

$$\begin{aligned}&\begin{aligned} {{n}_{1t}}&+{{n}_{1}}\left( r-k+{{\Lambda }^{\top }}{{\sigma }_{P}} \right) +{{n}_{1r}}\left[ a-br-{{\left( \Lambda -{{\sigma }_{P}} \right) }^{\top }}{{\sigma }_{R}} \right] \\&+\left( \frac{{{n}_{1rr}}}{2}-\frac{n_{1r}^{2}}{{{n}_{1}}} \right) {{\left\| {{\sigma }_{R}} \right\| }^{2}}=0, \end{aligned} \end{aligned}$$

(A9)

$$\begin{aligned}&\begin{aligned} {{n}_{2t}}&+{{n}_{2r}}\left( a-br \right) +\frac{\gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}}}{\gamma +\xi }{{\left( \Lambda -{{\sigma }_{P}} \right) }^{\top }}{{\sigma }_{R}} \\&+\frac{1}{2\left( \gamma +\xi \right) }\left[ {{\left( \frac{{{n}_{1r}}}{{{n}_{1}}}-{{\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) }^{2}} \right) }^{2}}-\frac{\gamma \bar{n}_{2r}^{2}+\xi n_{2r}^{2}}{n_{1}^{2}} \right] {{\left\| {{\sigma }_{R}} \right\| }^{2}} \\&+\frac{{{n}_{1r}}}{{{n}_{1}}\left( \gamma +\xi \right) }\sigma _{P}^{\top }{{\sigma }_{R}}+\frac{1}{2\left( \gamma +\xi \right) }{{\left\| \Lambda - {{\sigma }_{P}} \right\| }^{2}}=0, \end{aligned} \end{aligned}$$

(A10)

$$\begin{aligned}&\begin{aligned} {{{\bar{n}}}_{2t}}&+{{{\bar{n}}}_{2r}}\left( a-br \right) +\frac{1}{2}\left( {{{\bar{n}}}_{2rr}}-{{n}_{2r}}{{{\bar{n}}}_{2r}} \right) {{\left\| {{\sigma }_{R}} \right\| }^{2}} +\frac{1}{\gamma +\xi }\left[ 1+\frac{\xi }{2\left( \gamma +\xi \right) } \right] {{\left\| \Lambda - {{\sigma }_{P}} \right\| }^{2}} \\&+\left[ \frac{2{{n}_{1r}}}{{{n}_{1}}}-\frac{\xi }{2}\left( {{{\bar{n}}}_{2r}}+{{n}_{2r}} \right) -\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) -\xi {{n}_{1}}{{n}_{1r}}+\frac{\xi \left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) }{\gamma +\xi } \right] \\&\times \frac{1}{\gamma +\xi }{{\left( \Lambda -{{\sigma }_{P}} \right) }^{\top }}{{\sigma }_{R}}+\left[ 2{{n}_{1r}}+\xi {{n}_{1}}\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) \left( {{{\bar{n}}}_{2r}}+{{n}_{2r}} \right) \right. \\&\left. -2{{n}_{1r}}\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) -\xi {{n}_{1r}}\left( {{{\bar{n}}}_{2r}}+{{n}_{2r}} \right) \right] \times \frac{1}{2{{n}_{1}}\left( \gamma +\xi \right) } {{\left\| {{\sigma }_{R}} \right\| }^{2}} \\&+\frac{1}{2{{\left( \gamma +\xi \right) }^{2}}}{{\left[ \frac{{{n}_{1r}}}{{{n}_{1}}}-\left( \gamma {{{\bar{n}}}_{2r}}+\xi {{n}_{2r}} \right) \right] }^{2}}{{\left\| {{\sigma }_{R}} \right\| }^{2}}=0. \end{aligned} \end{aligned}$$

(A11)

To solve Eqs. (A9)–(A11), we further assume that

$$\begin{aligned} {{n}_{1}}={{e}^{{{q}_{1}}(t)r+{{q}_{2}}(t)}}, ~{{n}_{2}}={{q}_{3}}(t)r+{{q}_{4}}(t), ~{{{\bar{n}}}_{2}}={{{\bar{q}}}_{3}}(t)r+{{{\bar{q}}}_{4}}(t), \end{aligned}$$

(A12)

where ${{q}_{1}}(T)={{q}_{2}}(T)={{q}_{3}}(T)={{q}_{4}}(T)={{\bar{q}}_{3}}(T)={{\bar{q}}_{4}}(T)=0$.

Inserting Eq. (A12) into Eqs. (A9)–(A11) and separating the variables, we have

$$\begin{aligned}&{{q}_{1t}}-\frac{1}{2}\sigma _{r}^{2}q_{1}^{2}-\left[ b-\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}} \right] {{q}_{1}}+{{\lambda }_{r}}{{\sigma }_{P1}}+1=0, \end{aligned}$$

(A13)

$$\begin{aligned}&{{q}_{2t}}+a{{q}_{1}}+k-\left( {{\lambda }_{S}}{{\sigma }_{P2}}+{{\lambda }_{P}}{{\sigma }_{P3}} \right) =0, \end{aligned}$$

(A14)

$$\begin{aligned}&\begin{aligned} {{q}_{3t}}&-\left[ b+\frac{\xi }{\gamma +\xi }\left( {{\sigma }_{r}}\left( {{\lambda }_{r}}+{{\sigma }_{P1}} \right) +\frac{1}{2}\sigma _{r}^{2} \right) \right] {{q}_{3}}-\frac{1}{2}\sigma _{r}^{2}\xi q_{3}^{2} \\&-\frac{\gamma }{\gamma +\xi }\left[ {{\sigma }_{r}}\left( {{\lambda }_{r}}+{{\sigma }_{P1}} \right) +\frac{1}{2}\sigma _{r}^{2} \right] {{{\bar{q}}}_{3}}-\frac{1}{2}\sigma _{r}^{2}\gamma \bar{q}_{3}^{2} \\&+\frac{1}{\gamma +\xi }\left[ \left( {{\sigma }_{P1}}+\frac{1}{2}{{\sigma }_{r}} \right) {{\sigma }_{r}}{{q}_{1}}+\frac{1}{2}{{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) }^{2}} \right] =0, \\ \end{aligned} \end{aligned}$$

(A15)

$$\begin{aligned}&\begin{aligned} {{{\bar{q}}}_{3t}}&+\left[ \frac{\gamma \xi \sigma _{r}^{2}{{q}_{1}}}{{{\left( \gamma +\xi \right) }^{2}}}-b+\frac{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}}}{\gamma +\xi }\left( \frac{{{\gamma }^{2}}}{\gamma +\xi }+\frac{\xi }{2} \right) -\frac{\sigma _{r}^{2}{{q}_{1}}}{\gamma +\xi }\left( \gamma +\frac{\xi }{2} \right) \right] {{{\bar{q}}}_{3}} \\&+\left[ \frac{{{\xi }^{2}}\sigma _{r}^{2}{{q}_{1}}}{{{\left( \gamma +\xi \right) }^{2}}}+\frac{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}}}{\gamma +\xi }\left( \frac{\gamma \xi }{\gamma +\xi }+\frac{\xi }{2} \right) -\frac{3\xi \sigma _{r}^{2}{{q}_{1}}}{2\left( \gamma +\xi \right) } \right] {{q}_{3}} \\&+\frac{\gamma \xi \sigma _{r}^{2}}{2\left( \gamma +\xi \right) }\left( 1-\frac{1}{\gamma +\xi } \right) \bar{q}_{3}^{2}-\frac{\gamma {{\xi }^{2}}\sigma _{r}^{2}}{{{\left( \gamma +\xi \right) }^{2}}}{{q}_{3}}{{{\bar{q}}}_{3}}+\frac{\gamma {{\xi }^{2}}\sigma _{r}^{2}}{2{{\left( \gamma +\xi \right) }^{2}}}q_{3}^{2} \\&-\frac{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}}{{q}_{1}}}{\gamma +\xi }\left( 2+\frac{\xi }{\gamma +\xi } \right) +\frac{\sigma _{r}^{2}q_{1}^{2}}{\gamma +\xi } \\&+\frac{2-\xi }{2\left( \gamma +\xi \right) }{{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) }^{2}}-\frac{\xi \sigma _{r}^{2}q_{1}^{2}}{2{{\left( \gamma +\xi \right) }^{2}}}=0, \\ \end{aligned} \end{aligned}$$

(A16)

$$\begin{aligned}&{{q}_{4t}}+a{{q}_{3}}+\frac{1}{2\left( \gamma +\xi \right) }\left[ {{\left( {{\lambda }_{S}}-{{\sigma }_{P2}} \right) }^{2}}+{{\left( {{\lambda }_{P}}-{{\sigma }_{P3}} \right) }^{2}} \right] =0, \end{aligned}$$

(A17)

$$\begin{aligned}&{{\bar{q}}_{4t}}+a{{\bar{q}}_{3}}+\frac{2-\xi }{2\left( \gamma +\xi \right) }\left[ {{\left( {{\lambda }_{S}}-{{\sigma }_{P2}} \right) }^{2}}+{{\left( {{\lambda }_{P}}-{{\sigma }_{P3}} \right) }^{2}} \right] =0. \end{aligned}$$

(A18)

Considering the boundary conditions, the solutions are given by

$$\begin{aligned}&{{q}_{1}}(t)=\frac{{{\eta }_{1}}{{\eta }_{2}}\left( 1-{{e}^{\sqrt{{{\Delta }_{{{q}_{1}}}}}\left( T-t \right) }} \right) }{{{\eta }_{1}}-{{\eta }_{2}}{{e}^{\sqrt{{{\Delta }_{{{q}_{1}}}}}\left( T-t \right) }}},\\&{{q}_{2}}(t)=-\int _{t}^{T}{a{{q}_{1}}(s)\textrm{d}s}+\left( k- {{\lambda }_{S}}{{\sigma }_{P2}}-{{\lambda }_{P}}{{\sigma }_{P3}} \right) \left( T-t \right) ,\\&{{q}_{4}}(t)=-\int _{t}^{T}{a{{q}_{3}}(s)\textrm{d}s}+\frac{1}{2\left( \gamma +\xi \right) }\left[ {{\left( {{\lambda }_{S}}-{{\sigma }_{P2}} \right) }^{2}}+{{\left( {{\lambda }_{P}}-{{\sigma }_{P3}} \right) }^{2}} \right] \left( T-t \right) ,\\&{{\bar{q}}_{4}}(t)=-\int _{t}^{T}{a{{{\bar{q}}}_{3}}(s)\textrm{d}s}+\frac{2-\xi }{2\left( \gamma +\xi \right) }\left[ {{\left( {{\lambda }_{S}}-{{\sigma }_{P2}} \right) }^{2}}+{{\left( {{\lambda }_{P}}-{{\sigma }_{P3}} \right) }^{2}} \right] \left( T-t \right) , \end{aligned}$$

where

$$\begin{aligned} {{\Delta }_{{{q}_{1}}}}=&{{\left[ \left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}}-b \right] }^{2}}+2\sigma _{r}^{2}\left( {{\lambda }_{r}}{{\sigma }_{P1}}+1 \right) >0, \\&{{\eta }_{1,2}}=\frac{\left( {{\lambda }_{r}}-{{\sigma }_{P1}} \right) {{\sigma }_{r}}-b\pm \sqrt{{{\Delta }_{{{q}_{1}}}}}}{\sigma _{r}^{2}}. \end{aligned}$$

Moreover, ${{q}_{3}}(t)$ and ${{\bar{q}}_{3}}(t)$ are commonly determined by Eqs. (A15) and (A16). According to the Theorem on page 65 and the Lemma on the Extension of Solutions on page 67 in Walter (1998), the solutions for ${{q}_{3}}(t)$ and ${{\bar{q}}_{3}}(t)$ are existent and unique. Substituting Eqs. (A6), (A7) and (A12) into Eqs. (A4) and (A2), Eqs. (40) and (41) can be obtained, sequentially. Therefore, the proof of Theorem 3.6 is accomplished. $\square$