Time-Consistent Investment Strategies for a DC Pension Member with Stochastic Interest Rate and Stochastic Income

Abstract

This paper studies two multi-period mean-variance investment problems for a DC pension member before and after retirement. At any time, the pension manager can invest in a risk-free asset and multi-risky assets. Before retirement, the manager tries to optimize the mean-variance utility of the wealth in the member’s pension account at retirement. At retirement, the pension account wealth (or part of it) is used to purchase a paid-up annuity. After retirement, the manager has to pay the guaranteed annuity, continues to invest, and aims to optimize the mean-variance utility of the terminal wealth at a fix future time, to satisfy the pension member’s heritage and life needs in the next stage. Interest rate risk and income risk are introduced. Applying the game theory and the extended Bellman equation, the time-consistent investment strategies and the efficient frontiers before and after retirement are obtained explicitly. Obtained results indicate that the stochastic interest rate and the stochastic income have essential effects on the investment strategies.

Appendix: The Proof of Theorem 1

Proof

We prove this theorem by mathematical induction. For $t=T-1$, by equations (4), (12) and (13), we have

$$\begin{aligned}&V_{T\!{-}\!1}\left( x_{T-1},r_{T{-}1},y_{T-1}\right) {=}\max _{\pi _{T{-}1}}\left\{ \mathbb {E}_{x_{T{-}1},r_{T-1},y_{T{-}1}}\left( V_{T}\left( X_{T},R_{T},Y_{T}\right) \right) -\omega \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( g^{2}_{T}\left( X_{T},R_{T},Y_{T}\right) \right) \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \left. +\omega \left[ \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( g_{T}\left( X_{T},R_{T},Y_{T}\right) \right) \right] ^{2}\right\} \nonumber \\&\quad =\max _{\pi _{T-1}}\left\{ \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( X_{T}\right) -\omega \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( X_{T}^{2}\right) + \omega \left[ \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( X_{T}\right) \right] ^{2}\right\} \nonumber \\&\quad =\max _{\pi _{T-1}}\left\{ x_{T-1}r_{T-1}+\mathbb {E}(P'_{T-1})\pi _{T-1}+c_{T-1}y_{T-1}-\omega \pi _{T-1}'Var (P_{T-1})\pi _{T-1}\right\} . \end{aligned}$$

(A1)

Because $\omega >0$ and $Var (P_{T-1})$ is positive definite by Assumption 2, the first-order (necessary and sufficient) condition about $\pi _{T-1}$ in (A1) gives the optimal strategy

$$\begin{aligned} \pi ^{*}_{T-1}=\frac{1}{2\omega }Var ^{-1}(P_{T-1})\mathbb {E}(P_{T-1}). \end{aligned}$$

(A2)

Substituting equation (A2) into equations (A1) and (13), respectively, we can obtain

$$\begin{aligned} V_{T-1}\left( x_{T-1},r_{T-1},y_{T-1}\right)= & {} r_{T-1}x_{T-1}+c_{T-1}y_{T-1}+\frac{1}{4\omega }\mathbb {E} (P'_{T-1})Var ^{-1}(P_{T-1})\mathbb {E}(P_{T-1}), \end{aligned}$$

(A3)

$$\begin{aligned} g_{T-1}(x_{T-1},r_{T-1},y_{T-1})= & {} \mathbb {E}_{x_{T-1},r_{T-1},y_{T-1}}\left( g_{T}\left( X_{T},R_{T},Y_{T}\right) \right) \nonumber \\= & {} \mathbb {E}\left( x_{T-1}r_{T-1}+P'_{T-1}\pi _{T-1}+c_{T-1}y_{T-1}\right) \nonumber \\= & {} r_{T-1}x_{T-1}+c_{T-1}y_{T-1}+\frac{1}{2\omega }\mathbb {E}(P'_{T-1})Var ^{-1}(P_{T-1})\mathbb {E}(P_{T-1}).\nonumber \\ \end{aligned}$$

(A4)

On the other hand, by equations (14)-(25), we have

$$\begin{aligned}&\psi _{T-1}=1,\text { }w_{T-1}=A_{T}\mathbb {E}\left( b_{T-1}^{2\psi _{T}}\right) +B_{T}^{2}Var \left( b_{T-1}^{\psi _{T}}\right) =0,\\&\xi _{T-1}=A_{T}\mathbb {E}\left( b_{T-1}^{2\psi _{T}}\right) \mathbb {E}\left( P_{T-1}P'_{T-1}\right) +B_{T}^{2}Var \left( b_{T-1}^{\psi _{T}}P_{T-1}\right) =Var \left( P_{T-1}\right) ,\\&A_{T-1}=w_{T-1}-w_{T-1}^{2}\mathbb {E}(P'_{T-1})\xi ^{-1}_{T-1}\mathbb {E}(P_{T-1})=0,\\&B_{T-1}=B_{T}\left[ \mathbb {E}\left( b_{T-1}^{\psi _{T}}\right) -w_{T-1}\mathbb {E}\left( b_{T-1}^{\psi _{T}}\right) \mathbb {E}(P'_{T-1})\xi ^{-1}_{T-1}\mathbb {E}(P_{T-1})\right] =1,\\&f_{T-1}=B_{T}^{2}\mathbb {E}^{2}\left( b_{T-1}^{\psi _{T}}\right) \mathbb {E}(P'_{T-1})\xi ^{-1}_{T-1}\mathbb {E}(P_{T-1})+f_{T}=\mathbb {E}(P'_{T-1})Var ^{-1}(P_{T-1})\mathbb {E}(P_{T-1}).\\&Q^{T-1}_{T-1}=w_{T-1}\mathbb {E}(P_{T-1})=0,\ \ U^{T-1}_{T-1}=A_{T-1}=0,\ \ \beta ^{T-1}_{T-1}=B_{T-1}=1,\ \ \alpha ^{T-1}_{T-1}=A_{T-1}=0.\nonumber \\ \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \pi ^{*}_{T-1}=\xi _{T-1}^{-1}\left( \frac{B_{T}\mathbb {E}\left( b_{T-1}^{\psi _{T}}\right) \mathbb {E}(P_{T-1})}{2\omega r_{T-1}^{\varphi _{T-1}\psi _{T}}}-w_{T-1}r_{T-1}x_{T-1}\mathbb {E}(P_{T-1})-Q^{T-1}_{T-1}c_{T-1}y_{T-1}\right) , \end{aligned}$$

(A5)

$$\begin{aligned} V_{T-1}(x_{T-1},r_{T-1},y_{T-1})&=-\omega A_{T-1}r_{T-1}^{2\psi _{T-1}}x_{T-1}^{2}+B_{T-1}r_{T-1}^{\psi _{T-1}}x_{T-1}+\frac{1}{4\omega }f_{T-1}+\beta ^{T-1}_{T-1}r_{T-1}^{\varphi _{T-1}\psi _{T}}c_{T-1}y_{T-1}\nonumber \\&\qquad -2\omega U^{T-1}_{T-1}r_{T-1}^{\psi _{T-1} +\varphi _{T-1}\psi _{T}}c_{T-1}y_{T-1}x_{T-1} -\omega \alpha ^{T-1}_{T-1}r_{T-1}^{2\varphi _{T-1}\psi _{T}}c^{2}_{T-1}y^{2}_{T-1}\nonumber \\&\qquad -2\omega \sum _{T-1\leqslant l< h\leqslant T-1}S^{l,h}_{t} r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1} +\left( \prod _{m=t}^{h}\varphi _{m}\right) \psi _{h+1}}c_{l}c_{h}y^{2}_{t} \end{aligned}$$

(A6)

and

$$\begin{aligned} g_{T-1}(x_{T-1},r_{T-1},y_{T-1})&=B_{T-1}r_{T-1}^{\psi _{T-1}}x_{T-1}+\frac{1}{2\omega }f_{T-1}\nonumber \\&\quad +\beta ^{T-1}_{T-1}r_{T-1}^{\varphi _{T-1}\psi _{T}}c_{T-1}y_{T-1}. \end{aligned}$$

(A7)

Equations (A5)-(A7) show that equations (26)-(28) hold for $t=T-1$.

Now, suppose that equations (26)-(28) hold for $t=T-1,T-2,\cdots ,t+1$. Then, we consider the case for $t=t$. Substituting equations (27) and (28), the expressions of $V_{t+1}$ and $g_{t+1}$, into the extended Bellman equation (12), we have

$$\begin{aligned}&V_{t}(x_{t},r_{t},y_{t})=\max _{\pi _{t}}\left\{ \mathbb {E}_{x_{t},r_{t},y_{t}}\left( V_{t+1}\left( X_{t+1},R_{t+1},Y_{t+1}\right) \right) -\omega \mathbb {E}_{x_{t},r_{t},y_{t}}\left( g_{t+1}^{2}\left( X_{t+1},R_{t+1},Y_{t+1}\right) \right) \right. \nonumber \\&\left. \qquad \qquad \qquad \qquad +\omega \left[ \mathbb {E}_{x_{t},r_{t},y_{t}}\left( g_{t+1}\left( X_{t+1},R_{t+1},Y_{t+1}\right) \right) \right] ^{2}\right\} \nonumber \\&\quad =\max _{\pi _{t}}\left\{ \mathbb {E}_{x_{t},r_{t},y_{t}}\left( -\omega A_{t+1}R_{t+1}^{2\psi _{t+1}}X_{t+1}^{2}+B_{t+1}R_{t+1}^{\psi _{t+1}}X_{t+1} +\sum _{l=t+1}^{T-1}\beta ^{l}_{t+1}R_{t+1}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}Y_{t+1}\right. \right. \nonumber \\&\left. \left. \qquad -2\omega \sum _{l=t+1}^{T-1}U^{l}_{t+1} R_{t+1}^{\psi _{t+1}+\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}Y_{t+1}X_{t+1}-\omega \sum _{l=t+1}^{T-1} \alpha ^{l}_{t+1} R_{t+1}^{2\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c^{2}_{l}Y^{2}_{t+1} \right. \right. \nonumber \\&\qquad \left. \left. +\frac{1}{4\omega }f_{t+1}-2\omega \sum _{t+1\leqslant l< h\leqslant T-1}S^{l,h}_{t+1} R_{t+1}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1} +\left( \prod _{m=t+1}^{h}\varphi _{m}\right) \psi _{h+1}}c_{l}c_{h}Y^{2}_{t+1}\right) \right. \nonumber \\&\qquad \left. -\omega \mathbb {E}_{x_{t},r_{t},y_{t}}\left( \left( B_{t+1}R_{t+1}^{\psi _{t+1}}X_{t+1}+\frac{1}{2\omega }f_{t+1} +\sum _{l=t+1}^{T-1}\beta ^{l}_{t+1}R_{t+1}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}Y_{t+1}\right) ^{2}\right) \right. \nonumber \\&\qquad \left. +\omega \left[ \mathbb {E}_{x_{t},r_{t},y_{t}}\left( B_{t+1}R_{t+1}^{\psi _{t+1}}X_{t+1}+\frac{1}{2\omega }f_{t+1} +\sum _{l=t+1}^{T-1}\beta ^{l}_{t+1}R_{t+1}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}Y_{t+1}\right) \right] ^{2}\right\} \nonumber \\&\quad =-\omega w_{t}r_{t}^{2\psi _{t}}x_{t}^{2} +B_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) r_{t}^{\psi _{t}}x_{t} +\frac{1}{4\omega }f_{t+1}+B_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) r_{t}^{\varphi _{t}\psi _{t+1}}c_{t}y_{t}\nonumber \\&\qquad -\omega w_{t}r_{t}^{2\varphi _{t}\psi _{t+1}}c_{t}^{2}y_{t}^{2}-2\omega w_{t}r_{t}^{1+2\varphi _{t}\psi _{t+1}}c_{t}y_{t}x_{t}\nonumber \\&\qquad -2\omega \sum _{l=t+1}^{T-1}\left( U^{l}_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}+\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) +\beta ^{l}_{t+1}B_{t+1} cov \left( b_{t}^{\psi _{t+1}},b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right) \nonumber \\&\qquad \times r_{t}^{\psi _{t}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}x_{t} +\sum _{l=t+1}^{T-1}\beta ^{l}_{t+1}\mathbb {E}\left( b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}\nonumber \\&\qquad -2\omega \sum _{l=t+1}^{T-1}\left( U^{l}_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}+\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) +\beta ^{l}_{t+1}B_{t+1} cov \left( b_{t}^{\psi _{t+1}},b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right) \nonumber \\&\qquad \times r_{t}^{\varphi _{t}\psi _{t+1}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}c_{t}y^{2}_{t}\nonumber \\&\qquad -\omega \sum _{l=t+1}^{T-1}M_{t}^{l} r_{t}^{2\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c^{2}_{l}y_{t}^{2} -2\omega \sum _{t+1\leqslant l< h\leqslant T-1}N_{t}^{l,h} r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1} +\left( \prod _{m=t}^{h}\varphi _{m}\right) \psi _{h+1}}c_{l}c_{h}y^{2}_{t}\nonumber \\&\qquad +\max _{\pi _{t}}\left\{ -2\omega r_{t}^{1+2\varphi _{t}\psi _{t+1}}w_{t}\mathbb {E}\left( P'_{t}\right) \pi _{t}x_{t}-2\omega r_{t}^{2\varphi _{t}\psi _{t+1}}w_{t}\mathbb {E}\left( P'_{t}\right) \pi _{t}c_{t}y_{t} -\omega r_{t}^{2\varphi _{t}\psi _{t+1}}\pi '_{t}\xi _{t}\pi _{t}\right. \nonumber \\&\qquad \left. +B_{t+1}r_{t}^{\varphi _{t}\psi _{t+1}}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}\left( P'_{t}\right) \pi _{t}-2\omega \sum _{l=t+1}^{T-1} r_{t}^{\varphi _{t}\psi _{t+1}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}\left( Q_{t}^{l}\right) '\pi _{t}\right\} . \end{aligned}$$

(A8)

Because $\omega >0$, $r_{t}=R_{t}>0$ and $\xi _{t}$ is positive definite by Remark 3.2, then the first-order condition about $\pi _{t}$ in (A8) gives the optimal strategy

$$\begin{aligned} \pi ^{*}_{t}=\xi _{t}^{-1}\left( \frac{B_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P_{t})}{2\omega r_{t}^{\varphi _{t}\psi _{t+1}}}-w_{t}r_{t}x_{t}\mathbb {E}(P_{t})-\sum _{l=t}^{T-1}Q_{t}^{l}r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1} -\varphi _{t}\psi _{t+1}}c_{l}y_{t}\right) . \end{aligned}$$

(A9)

Substituting equation (A9) into equations (A8) and (13), we can obtain

$$\begin{aligned}&V_{t}(x_{t},r_{t},y_{t}) =-\omega \left( w_{t}-w_{t}^{2}\mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{2\psi _{t}}x_{t}^{2}\nonumber \\&\qquad +B_{t+1}\left( \mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) -w_{t}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{\psi _{t}}x_{t}\nonumber \\&\qquad +\frac{1}{4\omega }\left( B_{t+1}^{2}\mathbb {E}^{2}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})+f_{t+1}\right) -2\omega A_{t}r_{t}^{\psi _{t}+\varphi _{t}\psi _{t+1}}c_{t}y_{t}x_{t}\nonumber \\&\qquad -2\omega \sum _{l=t+1}^{T-1}\left( U^{l}_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}+\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right. \nonumber \\&\qquad \left. +\beta ^{l}_{t+1}B_{t+1} cov \left( b_{t}^{\psi _{t+1}},b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right. \nonumber \\&\qquad \left. -w_{t}\left( Q^{l}_{t}\right) '\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{\psi _{t}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}x_{t}\nonumber \\&\qquad +B_{t+1}\left( \mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) -w_{t}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{\varphi _{t}\psi _{t+1}}c_{t}y_{t}\nonumber \\&\qquad +\sum _{l=t+1}^{T-1}\left( \beta ^{l}_{t+1}\mathbb {E}\left( b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) -\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) B_{t+1}\left( Q^{l}_{t}\right) '\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) \nonumber \\&\qquad \times r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}\nonumber \\&\qquad -\omega A_{t}r_{t}^{2\varphi _{t}\psi _{t+1}}c^{2}_{t}y^{2}_{t}-\omega \sum _{l=t+1}^{T-1} \left( M_{t}^{l}-\left( Q^{l}_{t}\right) '\xi ^{-1}_{t}Q^{l}_{t}\right) r_{t}^{2\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c^{2}_{l}y^{2}_{t}\nonumber \\&\qquad -2\omega \sum _{l=t+1}^{T-1}\left( U^{l}_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}+\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right. \nonumber \\&\qquad \left. +\beta ^{l}_{t+1}B_{t+1} cov \left( b_{t}^{\psi _{t+1}},b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) \right. \nonumber \\&\qquad \left. -w_{t}\left( Q^{l}_{t}\right) '\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{\varphi _{t}\psi _{t+1}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}c_{t}y^{2}_{t}\nonumber \\&\qquad -2\omega \sum _{t+1\leqslant l< h\leqslant T-1}\left( N_{t}^{l,h} -\left( Q^{l}_{t}\right) '\xi ^{-1}_{t}Q^{h}_{t}\right) r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1} +\left( \prod _{m=t}^{h}\varphi _{m}\right) \psi _{h+1}}c_{l}c_{h}y^{2}_{t}\nonumber \\&\quad =-\omega A_{t}r_{t}^{2\psi _{t}}x_{t}^{2}+B_{t}r_{t}^{\psi _{t}}x_{t}+\frac{1}{4\omega }f_{t}-2\omega \sum _{l=t}^{T-1}U^{l}_{t} r_{t}^{\psi _{t}+\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}x_{t}\nonumber \\&\qquad +\sum _{l=t}^{T-1}\beta ^{l}_{t}r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}-\omega \sum _{l=t}^{T-1} \alpha ^{l}_{t} r_{t}^{2\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c^{2}_{l}y^{2}_{t}\nonumber \\&\qquad -2\omega \!\!\sum _{t\leqslant l< h{\leqslant } T-1}S^{l,h}_{t} r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l{+}1} +\left( \prod _{m=t}^{h}\varphi _{m}\right) \psi _{h{+}1}}c_{l}c_{h}y^{2}_{t}, \end{aligned}$$

(A10)

and

$$\begin{aligned}&\qquad g_{t}(x_{t},r_{t},y_{t})\nonumber \\&\quad =\mathbb {E}_{x_{t},r_{t},y_{t}}\left[ g_{t+1}\left( X_{t+1},R_{t+1},Y_{t+1}\right) \right] \nonumber \\&\quad =\mathbb {E}_{x_{t},r_{t},y_{t}}\left[ B_{t+1}R_{t+1}^{\psi _{t+1}}X_{t+1}+\frac{1}{2\omega }f_{t+1} +\sum _{l=t+1}^{T-1}\beta ^{l}_{t+1}R_{t+1}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}Y_{t+1}\right] \nonumber \\&\quad =B_{t+1}\left( \mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) -w_{t}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t}) \xi ^{-1}_{t}\mathbb {E}(P_{t})\right) r_{t}^{\psi _{t}}x_{t}\nonumber \\&\qquad +\frac{1}{2\omega }\left( B_{t+1}^{2}\mathbb {E}^{2}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})+f_{t+1}\right) \nonumber \\&\qquad +B_{t+1}\left[ \mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) -w_{t}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \mathbb {E}(P'_{t})\xi ^{-1}_{t}\mathbb {E}(P_{t})\right] r_{t}^{\varphi _{t}\psi _{t+1}}c_{t}y_{t}\nonumber \\&\qquad +\sum _{l=t+1}^{T-1}\left( \beta ^{l}_{t+1}\mathbb {E}\left( b_{t}^{\left( \prod _{m=t+1}^{l}\varphi _{m}\right) \psi _{l+1}}\theta _{t}\right) -B_{t+1}\mathbb {E}\left( b_{t}^{\psi _{t+1}}\right) \left( Q^{l}_{t}\right) '\xi ^{-1}_{t}\mathbb {E}(P_{t})\right) \nonumber \\&\qquad \times r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}\nonumber \\&\quad =B_{t}r_{t}^{\psi _{t}}x_{t}+\frac{1}{2\omega }f_{t} +\sum _{l=t}^{T-1}\beta ^{l}_{t}r_{t}^{\left( \prod _{m=t}^{l}\varphi _{m}\right) \psi _{l+1}}c_{l}y_{t}. \end{aligned}$$

(A11)

Equations (A9)-(A11) show that equations (26)-(28) hold for t. By the principle of mathematical induction, equations (26)-(28) hold for all $t=0,1,\cdots ,T-1$ and the theorem is thus proved.