Portfolio optimization with transaction costs: a two-period mean-variance model

Abstract

In this paper, we study a multiperiod mean-variance portfolio optimization problem in the presence of proportional transaction costs. Many existing studies have shown that transaction costs can significantly affect investors’ behavior. However, even under simple assumptions, closed-form solutions are not easy to obtain when transaction costs are considered. As a result, they are often ignored in multiperiod portfolio analysis, which leads to suboptimal solutions. To provide better insight for this complex problem, this paper studies a two-period problem that considers one risk-free and one risky asset. Whenever there is a trade after the initial asset allocation, the investor incurs a linear transaction cost. Through a mean-variance model, we derive the closed-form expressions of the optimal thresholds for investors to re-allocate their resources. These thresholds divide the action space into three regions. Some important properties of the analytical solution are identified, which shed light on solving multiperiod problems.

Appendices

Appendix 1: Value function for the first period

We apply the backward dynamic programming algorithm to obtain the value function. In particular, after we have obtained the piecewise value function for the latter period, we can derive the value function for the first period as follows:

$$\begin{aligned} V_0(\hat{h}^0_0, \hat{h}^1_0)&= \max \int _a^{B^\alpha _{0}} \frac{1}{b-a} \left[ \delta \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0} \right) -\lambda \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0}\right) ^2 \right. \nonumber \\&\left. +\,\,K^\alpha _1 \lambda \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\alpha _1 r^1_{0} \hat{h}^{1}_{0}\right) ^2\right] \,\mathrm {d}r^1_{0} \nonumber \\&+ \int _{B^\alpha _{0}}^{B^\beta _{0}} \frac{1}{b-a}{ \left[ \delta \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0}\right) -\lambda \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0}\right) ^2\right] }\,\mathrm {d}r^1_{0} \nonumber \\&+ \int _{B^\beta _{0}}^b \frac{1}{b-a} \left[ \delta \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0} \right) -\lambda \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0}\right) ^2 \right. \nonumber \\&\left. +\,\,K^\beta _1 \lambda \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\beta _1 r^1_{0} \hat{h}^{1}_{0}\right) ^2\right] \,\mathrm {d}r^1_{0} \nonumber \\&= \max \int _a^b \frac{1}{b-a}{ \left[ \delta \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0}\right) -\lambda \,\mathrm {E}\left( (r^0)^2 \hat{h}^{0}_{0} + R^1_{1} r^1_{0} \hat{h}^{1}_{0} \right) ^2\right] }\,\mathrm {d}r^1_{0} \nonumber \\&+ \int _a^{B^\alpha _{0}} \! \frac{1}{b-a} K^\alpha _1 \lambda \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\alpha _1 r^1_{0} \hat{h}^{1}_{0}\right) ^2 \,\mathrm {d}r^1_{0} \nonumber \\&+\int _{B^\beta _{0}}^b \frac{1}{b-a} K^\beta _1 \lambda \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\beta _1 r^1_{0} \hat{h}^{1}_{0}\right) ^2 \,\mathrm {d}r^1_{0}. \end{aligned}$$

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Since

$$\begin{aligned} \left. \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\alpha _1 r^1_{0} \hat{h}^{1}_{0}\right) ^3 \right| _{r^1_{T-2}=B^\alpha _{T-2}}&=0 \end{aligned}$$

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$$\begin{aligned} \left. \left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\beta _1 r^1_{0} \hat{h}^{1}_{0}\right) ^3 \right| _{r^1_{T-2}=B^\beta _{T-2}}&=0, \end{aligned}$$

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the value function can be simplified to

$$\begin{aligned} V_0(\hat{h}^0_0, \hat{h}^1_0)&= \max -\frac{1}{b-a} \left\{ \delta \left[ (r^0)^2 \hat{h}^{0}_{0} a + \frac{1}{2} \,\mathrm {E}(R^1_{1}) \hat{h}^{1}_{0}a^2\right] \right. \nonumber \\&\left. \! -\,\lambda \left[ \left( (r^0)^2 \hat{h}^{0}_{0}\right) ^2 a \!+\! (r^0)^2 \,\mathrm {E}(R^1_{1}) \hat{h}^{0}_{0} \hat{h}^{1}_{0} a^2 \!+\!\frac{1}{3} \,\mathrm {E}(R^1_{1})^2 \left( \hat{h}^{1}_{0}\right) ^2 a^3\right] \right. \nonumber \\&\left. -\,\,\frac{1}{3} K^\alpha _1 \lambda \left( J^\alpha _1 \hat{h}^{1}_{0}\right) ^{-1}\left[ \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\alpha _1 \hat{h}^{1}_{0} a\right] ^3 \right\} \nonumber \\&+\,\,\frac{1}{b-a} \left\{ \delta \left[ (r^0)^2 \hat{h}^{0}_{0} b + \frac{1}{2} \,\mathrm {E}(R^1_{1}) \hat{h}^1_{0}b^2\right] \right. \nonumber \\&\left. -\,\,\lambda \left[ \left( (r^0)^2 \hat{h}^{0}_{0}\right) ^2 b + (r^0)^2 \,\mathrm {E}(R^1_{1}) \hat{h}^{0}_{0} \hat{h}^{1}_{0} b^2 +\frac{1}{3} \,\mathrm {E}(R^1_{1})^2\left( \hat{h}^{1}_{0}\right) ^2 b^3\right] \right. \nonumber \\&\left. -\,\, \frac{1}{3} K^\beta _1 \lambda \left( J^\beta _1 \hat{h}^{1}_{0}\right) ^{-1} \left[ \frac{\delta }{2 \lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\beta _1 \hat{h}^{1}_{0} b\right] ^3 \right\} \end{aligned}$$

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$$\begin{aligned}&= \max \delta \left[ (r^0)^2 \hat{h}^{0}_{0} + \,\mathrm {E}(R^1_{1}) \hat{h}^{1}_{0} \frac{a+b}{2}\right] - \lambda \left[ \left( (r^0)^2 \hat{h}^{0}_{0}\right) ^2 \right. \nonumber \\&\left. +\,\, (r^0)^2 \,\mathrm {E}(R^1_{1}) \hat{h}^{0}_{0} \hat{h}^{1}_{0} (a+b) + \frac{1}{3} \,\mathrm {E}(R^1_{1})^2 \left( \hat{h}^{1}_{0}\right) ^2 (a^2+ab+b^2)\right] \nonumber \\&-\,\, \frac{\lambda }{b-a} \left\{ \frac{1}{3}\frac{{\left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0}\right) }^3}{\hat{h}^{1}_{0}} \left( \frac{K^\beta _1}{J^\beta _1}-\frac{K^\alpha _1}{J^\alpha _1}\right) \right. \nonumber \\&\left. -\,\,{\left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0}\right) }^2 \left( K^\beta _1 b - K^\alpha _1 a\right) \right. \nonumber \\&\left. +\,\, \left( \frac{\delta }{2 \lambda }- (r^0)^2 \hat{h}^{0}_{0}\right) \hat{h}^{1}_{0} \left( K^\beta _1 J^\beta _1 b^2 - K^\alpha _1 J^\alpha _1 a^2\right) \right. \nonumber \\&\left. -\,\, \frac{1}{3} \left( \hat{h}^{1}_{0}\right) ^2 \left( K^\beta _1 {(J^\beta _1)}^2 b^3 - K^\beta _1 {(J^\alpha _1)}^2 a^3 \right) \right\} . \end{aligned}$$

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We assume that $R^1_{0}\sim U(a,b)$, which implies $\,\mathrm {E}(R^1_{0})=\frac{a+b}{2}$ and $\,\mathrm {E}(R^1_{0})^2=\frac{a^2+ab+b^2}{3}$. Therefore, the value function can be further simplified to

$$\begin{aligned} V_0(\hat{h}^0_0, \hat{h}^1_0)=&\max \delta \,\mathrm {E}\left[ {(r^0)}^2 \hat{h}^{0}_{0} + R^1_{1} R^1_{0} \hat{h}^{1}_{0}\right] - \lambda \,\mathrm {E}\left[ (r^0)^2 \hat{h}^{0}_{0}+ R^1_{T-1} R^1_{0} \left( \hat{h}^{1}_{0}\right) \right] ^2 \nonumber \\&- \frac{\lambda }{b-a} \left\{ \frac{1}{3}\frac{\left( \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0}\right) ^3}{\hat{h}^{1}_{0}} \left( \frac{K^\beta _1}{J^\beta _1} \! -\! \frac{K^\alpha _1}{J^\alpha _1} \right) \! -\! \left( \frac{\delta }{2\lambda }\! -\! (r^0)^2 \hat{h}^{0}_{0}\right) ^2 \right. \nonumber \\&\left. \times \left( K^\beta _1 b \! - \! K^\alpha _1 a \right) + \left( \frac{\delta }{2 \lambda }- (r^0)^2 \hat{h}^{0}_{0}\right) \hat{h}^{1}_{0} \left( K^\beta _1 J^\beta _1 b^2 - K^\alpha _1 J^\alpha _1 a^2\right) \right. \nonumber \\&\left. -\, \frac{1}{3} \left( \hat{h}^{1}_{0}\right) ^2 \left( K^\beta _1 (J^\beta _1)^2 b^3 - K^\beta _1 (J^\alpha _1)^2 a^3 \right) \right\} \end{aligned}$$

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$$\begin{aligned} =&\max \delta \,\mathrm {E}\left[ (r^0)^2 \hat{h}^{0}_{0} + R^1_{T-1} R^1_{0} \hat{h}^{1}_{0}\right] - \lambda \,\mathrm {E}\left[ (r^0)^2 \hat{h}^{0}_{0}+ R^1_{T-1} R^1_{0} \left( \hat{h}^{1}_{0}\right) \right] ^2 \nonumber \\&\quad -\frac{\lambda }{3(b-a)\hat{h}^{1}_{0}} \left\{ \frac{K^\beta _1}{J^\beta _1} {\left[ \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\beta _1 b \hat{h}^{1}_{0}\right] }^3 \right. \nonumber \\&\left. -\,\frac{K^\alpha _1}{J^\alpha _1} {\left[ \frac{\delta }{2\lambda }- (r^0)^2 \hat{h}^{0}_{0} -J^\alpha _1 a \hat{h}^{1}_{0}\right] }^3 \right\} . \end{aligned}$$

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Appendix 2: Expressions for $\,\mathrm {E}(w_2(\gamma ))$ and $\,\mathrm {E}(w_2(\gamma ))^2$

The expectation of $w_2$ is

$$\begin{aligned} \,\mathrm {E}(w_2(\gamma ))&= \int _a^{B^\alpha _{T-2}} p(r^1_0) w^{(1)}_1 \,\mathrm {d}r^1_0 + \int _{B^\alpha _{T-2}}^{B^\beta _{T-2}} p(r^1_0) w^{(2)}_1 \,\mathrm {d}r^1_0 + \int _{B^\beta _{T-2}}^b p(r^1_0) w^{(3)}_1 \,\mathrm {d}r^1_0 \nonumber \\&= (r^0)^2 w_0+\rho \left( \frac{ \frac{\gamma }{2} - (r^0)^2 h^0_0}{J^\alpha _0 - {(r^0)}^2 (1+\alpha )}\right) , \end{aligned}$$

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where $\rho = \,\mathrm {E}(\xi ^\alpha _0) +\frac{1}{2(b-a)} \frac{ K^\alpha _1}{J^\alpha _1} \left( J^\alpha _0 -J^\alpha _1 a\right) ^2 - \frac{1}{2(b-a)} \frac{ K^\beta _1}{J^\beta _1}\left( J^\alpha _0 -J^\beta _1 b\right) ^2$.

The expectation of $(w_2)^2$ is

$$\begin{aligned}&\,\mathrm {E}(w_2(\xi ))^2\nonumber \\&\quad = \int _a^{B^\alpha _{T-2}} p(r^1_0) {(w^{(1)}_1)}^2 \,\mathrm {d}r^1_0 + \int _{B^\alpha _{T-2}}^{B^\beta _{T-2}} p(r^1_0) {(w^{(2)}_1)}^2 \,\mathrm {d}r^1_0 + \int _{B^\beta _{T-2}}^b p(r^1_0) {(w^{(3)}_1)}^2 \,\mathrm {d}r^1_0 \nonumber \\&\quad = \,\mathrm {E}\left[ {(r^0)}^2 \hat{h}^0_0 + R^1_1 R^1_0 \hat{h}^1_0 \right] ^2 + \frac{1}{(b-a)\hat{h}^1_0 } \left\{ -\frac{1}{3} \frac{ K^\alpha _1}{J^\alpha _1} \left( \frac{\gamma }{2}- {(r^0)}^2 \hat{h}^0_0- J^\alpha _1 a \hat{h}^1_0\right) ^3 \right. \nonumber \\&\left. +\,\, \frac{1}{3} \frac{ K^\beta _1}{J^\beta _1}\left( \frac{\gamma }{2}- {(r^0)}^2 \hat{h}^0_0- J^\beta _1 b \hat{h}^1_0\right) ^3 \right. \nonumber \\&\left. +\,\frac{K^\alpha _1}{J^\alpha _1} \left( \frac{\gamma }{2}- {(r^0)}^2 \hat{h}^0_0- J^\alpha _1 a \hat{h}^1_0\right) ^2 \frac{\gamma }{2} \right. \nonumber \\&\left. -\,\, \frac{K^\beta _1}{J^\beta _1} \left( \frac{\gamma }{2}- {(r^0)}^2 \hat{h}^0_0- J^\beta _1 b \hat{h}^1_0\right) ^2 \frac{\gamma }{2} \right\} \nonumber \\&\quad = \nu {\left( \frac{\frac{\gamma }{2}-{(r^0)}^2 w_0}{J^\alpha _0 - {(r^0)}^2 (1+\alpha )}\right) }^2 +2 \rho {(r^0)}^2 w_0 \left( \frac{\frac{\gamma }{2}-{(r^0)}^2 w_0}{J^\alpha _0 - {(r^0)}^2 (1+\alpha )}\right) + {\left( {(r^0)}^2 w_0\right) }^2,\nonumber \\ \end{aligned}$$

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where

$$\begin{aligned} \nu&= \,\mathrm {E}(\xi ^\alpha _0)^2 + \frac{1}{3(b-a)} \left[ \frac{ K^\beta _1}{J^\beta _1} \left( J^\alpha _0 - J^\beta _1 b\right) ^3 - \frac{ K^\alpha _1}{J^\alpha _1} \left( J^\alpha _0 - J^\alpha _1 a\right) ^3\right] \nonumber \\&-\,\, \frac{1}{b-a} \left[ \frac{ K^\beta _1}{J^\beta _1} \left( J^\alpha _0 - J^\beta _1 b\right) ^2 - \frac{ K^\alpha _1}{J^\alpha _1} \left( J^\alpha _0 - J^\alpha _1 a\right) ^2\right] . \end{aligned}$$

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