Robo-advising: a dynamic mean-variance approach

Abstract

In contrast to traditional financial advising, robo-advising needs to elicit investors’ risk profile via several simple online questions and provide advice consistent with conventional investment wisdom, e.g., rich and young people should invest more in risky assets. To meet the two challenges, we propose to do the asset allocation part of robo-advising using a dynamic mean-variance criterion over the portfolio’s log returns. We obtain analytical and time-consistent optimal portfolio policies under jump-diffusion models and regime-switching models.

Proofs for results in section 2

Proofs of proposition 1.

The value function is defined as $V\left( t,R_{t}\right) =\mathrm {E} _{t}(R_{T}^{*})-\frac{\gamma }{2}\mathrm {Var}_{t}(R_{T}^{*})$, where $R^{*}$ is the log return with equilibrium $\hat{\pi }$. Let $g\left( t,R_{t}\right) :=\mathrm {E}_{t}\left[ R_{T}^{*}\right]$. By the structures of $R_{T}^{*}$ and the conditional expectation and the conditional variance, we consider the forms

$$\begin{aligned} V\left( t,R_{t}\right)&=R_{t}+A\left( t\right) \text {, }A\left( T\right) =0; \end{aligned}$$

(18)

$$\begin{aligned} g\left( t,R_{t}\right)&=R_{t}+a\left( t\right) \text {, }a\left( T\right) =0. \end{aligned}$$

(19)

Furthermore, we have, via the general extended HJB in Björk et al. (2017), that

$$\begin{aligned} \underset{\pi _{t}}{\max }&\{\mathcal {D}V(t,R_{t})-\frac{\gamma }{2} \sigma ^{2}\pi _{t}^{2}\left( \frac{\partial g}{\partial R}\right) ^{2} +\int _{\mathbb {R}}\left[ g\left( t,R_{t^{-}}+\ln \left( 1+\beta \pi _{t}\right) \right) -g\left( t,R_{t^{-}}\right) \right] \lambda \left( dz\right) \\&+\int _{\mathbb {R}}\left[ g^{2}\left( t,R_{t^{-}}+\ln \left( 1+\beta \pi _{t}\right) \right) -g^{2}\left( t,R_{t^{-}}\right) \right] \lambda \left( \mathrm{d}z\right) \}=0, \end{aligned}$$

subject to $V(T,R)=R$, where $\mathcal {D}$ is the Dynkin operator. For a function F of $\left( t,R\right)$, $\mathcal {D}F\left( t,R\right) =\frac{\partial F}{\partial t}+\left( r+\left( \mu -r\right) \pi -\frac{1}{2}\sigma ^{2}\pi ^{2}\right) \frac{\partial F}{\partial R}+\frac{1}{2}\sigma ^{2}\pi ^{2}\frac{\partial ^{2} F}{\partial R^{2}}+\int _{\mathbb {R}}\left[ F\left( t,R_{t^{-}}+\ln \left( 1+\beta \pi \right) \right) -F\left( t,R_{t^{-}}\right) \right] \lambda \left( \mathrm{d}z\right)$. Incorporating the special forms (18) and (19), we have

$$\begin{aligned} \underset{\pi _{t}}{\max }&\{\frac{\mathrm{d}A}{\mathrm{d}t}+\left( r+\left( \mu -r\right) \pi _{t}-\frac{1}{2}\sigma ^{2}\pi _{t}^{2}\right) -\frac{\gamma }{2}\sigma ^{2}\pi _{t}^{2}+\int _{\mathbb {R}}\ln \left( 1+\beta \pi _{t}\right) \lambda \left( \mathrm{d}z\right) \\&-\frac{\gamma }{2}\int _{\mathbb {R}}\ln ^{2}\left( 1+\beta \pi _{t}\right) \lambda \left( \mathrm{d}z\right) \}=0. \end{aligned}$$

We then obtain the equilibrium strategy (5) by the first-order condition. $\square$

Proof of Proposition 2.

For any proportional portfolio $\pi$, by the dynamics of the log return process $\{R(t;\pi )\}$, we have, given $R(t,\pi )=R$, that

$$\begin{aligned} R(T;\pi )= & {} \int _{t}^{T}\left( r(s, X_s;\alpha (s))+\left( \mu (s, X_s;\alpha (s))-r(s, X_s;\alpha (s))\right) \pi _{s}-\frac{1}{2}(\sigma (s, X_s;\alpha (s))\pi _{s})^{2}\right) \mathrm{d}s\\&+\int _{t}^{T}\sigma _{s}^{i}\pi _{s}^{i}dB_{t}+R, \end{aligned}$$

and the objective function is to maximize

$$\begin{aligned} J(t,R,x;\pi , \alpha (t))=\mathbb {E}_{t}\left[ R(T;\pi )\ |X_t=x, \alpha (t)=i \right] -\frac{\gamma }{2}\mathrm {Var}_{t}(R(T;\pi )\ |X_t=x, \alpha (t)=i ). \end{aligned}$$

Let $\hat{\pi }(t, R, x, i)$ be an equilibrium feedback proportional allocation, $\{R(s)\}$ be the resulted log return process, $\{\hat{\pi }_s\}$ be the proportional portfolio process, and V(t, R, x, i) be the value function. Define

$$\begin{aligned} f(t,x,R,i)=\mathrm {E}_{t }\left[ \int _{t}^{T}\left( \left( \mu (s, X_s; \alpha (s))-r(s, X_s; \alpha (s))\right) \hat{\pi }_s-\frac{1}{2}(\sigma (s, X_s; \alpha (s))\hat{\pi }_s)^{2}\right) \mathrm{d}s \ |X_t=x, \alpha (t)=i \right] . \end{aligned}$$

It is easy to see that $J(t,R,X_t;\pi , \alpha (t))-R$ is not explicitly dependent on R, this gives us the conjecture that V(t, R, x, i) admits the form

$$\begin{aligned} V(t,R,x,i)=\tilde{V}(t,x,i)+R+{\mathbb {E}}_t\left[ \int _t^T r(s, X_s;\alpha (s))\mathrm{d}s\ |X_t=x, \alpha (t)=i \right] . \end{aligned}$$

By this form and the definition of equilibrium solution, we then obtain the PDE

$$\begin{aligned} \underset{\pi }{\max }\left\{ D\tilde{V}(t,x,i)+\left( \mu ^{i} -r^{i}\right) \pi -\frac{1}{2}(\sigma ^{i}\pi )^{2}-\frac{\gamma }{2}\left[ (\nu ^{i}\frac{\partial f^{i}}{\partial x})^{2}+2\rho \nu ^{i} \sigma ^{i}\pi \frac{\partial f^{i}}{\partial x}+(\sigma ^{i}\pi )^{2}+\sum _{j=1}^N q_{ij} f(t,x;j)^2\right] \right\} =0, \end{aligned}$$

(20)

subject to $\tilde{V}(T,x,i)=0$, where we used the simplified notations $\mu ^i:=\mu (t, x; i), \sigma ^i:=\sigma (t, x; i), m_i:=m( x; i), \nu _i=\nu (x;i)$, and

$$\begin{aligned} D\tilde{V}(t,x,i)=\frac{\partial \tilde{V}}{\partial t}+m_{i}\frac{\partial \tilde{V}}{\partial x}+\frac{1}{2}\nu _{i}^{2}\frac{\partial ^{2}\tilde{V}}{\partial x^{2}}+\sum _{j=1} ^{N}q_{ij}\tilde{V}(t,x,j). \end{aligned}$$

If $\tilde{V}$ is a solution to this equation, then the optimal $\pi ^*=\hat{\pi }(t,x;i)$ will be the equilibrium feedback strategy.

By the first-order condition, an equilibrium optimal trading strategy is as given in (7). Substituting $\hat{\pi }$ into $f^{i}$ gives

$$\begin{aligned} f(t,x,i)= & \mathrm {E}_{t}\left[ \int _{t}^{T}\left( \frac{( 2\gamma +1) ( \mu _s-r_s) ^{2}}{2(\gamma +1)^{2}\sigma _s^{2}}-\frac{\rho \gamma ^{2}\left( \mu _s-r_s\right) \nu _s}{\left( 1+\gamma \right) ^{2}\sigma _s}\frac{\partial f(s, X_s; \alpha (s))}{\partial x} \right. \right. \\&\left. \left. -\frac{1}{2}\frac{\left( \gamma \rho \nu _s \right) ^{2}}{\left( 1+\gamma \right) ^{2}}\left( \frac{\partial f(s, X_s; \alpha (s))}{\partial x}\right) ^{2}\right) \mathrm{d}s\right] . \end{aligned}$$

with some abusing of notations $r_s=r(s,X_s;\alpha (s)), \mu _s=\mu (s, X_s;\alpha (s)), \sigma _s=\sigma (s,X_s;\alpha (s))$ and $m_s=m(X_s;\alpha (s)), \nu _s=\nu (X_s;\alpha (s))$.

By the Feynman-Kac formula, we infer that $f^{i}$ satisfies (8). $\square$

Proof of Corollary 1: For conciseness, we use $f^i$ to mean the function $f(\cdot , \cdot ; i)$. For the Gaussian mean returns model, the PDE (8) becomes

$$\begin{aligned}&\frac{\partial f^{i}}{\partial t} +(\lambda (i)\bar{X}(i)-\lambda (i) x-\frac{\rho \gamma ^{2}\nu (i)}{\left( 1+\gamma \right) ^{2} }x)\frac{\partial f^{i}}{\partial x}+\frac{1}{2}\nu (i)^{2}\left( \frac{\partial ^{2}f^{i}}{\partial x^{2}}-\frac{\rho ^{2}\gamma ^{2}}{\left( 1+\gamma \right) ^{2}}(\frac{\partial f^{i}}{\partial x})^{2}\right) \nonumber \\&\quad +\frac{\left( 2\gamma +1\right) }{2\left( \gamma +1\right) ^{2} }x^{2}+\sum _{j=1}^{N}q_{ij}f(t,x,j)=0,\nonumber \\&\qquad f^{i} \left( T,x\right) =0. \end{aligned}$$

(21)

Consider a solution of the following quadratic form:

$$\begin{aligned} f(t,x,i)=A\left( t,i\right) x^{2}+B\left( t,i\right) x+C\left( t,i\right) . \end{aligned}$$

Substituting this into (21) and by separating variables, we obtain the following ODE system:

$$\begin{aligned}&\left\{ \begin{array} [c]{l} \frac{\mathrm{d}A}{\mathrm{d}t}\left( t,i\right) -2\left( \lambda (i)+\frac{\rho \gamma ^{2}\nu _{i}}{\left( 1+\gamma \right) ^{2}}\right) A\left( t,i\right) +\frac{2\rho ^{2}\gamma ^{2}\nu (i)^{2}}{\left( 1+\gamma \right) ^{2}}A^{2}\left( t,i\right) +\frac{\left( 2\gamma +1\right) }{2\left( \gamma +1\right) ^{2}}+\sum _{j=1}^{N} q_{ij}A\left( t,j\right) =0\text {,}\\ A\left( T,i\right) =0\text {,} \end{array} \right. \\&\left\{ \begin{array} [c]{l} \frac{\mathrm{d}B}{\mathrm{d}t}\left( t,i\right) -\left( \lambda (i)+\frac{\rho \gamma ^{2}\nu (i)}{\left( 1+\gamma \right) ^{2}}+\frac{2\rho ^{2}\gamma ^{2}\nu (i)^{2}}{\left( 1+\gamma \right) ^{2}}A\left( t,i\right) \right) B\left( t,i\right) +2\lambda (i)\bar{X}^{i}A\left( t,i\right) +\sum _{j=1}^{N}q_{ij}B\left( t,j\right) =0\text {,}\\ B\left( T,i\right) =0\text {,} \end{array} \right. \\&\left\{ \begin{array} [c]{l} \frac{\mathrm{d}C}{\mathrm{d}t}\left( t,i\right) +\nu (i)^{2}A\left( t,i\right) +\lambda (i)\bar{X}(i)B\left( t,i\right) -\frac{\rho ^{2}\gamma ^{2}\nu _{i}^{2} }{2\left( 1+\gamma \right) ^{2}}B^{2}\left( t,i\right) +\sum _{j=1} ^{N}q_{ij}C\left( t,j\right) =0\text {,}\\ C\left( T,i\right) =0\text {,} \end{array} \right. \end{aligned}$$

Hence, by Proposition 2, the equilibrium strategy is as given in (11). $\square$

Proof of Corollary 2: Note that in the stochastic volatility model, $\mu (i)-r_{i}=\delta _{i} x^{\frac{1+\beta }{2\beta }},\ \sigma (i)=x^{\frac{1}{2\beta }},\ m_{i} =\lambda (i)\bar{X}(i)-\lambda (i)x$, and $\nu (i)=\bar{\nu }(i)x^{\frac{1}{2}}$. Then PDE (8) becomes

$$\begin{aligned}&\frac{\partial f^{i}}{\partial t} +(\lambda (i)\bar{X}(i)-\lambda (i)x-\frac{\gamma ^{2}\rho \delta _{i}\bar{\nu }(i)}{\left( 1+\gamma \right) ^{2}}x)\frac{\partial f^{i}}{\partial x}+\frac{1}{2}\bar{\nu }(i)^{2}x\left( \frac{\partial ^{2}f^{i}}{\partial x^{2}}-\frac{\gamma ^{2}\rho ^{2}}{\left( 1+\gamma \right) ^{2}}(\frac{\partial f^{i} }{\partial x})^{2}\right) \nonumber \\&\quad +\frac{\left( 2\gamma +1\right) \delta (i)^{2}}{2\left( \gamma +1\right) ^{2}}x+\sum _{j=1}^{N}q_{ij}f(t,x,j)=0,\nonumber \\&\qquad f^{i} \left( T,x\right) =0. \end{aligned}$$

(22)

We try a solution of the following linear form:

$$\begin{aligned} f(t,x,i)=A\left( t,i\right) \cdot x+B\left( t,i\right) . \end{aligned}$$

Substituting this into (22) and by separating variables, we obtain the following ODE system:

$$\begin{aligned}&\left\{ \begin{array} [c]{l} \frac{\mathrm{d}A}{\mathrm{d}t}\left( t,i\right) -\left( \lambda (i)+\frac{\rho \gamma ^{2}\delta (i)\bar{\nu }(i)}{\left( 1+\gamma \right) ^{2}}\right) A\left( t,i\right) +\frac{\rho ^{2}\gamma ^{2}\bar{\nu }(i)^{2}}{2\left( 1+\gamma \right) ^{2}}A^{2}\left( t,i\right) +\frac{\left( 2\gamma +1\right) \delta (i)^{2}}{2\left( \gamma +1\right) ^{2}}+\sum _{j=1}^{N}q_{ij}A\left( t,j\right) =0\text {,}\\ A\left( T,i\right) =0\text {,} \end{array} \right. \\&\left\{ \begin{array} [c]{l} \frac{\mathrm{d}B}{\mathrm{d}t}\left( t,i\right) +\lambda (i)\bar{X}(i)A\left( t,i\right) +\sum _{j=1}^{N}q_{ij}B\left( t,j\right) =0\text {,}\\ B\left( T,i\right) =0\text {,} \end{array} \right. \end{aligned}$$

Hence, by Proposition 2, the equilibrium strategy is as given in (16). $\square$