Evaluation of Portfolio Decision Improvements by Markov Modulated Diffusion Processes: A Shapley Value Approach

Abstract

We use the Shapley value to evaluate the introduction of a second stochastic process to an optimization problem. We find that introducing a Time-Inhomogeneous Markov Modulated Diffusion process to an asset portfolio decision problem yields higher returns to the rational decision maker. These increases in returns can be diminished by a high volatility in the state change or a high discount factor.

Appendices

Appendix 1

In this appendix we derive the results presented in Sect. 15.5. The problem represented by Eq. (15.4) subject to the wealth constraints described by Eq. (15.20) are then reduced by HJB equation to

$$\displaystyle \begin{aligned} J(a_t,t) = \max_{c_s |{}_{s \in [t, t + \mathrm{d} t]}} E \left[ \int_t^{t+\mathrm{d} t} u(c_s) \mathrm{e}^{-\rho s}\mathrm{d} s + \mathrm{d}(J(a_t, t)) + J(a_t, t) + o(\mathrm{d} t) \right], \end{aligned}$$

which can be reduced to

$$\displaystyle \begin{aligned} 0 = \max_{c_s |{}_{s\in [t, t+\mathrm{d} t]}} E\left[ u(c_t) \mathrm{e}^{-\rho t} \mathrm{d} t + o(\mathrm{d} t) + \mathrm{d}(J(a_t,t))\right]. \end{aligned}$$

This, according to the problem at hand, can be expressed then as

$$\displaystyle \begin{aligned} \mathrm{d}( J(a_t, t)) &= \left( \frac{\partial J(a_t,t)}{\partial t} + \frac{\partial J(a_t,t)}{\partial a_t} a_t\mu_a + \frac{1}{2} \frac{\partial^2 J(a_t,t)}{\partial a_t^2} a_t^2 \sigma^2_a \right) \mathrm{d} t + \\ & +\frac{\partial J(a_t,t)}{\mathrm{d} a_t} a_t \sigma_a \mathrm{d} W_t + \left(\sum_{j\in \mathcal{S}} q_{ij}(t) [ J(a_t,t,j) - J(a_t,t,i)] \right) \mathrm{d} t. \end{aligned} $$

where

$$\displaystyle \begin{aligned} \mathrm{d} a_t = a_t \mu_a \mathrm{d} t + a_t \sigma_a \mathrm{d} W_t, \end{aligned}$$

$$\displaystyle \begin{aligned} \mu_a = r + \theta_{t,i} (\mu_i-r) - \frac{c_t}{a_t}, \end{aligned}$$

and

$$\displaystyle \begin{aligned} \sigma_a = \theta_t \sigma_i. \end{aligned}$$

These solutions are substituted in the HJB equation, thus

$$\displaystyle \begin{aligned} 0 &= \max_{c_s |{}_{s\in [t, t+\mathrm{d} t]}} E \left[ u(c_t) \mathrm{e}^{-\rho t} \mathrm{d} t + \left( \frac{\partial J(a_t,t)}{\partial t} + \frac{\partial J(a_t,t)}{\partial a_t} a_t \left(r + \theta_{t,i}(\mu_i-r) - \frac{c_t}{a_t}\right) + \right. \right. \\ & \left. \left. \frac{1}{2}\frac{\partial^2J(a_t,t)}{\partial a_t^2} a^2_t \theta_{t,i}^2\sigma^2_i \right) \mathrm{d} t + \left( \sum_{j\in \mathcal{S}} q_{ij}(t) [J(a_t,t,j)- J(a_t,t,i)] \right)\mathrm{d} t \right] \end{aligned} $$

(15.32)

Appendix 2

This annex, refers directly to Vallejo et al. [23], some results on this paper are required in order to understand construction and use of several elements from Eqs. (15.6) to (15.14).

Referred paper brings closed-form solutions of the utility maximization problem, solving a continuous decision making problem for an infinitely-lived rational consumer with a logarithmic utility function, when the risky asset is driven by a Time-Inhomogeneous Markov Modulated chain and a classical Geometric Brownian motion.

General solution can be simplified and is consistent with simplified versions of this model, providing solutions for classical problems with fixed variations. Those classical models and their solutions are required for this research.

The bond and risky asset are defined as

$$\displaystyle \begin{aligned} \frac{\mathrm{d} b_{t} }{b_{t} } =r\mathrm{d} t \end{aligned} $$

(15.33)

and

$$\displaystyle \begin{aligned} \frac{\mathrm{d} S_{t} }{S_{t} } =\mu _{i} \mathrm{d} t+\sigma \mathrm{d} W_{t} \end{aligned} $$

(15.34)

And the utility maximization problem is given by

$$\displaystyle \begin{aligned} {\mathrm{Maximize}\; E\left[\int _{0}^{\infty }u(c_{t} )\mathrm{e}^{-\rho t} \mathrm{d} t | \mathcal{F}_{t} \right]} \end{aligned} $$

(15.35)

$$\displaystyle \begin{aligned} {\; \quad s.t.\; \; \mathrm{d} a_{t} =a_{t} (r+\theta _{t,i} (\mu _{i} -r)-\frac{c_{t} }{a_{t} } )\mathrm{d} t+a_{t} \theta _{t,i} \sigma _{i} \; \mathrm{d} W_{t} } \end{aligned} $$

(15.36)

Requiring optimal control tools, such as Hamilton-Jacobi-Bellman equation, defining a value function as

$$\displaystyle \begin{aligned} J(a_{t} ,t)=\mathop{\mathrm{Maximize}}\limits_{c_{s} |{}_{s\in [t,\infty )} } \; E\left[\int _{t}^{\infty }u(c_{s} )\mathrm{e}^{-\rho s} \mathrm{d} s \right] \end{aligned} $$

(15.37)

Simplifying this and applying mean value theorem for integral calculus, it follows that

$$\displaystyle \begin{aligned} 0=\mathop{\mathrm{Maximize}}\limits_{c_{s} |{}_{s\in [t,\; t+\mathrm{d} t]} } \; E\left[u(c_{t} )\mathrm{e}^{-\rho t} \mathrm{d} t+o(\mathrm{d} t)+\mathrm{d} (J(a_{t} ,t))\right] \end{aligned} $$

(15.38)

Now, in order to continue, Ito’s Lemma is applied to expand d(J(a _t, t)), as

$$\displaystyle \begin{aligned} 0=\mathop{\mathrm{Maximize}}\limits_{c_{s} |{}_{s\in [t,\; t+\mathrm{d} t]} } \; E\left[\begin{array}{l} {u(c_{t} )\mathrm{e}^{-\rho t} \mathrm{d} t+o(\mathrm{d} t)+\frac{\partial J(a_{t} ,t)}{\partial a_{t} } a_{t} \theta _{t,i}^{} \sigma _{i} \mathrm{d} W_{t} } \\ {+\left(\frac{\partial J(a_{t} ,t)}{\partial t} +\frac{\partial J(a_{t} ,t)}{\partial a_{t} } a_{t} (r+\theta _{t,i} (\mu _{i} -r)-\frac{c_{t} }{a_{t} } )\right.}\\{\left. +\frac{1}{2} \frac{\partial ^{2} J(a_{t} ,t)}{\partial a_{t} ^{2} } a_{t} ^{2} \theta _{t,i}^{2} \sigma _{i}^{2} \right)\mathrm{d} t} \\ {+\left(\sum _{j\in E}q_{ij} (t)\left[J\left(a_{t} ,t,j\right)-J\left(a_{t} ,t,i\right)\right] \right)\mathrm{d} t} \end{array}\right] \end{aligned} $$

(15.39)

Under the assumption of independence between Brownian motion and Markov drift.

Equation (15.39) can be easily simplified by elimination of martingale elements as follows:

$$\displaystyle \begin{aligned} 0=\mathop{\mathrm{Maximize}}\limits_{c_{s} |{}_{s\in [t,\; t+\mathrm{d} t]} } \; E\left[\begin{array}{l} {u(c_{t} )\mathrm{e}^{-\rho t} \mathrm{d} t} \\ {+\left(\frac{\partial J(a_{t} ,t)}{\partial t} +\frac{\partial J(a_{t} ,t)}{\partial a_{t} } a_{t} (r+\theta _{t,i} (\mu _{i} -r)-\frac{c_{t} }{a_{t} } )\right.}\\{\left.+\frac{1}{2} \frac{\partial ^{2} J(a_{t} ,t)}{\partial a_{t} ^{2} } a_{t} ^{2} \theta _{t,i}^{2} \sigma _{i}^{2} \right)\mathrm{d} t} \\ {+\left(\sum _{j\in E}q_{ij} (t)\left[J\left(a_{t} ,t,j\right)-J\left(a_{t} ,t,i\right)\right] \right)\mathrm{d} t} \end{array}\right] \end{aligned} $$

(15.40)

Than under optimal conditions (assuming an optimal consumption), can be expressed as

$$\displaystyle \begin{aligned}0 &=u(c_{t}^{*} )\mathrm{e}^{-\rho t} +\frac{\partial J(a_{t} ,t)}{\partial {\kern 1pt} t} +\frac{\partial J(a_{t} ,t)}{\partial a_{t} } a_{t} (r+\theta _{t,i} (\mu _{i} -r)-\frac{c_{t}^{*} }{a_{t} } )+\frac{1}{2} \frac{\partial ^{2} J(a_{t} ,t)}{\partial a_{t} ^{2} } a_{t} ^{2} \theta _{t,i}^{2} \sigma _{i}^{2} \\ &+\sum _{j\in E}q_{ij} (t)\left[J\left(a_{t} ,t,j\right)-J\left(a_{t} ,t,i\right)\right] \end{aligned} $$

(15.41)

This kind of problems require both: an assumption for utility function and a proposed value function. By selecting a logarithmic utility, an admissible value function could be $J\left (a_{t} ,t,i\right )=\beta _{0} +\beta _{1} u(a_{t} )e^{-\rho t} +g(t,i)e^{-\rho t}$.

Simplifying, we have

$$\displaystyle \begin{aligned} 0 &=u(c_{t}^{*} )-\rho \, (\beta _{0} +\beta _{1} u(a_{t} ))+\frac{\partial g(t,i)}{\partial t} -\rho g(t,i) \\ & + \beta _{1} u'(a_{t} )a_{t} (r+\theta _{t,i} (\mu _{i} -r)-\frac{c_{t}^{*} }{a_{t} } )+\frac{1}{2} \beta _{1} u''(a_{t} )a_{t} ^{2} \theta _{t,i}^{2} \sigma _{i}^{2} \\ &+\sum _{j\in E}q_{ij} (t)\left[g\left(t,j\right)-g\left(t,i\right)\right] \end{aligned} $$

(15.42)

And therefore, with a logarithmic utility, and some partial derivatives, optimal portfolio and consumption decisions follow $\theta _{i} =\frac {\mu _{i} -r}{\sigma _{i}^{2} }$ and $c_{t}^{*} =\rho a_{t}$.