Statement: Let the dynamics of wealth of the agent satisfy $$dX_{t} = \pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$ where $(\pi,c)$ is an investment-consumption ($\pi$ - fraction of wealth to invest, $c$ - fraction of wealth to consume).

Under standard Merton optimization problem the agent is to maximize the expected utility $$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt + U(X_T)\Big],$$ under CRRA power utility $$U(x) = \frac{1}{1-\frac1\delta}x^{\frac1{1-\frac1\delta}}, \quad \delta > 0, \delta\neq 1,$$ so $\delta$ plays a role of risk-tolerance parameter.

The optimal plan is then given by $$\pi^* = \frac{\mu}{\sigma^2}\delta,\quad c^*_t=\Big( \frac1{\beta}-\big(1-\frac1\beta\big)e^{-\beta(T-t)}\Big)^{-1},$$ where $\beta = \frac{\mu^2}{2\sigma^2}\delta(1-\delta).$

Question: How can I interpret $\beta$ at this point? I have seen that $\frac{\mu^2}{2\sigma^2}\delta$ is sometimes referred to as expected portfolio return. But what is the meaning when I multiply it by $1-\delta$? If $\delta > 1$, it can be negative. I would call it effective expected portfolio return, but am not sure. If you can provide any reference, that would be perfect. Thanks in advance!

P.S. I was adapting my model to the standard Merton one, sorry for any discrepancies.

  • 1
    $\begingroup$ Looks like $\beta$ is expected portfolio return times a transformation of elasticity of intertemporal substitution (EIS). $\beta$ determines the savings rate. It depends on return and your preference to allocating wealth over time (EIS). $\endgroup$ – fesman Oct 30 '20 at 15:04

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