Interpreting the Merton Consumption Parameter Beta

Summary

The document presents a finite-horizon Merton investment-consumption problem with wealth invested in a risky asset and reduced by consumption. Under CRRA power utility, it states an optimal risky allocation proportional to risk tolerance and gives a time-dependent optimal consumption rate. The parameter beta combines expected excess return and volatility with the utility parameter, and appears in the formula governing how consumption varies as the terminal date approaches.

The question focuses on interpreting beta, especially why it can be negative when the stated risk-tolerance parameter exceeds one. It suggests interpreting one component as expected portfolio return but offers no answer, derivation, or empirical evidence. The setup is therefore useful as a prompt about optimal consumption and parameter interpretation, but conclusions depend on the model’s assumptions, including its utility specification, investment dynamics, and finite horizon; the document also flags possible discrepancies in its formulation.

Key ideas

• The model jointly chooses a risky investment fraction and a consumption rate over a finite horizon.
• The stated optimal risky allocation scales with expected return, inverse variance, and risk tolerance.
• Beta enters the time profile of optimal consumption and can be negative under the stated parameterization.
• The document poses but does not resolve the economic interpretation of beta.
• Any interpretation depends on the assumptions and notation used in the model.

Tags

# How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?


# How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?












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.

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