<p><strong>Statement:</strong> Let the dynamics of wealth of the agent satisfy
<span class="math-container">$$dX_{t} =
 \pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$</span>
where <span class="math-container">$(\pi,c)$</span> is an investment-consumption (<span class="math-container">$\pi$</span> - fraction of wealth to invest, <span class="math-container">$c$</span> - fraction of wealth to consume).</p>
<p>Under standard Merton optimization problem the agent is to maximize the expected utility <span class="math-container">$$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt +  U(X_T)\Big],$$</span>
under CRRA power utility <span class="math-container">$$U(x) = \frac{1}{1-\frac1\delta}x^{\frac1{1-\frac1\delta}}, \quad \delta &gt; 0, \delta\neq 1,$$</span> so <span class="math-container">$\delta$</span> plays a role of <em>risk-tolerance</em> parameter.</p>
<p>The optimal plan is then given by <span class="math-container">$$\pi^* = \frac{\mu}{\sigma^2}\delta,\quad c^*_t=\Big( \frac1{\beta}-\big(1-\frac1\beta\big)e^{-\beta(T-t)}\Big)^{-1},$$</span>
where <span class="math-container">$\beta = \frac{\mu^2}{2\sigma^2}\delta(1-\delta).$</span></p>
<p><strong>Question:</strong> How can I interpret <span class="math-container">$\beta$</span> at this point? I have seen that <span class="math-container">$\frac{\mu^2}{2\sigma^2}\delta$</span> is sometimes referred to as <em>expected portfolio return</em>. But what is the meaning when I multiply it by <span class="math-container">$1-\delta$</span>? If <span class="math-container">$\delta &gt; 1$</span>, it can be negative. I would call it <em>effective expected portfolio return</em>, but am not sure. If you can provide any reference, that would be perfect.
Thanks in advance!</p>
<p><strong>P.S.</strong> I was adapting my model to the standard Merton one, sorry for any discrepancies.</p>

CC BY-SA 4.0
2020-10-28T23:45:25.160Z
Your parametrization of CRRA via $\delta$ seems a bit strange. In any case, the $\delta(1-\delta)$ expression in consumption share $\beta$ should reflect the elasticity of intertemporal substitution.
Michael
CC BY-SA 4.0
2020-10-30T11:07:56.727Z
@Michael, thanks for your answer. That is an interesting perspective and thanks for pointing this out. However, as I look at the definition of the elasticity of intertemporal substitution, it follows that it should be equal to $\frac1\delta$, or risk-aversion coefficient in other words. At this point I cannot see, how it can relate to $\beta$.
If I am wrong, please, let me know. It would be nice as well, if you can refer me to some literature.
epine_se
CC BY-SA 4.0
2020-10-30T16:55:24.257Z
First, I meant $\delta$, sorry.
Do I understand correctly, that for running utility it essentially means the same. But that is interesting, as $\delta$ coming from terminal wealth optimization means *risk-tolerance*, but the one from the running integral optimization of consumption is an *elasticity of intertemporal substitution* and we require them to coincide for solvability.

Am super strange seeing it for the first time. Again, would be glad if you refer me somewhere.
epine_se
<optimization><consumption><risk-aversion><elasticity-of-substitution><portfolio-theory>
How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?
epine_se
# How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?

(Asked by: username_0 on 2020-10-28)

**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.

**Tags:** optimization, consumption, risk-aversion, elasticity-of-substitution, portfolio-theory
> **Comments:**
> **username_1**: Your parametrization of CRRA via $\delta$ seems a bit strange. In any case, the $\delta(1-\delta)$ expression in consumption share $\beta$ should reflect the elasticity of intertemporal substitution.  
> **username_0**: @Michael, thanks for your answer. That is an interesting perspective and thanks for pointing this out. However, as I look at the definition of the elasticity of intertemporal substitution, it follows that it should be equal to $\frac1\delta$, or risk-aversion coefficient in other words. At this point I cannot see, how it can relate to $\beta$. If I am wrong, please, let me know. It would be nice as well, if you can refer me to some literature.  
> **username_0**: First, I meant $\delta$, sorry. Do I understand correctly, that for running utility it essentially means the same. But that is interesting, as $\delta$ coming from terminal wealth optimization means *risk-tolerance*, but the one from the running integral optimization of consumption is an *elasticity of intertemporal substitution* and we require them to coincide for solvability.  Am super strange seeing it for the first time. Again, would be glad if you refer me somewhere.  

---

## 0 answers

