# Desired portfolio volume based on utility theory

I am working on a toy model, in part of which an investor has to decide (based on some utility theory) how much money to invest in a given portfolio. For simplicity, assume that the portfolio is already constructed, it has an expected return $\mu$ and the volatility $\sigma$ which are known to the investor. If the investor invests $x$ in the portfolio, he gets $(1+\rho)x$ on the next step where $$\rho \sim\mathscr N(\mu,\sigma^2)$$ is a stochastic return on the investment. Suppose, that at the current moment the investor has $X$ as his capital. Are there any formulas from the utility theory on how to compute the desired level of investments given $X,\mu$ and $\sigma$ - and perhaps some additional parameters such as risk aversion of the investor?

• Have you tried with exponential utility $U(x) = -e^{-\lambda x}$ and power utility $\frac{1}{p}x^p$? – quasi May 17 '13 at 13:36
• @quasi: I am not very familiar with utility theory, so can you elaborate on how to apply your advice? – Ilya May 17 '13 at 13:37

One approach is to use an exponential utility function: $U(x) = -e^{-\lambda x}$. Here, $\lambda$ records what is known as the absolute risk aversion. Exponential utility functions are nice because they have a wealth independence property (of course, this may be seen as a drawback). As we will see below, the initial capital $X$ plays no part in the optimal investment decision. This decision only depends on $\lambda$. Let's consider the agent's utility after investing $x$ dollars. This is
$$U(X + \rho x) = -e^{-\lambda (X + \rho x)} = e^{-\lambda X} \cdot \left( -e^{-\lambda \rho x} \right).$$ The first term above does not depend on $x$, and is positive. So, we only have to optimize (minimize) the second term over $x$. This is the wealth independence property. The second term is $$-\int_{-\infty}^\infty e^{-\lambda x y} e^{\frac{-(y - \mu)^2}{2 \sigma^2}}dy.$$ We can evaluate this analytically by completing the square, yielding $$-\int_{-\infty}^\infty e^{\frac{-(y - \mu + \sigma^2 \lambda x)^2}{2 \sigma^2}} e^{-\mu \lambda x + \frac{\sigma^2 \lambda^2 x^2}{2}} dy = -e^{-\mu \lambda x + \frac{\sigma^2 \lambda^2 x^2}{2}}.$$ The right hand side above is maximized when $-\mu \lambda x + \frac{\sigma^2 \lambda^2 x^2}{2}$ is minimized.
Differentiating, we achieve the optimal $x^* = \frac{\mu}{\sigma^2 \lambda}$. This makes sense at least qualitiatively. We invest more when $\mu$ is higher, lower when $\sigma^2$ is greater, and lower when $\lambda$ (our level of risk aversion), is higher.
• An another approach is to use the power utility functions. This family of functions (which includes $\log x$), will have the property of constant relative risk aversion, meaning that a constant proportion of wealth will be invested. – quasi May 17 '13 at 15:00
• thanks a lot for the answer, however I am more interested in case when the capital $X$ matters. Shall I look into power utility functions? – Ilya May 17 '13 at 15:02
• @quasi: If I do the same computation for $U(X)=\frac{1}{p}x^p$ then I will get: $$\int_{-\infty}^\infty U(X+\rho x) = \frac{1}{2}\left( (X+x \mu)^2+x^2\sigma^2 \right)$$ for $p=2$, which cannot be maximized. What do you suggest as values for $p$? – user5396 May 23 '13 at 14:22
• @Pepijn: Usually, models for utility functions are assumed to be concave in $x$. This leads to the requirement that $p<1$. Note the $p=0$ case corresponds to $\log x$. – quasi May 23 '13 at 21:55