Exponential Utility for Choosing Portfolio Investment Size

# Desired portfolio volume based on utility theory


# 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?

## Answer by quasi (score 3)

https://quant.stackexchange.com/a/8013

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.