Utility Theory and Mean Variance Analysis

Question

I was wondering if it's pertinent to use this interpretation of the expected utility function given by the Taylor series expansion,

$${E(U(W)}\approx{U[E(W)}]+\frac{U''[E(W)]\sigma^2_W}{2}\tag{1}$$

to delineate my optimal portfolio from the set of portafolios that lie on the efficient frontier?

So, let's say I were to have the power utility function,

$$\frac{W^{1-\gamma}}{1-\gamma}\tag{2}$$ and defined final wealth as $W=W_0(1+r)$, and $\gamma=.5$, I also assume that returns are normally distributed. Can I just plug this function and its second derivative back into 1, and then find the max utility among the set of efficient portfolios, by plugging each portfolio's $[E(r),\sigma^2]$? Am I allowed to do this? Is this completely wrong? Thanks !

Answer 1

Theoretically: no.

For most practical purposes: yes; given that risks are small risks, see these lecture notes on p76.

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Belows's the background and one example showing you why you can run into problems:

Given two portfolio compositions $P_i,P_j$ with $(\mu_i,\sigma_i)$ and $(\mu_j,\sigma_j)$, the agent prefers $i$ over $j$ if

$E(u(P_i))\geq E(u(P_j))$.

In your example, the Taylor approximation reads

$$\begin{align} E\left[u(1+x)\right]&=E\left[u(1+\mu+x-\mu)\right]\\ &=E\left[u(1+\mu)+\sum_{k=1}^{\infty}\left.\frac{\partial^k u}{\partial x^k}\right|_{x=(1+\mu)}\frac{(x-\mu)^k}{k!}\right]\\ &\approx u(1+\mu)+\frac{1}{2}u''(1+\mu)\sigma^2\\ &\equiv \tilde{E}_2[u(1+x)] \end{align} $$

This approximation holds well in most cases, but it will break in corner cases as higher order terms are lost in the approximation. Note that each contribution from the higher order terms is negative; thus we can have $E_2(u)$ induce a different ordering of investment opportunities than $E(u)$, as $E_2(u)$ does not consider all higher order effects.

Here's an (somewhat contrived) example: Given the two normal distributions in the table below, we see that the second order Taylor approximation (E2) yields a preference for distribution 2 over 1, the same holds when increasing the length of the Taylor approximation up to the sixth order (E4,E6). Only after incorporating the eighth order (or more), we see that the Taylor approximation reveals the true preference relation, i.e. that distribution 1 is preferred over 2.

i    mu        sigma    E2 E4 E6  E8 ..  E(u)
1    0.0795066 0.10     -  -  -   +  ..  +
2    0.0800000 0.11     +  +  +   -  ..  -

For most practical purposes, this error should be sufficiently small. Eg when comparing investment choices, the error bound should be so tight that you can work with approximate results.

HTH?