Mean-Variance Utility: Convenience, Approximations, and Limits

# Why do we assume quadratic utility in portfolio theory?


# Why do we assume quadratic utility in portfolio theory?












In my text (Investments by BKM), the investor's mean-variance utility (given as $U = E[R] - \frac12A\sigma^2$) is stated to be the objective function we wish to maximize. Upon further digging, it seems that this stems from the assumption of quadratic utility functions ($U = aW - bW^2$). This kind of bothers me since I see two unrealistic properties for quadratic utility functions. (1) They exhibit increasing absolute risk aversion, and (2) they achieve a satiation point, beyond which money/return begins to have negative value.

So why do we assume quadratic utility? Are there no other simple, more realistic functional forms for utility that would still lead to a reasonably clean portfolio optimization theory? Or are the issues I cited about the quadratic just negligible in practice?

## Answer by markowitz (score 4)

https://quant.stackexchange.com/a/26355

The assumption of quadratic utility function is convenient in portfolio theory because it is possible to demonstrate that if the portfolio returns are not normally distributed, the mean-variance approach is still best (best in the sense that any other distributional properties is amenable into mean and variance.) Conversely, if the return are normally distributed, the choice of utility function is irrelevant. More generally, if the portfolio return distributions are not known and we use a general utility function, the mean-variance approach is valid yet, but only as approximation.

## Answer by user20429 (score 2)

https://quant.stackexchange.com/a/25740

What you learn in school are models, meant to illustrate the concepts and methods of the field. Later you will learn about other forms of utility functions (power utility most prominently). With such families of utility functions the computations aren't as clean as with quadratic utility, but by then you will have understood the concepts and methods, and you will understand the approximate methods that you will need to use at that stage.

With that said, Markowitz tried, in a few papers, to explain why approximating one's utility function by a quadratic utility function makes sense in some cases. Not very convincing in my opinion, but it's out there.

## Answer by LazyCat (score 2)

https://quant.stackexchange.com/a/26359

In most settings, utility functions are defined up to an affine transformation: if $u(x)$ defines the preference of an investor, then so does $a*u(x)+b.$ This implies, you can normalize the Taylor expantion of any smooth utility function to $u(x)=x+a*x^2+\ldots$ around 0. So the next step is just to drop off higher order terms. The investor is also usually assumed to be risk-averse, which implies, that $a < 0.$ You can check the details, e.g. here: https://www.empiwifo.uni-freiburg.de/lehre-teaching-1/winter-term-10-11/materialien-portfolio-analysis/utility.pdf

## Answer by Fab (score 2)

https://quant.stackexchange.com/a/70866

The capital asset pricing model (CAPM) is based on mean-variance utility; investors choose their portfolio based only on its mean and variance.

This is an entirely different approach than expected utility maximisation.

You can express the expected utility using its Taylor approximation around the expected wealth, with $\mu = E[W], \Delta = W - \mu$ (note that $E[\Delta]=0$):

\begin{align} E[U(W)] &= E[U(\mu + \Delta)] \\ &= E\left[U(\mu) + \Delta \cdot U' + \frac 1 2 \Delta^2 U'' + \frac 1 6 \Delta^3 U''' + ...\right] \\ &= U(\mu) + \frac 1 2 \sigma^2 U''(\mu) + \frac 1 6 E[\Delta^3] U''' + ... \end{align}

Now, to turn this into a mean-variance utility (ie only a function of $\mu, \sigma$), there are 3 ways:

- Impose conditions on the utility function: Take an expected utility function $U$ whose higher derivatives (above second derivative) vanish. That is the idea with quadratic utility (which, as you point out, has many problems).

- Impose conditions on the returns, ie the distribution of $W$. It can be shown that one can express the above as a function of only $\mu, \sigma$ for elliptical distributions, no matter what $U$ is chosen. Those can have a restricted domain (ie limited liability, though then also limited upside, if I'm not mistaken).

- Impose joint conditions on both the expected utility function and the distribution of $W$. That's complicated.

> So why do we assume quadratic utility?

Easy to handle analytically.

> Are there no other simple, more realistic functional forms for utility that would still lead to a reasonably clean portfolio optimization theory?

Mean-variance utility has many problems. For example, a mean-variance investor could refuse a gift of a limited liability asset, if it's volatile enough. It doesn't make much sense.

Not aware of a clean portfolio optimisation theory under more realistic assumptions.

> Or are the issues I cited about the quadratic just negligible in practice?

Oh no, they are real. But mean-variance analysis or CAPM generally are hardly used in practice, but rather as conceptual tools, I'd say.