#economics

Modern Portfolio Theory

The One-Investment Model

Suppose you have utility function

$$ u = -x^{-\epsilon} $$

where $x$ is your net worth, and $\epsilon$ is a parameter characterizing your risk intolerance. Now suppose you have an investment that yields a random, lognormal return:

$$ e^R $$

where $R$ is a normal random variable:

$$ R \sim Norm(\mu, \sigma^2)$$

So, your expected utility is given by

$$ u = - \int_{-\infty}^{\infty}{e^{-\epsilon x} \frac{e^{\frac{(x-\mu)^2}{-2\sigma^2}}}{\sigma \sqrt{2\pi}} dx} $$

Some algebra and calculus show that, assuming $\epsilon$ is know, this is equivalent to

$$ u = - e^{- \mu + \epsilon \sigma^2 / 2} $$

which means your utility is a monotonic function of

$$ \mu - \frac{\epsilon}{2} \sigma^2 $$

The Multi-Investment Model

Suppose you have multiple investments with returns

$$ e^{R_1}, e^{R_2}, \cdots, e^{R_n} $$

where each $R_i$ is a normal random variable. We allow these variables to be non-independent.

By the properties of expected value and variance,

$$ \mu - \epsilon \sigma^2 / 2 $$

becomes

$$ \sum_{i=1}^{n}{p_i E[R_i]} - \frac{\epsilon}{2} \sum_{i,j=1}^{n}{p_i p_j Cov(R_i, R_j)} $$

Uncertainty

However, the above all assumes you actually know the true expected returns and covariances of all your investment options. In reality, you don't. This section will expand on the base theory by adding a way to account for this uncertainty.

Unfortunately, the final results from the last two sections are only monotonically related to utility - they aren't related linearly. This makes them break down when analyzing parameter uncertainty. For this reason, we have to return to the function linearly related to utility:

$$ u = - e^{- \mu + \epsilon \sigma^2 / 2} $$

Suppose have uncertainty for our estimate of $\mu$ and $\sigma^2$. For simplicity, let's assume $\mu$ comes from a normal distribution with mean $a$ and standard deviation $b$, and that $\sigma^2$ comes from a normal distribution with mean $c$ and standard deviation $d$.

The our expected utility is

$$ u = \int_{-\infty}^{+\infty}{\int_{-\infty}^{+\infty}{- e^{- x + \epsilon y / 2} \frac{e^{\frac{(x-a)^2}{-2b^2}}}{b\sqrt{2\pi}} \frac{e^{\frac{(y-c)^2}{-2d^2}}}{d\sqrt{2\pi}} dx}dy} $$

which is equivalent (under linear transportation) to

$$ u = -\frac{1}{bd} \int_{-\infty}^{+\infty}{\int_{-\infty}^{+\infty}{e^{- x + \epsilon y / 2} e^{\frac{(x-a)^2}{-2b^2}} e^{\frac{(y-c)^2}{-2d^2}} dx}dy} $$

Some more involved calculus and algebra shows this means utility is equivalent (up to linear transformation) to

$$ -e^{-a + \frac{1}{2}b^2 + \frac{\epsilon}{2}c + \frac{\epsilon^2}{8}d^2} $$

In practice the two uncertainty terms ($b$ and $d$) are frequently negligible. For instance, if look at the SP500's real returns from the last 125 years and assume $\epsilon = 0.35$, those two terms collectively reduce our utility by the equivalent of 0.02% reduced annual returns.

Both measures of risk shrink inversely as our sample grows. So, for instance, reducing our analysis to 10 years would make these terms grow to about 0.25%. From this we can see that these terms can matter for assets with significant uncertainty, but for old, broad asset classes, they typically are negligible.

Finally, the above obviously can be monotonically transformed to

$$ a - \frac{1}{2}b^2 - \frac{\epsilon}{2}c - \frac{\epsilon^2}{8}d^2 $$

Now all of this is a little bit bullshit. In practice, your PDFs over expected returns and risk will almost certainly not be normal distributions (T-student and chi-square distributions are much more likely). However, I'd be surprised if the results differed much.

One last tidbit, suppose you were estimating these parameters based on i.i.d. historical returns, then

$$ a = \bar{x} $$ $$ b^2 = s^2/(n-1) $$ $$ c = s^2 $$ $$ d^2 = 2 s^2 / (n-1) $$

So the formula becomes

$$ a - \frac{\epsilon}{2}s^2 - \frac{2 + \epsilon^2}{4(n-1)}s^2 $$

or

$$ a - \frac{\epsilon}{2} s^2 \left( 1 + \frac{2 + \epsilon^2}{2\epsilon(n-1)} \right) $$

which is just the original one-investment formula but with a small uncertainty adjustment

Wikipedia contributors. (2020, November 11). Estimation of covariance matrices. In Wikipedia, The Free Encyclopedia. Retrieved 20:58, December 20, 2020, from https://en.wikipedia.org/w/index.php?title=Estimation_of_covariance_matrices&oldid=988126642