Markowitz model

Modern Portfolio Theory

Suppose we have a utility function, $u$ and some random variable, $R$ representing investment returns. If we use a Taylor series to approximate a utility function, we can show that utility is approximately equal to

Research has confirmed that this approximation is quite close for a wide variety of utility functions and return distributions Portfolio Selection: Efficient Diversification of Investments Portfolio theory: as I still see it Levy, though it can obviously be made more accurate by including more terms in the Taylor expansion. The main limitation in practice is when significant leverage is used - more on that later.

Using some mathematical properties of expectation and variance, we adjust this equation to allow for multiple linearly-related investments to achieve

where $R_i$ represents the distribution of returns for an asset and $p_i$ represents the percent of our portfolio we have invested in that asset. The $\epsilon$ coefficient represents aversion to risk. It is 0 if utility grows linearly with wealth, 1 if utility grows logarithmically with wealth, and $e+1$ if utility is $-1/x^e$.

This formula is just a multivariate quadratic, which means it can be efficiently solved with quadratic programming Quadratic programming.

This formula was derived decades ago by Harry Markowitz Harry Markowitz in his seminal paper Portfolio Selection. that virtually founded modern portfolio theory as a field.

Short v Long Run

Fine, but all of this is for a investments in a single time period return. How does this work over longer periods of time?

Well, if you assume a sequence of time periods with i.i.d. returns, then expanding the above results is relatively straightforward mathematically speaking. The result? The return, variance, and covariance parameters of the joint log-normal distribution all simply scale linearly with the number of time periods.

This clear mathematical result actually runs counter conventional wisdom: the optimal strategy for short- and long-run investing is actually the same.

Note, however, that this result is only as strong as its assumptions: that you're investing a single pot of money with no new contributions or withdrawals, and with i.i.d. investment returns.

Age & Risk

Let's relax one of those assumptions by considering a person who is still working, and so has much of their wealth tied up in human capital rather than financial capital. Let $p$ be the percent of their wealth in financial capital. Then their utility function is

$$u(R) = - ((1-p) + p \cdot e^R)^{-\epsilon}$$

where $R$ is the log of their investment returns.

Now, solving this for real is mathematically challenging, so we'll instead solve a second-order approximation:

$$u(0) = -1$$ $$u'(0) = p \epsilon $$ $$u''(0) = - p \epsilon \left( p \epsilon + p - 1 \right)$$

and so utility can be approximated with a Taylor series Taylor series. In Wikipedia:

$$u(x) \approx -1 + \epsilon p R - p \epsilon \left( p \epsilon + p - 1 \right) R^2 / 2$$

So finally, let's assume $R$ comes from a normal distribution with parameters $\mu$ and $\sigma$. Then we can approximate expected utility with

$$ E[u(x)] = -1 + \epsilon p \mu - p \epsilon \left( p \epsilon + p - 1 \right) \left( \mu^2 + \sigma^2 \right) / 2 $$

In realistic settings, $\mu^2 \lt \lt \sigma^2$, which makes this just

If we use the fact utility functions are only unique up to linear transformation, we can simplify this further:

As a sanity check, note that if $p = 1$, then we return to the original formula:

Conversely, if $p = 0$:

This may seem bizarre since this appears to suggest young people should be risk-seeking, but this is merely the result of us taking the logarithm of returns - this formula is actually equivalent to assuming $u(x) = x$. This makes perfect sense, since if you only invest a tiny portion of your wealth, your utility function is essentially linear.

This model essentially says that if you're using the more naive model we started with, then your $\epsilon$ increases from -1 early in your career, approaches your actual consumption-utility elasticity.

The above is proof is based on a second-order approximation of the utility function. My attempts to verify it via simulation have led to mixed results, so while I'm pretty confident in these broad strokes (i.e. the above paragraph), I'm less confident in th literal formula.

Criticism

This approach to investment has been criticized as "error maximization" Scherer:

This critique is fair but hardly fatal since we can make corrections to account for the measurement error. Consider first the error in our estimates of $E[R_i]$. This adds a variance equal to the square of the standard error of the estimate, so our equation becomes

However, the above only holds if our estimates of $E[R_i]$ have normal errors. Moreover, our estimates of these parameters aren't independent Estimation of covariance matrices. In Wikipedia. We can work around these problems using Monte Carlo simulations by sampling parameter values and weighing by their joint likelihood.

Still, this doesn't solve the fundamental issue: where are we getting the PDFs for these parameters? Traditionally, this has been done using historical data, but as the SEC says, "past returns are not indicative of future performance."

The other criticism is that returns don't, in fact, follow a log-normal distribution, but, as far as I can tell, this criticism is generally viewed as non-fatal in that (a) it can be overcome with more complicated math and (b) that more complicated math is closely approximated by the Markowitz model anyway.