Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

[PGP-I FM] Markowitz’s Portfolio Theory: I

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In our journey so far – from giving a justification for NPV for valuing investments to valuation of common stocks – we have been talking about risk all along, but we haven’t really done full justice to it. Fair enough, it’s been at the back of our mind in all our discussions all along, but we still don’t quite have a way to measure it.

Like for most fundamental ideas in finance, the first systematic treatment of risk can also be traced back to early/mid-twentieth century, which brings us to the third protagonist in our story.

In the early ’50s, a precocious graduate student at the University of Chicago named Harry Markowitz was looking for a suitable topic for his dissertation. After an encounter with a trader sparked an interest in financial markets, his would-be adviser suggested he read John Burr Williams’ Theory of Investment Value, whose Dividend Discount Model we learnt while talking about common stock valuation.

What struck Markowitz was that if John Burr Williams’ theory is correct, then people should buy just one stock – the one that offered the maximum possible expected return and nothing else. But, he noticed, obviously it’s not what people did (or do). To quote Peter Bernstein in his entertaining biography of finance, Markowtiz

…was stuck with the notion that people should be interested in risk as well as return.

This, of course, is nothing new to us now. We learnt that while talking about the St. Petersburg Paradox: that while valuing risky gambles we should not be looking at expected return from the gamble, but expected utility of returns from the gamble. One of the things that we noticed then was that for any risky gamble $X$, expected utility $E[U(X)]$ is always less than the utility of the sure thing $U(E[X])$ (mathematically known as the Jensen’s inequality for concave functions), i.e.:

$\mathbb{E}[U(X)] < U(\mathbb{E}[X])$

Markowitz had this figured this out too, and he exploited this idea to come up with a way to quantify the trade-off between risk and return.

Given a certain starting wealth $W$, Markowitz studied the change in expected utility to marginal investments in risky gambles.

That is, he considered the quantity $\mathbb{E}[U(W(1 + h))]$ for any small risky gamble $Wh$ relative to the starting wealth $W$.

As you may have done in your statistics courses, a useful way to think about a risky gamble is as a random variable (something that takes a different value depending on the ‘state of the world’). Since we can talk about $h$ as a random variable, we can talk about its expected value, say $\mu$, and variance, say $\sigma^2$.

With $h$ small we can evaluate $U(W(1 + h))$ as a Taylor series, and then the expected utility from wealth including the gamble can be written as:

\begin{aligned}\mathbb{E}[U(W(1 + h))] &=\mathbb{E}[U(W(1 + h))] \\& =\mathbb{E}[U(W) + WhU'(W) + \frac{1}{2}W^2h^2U''(W) + ...] \end{aligned}

With $h$ small we can ignore the the exponents of $h$ greater than $2$, and this gives us:

$\mathbb{E}[U(W(1 + h))] \approx\mathbb{E}[U(W) + WU'(W)h + \frac{1}{2}W^2U''(W){h}^2]$

Given a certain (sure) starting $W$, we can write the above as:

$\mathbb{E}[U(W(1 + h))] -\mathbb{E}[U(W)] \approx WU'(W)\mathbb{E}[\tilde{h}] + \frac{1}{2} W^2U''(W)\mathbb{E}[h^2]$

that is, the change in expected utility:

$\Delta\mathbb{E}[U(W(1 + h))] \approx WU'(W) \mu + \frac{1}{2}W^2 U''(W) (\mu^2 + \sigma^2)$

Given that $h$ is small, its expected value $\mu$ would be smaller still and we can ignore the higher powers of $\mu$ to give:

$\Delta\mathbb{E}[U(W(1 + h))] \approx WU'(W)\mu + \frac{1}{2}W^2U''(W) \sigma^2$

Since $U(W)$ is known, so are $W \mbox {and } U'(W)$, and we have:

\begin{aligned} \frac{\Delta\mathbb{E}[U(W(1 + h))]}{WU'(W)} & \approx \mu + \frac{WU''(W)}{2U'(W)} \\& \approx \mu - A \sigma^2 \end{aligned}

where $A = - \displaystyle \frac{WU''(W)}{2U'(W)} > 0$ because concavity of $U(W)$ implies $U'(W) > 0$ and $U''(W) < 0$. The coefficient $A$ defines a measure of relative risk aversion. That is, higher the value of $A$, more risk-averse the person, and lower its value more risk-loving the person. (What would be the value of $A$ for a risk-neutral person?)

That is, the change in expected utility from a marginal gamble depends only on the expected return and variance of the gamble. And expected utility goes up as the expected return from the gamble increases and goes down as variance increases.

For small gambles, then according to Markowitz people should only consider a single number to talk about risk, i.e. its variance $\sigma^2$irrespective of the number of states of the world. This turned out to be a revolutionary idea in the history of finance, and is a cornerstone in the theory of portfolio choice and asset pricing. For his efforts Markowitz was awarded the Nobel Prize in Economics in 1990.

This result, that however complex the world maybe, for small gambles people need only consider the expected return $\mu$ and variance of the gamble $\sigma^2$ will form the basis for our further discussions.

Not only did Markowitz notice that people care both about risk and return, he also observed that people held not one but a portfolio of stocks. Just on its own, the fact that people should care about expected return and variance of gambles doesn’t necessarily imply that people would hold multiple stocks. If they knew their degree of risk-aversion, they would just want to pick one that offered the ‘right’ trade-off for them.

The fact that people could and did hold a portfolio of stocks made ample economic sense. Consider the following two risky gambles:

It should be clear that by holding half of each $X_1$ and $X_2$, an investor could make his end-of-period payoff the same (= 100), irrespective of the end-of-period state of the world, i.e. the portfolio of $X_1$ and $X_2$ with equal percentage invested in each is completely risk-less.

This, of course, is an extreme example and in general such gambles would be rare that offered perfectly negatively correlated payoffs. However, Markowitz’s point had been made. As long as end-of-period payoffs are not perfectly positively correlated investors could reduce the variance or risk associated with the end-of-period payoffs by holding multiple stocks. We’ll generalize this idea in the next post.