# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

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

With Markowitz having shown us that only expected return and variance of the gambles matter, we need not restrict ourselves to  considering 120-80 kind of gambles with only two possible states of the world. The only thing we need is estimates of expected return and variance of the gambles – and we can study their combinations more generally. Let’s do that now.

Consider two stocks $X$ and $Y$ with expected returns $\mu_X \mbox{ and } \mu_Y$, variances $\sigma^2_X \mbox{ and } \sigma^2_Y$ and correlation $\rho_{XY}$ between them. If we consider an investor with unit wealth,  with amount $\omega_X$ invested in stock $X$, and $\omega_Y = 1 - \omega_X$ invested in stock $Y$  then the portfolio expected return $latex\mathbb{E}[R_p]$ and portfolio variance $\sigma^2$ are easily obtained using some basic results from probability theory as:

\begin{aligned}\mathbb{E}[R_p] &= \omega_X \mu_X + \omega_Y \mu_Y \\ &= \omega_X \mu_X + (1 - \omega_X) \mu_y \\& = \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \\ \sigma^2 &= \omega^2_X \sigma^2_X + \omega^2_Y \sigma^2_Y + 2 \omega_X \omega_Y \rho \sigma_X \sigma_Y \\& = \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \\& < (\omega_X \sigma_X + (1 - \omega_X) \sigma_Y)^2 \end{aligned}

It should be clear that the relationship between the portfolio weight in any asset and portfolio expected return is linear, and that between portfolio weight and variance is quadratic. Our purpose, however, is to look at the trade-off between expected return and variance of the portfolio.

We are lucky that the relationship between expected return and weight in any asset is linear so we can eliminate the weights and express expected return as a quadratic function of variance. The algebraic expression is messy and lacks intuition, but for any given level of $\rho$, it can shown (and as we did in the class using Excel) that shape of the trade-off is something like this:

Efficient Frontier: Two Assets

That is, the opportunities available to an investor is a concave envelope. And this envelope captures the trade-off between expected return and risk available from the portfolio (geometrically speaking, it is a conic section – you can do the math and check which one!).

It should be clear that to a rational investor all the points below the minimum variance point should be inferior – as all those points represent a lower expected return for any given level of risk. That is, no rational investor would prefer to choose a portfolio that lie below the minimum variance point.

The envelope traced by the upper arm of the curve above the minimum variance point is called the Opportunity Set (or Efficient Set or Efficient Frontier). This is the set of opportunities available to a rational investor given the securities available in the market.

So, according to Markowitz all investors should choose one of the portfolios lying on the Efficient Frontier depending on their degree of risk aversion. So, if an investor is risk loving he/she should choose one of the points on the top right end (high risk, high expected return), or if he/she is risk-averse choose one of the points on the bottom left part of the frontier, but never below  the Frontier.

In a two-asset world it was easy to visually identify the Efficient Frontier – but for an $N$ asset world, the Markowitz portfolio selection boils to solving the following Quadratic Programming problem:

$\displaystyle \max_{\omega} \sum_{i = 1}^{N} \omega_i \mu_i$

$\mbox{ s.t. } \displaystyle \sigma^2 = \sum_{i=1}^{N} \sum_{j = 1}^{N} \omega_i \omega_j \sigma_i \sigma_j = c$

or alternatively,

$\displaystyle \min_{\omega} \sum_{i=1}^{N} \sum_{j = 1}^{N} \omega_i \omega_j \sigma_i \sigma_j$

$\displaystyle \mbox{ s.t. } \sum_{i = 1}^{N} \omega_i \mu_i = c$

And how does the Efficient Frontier looks like for the case of $N$ assets? As expected, all the set of opportunities available increase. But, importantly, luckily for us, Robert Merton showed that the Efficient Frontier retains the same concave shape whatever be the number of securities in the market. In general, the shape looks something like the following (from your book):

Efficient Frontier: Multiple Assets (Click on the graph to zoom; Source: Brealey-Myers, 9th Ed.)

Again, according to Markowitz, no investor should be on any point below the “pink line” (Efficient Frontier, traced by ABCD), i.e. in the shaded region, as all points on the curve ABCD offer a higher expected return for any given level of risk / variance.

Given that expected utility is also a function of expected return and variance, by clubbing the two together Markowitz had solved the Portfolio Selection problem for a rational investor. So, if an investor were risk-averse he/she would choose a portfolio like C or D, and if one were risk-loving then he/she would choose a portfolio like B or A, but never anything below the curve ABCD.

Special Cases

In the two-stock world two special cases are interesting.

Consider two stocks $X$ and $Y$ with expected returns $\mu_X \mbox{ and } \mu_Y$, variances $\sigma^2_X \mbox{ and } \sigma^2_Y$ and correlation $\rho_{XY}$ between them. We had the following relationships for the portfolio variance from the two stocks:

\begin{aligned} \mathbb{E}[R_p] &= \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \\ \sigma^2 &= \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \end{aligned}

1. Perfect positive correlation: $\rho = 1$

Setting $\rho = 1$ doesn’t change the expected return $latex\mathbb{E}[R_p]$, but simplifies the portfolio variance to:

\begin{aligned} \sigma^2 &= (\omega_X \sigma_X + (1 - \omega_X) \sigma_Y)^2 \\ \Rightarrow \sigma &= \omega_X \sigma_X + (1 - \omega_X) \sigma_Y \end{aligned}

i.e. the portfolio standard deviation is just a weighted average of the standard deviation of the two assets. If the two stocks are perfectly positively correlated, that is they move in lock-step in the same direction all the time, it’s as if they are the two same stocks.

2. Perfect negative correlation:$\rho = -1$

Again, setting $\rho = -1$ doesn’t change the expected return $latex\mathbb{E}[R_p]$, but simplifies the portfolio variance to:

\begin{aligned} \sigma^2 &= (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y)^2\\ \Rightarrow \sigma &= \pm (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y) \end{aligned}

While even in this case the portfolio standard deviation is just a weighted average of the standard deviation of the two assets, there are two possibilities (two roots) given the magnitude of $\sigma_X \mbox{ and } \sigma_Y$. While mathematically there are two possibilities, as the graph below shows us, economically there is only one possibility.

What’s more interesting, however, is that when $\rho = -1$we can reduce the portfolio variance to zero. How is that? We have:

$\sigma = \pm (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y)$

And setting,

\begin{aligned} \omega_X &= \frac{\sigma_Y}{\sigma_X + \sigma_Y} \\ \\ \Rightarrow \sigma &= 0 \end{aligned}

That is, when there is perfect negative correlation (recall our earlier 120-80 example), by appropriately allocating our wealth in the two stocks we can reduce our portfolio variance to 0, i.e. remove all risk.

In general, depending on whether the value of the correlation is such that $\rho = 1$ or $\rho = -1$, or $|\rho| < 1$, the efficient frontier changes as below:

Efficient Frontier: Two Assets (Special Cases)

And here is Markowitz on Markowitz.