Modern Portfolio Theory

MPT is a perfect example of relevant applications of statistics for you career-minded individuals. It also provides a constant reminder that the worthwhile parts of Finance only exist because of statistical and mathematical tools (just kidding). Whatever the case, this concept remains one of the most financial focused sections of this book.

The key to Modern Portfolio Theory is understanding that for our purposes it really boils down to two equations. The motivation behind this theory is trying to find the best portfolio allocation - how much we invest in each stock - given multiple stocks (in case you need a refresher, portfolios are made up of various stocks). These stocks have two metrics we are concerned with: returns and risk (measured here as variance). We want, then, to optimize the portfolio based on the possible stock allocations (\(\%\) invested in various stocks).

The most important thing to us in this book is the total return and risk of the portfolio as a whole. This is just an application of statistical algebra, derived from the following equations:

\[E(aX + bY) = aE(X) + bE(Y)\] \[Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)\]

Where \(a\) and \(b\) are constants, and \(X\) and \(Y\) are random variables.

We can apply this concept to portfolios via the following equations:

\[E(P) = w_1r_1 + w_2r_2\]

Where \(P\) is portfolio returns, \(w_i\) gives the weight of the \(i^{th}\) stock (how much of the portfolio is invested into one stock, could be 0 to 100\(\%\). In this case, we know \(w_1 + w_2 = 1.00\), so we could re-write the above equation using \(w_2 = 1 - w_1\), and, indeed you may often see it in this format) and \(r_i\) gives the return of the \(i^{th}\) stock (how much it makes).

This is pretty intuitive then: the overall expected return of the portfolio is the weighted expected returns of the stocks that make it up (weighted by how much they are represented in the portfolio). So, if we held a portfolio that was invested \(70\%\) in a stock that returned \(\$10\) and \(30\%\) in a stock that returned \(\$5\), the expected return of my portfolio would be \(E(P) = (.7)10 + (.3)5 = \$8.5\).

Slightly trickier is the total variance of the portfolio. Applying the second equation from above, we get:

\[Var(P) = w_1^2s_1^2 + w_2^2s_2^2 + 2w_1w_2s_{1,2}\]

Where \(w_i\) still gives weight, \(s_i^2\) gives the variance of the \(i^{th}\) portfolio and \(s_{i,j}\) gives the covariance of portfolios \(i\) and \(j\).

Take time to convince yourself that this is the same equation from above (hint: \(w_1\) and \(w_2\) are just \(a\) and \(b\)). The intuition here is that the total risk of the portfolio consists of the (weighted) risks of each stock (the first two terms) AND the interactive risks of the stock (the third term). This interactive risk is what we discussed earlier, with the oil and gas station companies example. If the two stocks are highly related (have a large covariance) then when one does poorly the other tends to do poorly (and vice versa). This risk is captured by the third term in the above equation.

So, if we have a portfolio that is \(70\%\) invested in a stock with \(\$3\) of variance and the other \(30\%\) invested in a stock with \(\$1\) variance and the two stocks have a covariance of \(\$5\), our portfolio variance would be \(.7^23^2 + .3^21^2 + 2(.7)(.3)(5) = \$6.6\).

we will only work with portfolios made up of two stocks. However, this can be applied to any number of stocks, and the general form for return and risk (as you can probably guess) is as follows:

\[E(P) = \sum\limits_{i=1}^n w_ir_i\] \[Var(P) = \sum\limits_{i=1}^n w_i^2s_i^2 + \sum\limits_{i=1}^n \sum\limits_{j=1}^n w_iw_jCov_{i,j}\]

For all \(n\) stocks.

For those following along at home, it only seems like we’ve done half the work. While it is nice to know the mean and variance of a portfolio, these are only useful if we can optimize them (pick the best allocation/weights given the stocks that make up the portfolio). This is not a book on optimization, but for the sake of interest, here is the rest of the process. If you are not interested in optimization, please feel free to skip this section.

Let’s take, for example, minimizing risk. The method would be to take the above equation and set a constraint, from which you could optimize (usually via a Lagrangian). Our constraint here would be that all the weights must add up to \(1\) (since we can’t have more than \(100\%\) of in stocks). So, the problem for two stocks becomes:

\[min \; (w_1^2s_1^2 + w_2^2s_2^2 + 2w_1w_2s_{1,2})\] \[s.t. \; \; w_1 + w_2 = 1\]

Using the Lagrangian, we would then optimize:

\[L = (w_1^2s_1^2 + w_2^2s_2^2 + 2w_1w_2s_{1,2}) - \lambda(w_1 + w_2 - 1)\] \[\frac{\partial L}{\partial w_1} = 2w_1s_1^2 + 2w_2s_{1,2} = \lambda\] \[\frac{\partial L}{\partial w_2} = 2w_2s_2^2 + 2w_1s_{1,2} = \lambda \] \[\frac{\partial L}{\partial \lambda} =w_1 + w_2 = 1\]

By setting the first two partials equal and following through with some algebra, we obtain:

\[\frac{w_1}{w_2} = \frac{s_2^2 - s_{1,2}}{s_1^2 - s_{1,2}}\]

Which we can then combine with the final partial solve the optimized value:

\[w_2^{*} = \frac{1}{\frac{s_2^2 - s_{1,2}}{s_1^2 - s_{1,2}} + 1}\]

And, of course, using our constraint:

\[w_1^{*} = 1 - w_2^{*}\]

Where \(w_i^{*}\) gives the optimal weight of the \(i^{th}\) portfolio.

Again, this sort of exercise is beyond the scope of this book; if you find it interesting, feel free to investigate an optimization-oriented economics course. For now, we will stick with our applications of the Modern Portfolio Theory: finding risk and return of a portfolio.