This article is one in a long series which I hope will help explain the ins and outs of DFA – Dimensional Fund Advisors. NOTE: This is my interpretation and explanation only. For the final word, please refer to the DFA Canada Website.
Continuing from Part VI, where we saw the formula for CAPM (the Capital Asset Pricing Model), we found that the model suggests that, given a diversified portfolio, your sensitivity (β, or “beta”) to the market factor (also known as the “equity factor”) should explain your expected return.
As a refresher, here is the formula:
E(Rp) = Rf + β(Rm – Rf)
Let’s look at a hypothetical 10 year period where the risk free rate was 3%, the market returned 10% and a mutual fund manager we selected to run our portfolio earned 15% on our portfolio. The portfolio was 1.5 times as volatile as the market. According to CAPM, we can calculate what performance we should have expected, given the level of sensitivity to the market:
E(Rp) = 3% + 1.5(10% – 3%)
E(Rp) = 3% + 1.5(7%)
E(Rp) = 3% + 10.5%
E(Rp) = 13.5%
So we see that CAPM says that we should have expected an annualized return of 13.5%. BUT – you’ll note that I indicated that the manager returned 15%. (N.B.: returns are assumed to be after fees.) So in other words, the expected return (Rp) was less than the actual return. This is where Alpha comes in….
Alpha is represented by the symbol “α”. If you go by some standard definitions, α represents the excess return of a manager over and above that which is expected by a benchmark or predicted return of a model. Calculating α is very simple. In the example above it’s 1.5%, which is calculated by taking the actual return (15%) and subtracting the return of the benchmark (which in our case is the return predicted by the model to account for the fact that the portfolio is 1.5 times as volatile as the market) which is 13.5%. This is where we get 1.5%. Note that it is possible to have negative alpha (in fact that seems to be the norm) – this indicates that the manager is underperforming what is expected, after having taking into account the risk adjusted return of the portfolio. To put it into a formula, the actual return of the portfolio (Rp) (and not E(Rp) which is the expected return) is as follows:
Rp = Rf + β(Rm – Rf) + α
I’m going to write it again and highlight a few terms:
Rp = Rf + β(Rm – Rf) + α
What I’ve highlighted is E(Rp), the expected return and is simply the CAPM formula from the top of this post. If we simply substitute E(Rp) into the above formula we get:
Rp = E(Rp) + α
Which is basically saying that the return on the portfolio equals the expected return plus alpha. If we re-arrange the terms to bring E(Rp) to the left hand side we get:
Rp – E(Rp) = α
Which is exactly what we defined alpha above when we said alpha “is calculated by taking the actual return and subtracting the return of the benchmark or expected return predicted by the model”.
Prevalence of CAPM
CAPM is very prevalent. If you go to morningstar, globefund, etc. you will see Beta numbers (usually 3 year Beta numbers) for every fund. Alpha is harder to find (my guess is because it’s usually negative). And when you do see a fund with positive alpha, it gets trumpeted by advisors (or by wholesalers to advisors) endlessly. Positive alpha is a good thing and is supposed to measure the excess return earned on a portfolio over and above what is predicted by CAPM.
Now that you are up to speed on CAPM, I regret to inform you that it has been all but invalidated. However, it is still used extensively in the retail financial services (and even institutionally) and is still taught in MBA schools.
Part VIII will begin to look at some data that is behind part of the research and theories behind DFA (Dimensional Fund Advisors).