Continuing from Part I in our discussion 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

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 underline a few terms:

Rp = Rf + β(Rm - Rf) + α

What I’ve underlined 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”.

If you have found any information on this site to be useful, please feel free to sign up to our newsletter by filling out the form located on the right hand side of the page. The newsletter is free and will alert you to site updates, including modified tools and articles that you can use with your clients and for prospecting.

Commissions, trailing commissions, management fees and expenses all may be associated with mutual fund investments. Please read the prospectus before investing. Any indicated rates of return are historical annual compounded total returns, including changes in share or unit values and reinvestment of all dividends or distributions, and do not take into account sales, redemption, distribution or optional charges or income taxes payable by any security holder that would have reduced returns. Mutual funds are not covered by the Canada Deposit Insurance Corporation or by any other government insurer. Mutual funds are not guaranteed and their values may change frequently. Past performance may not be repeated.