Calculating alpha in investing: A worked example
The formula for alpha is as follows:
Alpha = Actual return of the portfolio (Rp) - risk-free rate (Rf) - beta * market risk premium (Rm-Rf)
Rm is the return on the market, in this case, the S&P 500.
I'll work through these terms with an example. Say a fund returns 20%, and the S&P 500 returns 12%, but it invests in relatively risky stocks, so its beta is 1.1. The risk-free interest rate is 5%.
In this case, Rp=20%, Rf=5%, beta=1.1, and Rm=12%, so the market risk premium (Rm-Rf) works out to 8%.
A quick look at these numbers might lead an investor to conclude that the manager beat the index by 8% (taking away the market return of 12% from the 20% in the fund), but in reality, they took on some beta on board to do it.
Using the formula for alpha above gives alpha=20% - 5% - 1.1 * (12% - 5%), which results in an alpha of 7.3%.
In other words, the manager has added value by actively managing the portfolio, and the 20% return is solely down to taking on risk by buying high-beta stocks.
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