Alpha and beta

A common way to determine the effectiveness of a fund manager is through the use of alpha. When calculated, alpha determines whether a fund is over or underperforming expectations. If a test question provides the expected return of a fund, the calculation is fairly simple:

$\text{Alpha}=\text{actual return - expected\hspace{0.33em}return}$

A question could sound something like this:

An investor determines the expected return of a large-cap stock mutual fund over a year to be +14%. At the end of the year, the actual return was +17%. What is the alpha of the fund?

$\text{Alpha}=\text{17\% - 14\%}$
$\text{Alpha}=\text{3}$

A positive alpha of 3 means the fund is over-performing expectations by 3%. This is a sign the fund manager is doing a good job of managing the portfolio. If the alpha was negative, the fund would be underperforming expectations by the amount of the alpha. If the alpha was zero, the fund would be meeting expectations.

Math-based alpha questions can be more complicated, and typically involve another figure - beta.

A portfolio with a beta of 1.0 has the same volatility as the market, historically speaking. Meaning, this portfolio has generally followed the market in the past. If the S&P 500 was up 10% last year, this portfolio was up 10% (10% x 1.0) as well.

A portfolio with a beta above 1.0 is more volatile than the market. A portfolio with a beta of 1.5 moves 1.5 times faster than the market. If the S&P 500 was up 10% last year, this portfolio was up 15% (10% x 1.5).

A portfolio with a beta between zero and 1.0 is less volatile than the market. A portfolio with a beta of 0.5 moves at half the speed of the market. If the S&P 500 was up 10% last year, this portfolio was up 5% (10% x 0.5).

Last, a portfolio with a negative beta moves opposite to the market. A portfolio with a beta of -2.0 moves at twice the speed of the market, but in the opposite direction. If the S&P 500 was up 10% last year, this portfolio was down 20% (10% x -2.0).

Here’s a table summarizing what we just discussed:

There are two types of math-based questions involving both alpha and beta to be aware of. First, let’s explore this question:

An investor is comparing two different funds in an investment analysis. BCD stock fund maintains a beta of 1.0, while TUV stock fund maintains a beta of 1.5. Last year, BCD stock fund’s performance was +14%, while TUV stock fund’s performance was +19%. What is TUV stock fund’s alpha last year?

Given alpha is a measurement of over or underperformance, we must compare the performance of TUV stock (+19%) to its expected performance. The expected performance is not explicitly provided, but we can make an assumption based on the information provided on BCD stock fund. The only reason it’s included in the question is to tell you the performance of the market in a sneaky way. Remember, a beta of 1 means the investment’s volatility is equal to the market. We can safely assume the market (assumptively the S&P 500) performed equally to BCD stock fund, therefore the market return last year was +14%.

TUV stock fund maintains a beta of 1.5, meaning it historically has moved 1.5 times faster than the market. Because beta is positive, we can assume it’s moving in the same direction as the market. With that information, we can take the beta (1.5) and multiply it times the assumptive market return (14%). This tells us the expected return of TUV stock is 21% (1.5 x 14%).

Now, we can use the original alpha formula:

$\text{Alpha}=\text{actual return - expected\hspace{0.33em}return}$
$\text{Alpha}=\text{19\% - 21\%}$
$\text{Alpha}=\text{-2}$

An alpha of -2 means the TUV stock fund underperformed expectations by 2%. The fund manager hopefully will do a better job the following year!

There’s another formula you can utilize to calculate alpha involving a few new components. Here it is:

$\text{Alpha}=\text{(PR - RF) - (Beta\hspace{0.33em}x (MR - RF))}$

$\begin{array}{rl}\text{Where:}& \\ PR& =\text{portfolio\hspace{0.33em}return}\\ RF& =\text{risk-free\hspace{0.33em}return}\\ MR& =\text{market\hspace{0.33em}return}\end{array}$

The portfolio return and market return should be self-explanatory. The risk-free rate of return measures the return on a relatively risk-free security. The most commonly cited risk-free security is the 3 month Treasury bill. It’s very close to being completely free of risk due to its short-term nature and US government backing, although all securities come with at least some risk potential.

Here’s an example of a question involving this formula:

An investor is analyzing the market and the returns of a small-cap stock fund held in their portfolio. The fund was up 28% while maintaining a beta of 2.5 last year. During the same year, the S&P 500 was up 10%, the Russell 2000 was up 14%, and the 3-month Treasury bill gained 2%. What is the small-cap stock fund’s alpha?

This is a tough question, but can you figure it out using the formula above?

Alpha is most relevant in determining the effectiveness of an actively managed fund because these fund types aim to outperform their benchmarks (their relevant market index). If a small-cap stock fund manager seeks to pick the top small-cap stocks in the Russell 2000, alpha serves as a good measure of their successes or failures. If the stocks they choose outpace the index on average, they’ll attain a positive alpha. The higher the alpha, the better their investments are performing. And vice versa.

Passively managed funds are built to match the performance of their benchmarks, and therefore should maintain alpha values near zero (meaning they don’t over or underperform the market). The same concept applies to beta as well; passively managed funds should maintain a beta near 1 (meaning they move at the same volatility as the market).