Alpha and beta

A common way to evaluate a fund manager’s performance is by using alpha. Alpha tells you whether a fund’s return was higher or lower than what was expected.

When a question gives you the expected return directly, the calculation is straightforward:

$\text{Alpha}=\text{actual return - expected return}$

A question could sound 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 outperformed expectations by 3%. If the alpha were negative, the fund would have underperformed expectations by that amount. If the alpha were zero, the fund would have met expectations.

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

A portfolio with a beta of 1.0 has (historically) had the same volatility as the market. In other words, it has generally moved in line with the market. If the S&P 500 was up 10% last year, this portfolio would be expected to be up about 10% as well (10% x 1.0).

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 as much as the market. If the S&P 500 was up 10% last year, this portfolio would be expected to be 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 about half as much as the market. If the S&P 500 was up 10% last year, this portfolio would be expected to be up 5% (10% x 0.5).

Last, a portfolio with a negative beta tends to move opposite the market. A portfolio with a beta of -2.0 moves about twice as much as the market, but in the opposite direction. If the S&P 500 was up 10% last year, this portfolio would be expected to be 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?

Because alpha measures over- or underperformance, we need to compare TUV’s actual return (+19%) to its expected return.

The expected return isn’t stated directly, but we can infer the market return from BCD stock fund. The key clue is that BCD has a beta of 1.0, meaning it moves in line with the market. So we can treat BCD’s +14% return as the market return for the year.

Next, use TUV’s beta to estimate its expected return:

• Market return (from BCD): 14%
• TUV beta: 1.5
• Expected return for TUV: $1.5\times 14\%=21\%$

Now apply the alpha formula:

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

An alpha of -2 means the TUV stock fund underperformed expectations by 2%.

There’s another formula you can utilize to calculate alpha that includes a few additional components:

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

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

The portfolio return and market return are the returns for the portfolio and its benchmark. The risk-free rate of return is the return on a relatively risk-free security. The most commonly cited risk-free security is the 3 month Treasury bill. It’s considered close to risk-free because of its short maturity and U.S. government backing, although all securities carry at least some risk.

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 when evaluating an actively managed fund because active managers aim to outperform a benchmark (a relevant market index). For example, if a small-cap stock fund manager tries to select small-cap stocks that will beat the Russell 2000, alpha is a direct measure of whether that goal was achieved. If the fund outpaces the index (after adjusting for beta, and possibly the risk-free rate), alpha will be positive. If it lags, alpha will be negative.

Passively managed funds are designed to match their benchmarks, so their alpha should be near zero (they shouldn’t consistently over- or underperform). A similar idea applies to beta: a passive fund designed to track the overall market should typically have a beta near 1.