A common way to evaluate a fund manager’s effectiveness is by using alpha. Alpha tells you whether a fund’s return was higher or lower than what you would have 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 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 outperformed expectations by 3%. If alpha is negative, the fund underperformed expectations by that amount. If alpha is zero, the fund performed exactly as expected.

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 is expected to move 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 is expected to move 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 is expected to move opposite the market. A portfolio with a beta of -2.0 is expected to move 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:

S&P 500 return	Portfolio beta	Portfolio return
Up 10%	1.0	Up 10%
Up 10%	1.5	Up 15%
Up 10%	0.5	Up 5%
Up 10%	-2.0	Down 20%

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?

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

The expected return isn’t stated directly, but the question gives you enough information to infer it:

• BCD has a beta of 1.0, which means it’s expected to move in line with the market.
• So BCD’s return (+14%) is being used as a proxy for the market return.

That means the market return last year was +14%.

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

• TUV’s beta is 1.5, so it’s expected to move 1.5 times as much as the market.
• Expected return = $1.5\times 14\%=21\%$

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 involving a few new components. Here it is:

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

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

The portfolio return and market return are usually given in the question. 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 very close to risk-free due to its short-term nature 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 is trying to beat the Russell 2000 by selecting small-cap stocks, alpha is a useful way to measure whether those choices added value.

• If the fund outpaces its benchmark (after adjusting for risk), alpha is positive.
• If it lags its benchmark, alpha is negative.

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