Alpha and beta

A common way to evaluate a fund manager’s performance is by using alpha. Alpha tells you whether a fund overperformed or underperformed its expected return.

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 tended to move in line with the market. If the S&P 500 was up 10% last year, this portfolio would be expected to be 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 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, you need to compare TUV’s actual return (+19%) to its expected return.

The expected return isn’t stated directly, but you can infer the market return from BCD:

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

So, assume 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 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 fund 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 due to its short-term nature and U.S. government backing, although all securities carry some level of 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 these funds aim to outperform their benchmarks (their 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 measures how successful that effort was. If the fund outpaces the index (after adjusting for risk using beta), alpha is positive. If it lags, alpha is negative.

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