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:
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?
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:
That means the market return last year was +14%.
Next, use TUV’s beta to estimate TUV’s expected return:
Now apply the alpha formula:
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:
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?
Answer: -4
This fund manager underperformed expectations by 4%, leading to an alpha of -4.
One note to point out in the question: both the S&P 500 and the Russell 2000 returns were provided, but only the Russell 2000 was used.
Because the fund is a small-cap stock fund, you want the market return (MR) to come from the index that best matches (is most correlated with) small-cap performance. The S&P 500 contains large- and mid-cap stocks, while the Russell 2000 is a small-cap stock index. Therefore, the S&P 500 return should be disregarded.
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.
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.
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