Alpha, Beta & Management Style
What alpha measures
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.
🔑 Simple alpha formula (used when a question gives you the expected return directly):
Alpha = actual return - expected return
Worked example 1 — expected return given
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?
Alpha = 17% - 14% Alpha = 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.
🔑 Interpreting alpha values
| Alpha value | Meaning (as the page describes it) |
|---|---|
| Positive alpha | The fund outperformed expectations by that amount (e.g. alpha of 3 = outperformed expectations by 3%) |
| Zero alpha | The fund met expectations |
| Negative alpha | The fund underperformed expectations by that amount |
Math-based alpha questions can be more complicated and typically involve another figure: beta.
Beta — the volatility scale
Beta measures volatility as compared to the market (benchmark index).
🔑 Beta interpretation scale (exactly as described on the page):
| Beta value | Volatility vs. the market | Behavior described | Example if S&P 500 is up 10% |
|---|---|---|---|
| Beta = 1.0 | Same volatility as the market (historically) | Has generally moved in line with the market | Expected to be up about 10% (10% x 1.0) |
| Beta above 1.0 | More volatile than the market | A beta of 1.5 moves 1.5 times as much as the market | Expected to be up 15% (10% x 1.5) |
| Beta between zero and 1.0 | Less volatile than the market | A beta of 0.5 moves about half as much as the market | Expected to be up 5% (10% x 0.5) |
| Negative beta | Tends to move opposite the market | A beta of -2.0 moves about twice as much as the market, but in the opposite direction | Expected to be down 20% (10% x -2.0) |
The page’s own summary table:
| 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.
Question type 1 — inferring the market return from a beta-1.0 fund
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 × 14% = 21%
Now apply the alpha formula: Alpha = actual return - expected return Alpha = 19% - 21% Alpha = -2
An alpha of -2 means the TUV stock fund underperformed expectations by 2%.
Question type 2 — the full (risk-free-adjusted) alpha formula
Full alpha formula:
Alpha = (PR - RF) - (Beta x (MR - RF))
Where:
| Variable | Meaning |
|---|---|
| PR | Portfolio return |
| RF | Risk-free return |
| MR | Market return |
Reconstructed layout. The page’s text extraction flattened the “Where:” block into a single run (“PR RF MR = portfolio return = risk-free return = market return”). The variables and their definitions are paired in order above, per the page’s own usage in the worked example.
| Component | Explanation from the text |
|---|---|
| Portfolio return / market return | The returns for the portfolio and its benchmark |
| Risk-free rate of return | 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. |
Worked example 2 — full 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?
Answer: -4
Alpha = (PR - RF) - (Beta x (MR - RF)) Alpha = (28% - 2%) - (2.5 x (14% - 2%)) Alpha = 26% - (2.5 x 12%) Alpha = 26% - 30% Alpha = -4
This fund manager underperformed expectations by 4%, leading to an alpha of -4.
Benchmark trap. 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 should use the index most closely correlated with small-cap performance. The S&P 500 is primarily large-cap (and some mid-cap), while the Russell 2000 is a small-cap stock index. So the S&P 500 return should be disregarded for this calculation.
Alpha, beta, and management style
| Fund type | Expected alpha | Expected beta |
|---|---|---|
| Actively managed fund | Alpha is most relevant here, because active managers aim to outperform a benchmark (a relevant market index). 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 fund | Designed to match their benchmarks, so alpha should be near zero (they shouldn’t consistently over- or underperform) | A passive fund designed to track the overall market should typically have a beta near 1 |
Example given: 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.
Key points
Alpha
- Measures over or underperformance of a portfolio or security
- Positive alpha = overperformance
- Zero alpha = meeting expectations
- Negative alpha = underperformance
Beta
- Volatility measure as compared to the market (benchmark index)
Sources
Primary/official references for the material in this chapter. Every link was fetched and returned HTTP 200 on 2026-08-15.
| # | Source | Publisher |
|---|---|---|
| 1 | Markowitz/Sharpe — portfolio theory and CAPM, the source work | Nobel Prize |
| 2 | e-Handbook of Statistical Methods — mean, median, dispersion | NIST/SEMATECH |
| 3 | Fama/Shiller — efficient markets and asset-price evidence | Nobel Prize |
| 4 | Achievable Series 65 — chapter 1.3.10 | Achievable (course text) |