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 |
|---|---|
| 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:
| 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) |
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 |
| Component | Explanation |
|---|---|
| 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)
Practice questions
Q1. What does alpha measure?
Show answer
Whether a fund’s return was higher or lower than expected. Alpha = actual return − expected return.
Q2. A fund’s expected return was 14% and actual return was 17%. What is alpha?
Show answer
+3. The fund outperformed expectations by 3%.
Q3. If the S&P 500 rises 10%, what return would you expect from a fund with beta 1.5?
Show answer
Up 15% (10% × 1.5). Beta above 1.0 means more volatile than the market.
Q4. BCD fund has beta 1.0 and returned +14%. TUV fund has beta 1.5 and returned +19%. What is TUV’s alpha?
Show answer
−2. Market return = 14% (from beta-1.0 BCD). Expected TUV return = 1.5 × 14% = 21%. Alpha = 19% − 21% = −2 (underperformed by 2%).
Q5. What is the full risk-adjusted alpha formula?
Show answer
Alpha = (PR − RF) − [Beta × (MR − RF)], where PR = portfolio return, RF = risk-free return (commonly 3-month T-bill), MR = market return.
Q6. A small-cap fund returned 28% with beta 2.5. Russell 2000 was up 14%, S&P 500 up 10%, and the 3-month T-bill gained 2%. What is alpha?
Show answer
−4. Use Russell 2000 as MR (small-cap benchmark, not S&P 500). Alpha = (28%−2%) − [2.5×(14%−2%)] = 26% − 30% = −4.
Q7. What beta would you expect from a passive fund tracking the overall market?
Show answer
Near 1.0 — same volatility as the benchmark.
Q8. For actively vs. passively managed funds, how should alpha typically behave?
Show answer
Active managers aim to beat the benchmark, so alpha can be positive or negative. Passive funds designed to match benchmarks should have alpha near zero.
Q9. A fund has beta −2.0 and the market rises 10%. What is the expected portfolio return?
Show answer
Down 20% (10% × −2.0). Negative beta moves opposite the market.
Sources
Primary/official references for the material on this page. 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) |