Alpha Calculator
Calculate Jensen's Alpha — the excess return a portfolio generates beyond what CAPM predicts for its level of risk.
Alpha (Jensen's α)
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CAPM Expected Return
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Excess Return (above risk-free)
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Equity Risk Premium
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Calculation Details
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How to use this calculator
Enter the Portfolio Return, Risk-Free Rate, Portfolio Beta, and Market Return (benchmark return for the same period).
The calculator computes:
- Jensen’s Alpha: the excess return above what CAPM predicts given the beta
- CAPM Expected Return: what CAPM says the portfolio should have earned
- Excess Return: total return above the risk-free rate (before beta adjustment)
- Equity Risk Premium: market return minus risk-free rate
A fund returned 18%, market returned 12%, risk-free rate is 5%, portfolio beta is 1.4
Market risk premium = 12% − 5% = 7% CAPM expected return = 5% + 1.4 × 7% = 5% + 9.8% = 14.8% Alpha = 18% − 14.8% = +3.2%
The fund earned 3.2% more than its risk (beta 1.4) would have predicted. This is positive alpha, meaning genuine value added over what a levered passive index would have delivered.
The Jensen’s Alpha formula
Where:
- α = Jensen’s Alpha (the value-add above CAPM)
- Rp = Portfolio return
- Rf = Risk-free rate
- β = Portfolio beta (sensitivity to market)
- Rm = Market / benchmark return
The term in brackets is the CAPM expected return, which is what the portfolio should have earned given its risk level. Alpha is everything above (or below) that expectation.
Positive alpha: the portfolio manager (or strategy) added value beyond passive risk-taking. Negative alpha: the manager destroyed value relative to a passive beta-equivalent allocation.
What the data actually shows about alpha generation
Despite widespread claims of alpha generation, large-scale data is sobering:
| Category | % With Positive Alpha (10-yr, after fees) |
|---|---|
| U.S. large-cap active funds | ~12–18% |
| U.S. small-cap active funds | ~25–35% |
| International equity active funds | ~20–30% |
| Fixed income active funds | ~30–40% |
| Hedge funds (reported) | ~35–45% |
SPIVA data (S&P Global, annual) consistently shows that after fees, 80–88% of U.S. large-cap active managers underperform their benchmark over 15-year periods.
Alpha is like athletic talent; it exists, but only in a minority of practitioners, and it is not distributed randomly. Identifying it prospectively from past performance alone is statistically very difficult. A 5-year track record of positive alpha is statistically consistent with luck.
Why outperforming the market does not always mean positive alpha
A fund can outperform its benchmark in absolute return terms while still having negative alpha if the outperformance is explained by extra risk (higher beta).
Example: Two funds in a year the S&P 500 returns 15%, risk-free rate is 5%
| Fund | Return | Beta | CAPM Expected | Alpha | True Performance |
|---|---|---|---|---|---|
| Fund A | 20% | 1.5 | 5% + 1.5×10% = 20% | 0.0% | Exactly CAPM-predicted |
| Fund B | 18% | 1.0 | 5% + 1.0×10% = 15% | +3.0% | Genuine outperformance |
| Fund C | 22% | 2.0 | 5% + 2.0×10% = 25% | −3.0% | Underperformed for risk |
Fund C had the highest raw return (22%) but negative alpha (−3%). It underperformed what a 2× leveraged S&P 500 position would have returned. Fund B had a lower absolute return but the best alpha; it outperformed without taking extra beta risk.
How CAPM breaks down every portfolio return into its components
Every portfolio return can be decomposed into three components:
| Component | Source | Controllable? |
|---|---|---|
| Risk-free rate | T-bill / short-term yield | No |
| Beta × Market Premium | Exposure to market risk | Yes, through allocation |
| Alpha | Skill, strategy, or luck | Limited |
For most long-term investors, the beta component explains 90%+ of returns. Alpha (genuine manager skill) is small, rare, and erodes rapidly as the manager’s assets under management grow.
This is the core case for index investing: capture the beta component cheaply (0.03% expense ratio), and avoid paying high fees for alpha that statistically does not persist.
Alpha across market cycles
Alpha isn’t constant; it varies by market environment and is often cyclical:
Growth environments (rising markets): High-beta active managers appear to have negative alpha because their beta alone explains the return. Passive, market-return-focused managers look best.
Value environments (cheap markets after crashes): Skilled stock pickers can identify mispriced securities, often generating genuine positive alpha. The 2009–2011 period saw significant active alpha for value managers.
Low-return environments: When market returns are low (near the risk-free rate), the equity risk premium is small. Alpha matters more because it is a larger fraction of total return. Small absolute alpha (1–2%) is more significant when the market only returns 4%.
| Market Environment | Active Alpha Frequency | Notes |
|---|---|---|
| Bull market (low dispersion) | Low | Passive hard to beat |
| Bear market / recovery | Higher | Stock selection rewarded |
| High dispersion markets | Higher | Individual stock moves diverge |
| Low volatility, rising | Low | Beta drives everything |
How Treynor ratio and information ratio complement Jensen’s alpha
Treynor Ratio = (Rp − Rf) / Beta
Similar to Sharpe but uses beta (systematic risk) instead of total standard deviation. It measures efficiency of beta use. Useful for evaluating one component of a diversified portfolio.
Information Ratio = Alpha / Tracking Error
Measures alpha per unit of tracking error (deviation from benchmark). Used to evaluate active managers. A ratio above 0.5 is considered good; above 1.0 is exceptional.
| Metric | Numerator | Denominator | Best Use |
|---|---|---|---|
| Sharpe | Excess return | Total SD | General risk-adjusted comparison |
| Sortino | Excess return | Downside SD | Asymmetric strategies |
| Treynor | Excess return | Beta | Systematic risk evaluation |
| Alpha (Jensen’s) | Excess above CAPM | (absolute) | Manager skill assessment |
| Information Ratio | Alpha | Tracking error | Active vs. passive decision |
The bottom line
Jensen’s Alpha answers the question every investor should ask of any active manager or strategy: “Did you earn your return through skill, or through taking more risk?”
A strategy with high alpha in a good period could be:
- Genuine skill (hard to find, worth paying for)
- Beta mislabeled as alpha (common)
- Survivor bias (many funds closed; you only see the winners)
- Luck (5 years of good performance in 80% of random strategies)
Use alpha alongside Sharpe ratio, beta, and long-run track records to build a more complete picture. The Beta Calculator and Portfolio Beta Calculator are the natural companions for this analysis.
Frequently Asked Questions
What is Jensen's Alpha?
Jensen's Alpha (α) is the excess return a portfolio generates beyond what is predicted by CAPM for its level of systematic risk. α = Rp − [Rf + β × (Rm − Rf)]. Positive alpha indicates outperformance; negative alpha indicates underperformance on a risk-adjusted basis.
What is a good alpha?
Any positive alpha is technically good — it means the manager added value above what the risk warrants. Sustained alpha above 2–3% per year is considered exceptional and rare. Most actively managed funds deliver negative alpha after fees over 10-year periods.
What is the difference between alpha and total return?
Total return is the raw performance number. Alpha is the risk-adjusted excess return — what the portfolio earned above and beyond what a passive investment with the same beta would have earned. A fund returning 15% might have negative alpha if its beta implied a 17% CAPM return.
Can alpha be negative?
Yes. A fund with negative alpha underperformed its CAPM benchmark. This is common among active managers after fees. A fund returning 10% with beta 1.2 when CAPM predicted 12% has alpha of −2% — it took more risk than the market but was rewarded less.
What is the risk-free rate to use?
Use the 3-month U.S. Treasury bill rate for U.S. portfolios, or the relevant short-term government rate for other countries. The risk-free rate should match the portfolio's base currency and time horizon.
What market return should I use?
Use the return of the benchmark most relevant to the portfolio's strategy — S&P 500 for U.S. large-cap, Russell 2000 for small-cap, MSCI World for global equity. The benchmark should be the portfolio's primary source of beta.
How is alpha related to the Sharpe ratio?
Both are risk-adjusted metrics but measure different things. Alpha measures return relative to CAPM using beta (market risk). Sharpe measures return relative to the risk-free rate per unit of total volatility. A high-Sharpe portfolio may or may not have positive alpha depending on its beta.
Why do most fund managers have negative alpha?
After management fees (0.5–1.5% annually), transaction costs, and market impact, it is mathematically difficult to beat a passive benchmark consistently. The aggregate of all active managers must equal the market average, meaning half underperform before fees — and most underperform after fees.
Is alpha the same as "outperformance"?
Not exactly. Outperformance is simply beating the benchmark return. Alpha is outperformance adjusted for risk (beta). A fund that returned 20% in a year the S&P 500 returned 15% outperformed, but if its beta was 2.0, CAPM expected 25% — giving it negative alpha of −5%.
What is the Treynor ratio and how does it relate to alpha?
The Treynor ratio = (Portfolio Return − Risk-Free Rate) / Beta. It measures excess return per unit of systematic risk (beta), similar to how Sharpe uses total risk. Alpha measures absolute outperformance above CAPM; Treynor measures efficiency of beta use per unit of return.
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