Jensen’s Alpha Calculator
Use this Jensen’s Alpha Calculator to compare the entered strategy mean with a single-factor CAPM-implied return for one selected benchmark and period. Historical beta, benchmark premium, covariance and every aligned contribution remain visible, while the page keeps statistical significance, manager skill and future outperformance outside the result.
Enter aligned strategy and benchmark returns
Use signed equal-frequency percentage returns in matching oldest-to-newest order, plus one risk-free return in the same per-period units.
Use the same frequency and percentage-point units as both return series.
Enter percentage numbers without percent signs. Separate with spaces, commas, semicolons or new lines. Maximum 500.
Enter exactly one benchmark return for every strategy return, in the same order and per-period percentage units.
Entered CAPM residual arithmetic
Entered Benchmark-Adjusted Performance 1.0.0.
| Index | Strategy | Benchmark | Strategy deviation | Benchmark deviation | Cross-product |
|---|
How Jensen’s alpha is calculated
CAPM-implied return = Risk-free + Beta × (Benchmark mean − Risk-free)
Jensen’s alpha = Strategy mean − CAPM-implied return
Version 1.0.0 estimates historical beta from N-minus-one sample covariance and benchmark variance, then applies the single-factor CAPM equation to the entered mean returns.
Alpha is reported in the same per-period percentage-point units as the inputs. No statistical significance test, multifactor adjustment, transaction-cost model or annualization is added.
Worked example from the audited fixture
The audited 30-period example has a strategy mean of 0.110000%, benchmark mean of 0.043333%, risk-free return of 0.020000% and historical beta of 1.639845.
- CAPM-implied return = 0.020000% + 1.639845 × (0.043333% − 0.020000%) = 0.058263%.
- Jensen’s alpha = 0.110000% − 0.058263% = 0.051737 percentage points per entered period.
Reproduce it: select “Load audited example” above. The calculator loads the same 30 aligned observations used by the independent fixture.
How to interpret the result
- A positive result means the entered strategy mean exceeded this model’s benchmark- and beta-adjusted expected return in the selected sample; a negative result means it fell below it.
- Changing the benchmark, return window, risk-free series, fee treatment or factor model can change the result.
- Alpha alone does not establish skill or persistence. This calculator does not produce a t-statistic, confidence interval or multifactor attribution.
Assumptions and limits
- Enter 3 to 500 strategy returns and the same number of aligned benchmark returns.
- Returns and the risk-free input must share one frequency and percentage-point convention.
- A constant benchmark has zero variance, so historical beta and alpha are rejected.
- The result is sensitive to the benchmark, sample window, fee treatment and single-factor CAPM assumptions.
- No annualized alpha, significance test, manager-skill label, forecast, signal or recommendation is produced.
Treynor vs Jensen’s alpha vs M² vs Information Ratio
These measures are not interchangeable. Choose the one whose denominator and question match the evidence you intend to inspect, then keep the benchmark, frequency, window and return treatment consistent.
| Measure | Reference or denominator | What the output expresses | Main boundary |
|---|---|---|---|
| Treynor Ratio | Historical beta | Excess return per unit of benchmark beta | Ignores total and idiosyncratic volatility. |
| Jensen’s alpha | Single-factor CAPM expectation | Per-period residual return | Sensitive to benchmark, beta and factor-model choice. |
| M² performance | Benchmark sample volatility | Volatility-matched return and active difference | Assumes hypothetical costless scaling. |
| Information Ratio | Tracking error | Active return per unit of active-return variability | Answers a different benchmark-relative consistency question. |
Frequently asked questions
- Enter 3 to 500 aligned strategy and benchmark percentage returns plus one risk-free return in the same per-period units.
- N-minus-one sample covariance between strategy and benchmark returns is divided by N-minus-one sample benchmark variance.
- Risk-free return plus historical beta times benchmark mean return minus risk-free return.
- The CAPM-implied return is subtracted from the entered strategy mean return.
- Version 1.0.0 reports alpha in the same per-period percentage-point units as the entered returns.
- Historical beta is undefined, so the calculation is rejected.
- No. It reports one deterministic sample residual without a t-test, multifactor model or annualization.
- No. It is sensitive to the sample, benchmark and CAPM assumptions and does not prove skill, persistence or future outperformance.
Sources and methodology
- CFA Institute — Treynor and Jensen — Official single-period Jensen equation, beta definition and benchmark cautions.
- Michael Jensen — Mutual Fund Performance — Original 1968 Jensen performance-measure paper.
Inspect benchmark-relative performance
Compare the trading records behind your sample
Keep one broker account, benchmark, return convention, fee treatment and sampling rule across the entered observations before comparing arithmetic.
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