M² Performance Calculator
Use this M² Performance Calculator to restate the entered strategy return at the selected benchmark’s sample volatility. It shows both sample standard deviations, the scaling factor, Sharpe component and benchmark-relative difference, while making clear that the adjustment assumes costless scaling and is neither a forecast nor an executable portfolio.
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 volatility-matched arithmetic
Entered Benchmark-Adjusted Performance 1.0.0.
| Index | Strategy | Benchmark | Strategy deviation² | Benchmark deviation² |
|---|
How M² performance is calculated
M² adjusted return = Risk-free + Portfolio Sharpe component × Benchmark sample SD
M² active difference = M² adjusted return − Benchmark mean
Both standard deviations use the N-minus-one sample convention. The portfolio’s entered excess return per unit of sample volatility is rescaled to the benchmark’s sample volatility and returned to percentage-point units.
Version 1.0.0 treats the adjusted return as the primary M² output and separately shows adjusted return minus benchmark mean. It applies no annualization or performance grade.
Worked example from the audited fixture
The audited 30-period example has a strategy mean of 0.110000%, risk-free return of 0.020000%, strategy sample standard deviation of 0.783163% and benchmark sample standard deviation of 0.475382%.
- Sharpe component = (0.110000% − 0.020000%) ÷ 0.783163% = 0.114919. M² adjusted return = 0.020000% + 0.114919 × 0.475382% = 0.074630%.
- M² active difference = 0.074630% − 0.043333% benchmark mean = 0.031297 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
- The adjusted return answers a hypothetical question: what return corresponds to the strategy’s entered Sharpe component if total volatility is scaled to the benchmark’s sample volatility?
- The active difference compares that hypothetical adjusted return with the entered benchmark mean. It is not an expected live-trading profit.
- Financing, leverage, liquidity, fees and serial dependence are absent, so a positive result does not prove the scaling could be implemented or repeated.
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 strategy or benchmark makes the required volatility normalization undefined and is rejected.
- The hypothetical scaling does not include borrowing, financing, trading, liquidity or leverage constraints.
- No annualized return, quality label, skill claim, 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.
- Version 1.0.0 uses the N-minus-one sample standard deviation for both strategy and benchmark returns.
- It is strategy mean minus risk-free return, divided by strategy sample standard deviation.
- Risk-free return is added to the portfolio Sharpe component multiplied by benchmark sample standard deviation.
- It is the adjusted return minus the entered benchmark mean return, shown as a separate supporting output.
- The required sample volatility is zero, so the volatility normalization is undefined and rejected.
- No. It is a hypothetical costless volatility normalization and does not model leverage limits, financing, liquidity or transaction costs.
- No. It describes the entered sample under disclosed assumptions and creates no quality grade, skill claim, forecast, signal or recommendation.
Sources and methodology
- PerformanceAnalytics — Modigliani-Modigliani Measure — Documented benchmark-volatility adjusted-return definition and same-period risk-free input.
- NIST/SEMATECH — Standard Deviation — N-minus-one sample standard-deviation convention used by version 1.0.0.
Compare volatility-normalized evidence
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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