M-squared keeps the Sharpe ranking and turns it into percentage points of return at the benchmark's risk, which is the unit a client reads.
M-squared rescales a fund to the benchmark's volatility and reports the return it would then have earned: M² = risk-free rate + Sharpe ratio of the fund × volatility of the benchmark. A fund returning 9.6 per cent at 16 per cent volatility, against a benchmark returning 7.5 per cent at 12 per cent and a 3.0 per cent risk-free rate, has an M² of 7.95 per cent: it beat the benchmark by 0.45 points on equal risk, not by the 2.10 points the raw returns show.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The Sharpe ratio ranks funds correctly but produces a number no client understands: 0.41 against 0.375. M², proposed by Franco Modigliani and Leah Modigliani, ranks them identically and expresses the answer in percentage points of return, which is the unit a trustee or an investment committee actually reads.
| Portfolio | Return | Volatility | Excess over cash |
|---|---|---|---|
| Benchmark | 7.5% | 12% | 4.5% |
| Fund A, concentrated equity | 9.6% | 16% | 6.6% |
| Fund B, low-volatility | 7.0% | 8% | 4.0% |
| Fund C, aggressive growth | 11.0% | 22% | 8.0% |
The risk-free rate is 3.0 per cent. Returns and volatilities must cover the same period and frequency, typically 36 or 60 monthly returns, annualised. The mechanics of the inputs are in how to calculate the Sharpe ratio of a fund.
Sharpe = (return − risk-free) ÷ volatility
Benchmark: 4.5% ÷ 12% = 0.3750
Fund A: 6.6% ÷ 16% = 0.4125 Fund B: 4.0% ÷ 8% = 0.5000 Fund C: 8.0% ÷ 22% = 0.3636
M² = risk-free + Sharpe × benchmark volatility
Fund A: 3.0% + 0.4125 × 12% = 7.95%
Fund B: 3.0% + 0.5000 × 12% = 9.00%
Fund C: 3.0% + 0.3636 × 12% = 7.36%
Excel, with fund return in B2, fund volatility in C2, risk-free in F1 and benchmark volatility in F2: =$F$1+(B2-$F$1)/C2*$F$2
M² alpha = M² − benchmark return = (Sharpe of fund − Sharpe of benchmark) × benchmark volatility
Fund A: 7.95% − 7.5% = 0.45 points, the same as (0.4125 − 0.3750) × 12%.
M² is not an abstraction. It is the return of a real portfolio mixing the fund with cash so that the mix has the benchmark's volatility.
| Fund | Raw return | Raw excess over benchmark | Sharpe | M² | M² alpha |
|---|---|---|---|---|---|
| C, aggressive growth | 11.0% | 3.50 | 0.364 | 7.36% | −0.14 |
| A, concentrated equity | 9.6% | 2.10 | 0.413 | 7.95% | 0.45 |
| B, low-volatility | 7.0% | −0.50 | 0.500 | 9.00% | 1.50 |
On raw returns, C wins by 3.50 points and B trails the benchmark. On equal risk the order reverses exactly: B adds 1.50 points, A adds 0.45, and C destroys 0.14 points. C's return was paid for with risk the client could have bought more cheaply by holding the benchmark itself.
Every fund's M² depends on the rate used, and funds with different volatilities move by different amounts. This is the sensitivity most reports omit.
| Risk-free rate | Benchmark Sharpe | Fund A | Fund B | Fund C |
|---|---|---|---|---|
| 1% | 0.542 | −0.05 | 2.50 | −1.05 |
| 3% | 0.375 | 0.45 | 1.50 | −0.14 |
| 5% | 0.208 | 0.95 | 0.50 | 0.77 |
At 5 per cent cash, the ranking becomes A, C, B. A higher risk-free rate shrinks the excess return of the low-volatility fund proportionately more, and the leverage that M² applies to it is costed at that higher rate. Use the cash rate actually earned over the measurement period, month by month, and state it next to the result.
Report M² beside the raw return whenever funds with different risk are compared: it keeps the Sharpe ranking and puts it in percentage points. Here it turns a 2.10-point outperformance into 0.45 points and shows the dullest fund was the best one. The mandate workbooks, with the time-weighted and money-weighted returns behind such comparisons, are on the Asset Management companion page.
They rank funds identically for a given risk-free rate, but M² is expressed in percentage points of return. A Sharpe of 0.4125 against a benchmark's 0.3750 becomes an M² of 7.95 per cent against 7.5 per cent: the fund, scaled to the benchmark's 12 per cent volatility, would have beaten it by 0.45 points. Clients and trustees read the second form far more easily.
Yes, when the fund is less volatile than the benchmark. A fund returning 7.0 per cent at 8 per cent volatility is scaled up 1.50 times with borrowing at the risk-free rate, giving an M² of 9.00 per cent. If the borrowing actually costs 4.5 per cent, the achievable figure falls to 8.25 per cent, so state the borrowing assumption.
The cash rate over the same period as the returns, compounded month by month, and the same rate for every fund compared. The choice is not neutral: in the illustrative case, raising it from 3 to 5 per cent cuts the low-volatility fund's M² alpha from 1.50 to 0.50 points and lifts the aggressive fund from −0.14 to 0.77, reordering the ranking.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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