M² measure

The Sharpe ratio ranks portfolios correctly, but "0.69 against 0.73" means little to a client. M² turns the same idea into a return.

It asks: if this portfolio had been run at exactly the market's level of risk, what would it have returned? You get there by mixing the portfolio with the risk-free asset (to reduce risk) or borrowing (to increase it) until its standard deviation equals the market's.

Once the risks are equal, the returns can be compared head-on. If the risk-adjusted return is above the market's, the portfolio outperformed; below it, there is a shortfall.

The measure was derived by Franco Modigliani and his granddaughter (1997), hence M-squared.

The workbook's example: portfolio return 35%, market return 28%, Treasury bill rate 6%, portfolio standard deviation 42%, market standard deviation 30%.

Intuitive approach

1. Excess return of the portfolio: 35 − 6 = 29%.
2. Scale it to the market's risk: 29 × (30 ÷ 42) = 20.71%.
3. Add back the risk-free rate: 20.71 + 6 = 26.71%.
4. Compare with the market's 28%: 26.71 − 28 = −1.29% — a shortfall.

Capital Market Line approach

• Weight in the portfolio = 30 ÷ 42 = 0.714; weight in T-bills = 0.286.
• M² = 0.714 × 35% + 0.286 × 6% = 26.7%, a shortfall of about 1.3%.

The portfolio earned 7 percentage points more than the market in raw terms, and still underperformed once its extra risk is accounted for.

M² = (σ_benchmark ÷ σ_portfolio) × (Rp − Rf) + Rf

   = Sharpe ratio of portfolio × σ_benchmark + Rf

Outperformance (or shortfall) = M² − R_benchmark

The workbook's Chapter 20 caselet, applied to an illustrative ₹1 crore PMS account. The portfolio returned 25% with volatility 18%. The benchmark returned 22% with volatility 15%. The risk-free rate is 8%.

M² = (15 ÷ 18) × (25 − 8) + 8
   = 0.8333 × 17 + 8
   = 14.167 + 8
   = 22.167%

Outperformance = 22.167% − 22% = +0.167%.

The client sees a ₹3 lakh beat. After adjusting for the extra volatility, the manager's genuine edge is worth about ₹16,700 — positive, but a sixth of a percentage point, not three.

Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers), section 20.4.6, derives M² both intuitively and through the Capital Market Line, and a Chapter 20 caselet asks whether the portfolio "actually beat the market adjusting its risk to that of the market". Expect a computation and a Yes/No on outperformance, and possibly a question matching M² to its description ("adjusts portfolio risk to match benchmark risk").

• Add the risk-free rate back. The caselet lists a "Formula One" without Rf; comparing that number with the market return is wrong. M² comparable to the market return must include Rf.
• The ratio is benchmark risk ÷ portfolio risk, not the other way round. A riskier portfolio is scaled down.
• M² and the Sharpe ratio always rank portfolios the same way — M² is the Sharpe ratio expressed as a return. Caselet 2 in the same chapter confirms it: Sharpe 0.944 against 0.933, the same verdict as M².
• Rounding: the two approaches give 26.71% and 26.7%, a shortfall of 1.29% and 1.3%. Both are the workbook's figures.
• M² uses total risk (standard deviation), like Sharpe — not beta, which is Treynor's and Jensen's measure of risk.