Financial
Modigliani M-Squared (M2) Return & Grinold's Fundamental Law (IC * SQRT(BR))
Calculate a hypothetical volatility-scaled return and a simplified skill/breadth model using fictional, consistently measured inputs.
Separate a volatility-scaled return from its benchmark difference
This worksheet uses M2 return to mean Rf+((Rp-Rf)/portfolio volatility)*benchmark volatility. Some sources instead call the difference from benchmark return M2. Label both outputs explicitly: the example gives a scaled return of 13% and a difference of 1 percentage point against the 12% benchmark. This difference is not regression alpha.
Use returns, risk-free rate and volatilities measured over compatible periods and conventions, entered as decimals. The illustration assumes a constant risk-free return and frictionless scaling; borrowing costs, leverage constraints and changing risk can prevent realizing the hypothetical result. Positive portfolio volatility is required.
The separate IC*SQRT(breadth) calculation is a simplified expected information-ratio model. Breadth means independent decisions, not transaction count. Correlated bets, unstable forecast skill and implementation constraints can invalidate the simple scaling. A transfer coefficient is used in extended formulations. No ranking or future-return claim follows from these fictional inputs.
| Portfolio / Active Metric | Cells & Inputs | Calculation | Institutional Output |
|---|---|---|---|
| Fund Return Rp & Vol Sigma_p | B2 = 16%, B3 = 20% | Rf B4 = 4% -> Sharpe = 0.60 | Higher raw risk than index |
| Index Return Rb & Vol Sigma_B | B5 = 12%, B6 = 15% | M2 = 4% + 0.60 * 15% | Scaled return 13%; benchmark difference +1 percentage point |
| Skill IC & Bet Breadth BR | B7 = 0.05, B8 = 100 bets | 0.05 * SQRT(100) | Information Ratio IR = 0.50 |
| 4x Breadth (400 bets/yr) | B8 set to 400 | 0.05 * SQRT(400) | Model IR = 1.00 |
Modigliani Risk-Adjusted Return M2 = Rf + Sharpe * Sigma_Benchmark
This is a model calculation at the supplied benchmark volatility, not a guaranteed achievable return. Zero benchmark volatility yields the risk-free rate under this formula.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B3<=0,B6<0),"Invalid inputs",ROUND(B4 + ((B2 - B4) / B3) * B6, 4)))Grinold Fundamental Law of Active Management IR = IC * SQRT(Breadth)
IC must be between -1 and 1; breadth must be positive. Effective independent breadth may be fractional. Increasing supplied breadth from 100 to 400 doubles this model output only when skill and the other assumptions remain unchanged.
=IF(COUNT(B7:B8)<>2,"Invalid inputs",IF(OR(ABS(B7)>1,B8<=0),"Invalid inputs",ROUND(B7 * SQRT(B8), 4)))Try the example
Use the original values or change them to test your understanding.