The Modigliani-Modigliani measure is the portfolio return adjusted upward or downward to match the benchmark's standard deviation. This puts the portfolio return and the benchmark return on 'equal footing' from a standard deviation perspective. $$MM_{p}=\frac{E[R_{p} - R_{f}]}{\sigma_{p}}=SR_{p} * \sigma_{b} + E[R_{f}]$$ where \(SR_{p}\) - Sharpe ratio, \(\sigma_{b}\) - benchmark standard deviation
Modigliani(Ra, Rb, Rf = 0, ...)
| Ra | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns |
|---|---|
| Rb | return vector of the benchmark asset |
| Rf | risk free rate, in same period as your returns |
| … | any other passthrough parameters |
This is also analogous to some approaches to 'risk parity' portfolios, which use (presumably costless) leverage to increase the portfolio standard deviation to some target.
J. Christopherson, D. Carino, W. Ferson. Portfolio Performance Measurement and Benchmarking. 2009. McGraw-Hill, p. 97-99. Franco Modigliani and Leah Modigliani, "Risk-Adjusted Performance: How to Measure It and Why," Journal of Portfolio Management, vol.23, no., Winter 1997, pp.45-54
data(managers) Modigliani(managers[,1,drop=FALSE], managers[,8,drop=FALSE], Rf=.035/12)#> [1] 0.01678381Modigliani(managers[,1:6], managers[,8,drop=FALSE], managers[,8,drop=FALSE])#> HAM1 HAM2 HAM3 #> Modigliani-Modigliani measure: SP500 TR 0.01281799 0.01505458 0.0131509 #> HAM4 HAM5 HAM6 #> Modigliani-Modigliani measure: SP500 TR 0.01057959 0.01053081 0.01844616Modigliani(managers[,1:6], managers[,8:7], managers[,8,drop=FALSE])#> HAM1 HAM2 HAM3 #> Modigliani-Modigliani measure: SP500 TR 0.01281799 0.01505458 0.01315090 #> Modigliani-Modigliani measure: EDHEC LS EQ 0.01062640 0.01168261 0.01078361 #> HAM4 HAM5 HAM6 #> Modigliani-Modigliani measure: SP500 TR 0.01057959 0.010530812 0.01844616 #> Modigliani-Modigliani measure: EDHEC LS EQ 0.00956933 0.009546295 0.01328426