CFAI Text Vol 4, P375 (Reafing 44)
In 2nd last paragraph, it is stated :
A portfolio that matches the performance of the market will have an M-squared of zero, whereas a portfolio that outperform the market will have an M-square that is positive.
I can not see the points from the equation above the 2nd last paragraph. Anyone can help ?
If the portfolio has the same Sharpe ratio as the market, then,
(Rp – Rf)/σp = (RM – Rf)/σM
and,
M² = (Rp – Rf)(σM/σp) – (RM – Rf)
M² = σM[(Rp – Rf)/σp] – (RM – Rf)
M² = σM[(RM – Rf)/σM] – (RM – Rf)
M² = (RM – Rf) – (RM – Rf)
M² = 0
If the portfolio has a higher Sharpe ratio than the market, then,
(Rp – Rf)/σp > (RM – Rf)/σM
and,
M² = (Rp – Rf)(σM/σp) – (RM – Rf)
M² = σM[(Rp – Rf)/σp] – (RM – Rf)
M² > σM[(RM – Rf)/σM] – (RM – Rf)
M² > (RM – Rf) – (RM – Rf)
M² > 0
Thank you very much !
My pleasure.