Hi,
Infact you can derive all the other performance measures from the Sharpe measure that is Sharpe can be viewed as building block for other measures.
- Sharpe=(E(Rp)-Rf)/σ is for the appraisal of a non diversified portfolio where σ the volatility of the portfolio=sqrt(systematic risk +non systematic risk)=sqrt(β^2*σm^2+σe^2) where systematic risk=β^2*σm^2, σm is the volatility of the market and β is the systematic risk of the portfolio.non systematic risk=σe^2,if you diversify away all the non systematic risk of the portfolio then the non systematic risk diversifies away to 0 then, σe^2=0 =>σ=sqrt(β^2*σm^2+0)=sqrt(β^2*σm^2)=β*σm then what remains in terms of risk is the β which determines the risk of the portfolio. Therefore we replace σ with β to change the Sharpe from (E(Rp)-Rf)/σ to (E(Rp)-Rf)/β which is nothing but the Treynor measure.
- Now, Treynor for market=(E(Rm)-Rf)/βm=(E(Rm)-Rf)/1=(E(Rm)-Rf)(since βm=1 the beta for the market is 1).If the portfolio generates same return per unit of systematic risk as compared to market then Treynor for market=Treynor for portfolio =>(E(Rm)-Rf)=(E(Rp)-Rf)/β =>E(Rp)-Rf=β*(E(Rm)-Rf) =>E(Rp)=Rf+β*(E(Rm)-Rf) (CAPM,alpha=0) this means that there is no extra return that is being generated by portfolio when compared to market in terms of per unit of systematic risk alpha=0,you shall get in the market the return Rf+β*(E(Rm)-Rf) for taking the systematic risk β, when alpha is being generated that is portfolio generates extra return for taking the same amount of risk β then E(Rp)=Rf+β*(E(Rm)-Rf)+alpha that is your portfolio is earning more than what market reward for bearing the same systematic risk β,rearrange the preceding equation to get alpha =E(Rp)-(Rf+β*(E(Rm)-Rf)) where ERP=Expected risk premium=(E(Rm)-Rf) thus we can rewrite alpha =E(Rp)-(Rf+β*ERP).alpha>0=> E(Rp)-(Rf+β*ERP)>0 or E(Rp)-Rf/β >ERP=(E(Rm)-Rf)/βm =>Treynor for portfolio>Treynor for market also if alpha<0=> E(Rp)-(Rf+β*ERP)<0 or E(Rp)-Rf/β <ERP=(E(Rm)-Rf)/βm =>Treynor for portfolio<Treynor for market that is we shall arrive at the same conclusions when appraising a portfolio using either the Treynor or the alpha,therefore Treynor and alpha are good substitutes for each other.
- To measure performance of the managers that is how much extra they are earning for bearing the same systematic risk that is how much extra return(alpha) they could generate from market.(this shows their breadth and depth of Information about the asset they are investing in),just as Sharpe is (E(Rp)-Rf)/Volatility of (E(Rp)-Rf) measure the excess return per unit of volatility of excess return similarly we measure the alpha per unit of volatility of alpha which is nothing but the Information ratio(IR)=alpha/σ(alpha).
Other measure are just the extensions of the Sharpe,
- Just replace benchmark Rf with MAR(Minimum acceptable return) in numerator and the volatility of portfolio σ with the downside volatility of portfolio σd in the denominator, the downside volatility measure the standard deviation of returns that are less than the MAR.So that Sharpe changes from (E(Rp)-Rf)/σ to (E(Rp)-MAR)/σd which is nothing but the Sortino measure.Thus Sortino is just a manipulation of Sharpe.
- Sharpe for market=(E(Rm)-Rf)/σm.If the portfolio generates same return per unit of volatility as compared to market then Sharpe for market=Sharpe for portfolio =>(E(Rm)-Rf)/σm=(E(Rp)-Rf)/σ =>E(Rp)-Rf=(σ/σm)*(E(Rm)-Rf) =>E(Rp)=Rf+(σ/σm)*(E(Rm)-Rf) (M^2=0) rearrange preceding equation to get SR=(E(Rp)-Rf)/σ = (1/σm)*(E(Rm)-Rf=>SR*σm-(E(Rm)-Rf)=0=M^2(SR is sharpe ratio of portfolio) , this means that there is no extra return that is being generated by portfolio when compared to market in terms of per unit of volatility M^2=0 , when there are extra return generated per unit of volatility by portfolio as compared to the market then M^2 >0 =>SR*σm-(E(Rm)-Rf)>0 =>SR>(E(Rm)-Rf)/σm that is sharpe of portfolio is greater than that of market,when there are lesser return generated per unit of volatility by portfolio as compared to the market then M^2 <0 =>SR*σm-(E(Rm)-Rf)<0 =>SR<(E(Rm)-Rf)/σm that is sharpe of portfolio is lesser than that of market.So we arrive at the same conclusions when appraising a portfolio using either the Sharpe or the M^2,therefore Sharpe and M^2 are good substitutes for each other.
- Just replace benchmark Rf with EL(Lowest benchmark return) in numerator .So that Sharpe changes from (E(Rp)-Rf)/σ to (E(Rp)-EL)/σ which is nothing but the Roy's SF ratio.Thus Roy's SF is just a manipulation of Sharpe.
Thanks
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