1.4.4 Performance Measures

1. Sharpe Performance Index (SPI, Sharpe Ratio)

The ratio of the mean excess return on portfolio i to the standard deviation of the returns of portfolio i.
SPI=E(Ri)−RfσiSPI=\frac{E(R_i)-R_f}{\sigma_i}SPI=σiE(Ri)−Rf
A measure of excess return per unit of risk (total risk), the higher is better.
- SPI greater than the slope of the CML: superior performance to equilibrium return.
- SPI below the slope of the CML: inferior performance to equilibrium return.

2. Treynor Performance Index (TPI)

The ratio of the mean excess return on portfolio i to the Beta(β\betaβ) of portfolio i.
TPI=E(Ri)−RfβiTPI=\frac{E(Ri)-R_f}{\beta_i}TPI=βiE(Ri)−Rf
A measure of excess return per unit of risk (systematic risk), the higher is better.
For a well-diversified portfolio, beta is widely accepted as an appropriate measure of risk.
According to CAPM, then
TPI=E(Ri)−Rfβi=E(Rm)−RfTPI = \frac{E(R_i)-R_f}{\beta_i}=E(R_m)-R_fTPI=βiE(Ri)−Rf=E(Rm)−Rf
- E(Rm)−RfE(R_m)-R_fE(Rm)−Rf also called the alpha measure.
- TPI is greater than the alpha measure is considered to have a positive alpha (indicting superior performance) and vice versa.
- TPI>E(Rm)−RfTPI>E(R_m)-R_fTPI>E(Rm)−Rf, superior performance
- TPI<E(Rm)−RfTPI<E(R_m)-R_fTPI<E(Rm)−Rf, inferior performance

3. Jensen’s Performance Index (JPI)

The difference between actual return and return required to compensate for systematic risk (CAPM), also called Jensen’s Alpha.
JPI is like TPI, as both measures assume investors hold well-diversified portfolios.
αi=Ri−[Rf+β(Rm−Rf)]\alpha_i =R_i-[R_f+\beta(R_m-R_f)]αi=Ri−[Rf+β(Rm−Rf)]

4. Sortino Ratio

The Sortino ratio (SR) is modification of SPI. Both ratios measure the risk-adjusted return of an asset or portfolio.
However, fi the primary focus is on downside risk, then SR is considered to be an improvement over SPI.
Sortino ratio is more appropriate for a case where returns are not symmetric.
SR=Rp−T1N∑t=1Nmin(0,Rpt−T)2SR=\frac{R_p-T}{\sqrt{\frac{1}{N}\sum^N_{t=1}min(0,R_{pt}-T)^2}}SR=N1∑t=1Nmin(0,Rpt−T)2Rp−T
The denominator is the downside deviation, as measured by the standard deviation of negative returns.
TTT is the target or required rate of return fro the investment strategy, also known as MAR or minimum accepted rate of return.
TTT may be set to the risk free rate or another hurdle rate.

5. Tracking Error

The standard deviation of return difference between the portfolio and the benchmark.
TE=σRp−RbTE=\sigma_{R_p-R_b}TE=σRp−Rb

6. Information Ratio

The residual return of the managed portfolio relative to its benchmark divided by the tracking error.
IR=E(Rp)−E(RB)σRp−RBIR=\frac{E(R_p)-E(R_B)}{\sigma_{R_p-R_B}}IR=σRp−RBE(Rp)−E(RB)

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1.4.4 Performance Measures

1. Sharpe Performance Index (SPI, Sharpe Ratio)

2. Treynor Performance Index (TPI)

3. Jensen’s Performance Index (JPI)

4. Sortino Ratio

5. Tracking Error

6. Information Ratio

1.4.4 Performance Measures相关推荐

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