Risk adjusted Return for a portfolio – Part 2
As time progressed, three major composite measures of risks for a Portfolio were adopted by the risk professionals.
- Treynor Index
- Sharpe Performance Index
- Jensen Performance Index
Treynor Index :
In order to use the Treynor Index, you must know three things; the portfolio return, the risk-free rate of return, and the beta of the portfolio. For the risk-free rate of return, you may use the average return (over the period of time) of some government bond or note. The beta of the portfolio is a measure of the systematic risk of the portfolio. Using the beta, rather than the standard deveiation, you are assuming that the portfolio is a well diversified portfolio. If you are looking at the return of a mutual fund, this figure is typically available from the fund company itself.
For those of you who want to know the formula for the index;
Treynor = (Portfolio Return – Risk-Free Return) / Beta
Let’s use the same example information. A portfolio manager achieved a return of 15.0%, his portfolio had beta measurement of 1.1 and the market achieved a return of 14.6% vs. a risk free rate of return of 7%. To calculate the Jensen Index:
index = (.15 – .07) / 1.1 = 0.0727
To compare, another portfolio manager achieved a return of 13.5% with a beta of .81. The Treynor index for this porfolio manager is:
index = (.135 – .07) / 0.81 = 0.0802
This means that the 2nd portfolio manager out performed the first portfolio manager on a risk-adjusted basis.
Sharpe Performance Index :
In order to use the Sharpe Index, you must know three things; the portfolio return, the risk-free rate of return, and the Standard Deviation of the portfolio. For the risk-free rate of return, you may use the average return (over the period of time) of some government bond or note. The Standard Deviation of the portfolio is a measure of the systematic risk of the portfolio. Using the Standard Deviation, rather than the beta (as in the Treynor Index), you are assuming that the portfolio is NOT a deversified portfolio.
Because the numerator is the Portfolio’s Risk Premium, the Sharpe Index measures the Risk Premium earned per unit of total risk.
Si = Ri – RFR / oi
Ri = Average Rate of Return for Portfolio
RFR = Average Rate of Return for Risk-Free Assets
oi = Standard Deviation of Rate of Portfolio’s Return
Jensen Peformance Measure:
Superior portfolio managers who accurately predict market turns or who identify undervalued investments earn higher have consistently positive random error terms
rjt = rfrt + bj[rmt - rfrt ] + ujt
this version of capm suggests that return is a function of risk-free rate, plus risk premium that depends on systematic risk, plus, random error term with an intercept to measure the positive or negative difference,
the equation becomes rjt – rfrt = alpha + bj[rmt - rfrt ] + ujt
rearranged —– alpha = (rjt – rfrt) – bj (rmt – rfrt)
where alpha measures how much of the rate of return on the portfolio is attributable to the manager’s ability to derive above-average returns adjusted for risk
