In order to determine the risk-adjusted returns of investment portfolios, several eminent authors have worked since 1960s to develop composite performance indices to evaluate a portfolio by comparing alternative portfolios within a particular risk class. The most important and widely used measures of performance are:
Ø The Treynor Measure
Ø The Sharpe Measure
Ø Jenson Model
Ø Fama Model
The Treynor Measure
Developed by Jack Treynor, this performance measure evaluates funds on the basis of Treynor's Index. This Index is a ratio of return generated by the fund over and above risk free rate of return (generally taken to be the return on securities backed by the government, as there is no credit risk associated),b during a given period and systematic risk associated with it (beta). Symbolically, it can be represented as:
Treynor's Index (Ti) = (Ri - Rf)/Bi.
Where, Ri represents return on fund, Rf is risk free rate of return and Bi is beta of the fund.
The Sharpe Measure
In this model, performance of a fund is evaluated on the basis of Sharpe Ratio, which is a ratio of returns generated by the fund over and above risk free rate of return and the total risk associated with it. According to Sharpe, it is the total risk of the fund that the investors are concerned about. So, the model evaluates funds on the basis of reward per unit of total risk. Symbolically, it can be written as:
Sharpe Index (Si) = (Ri - Rf)/Si
Where, Si is standard deviation of the fund.
While a high and positive Sharpe Ratio shows a superior risk-adjusted performance of a fund, a low and negative Sharpe Ratio is an indication of unfavorable performance.
Comparison of Sharpe and Treynor
Sharpe and Treynor measures are similar in a way, since they both divide the risk premium by a numerical risk measure. The total risk is appropriate when we are evaluating the risk return relationship for well-diversified portfolios. On the other hand, the systematic risk is the relevant measure of risk when we are evaluating less than fully diversified portfolios or individual stocks.
Jenson's model proposes another risk adjusted performance measure.
This measure was developed by Michael Jenson and is sometimes referred to as the Differential Return Method. This measure involves evaluation of the returns that the fund has generated vs. the returns actually expected out of the fund given the level of its systematic risk. The surplus between the two returns is called Alpha, which measures the performance of a fund compared with the actual returns over the period. Required return of a fund at a given level of risk (Bi) can be calculated as:
Ri = Rf + Bi (Rm - Rf)
Where, Rm is average market return during the given period.
The Eugene Fama model is an extension of Jenson model. This model compares the performance, measured in terms of returns, of a fund with the required return commensurate with the total risk associated with it. The difference between these two is taken as a measure of the performance of the fund and is called net selectivity.
Required return can be calculated as:
Ri = Rf + Si/Sm*(Rm - Rf)
Where, Sm is standard deviation of market returns. The net selectivity is then calculated by subtracting this required return from the actual return of the fund.