Difference between Sharpe, Treynor and Jensen Portfolio Evaluation Measures

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 (S_t) = (R_t - R_f)/Sd

Where, S_t = Sharpe’s Index

R_t= represents return on fund and

R_f= is risk free rate of return.

S_d= is the standard deviation

The Treynor Measure

Jack L. Treynor based his model on the concept of characteristic line. This line is the least square regression line relating the return to the risk and beta is the slope of the line. The slope of the line measures volatility. A steep slope means that the actual rate of return for the portfolio is highly sensitive to market performance whereas a gentle slope indicates that the actual rate of return for the portfolio is less sensitive to market fluctuations.

The Treynor index, also commonly known as the reward-to-volatility ratio, is a measure that quantifies return per unit of risk. This Index is a ratio of return generated by the fund over and above risk free rate of return, during a given period and systematic risk associated with it (beta). The portfolio beta is a measure of portfolio volatility, which is used as a proxy for overall risk – specifically risk that cannot be diversified. A beta of one indicates volatility on par with the broader market, usually an equity index. A beta of 0.5 means half the volatility of the market. Portfolios with twice the volatility of the market would be given a beta of 2. Symbolically, Treynor’s ratio can be represented as:

Treynor's Index (T_t) = (R_t – R_f)/B_t

Whereas,

T_t = Treynor’ measure of portfolio

R_t = Return of the portfolio

R_f = Risk free rate of return

B_t = Beta coefficient or volatility of the portfolio

Jensen Model

Jensen's model proposes another risk adjusted performance measure. This measure was developed by Michael Jensen 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 (b) can be calculated as:

R_t – R = a + b (R_m – R)

Where, R_t = Portfolio Return

R = Risk less return

a = Intercept the graph that measures the forecasting ability of the portfolio manager.

b = Beta coefficient, a measure of systematic risk

R_m = Return of the market portfolio