# Difference between Sharpe, Treynor and Jensen Portfolio Evaluation Measures

## Portfolio Performance EvaluationUnit 4 SAPM Notes Difference between Sharpe, Treynor and Jensen

### 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 (St) = (Rt - Rf)/Sd

Where, St = Sharpe’s Index

Rt= represents return on fund and

Rf= is risk free rate of return.

Sd= 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 (Tt) = (Rt – Rf)/Bt

Whereas,

Tt = Treynor’ measure of portfolio

Rt = Return of the portfolio

Rf = Risk free rate of return

Bt = 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:

Rt – R = a + b (Rm – R)

Where, Rt = 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

Rm = Return of the market portfolio

### Comparison of Sharpe, Treynor and Jensen

 Basis Sharpe Treynor Jensen Risk Sharpe used standard deviation as the risk measure to capture the overall risk of the portfolio. Treynor used beta as the risk measure to capture the volatility of the portfolio relative to the market. Jensen's alpha takes into consideration the capital asset pricing model (CAPM) market theory and includes a risk-adjusted component in its calculation. Applicability Sharpe ratio is applicable to all portfolios. Treynor is applicable to well-diversified portfolios. Jensen is also informative in case of well-diversified portfolios. Performance measurement Sharpe is a more forward-looking performance measure. Treynor is used to measure historical performance. Jensen Alpha measures excess of actual return over CAPM expected return. Risk According to Sharpe, investor is concerned about the total risk. According to Treynor, investor is concerned about the systematic risk. According to Jensen, investor is concerned about the systematic risk. Formula Sharpe Index (St) = (Rt - Rf)/Sd Treynor's Index (Tt) = (Rt – Rf)/Bt Rt – R = a + b (Rm – R)