# Sharpe and Treynor Portfolio Performance Measures, Meaning, Advantages and Limitations

## Portfolio Performance EvaluationUnit 4 SAPM Notes Sharpe and Treynor Portfolio Performance 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 (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

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. This index gives a measure of portfolios total risk and variability of returns in relation to the risk premium. This method ranks all portfolios on the basis of St. Larger the value of St, the better the performance of the portfolio.

The following figure gives a graphic representation of Sharpe’s index. Sd measure the slope of the line emanating from the risk less rate outward to the portfolio in question.

Example
 Portfolio Average return S.D. Risk Free Rate A 15% 3% 9% B 20% 8% 9%

S= (15 – 9)/3 = 2
S= (20 – 9)/8 = 1.375
Thus, portfolio A is ranked higher because its index i.e. 2.0 is higher as compare to B’s index i.e. 1.375. This is despite the fact that B has a higher return (20% >15%)

a) The main advantage of this ratio is that it is easy to calculate and it is used widely.

b) This index gives a measure of portfolios total risk and variability of returns in relation to the risk premium.

c) The Sharpe ratio also standardizes the relationship between risk and return and therefore can be used to compare different asset classes i.e., comparison of stocks with commodities.

d) An advantage of Sharpe ratio is that a beta estimate is not required.

a) When risk free rate is known, it is very difficult to find the right expected return and standard deviation. In a stable market, it is very easy to predict expected return but in today’s dynamic market it is very difficult to predict future expected return.

b) This ratio is not appropriate when evaluating individual stocks because it uses total risk rather than systematic.

c) It is overstated if the return are smoothen and historical prices are used.

d) It can be manipulated by the fund managers if non-linear derivatives are used.

### 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

All risk-averse investors would like to maximize this value. While a high and positive Treynor's Index shows a superior risk-adjusted performance of a fund, a low and negative Treynor's Index is an indication of unfavorable performance. Treynor ratios can be used in both an ex-ante and ex-post sense. The ex-ante form of the ratio uses expected values for all variables, while the ex-post variation uses realized values.

Graphically Treynor’s measure is depicted as:

Example
 Portfolio Return Volatility Risk less Rate A 20% 5% 8% B 24% 8% 8%
Treynor’s index has ranked portfolio A as the better performer because value is higher (2.4 > 2.0) despite the fact that portfolio B has a higher return (24% > 20%). It is due to the difference in volatility of two portfolios.

a) The main advantage to the Treynor Ratio is that it indicates the volatility a stock brings to an entire portfolio.

b) The Treynor Ratio should be used only as a ranking mechanism for investments within the same sector.  In a situation where rate of return from various investments alternatives are same, investments with higher Treynor Ratios are less risky and better managed.

c) It is proper measure for diversified portfolio.

d) This method is easy to calculate and simple to understand.

#### Limitations of Treynor’s ratio:

a) It is only a ranking criterion. It does not consider any values or metrics calculated by means of the management of portfolios or investments.

b) A Treynor ratio is a backward-looking design. This ratio gives importance to how the portfolio behaved in pas. It is possible that a portfolio may perform differently in future from how it has done in the past.

c) Weakness of Treynor’s ratio is that it requires an estimate of beta, which can differ a lot depending on the source which in turn can lead to mis-measurement of risk adjusted return. Many investors accomplish that a beta cannot give a clear picture of risk involved with the investment.

d) It can be overstated if market neutral strategies are used and assets used in the portfolio are highly leveraged.