## Wednesday, April 13, 2016

### 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%)

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), during a given period and systematic risk associated with it (beta). Symbolically, it can be represented as:
Treynor's Index (Tt) = (Rt - R)/Bt
Whereas,
Tt = Treynor’ measure of portfolio
Rt = Return of the portfolio
R = Risk less 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.
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.

Comparison of Sharpe and Treynor
Treynor and Sharpe measures are pretty much similar performance measures with very few differences:
a)      Risk measure: 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.
b)      Applicability: While Sharpe ratio is applicable to all portfolios, Treynor is applicable to well-diversified portfolios.
c)       Performance measurement: While Sharpe is used to measure historical performance, Treynor is a more forward-looking performance measure.
d)      Portfolio diversification: If a portfolio is perfectly diversified (does not contain any unsystematic risk), the two measures would give identical rankings because the total variance of the portfolio would be a systematic variance. If a portfolio is poorly diversified, it is possible for it to have a high ranking on the basis of Treynor measures, but a much lower ranking on the basis of Sharpe measure. Any difference should be directly attributable to the poor diversification of the portfolio.

e)      Risk: According to Sharpe, investor is concerned about the total risk. But, according to Treynor, investor is concerned about the systematic risk.