Portfolio Performance Evaluation
Unit 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 (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
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 (S_{t}) = (R_{t} - R_{f})/Sd |
Treynor's Index (T_{t})
= (R_{t} – R_{f})/B_{t} |
R_{t} – R = a + b (R_{m}
– R) |
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