Table of Contents:
 Meaning of Portfolio Evaluation
 Portfolio Evaluation Methods
 Sharpeâ€™s Measure/Sharpeâ€™s Ratio
 Treynorâ€™s Measure/Treynorâ€™s Ratio
 Jensen Measure/Jensen Ratio
 Modigliani & Modigliani Measure (MÂ² Measure)
Meaning of Portfolio Evaluation
Portfolio evaluation involves the evaluation of the performance of the portfolio. It is the fundamental process that involves comparing the returns generated by a portfolio with the return earned on one or more other portfolios or a benchmark portfolio. Portfolio evaluation consists of two major functions:
 Performance measurement and
 Performance evaluation.Â
Performance measurement is an accounting function which measures the return earned on a portion during the holding period or investment period.
Performance evaluation, on the other hand,Â determinesÂ whether the performance was superior or inferior and whether the performance wasÂ a result ofÂ skill or luck.
When assessing a portfolio’s performance, the return earned on the portfolio must be essentially evaluated concerning the risk associated with that portfolio. One approach would be to group portfolios into equivalent risk classes and then compare the returns of portfolios within each risk category. An alternative approach would be to specifically adjust the return for the riskiness of the portfolio by developing riskadjusted return measures and using these for evaluating portfolios across differing risk levels.
Portfolio Evaluation Methods
Portfolio evaluation methods are as follows:
 Sharpe ratio
 Treynor ratio
 Jensen’s Ratio
 Modigliani and Modigliani
Sharpe’s Measure/Sharpe’s Ratio
Sharpe’s measure is also known as the ‘Reward to Variability’ Ratio. The returns from a portfolio are firstly adjusted for riskfree return. These excess returns,Â which serve as aÂ reward for investing in risky assets areÂ then evaluated in terms of the return per unit of risk.
Sharpe’s Ratio Formula:
$$\mathrm{Sharpe}’\mathrm s\;\mathrm{Ratio}\;(\mathrm{SR})\;=\;\left[\frac{{\mathrm R}_{\mathrm P}{\mathrm R}_{\mathrm F}}{{\mathrm\sigma}_{\mathrm P}}\right]$$
Where,
R_{p} = Realized Return on a portfolio during a holding period
R_{F} = Riskfree rate of Return
Ïƒ_{P }= Standard deviation of the Portfolio
Example:Â Let usÂ considerÂ two portfolios and calculate the Sharpe ratio for each.
Portfolio  Return (R_{P})  RiskFree (R_{F})  Excess return (R_{P} – P_{F} ) 
Portfolio risk (SD) 
A  32  10  22  12 
B  20  11  9  7 
Solution: $$\mathrm{Sharpe}’\mathrm s\;\mathrm{Ratio}\;:\;\mathrm{Portfolio}\;\mathrm A\;\;=\;\left[\frac{3210}{12}\right]\;=\;1.8333\\\mathrm{Portfolio}\;\mathrm B\;=\;\left[\frac{20\;\;11}7\right]\;=\;1.285$$
In the case of Portfolio A the reward per unit of risk is relatively higher. Hence it indicates that its performance is good.
Example: Calculate the Sharpe Ratio for the four different portfolios based on the data given below.
Portfolio  Expected Rate of Return 
SD of Returns from Portfolios 
P  15%  9.00 
Q  10%  3.00 
R  14%  7.50 
S  11%  6.00 
The expected rate of return on the market portfolio is 8.50% with an SD of 3. The RF(riskfree rate) is 5%. Which portfolio has performed the best?Â
Portfolio 
Sharpe Ratio 
P  (15 – 5)/6.00 = 1.000 
Q  (10 – 5)/ 3.00 = 1.6667 
R  ( 14 – 5)/ 7.50 = 1.2000 
S  ( 11 – 5)/6.00 = 1.000Â 
Therefore, Portfolio R is the strongest performer.
Treynor’s Measure/Treynor’s Ratio
Treynor’s measure is also known as the ‘Reward to Volatility ratio’. Treynor considers portfolio beta as a measure of risk Portfolio beta is the average beta of individual assets in the given Portfolio. This beta designates the market risk of the given portfolio.
$$\mathrm{Treynor}’\mathrm s\;\mathrm{Measure}\;(\mathrm{TR})\;=\;\left[\frac{{\mathrm R}_{\mathrm P}\;\;{\mathrm R}_{\mathrm F}}{{\mathrm\beta}_{\mathrm P}}\right]$$
where,
R_{PÂ } = Realized Return on a Portfolio
R_{F} = Riskfree Rate of Return
Î²_{P}Â = Portfolio
Example: Assume that you are an administrator of a pension fund,Â such as ICICI Prudential Life Time Pension FundsÂ and you have to decide whether to renew your contracts with your three money managers. You must measure how they have performedÂ to make an informed decision.
Consider the following performance results for each individual:
 Market return of 14%,
 Riskfree rate of 8%, and
 Beta of 1.
Investment Manager 
Average Annual Rate of Return 
Beta 
W  0.14  1.00 
X  0.18  1.20 
Y  0.16  1.05 
Z  0.12  0.90 
Solution: The T values for each investment manager can be calculated as :
$${\mathrm T}_{\mathrm W\;\;}=\;\left[\frac{0.14\;\;0.08}{1.00}\right]\;=\;0.06\\\\{\mathrm T}_{\mathrm X}\;=\;\left[\frac{0.18\;\;0.08}{1.20}\right]\;=\;0.083\\\\{\mathrm T}_{\mathrm Y}\;=\;\left[\frac{0.16\;\;0.08}{1.05}\right]\;\;=\;0.076\\\\{\mathrm T}_{\mathrm Z\;}\;=\;\left[\frac{0.12\;\;0.8}{0.90}\right]\;=\;0.044\\\\$$
These results show that Z did not outperform the market. X had the strongest performance and both Y and X surpassed market expectations.
Example: Calculate the Treynor Ratio based on the performance of four portfolio managers over five years. Use the data given below
 The riskfree rate (RF) is 10% and
 The market return (RM) is 16%.
Portfolio Manager  Average Return (%)  Beta 
P  15  0.90 
Q  17  1.25 
R  17  1.05 
S  14  0.80 
Choose the portfolio manager that has performed best.
Solution:
Portfolio Manager  Treynor Ratio  Treynor Ratio 
P  $$\frac{(15\;\;10)}{0.90}=\;5.555$$  5.555 
Q  $$\frac{(17\;\;10)}{1.25}=\;5.600$$  5.600 
R  $$\frac{(17\;\;10)}{1.05}=\;6.666$$  6.666 
S  $$\frac{(14\;\;10)}{0.80}=\;5.000$$  5.000 
Therefore, Q is the strongest performer among all.
Jensen Measure/Jensen Ratio
The Treynor and Sharpe Indexes provide measures forÂ evaluatingÂ the relative performances of various portfolios,Â taking into account the level of risk involved. Jensen attempts to construct a measure of absolute performance on a riskadjusted basis Le, a definite standard against which performances of various funds can be measured.
This standard is based on the measurement of the “portfolio manager’s predictive ability i.e., his ability to earn returns through successful prediction of security prices which are higher than those which we would expect given the level of riskiness of his portfolio”.
A simplified version of his basic model is given as follows:
$${\overline{\mathrm R}}_{\;\mathrm{jt}}\;\;{\mathrm R}_{\mathrm{ft}}\;=\;{\mathrm\alpha}_{\mathrm j}\;+\;{\mathrm\beta}_{\mathrm j}\;({\overline{\mathrm R}}_{\mathrm{mt}}\;\;{\mathrm R}_{\mathrm{ft}})\\$$
Where,
 $${\overline{\mathrm R}}_{\;\mathrm{jt}}\;=\;\mathrm{Average}\;\mathrm{return}\;\mathrm{on}\;\mathrm{portfolio}\;\mathrm j\;\mathrm{for}\;\mathrm{period}\;\mathrm t,$$
 $${\mathrm R}_{\mathrm{ft}}\;=\;\mathrm{Risk}\mathrm{free}\;\mathrm{rate}\;\mathrm{of}\;\mathrm{interest}\;\mathrm{for}\;\mathrm{period}\;\mathrm t,$$
 $${\mathrm\beta}_{\mathrm j}\;=\;\mathrm{Measure}\;\mathrm{of}\;\mathrm{systematic}\;\mathrm{risk},$$
 $${\overline{\mathrm R}}_{\mathrm{mt}}\;=\mathrm{The}\;\mathrm{average}\;\mathrm{return}\;\mathrm{of}\;\mathrm a\;\mathrm{market}\;\mathrm{portfolio}\;\mathrm{for}\;\mathrm a\;\mathrm{period}\;\mathrm t.\;$$
 $$\alpha_j\;=\;Measures\;forecasting\;ability$$
An implication of forms of the Sharpe and Treynor models is that the intercept of the line is at the origin. In the Jensen model, the intercept can be at any point, including the origin.
The line Î±_{j} = 0 indicates neutral performance by management; i.e., management has done as well as an unmanaged market portfolio or a large, randomly selected portfolio managed with a naive buyandhold strategy.
The lower line, Î±_{j} = a negative value, indicates inferior management performance because management did not do as well as an unmanaged portfolio of equal systematic risk. This situation could arise in part because portfolio returns were not sufficient to offset the expenses incurred in the selection and management process.
The intercept may be interpreted in this fashion by examining the above formula. This occurs because if the portfolio manager is performing in a superior fashion, his intercept will have a positive value. After all, it will indicate that his portfolio is consistently overperforming the overall market. This would happen if the manager either had a superior ability to select undervalued securities or had a superior ability to recognize turning points in the market.
On the other hand,Â a negative interceptÂ would indicate that the manager consistently underperformed the overall market.Â In other words,Â the riskadjusted returns of his portfolio were consistently lower than the riskadjusted returns of the marketÂ duringÂ the same period.
Example: Calculate the Actual Return and Risk
Funds  R_{ft}  R_{jt}  R_{mt}  Beta 
Fund X  5Â  12  15  0.5 
Fund Y  5  20  15  1.0 
Fund Z  5  14  15  1.10 
Solution: From equation 1 return on the portfolio is:

$${\overline{\mathrm R}}_{\;\mathrm{jt}}\;\;{\mathrm R}_{\mathrm{ft}}\;=\;{\mathrm\alpha}_{\mathrm j}\;+\;{\mathrm\beta}_{\mathrm j}\;({\overline{\mathrm R}}_{\mathrm{mt}}\;\;{\mathrm R}_{\mathrm{ft}})\\$$
 $$\alpha\;=r_p\;\;r_{jt}$$
Fund X:
R_{jtÂ }= 5 + 0.5 (15 – 5) = 10
Î± = 1210 = 2% (Excess Positive Return)
Fund Y:
R_{jtÂ }= 5 + 1.5 (15 – 5) = 15
Î± = 2015 = 5% (Excess Positive Return)
Fund Z:
R_{jtÂ }= 5 + 1.10 (15 – 5) = 16
Î± = 1416 = 20% (Negative Return)
Example: Calculate Jenson’s Alpha based on the results of four portfolio managers for 5 years. Use the data given below to choose the manager with the best performance.
 The riskfree rate (RF) is 10% and
 The market return (RM) is 16%.
Portfolio Manager  Average Return (%)  Beta 
P  15  0.90 
Q  17  1.25 
R  17  1.05 
S  14  0.80 
Solution: Q is the best performer.
Portfolio Manager  Treynor Ratio  Treynor Ratio 
P  15 – [10 + 0.80(16 – 10)] = 0.40  0.40 
Q  15 – [10 + 0.80(16 – 10)] = 0.50  0.50 
R  15 – [10 + 0.80(16 – 10)] = 0.70  0.70 
S  15 – [10 + 0.80(16 – 10)] = 0.80  0.80 
Modigliani & Modigliani Measure (MÂ² Measure)
Modigliani Modigliani measure, which is referred to as M’ provides a riskadjusted measure of performance that has an economically meaningful interpretation.
The M^{2} is given by M^{2Â }= r_{p*Â }– r_{m}
where, M^{2} Modigliani – Modigliani measure,
r_{p* = }return on the adjusted portfolio,
r_{m = }return on the market portfolio.
The above image shows that P will have a negative M^{2} measure when its capital allocation line is less steep than the capital market line, i.e., when its Sharpe ratio is lower than that of the market index.
Example: Calculate the following performance measures for portfolio P and the market M. The Tbill rate during the period is 6%. By which measures did the P portfolio outperform the market?
Use the following data for a particular sample period.
Â  Portfolio P  Market M 
Average return  35%  28% 
Beta  1.20%  1.00% 
Standard Deviation  42%  30% 
Tracking error (nonsystematic risk), Ïƒ(e)  18%  0 
Solution: P has a standard deviation of 42% versus a market standard deviation of 30%.
Therefore, the adjusted portfolio P would be formed by mixing bills and portfolio P with weights 30/42 = 0.714 in P and 1 – 0.714 = 0.286 in bills.
The return on this portfolio would be (0. 286 Ã— 6%) + (0.714 Ã— 35%) = 26.7%, which is 1.3% less than the market return.
Thus, portfolio P has an M^{2} measure of 1.3 %.