We have two major requirements of a PM:
1. The ability to derive above average returns for a given risk class (large risk-adjusted returns); and
2. the ability to completely diversify the portfolio to eliminate all unsystematic risk.
May also desire large real (inflation-adjusted) returns, maximization of current income, high after-tax rate of return, preservation of capital.
Requirement #1 can be achieved either through superior timing or superior security selection. A PM can select high beta securities during a time when he thinks the market will perform well and low (or negative) beta stocks at a time when he thinks the market will perform poorly.
Conversely, a PM can try to select undervalued stocks or bonds for a given risk class.
Requirement #2 argues that one should be able to completely diversify away all unsystematic risk (as you will not be compensated for it). You can measure the level of diversification by computing the correlation between the returns of the portfolio and the market portfolio. A completely diversified portfolio correlated perfectly with the completely diversified market portfolio because both include only systematic risk.
Some portfolio evaluation techniques measure for one requirement (high risk-adjusted returns) and not the other; some measure for complete diversification and not the other; some measure for both, but don't distinguish between the two requirements.
Composite Equity Portfolio Performance Measures
As late as the mid 1960s investors evaluated PM performance based solely on the rate of return. They were aware of risk, but didn't know how to measure it or adjust for it. Some investigators divided portfolios into similar risk classes (based upon a measure of risk such as the variance of return) and then compared the returns for alternative portfolios within the same risk class.
We shall look at some measures of composite performance that combine risk and return levels into a single value.
Treynor Portfolio Performance Measure (aka: reward to volatility ratio)
This measure was developed by Jack Treynor in 1965. Treynor (helped developed CAPM) argues that, using the characteristic line, one can determine the relationship between a security and the market. Deviations from the characteristic line (unique returns) should cancel out if you have a fully diversified portfolio.
Treynor's Composite Performance Measure: He was interested in a performance measure that would apply to ALL investors regardless of their risk preferences. He argued that investors would prefer a CML with a higher slope (as it would place them on a higher utility curve). The slope of this portfolio possibility line is:
A larger T
i value indicates a larger slope and a better portfolio for ALL INVESTORS REGARDLESS OF THEIR RISK PREFERENCES. The numerator represents the risk premium and the denominator represents the risk of the portfolio; thus the value, T, represents the portfolio's return per unit of systematic risk. All risk averse investors would want to maximize this value.The Treynor measure only measures systematic risk--it automatically assumes an adequately diversified portfolio.
You can compare the T measures for different portfolios. The higher the T value, the better the portfolio performance. For instance, the T value for the market is:
In this expression, b
m = 1.Demonstration of Comparative Treynor Measures: Assume that you are an administrator of a large pension fund (i.e. Terry Teague of Boeing) and you are trying to decide whether to renew your contracts with your three money managers. You must measure how they have performed. Assume you have the following results for each individual's performance:
| Investment Manager |
Average Annual Rate of Return | Beta |
| Z | 0.12 | 0.90 |
| B | 0.16 | 1.05 |
| Y | 0.18 | 1.2 |
You can calculate the T values for each investment manager:
| Tm | (0.14-0.08)/1.00=0.06 |
| TZ | (0.12-0.08)/0.90=0.044 |
| TB | (0.16-0.08)/1.05=0.076 |
| TY | (0.18-0.08)/1.20=0.083 |
These results show that Z did not even "beat-the-market." Y had the best performance, and both B and Y beat the market. [To find required return, the line is: .08 + .06(Beta).
One can achieve a negative T value if you achieve very poor performance or very good performance with low risk. For instance, if you had a positive beta portfolio but your return was less than that of the risk-free rate (which implies you weren't adequately diversified or that the market performed poorly) then you would have a (-) T value. If you have a negative beta portfolio and you earn a return higher than the risk-free rate, then you would have a high T-value. Negative T values can be confusing, thus you may be better off plotting the values on the SML or using the CAPM (in this case, .08+.06(Beta)) to calculate the required return and compare it with the actual return.
Sharpe Portfolio Performance Measure (aka: reward to variability ratio)
This measure was developed in 1966. It is as follows:
It is VERY similar to Treynor's measure, except it uses the total risk of the portfolio rather than just the systematic risk. The Sharpe measure calculates the risk premium earned per unit of total risk. In theory, the S measure compares portfolios on the CML, whereas the T measure compares portfolios on the SML.
Demonstration of Comparative Sharpe Measures: Sample returns and SDs for four portfolios (and the calculated Sharpe Index) are given below:
Portfolio |
Avg. Annual RofR |
SD of return |
Sharpe measure |
B |
0.13 |
0.18 |
0.278 |
O |
0.17 |
0.22 |
0.409 |
P |
0.16 |
0.23 |
0.348 |
Market |
0.14 |
0.20 |
0.30 |
Thus, portfolio O did the best, and B failed to beat the market. We could draw the CML given this information: CML=.08 + (0.30)SD
Treynor Measure vs. Sharpe Measure. The Sharpe measure evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk in the denominator). If we had a fully diversified portfolio, then both the Sharpe and Treynor measures should given us the same ranking. A poorly diversified portfolio could have a higher ranking under the Treynor measure than for the Sharpe measure.
Jenson Portfolio Performance Measure (aka differential return measure)
This measure (as are all the previous measures) is based on the CAPM:
We can express the expectations formula (the above formula) in terms of realized rates of return by adding an error term to reflect the difference between E(R
j) vs actual Rj:
By subtracting the risk free rate from both sides, we get:
Using this format, one would not expect an intercept in the regression. However, if we had superior portfolio managers who were actively seeking out undervalued securities, they could earn a higher risk-adjusted return than those implied in the model. So, if we examined returns of superior portfolios, they would have a significant positive intercept. An inferior manager would have a significant negative intercept. A manager that was not clearly superior or inferior would have a statistically insignificant intercept. We would test the constant, or intercept, in the following regression:
This constant term would tell us how much of the return is attributable to the manager's ability to derive above-average returns adjusted for risk.
Applying the Jenson Measure. This requires that you use a different risk-free rate for each time interval during the sample period. You must subtract the risk-free rate from the returns during each observation period rather than calculating the average return and average risk-free rate as in the Sharpe and Treynor measures. Also, the Jensen measure does not evaluate the ability of the portfolio manager to diversify, as it calculates risk premiums in terms of systematic risk (beta). For evaluating diversified portfolios (such a most mutual funds) this is probably adequate. Jensen finds that mutual fund returns are typically correlated with the market at rates above .90.
Application of Portfolio Performance Measures
Calculated Sharpe, Treynor and Jenson measures for 20 mutual funds. Using the Jenson measure, only 3 managers had superior performance (Fidelity Magellan, Templeton Growth Funds, and Value Line Special Situations Fund) while 2 managers had inferior performance (Oppenheimer Fund and T. Rowe Price Growth Stock Fund).
Relationship among Portfolio Performance Measures
For all three methods, if we are examining a well-diversified portfolio, the rankings should be similar. A rank correlation measure finds that there is about a 90% correlation among all three measures. Reilly recommends that all three measures. [In my opinion the Jensen measure is the most stringent. It is testing for statistical significance, whereas the other methods are not. The other methods are also examining average returns, whereas the Jensen measure uses actual returns during each observation period.]
Factors that Affect Use of Performance Measures
You need to judge a portfolio manager over a period of time, not just over one quarter or even one year. You need to examine the manager's performance during both rising and falling markets. There are also other problems associated with these measures:
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Measurement Problems: All of these measures are based on the CAPM. Thus, we need a real world proxy for the theoretical market portfolio. Analysts typically use the S&P500 Index as the proxy; however, it does not constitute a true market portfolio. It only includes common stocks trading on the NYSE. Roll, in his 1980/1981 papers, calls this benchmark error.We use the market portfolio to calculate the betas for the portfolios. Roll argues that if the proxy used for the market portfolio is inefficient, the betas calculated will be inappropriate. The true SML may actually have a higher (or lower) slope. Thus, if we plot a security that lies above the SML it could actually plot below the "true" SML.
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Global Investing: Incorporating global investments (with their lower coefficients of correlation) will surely move the efficient frontier to the left, thus providing diversification benefits. It may also shift the efficient frontier upward (increasing returns). [However, we have no proxy to measure global markets.]