Performance Measures Here are some performance measures that have been used (Refer Chapter 24 of text): 1. Sharpe’s Measure: (Rp - Rf)/Sigma_p 2. M-Square (an economic interpretation of the Sharpe ratio) 3. Jensen’s alpha: Alpha_p = Rp - [Rf + Beta_p(Rm-Rf)] 4. Treynor’s Square : Alpha_p/Beta_p Treynor’s Measure: (Rp-Rf)/Beta_p 5. Appraisal Ratio: (Rp-Rf)/(volatility of non-market risk in portfolio)
Jensen’s Alpha (1/6) Jensen’s alpha measures the extra return that the portfolio earns after adjusting for its “beta” risk. The beta here does not have to refer to only the market beta, but to all factors that are important to understanding the allocation of the fund. An example: Suppose an actively managed fund has the following allocation. It allocates 20% to small cap, and 80% to large cap. How do we evaluate the manager?
Jensen’s Alpha (2/6) 1. Get the historical return series for the fund, and calculate the excess return (Rp-Rf). 2. Get the passive portfolio’s that can serve as the benchmark. You could use one passive portfolio (say, S&P 500). Or you may even want to use multiple passive portfolio’s (for example, add a small-cap index like the Rusell 2000). Calculate the excess return for each of these benchmarks. 3. Run a regression of (Rp-Rf) on the excess returns on the benchmarks, eg. for our example, when we know the portfolio manager is investing in both large cap and small cap stocks, we want to two passive indices, (S&P500-Rf) and (Rusell2000-Rf). 4. Examine the intercept. If the intercept is positive and statistically significant, the manager has outperformed his benchmark. If its negative, he has underperformed his benchmark.
Problem with Jensen’s Alpha (3/6) Although Jensen’s alpha is theoretically a very appealing performance evaluation method, and also adjusts for risk - in practice, it is difficult to use in practice. The reason why it doesn’t work is because, even if the manager is skillful, the “alpha” is likely to be small, and therefore it is difficult to statistically prove that the alpha is positive. When the “alpha” is small, we require either large amounts of data, or we require the manager to have a very low volatility in his excess returns. For typical fund managers, we will thus not be able to conclude that the manager has an alpha different from zero. Consider, in the next slide, an example of the alpha of PEP. We will conclude that PEP would have to beat the market by 1.5% a month for 5 years before we are certain it has a positive alpha.
Jensen’s Alpha and PEP (4/6) Over the period, , the cumulative return on PEP over this period was 21%, in comparison with an S&P 500 return of –10%. A regression of the monthly excess return against that of S&P 500 gives an alpha of 0.56% (or about 7% annualized) with a t statistic of As the t-statistic is less than 2, we cannot say with confidence that PEP has a positive alpha. Despite the fact that has beaten the market by 7% annualized over 5 years, we still cannot say statistically that PEP has outperformed the market. There are two questions we need to ask: –Why is it statistically so difficult to conclude that PEP has beaten the market? –By how much would PEP have to outperform the S&P for us to be certain of the result?
Cumulative Returns (5/6)
Jensen’s Alpha (6/6) The cumulative return graph on the previous page illustrates graphically why it is so difficult to statistically conclude that PEP outperformed the market. In particular, PEP is much more volatile than the S&P 500. The annualized volatility of PEP is 28%, in comparison with S&P’s volatility of 19%. The higher the volatility, the greater the alpha would have to be before we can conclude that PEP has beaten the market. Given the volatility, we can observe (by experimentation) that PEP would have to beat the market by about 1.5% a month for 5 years for us to be certain that PEP has outperformed. However, given the last five year return history, we cannot say with any certainty that PEP has truly outperformed the market.
Treynor’s Square We define this as the ratio of the alpha of the portfolio to the beta of the portfolio: Treynor’s Square = Alpha_p/Beta_p. The logic behind this ratio is that we should require a higher alpha from a portfolio of a higher beta. This measure is useful as it allows us to rank managers by their risk-adjusted performance, after adjusting for the beta risk they take.
Treynor’s Measure An alternative way of expressing this measure is: (Rp- Rf)/Beta_p). This measure is called Treynor’s Measure. Treynor’s Measure = (Rp-Rf)/Beta_p. This is equivalent to calculating: Alpha_p/Beta_p + (Rm- Rf). Thus, if you use either the Treynor’s square, or Treynor’s measure to rank portfolios, you will get the same results. As the measure uses the Jensen’s Alpha, it has the same limitations that we already discussed regarding Jensen’s alpha.
An Example for Treynor Measures Suppose you have a choice between investing in two managers. Which would you prefer? Manager A: Alpha= 2%, Beta =0.9. Manager B: Alpha=3%, Beta=1.6. The Treynor Square measure for A is 2/0.9=2.22. The Treynor Square measure for B is 3/1.6= Because A has a higher measure, you’ll prefer A. Intuition: Suppose you combine B with t-bills, with weights (0.9/1.6= ) and , respectively. In this case, this combined portfolio will have a beta of 0.9, but an alpha of Clearly, you will prefer A.
Appraisal Ratio (1/2) Appraisal Ratio: Alpha_p/(Vol of non-market risk). Suppose you run the following regression: Rp-Rf = Alpha + Beta (Rm-Rf) + e. Then, the volatility of “e” represents the non-market risk or residual risk - or the extra risk you take over the benchmark. Intuitively, the appraisal ratio trades off the extra return you receive by investing in the active portfolio, versus the extra risk you take. You can calculate the volatility of the idiosyncratic risk by: volatility of the non-market risk = sqrt[ (Vol of Portfolio)^2 - (Beta*Vol of Market)^2 ].
Appraisal Ratio:An Example (2/2) Portfolio P: Alpha=1.63, Beta=0.69, Vol = Portfolio Q: Alpha=5.28, Beta=1.40, Vol= Benchmark: Vol =8.48. The non-market risk taken by P is: SQRT(6.17* *0.69*8.48*8.48)=1.95. The appraisal ratio for P is: 1.63/1.95 = The non-market risk taken by Q is: SQRT(14.89* *1.4*8.48*8.48)=8.98. The appraisal ratio for Q is: 5.28/8.98=0.59. Thus, P has a higher appraisal ratio and should be preferred.
Final Comments (1/2) Note that the performance measures differ from each other, and it is possible that they may also give different rankings. The appropriate measure to use will depend on your total portfolio. Here are some thumb rules to follow:
Thumb Rules (2/2) 1. Suppose you are only investing in 1 portfolio: P or Q. In that case, choose the one with the highest Sharpe Ratio or M Square. 2. Always choose a portfolio with positive alpha. 3. Comparison between two funds with the same alpha: – Suppose you already have an index portfolio, and you want to add one actively managed portfolio, P or Q. In this case, choose the one with the higher appraisal ratio. –If your portfolio consists of several actively managed portfolios, then choose between P and Q by the Treynor measure.
The TIAA-CREF Social Choice Fund As another example, consider the TIAA-CREF Social Choice fund (http://www.tiaa-cref.org).http://www.tiaa-cref.org The Social Choice fund invests in firms that do not violate certain socially desirable objectives. The fund typically has a mix of equity and fixed income instruments.
Sharpe Ratio and M-Square Comparing the TIAA-CREF fund to the index fund over the last five years, we get. Riskfree rate = 2%. Social Choice Fund: –Sharpe Ratio = 0.31, Vol = 12%, –Cumulative 5-yr Fund Return = 44%. Index Fund: –Sharpe Ratio = 0.09, Vol = 20.5%, – Cumulative 5-yr Fund Return = 28%. Thus, the M Square for the Stock Fund is ( )(20.5) = 4.53%/year. If we leveraged the fund by borrowing at a rate of 2% to create a portfolio of the same volatility as the index fund, then this portfolio would have outperformed this index by 4.53% /year.
Alpha, Treynor and Appraisal Ratio We estimate the alpha of the fund by regressing the excess returns on the fund, (R p – R f ), on the excess returns of the benchmark, (R m – R f ). From the regression results: –Alpha of the Fund: 2.13% per year. –Beta of the Fund = –Treynor measure: alpha/beta = –Appraisal Ratio = 1.08.
Market Timing What is market timing? Does timing work? How do we do performance evaluation when the fund manager times the market? How well do the timing measures work in practice?
Market Timing Market timing involves: 1. Shifting funds between a market-index and cash. 2. Shifting funds between high beta stocks and low beta stocks. Essentially, market timing attempts to anticipate market up and down movements. Perhaps the most famous timing strategy is the “Dow Theory”.
The Dow Theory The Dow Theory is named after the founding editor of the WSJ, Charles Henry Dow. But we know of the Dow Theory, not from Dow himself (who died in 1902), but from William Peter Hamilton. Hamilton was the editor of the journal for 27 years after Dow (taking over in 1902, after the death of Charles Dow), and he wrote a series of editorials forecasting major trends. The theory, that Hamilton attributes to Dow, was further elaborated in Hamilton’s book, “The Stock Market Barometer”, published in However, much of what we know comes from a book by Robert Rhea (1932, The Dow Theory, Barron’s, New York.). For a brief history, see: –http://www.e-analytics.com/cd.htm.
The Dow Theory and Hamilton Hamilton believed both in informationally efficient markets (“The market movement reflects al the real knowledge available..”) as well as in the “irrational exuberance” of individual investors, “..the pragmatic basis for the theory, a working hypothesis, if nothing more, lies in human nature itself. Prosperity will drive men to excess, and repentance for the consequences of those excesses will produce a corresponding depression.”
The Dow Theory: Basics Market movements may be decomposed into primary, secondary and tertiary trends. The primary trend is the long-term movement of prices, lasting from several months to several years. Secondary or intermediary trends are caused by short- term deviations of prices from the underlying trend line. These deviations are eliminated via corrections, and prices revert back to trend values. Tertiary are daily fluctuations of little importance. Primary trends are further classified into bull and bear markets.
The Bull Market Bull Markets have three stages: “first, is the revival of confidence in the future of business, second is the response of stock prices to the known improvement in corporate earnings, and the third is the period when speculation is rampant and inflation apparent.”
The Bear Market Bear Markets also have three stages, “the first represents the abandonment of the hopes on which the stocks were purchased at inflated prices; the second reflects selling due to decreased business and earnings, and the third is caused by distressed selling of sound securities, regardless of value”, (Rhea, The Dow Theory, 1932).
On Identifying the Primary Trend The objective of market timing is to identify the primary trend (bull or bear market). Here are some rules: 1. The trend must be confirmed by movement in two different market sectors. Movement in one sector alone is not reliable. 2. A big move followed by a period of quiescence usually identifies the beginning of a primary trend in that direction.
How well does the theory do? Comparing the strategy to a buy and hold strategy over 27 years, Hamilton beats the market until 1926, when the strong bull market took over. However, on a risk adjusted basis, his portfolio has a higher Sharpe ratio (0.559 vs 0.456) over the entire period. His average arithmetic return is 10.73% vs. the market’s 10.75%. Hamilton died in How well would the theory do today? This was investigated by researchers at NYU and Yale [see
Duplicating Hamilton’s Strategy Neural network: train a program to “learn” the strategy. This strategy is then applied to Over the whole period, 5.48 vs : Buy-Hold=1.48, Ham= : Buy-Hold=3.21, Ham= : Buy-Hold=9.64, Ham= : Buy-Hold=7.71, Ham= : Buy-Hold=0.41, Ham= : Buy-Hold=12.63, Ham= : Buy-Hold=15.44, Ham=16.24
Measuring Market Timing Analyzing the market timing ability of Hamilton was easy, because we knew his calls from his editorials. However, the typical fund manager does not advertise how he is timing - so how do we infer his timing ability? The first question: how do we model the value added by the manager via market timing? We can view the manager’s ability to market time as an embedded put option. In other words, by investing in a manager with timing ability, we are buying an index fund with an embedded put. –In fact, it may be argued that the real test of a market timer is in a volatile (non-trending) market.
Two Tests 1.Treynor and Mazuy Test: Add a square term to the usual regression: Rp - Rf = a + b(Rm-Rf) + c(Rm-Rf)^2 + e. 2. Henriksson and Merton: Rp - Rf = a + b(Rm-Rf) + c(Rm-Rf)D + e. We will mainly consider the second test.
Manager’s Timing Ability Because a manager’s timing ability allows you to avoid a downturn in a bear market, we consider the following regression: Rp-Rf = a + b (Rm-Rf) + c(Rm-Rf)D + e, where D=1, if market gives a positive return, and D=0 if the market gives a negative return. In a bull market, the manager’s beta will be (b+c), but in a bear market, the beta will be (b). In other words, we test whether the manager increases his ‘beta’ or the weight on the market in a bull market, and decreases it in a bear market. If we run a regression and estimate a positive “c”, then it shows that the manager has timing abilities. See the attached spreadsheet for an example of the test on a sample strategy.
Implementation Problems The typical problem with implementation is that the data we use may not match with the timing horizon of the portfolio manager. If the manager times on a daily basis, but we only have monthly data it will be difficult to capture the timing ability. Suppose, for example, over the month the market moved up. Even if the timer was successful, it is difficult to distinguish the timing ability from the buy and hold strategy.
Some Recent Results Goetzmann, Ingersoll and Ivkovic at Yale recently applied the test to asset allocation funds. [“Monthly Measurement of Daily Timers”, 1998]. Of the 23 Asset Allocation Funds, they found 2 funds that showed significant market timing skills.