Abstract
A central problem for performance-oriented portfolio formation modeling is to backtest the performance of a forecast model/method over a large sample of securities for a relatively long time period. In a multifactor, multistyle, multirisk modeling framework, backtest assessments of return forecast performance potential are complicated by measurement error, pervasive multicollinearity, specification error, and even omitted variable error. This chapter shows how the matched control methodology commonly used in controlled and partially controlled studies can be adapted to isolate well the realized return response to a return forecast with complete suppression of covariation distortion. This chapter formulates and then illustrates the use of a power optimizing mathematical program that transforms a cross section of forecast rank-ordered fractile portfolios into an associated relative rank-ordered cross section of control-matched portfolios. In contrast to the linearity/bilinearity assumption in most multivariate stock return models and the strong distribution assumptions in regression estimation, the matched control methodology requires no distribution or functional form assumptions. Estimation benefits relative to conventional multivariate regression assessments include (1) complete elimination of collinearity distortion, (2) mitigation of measurement error, (3) and mitigation of specification errors for known variable dependencies of unknown functional form. Particularly important in the test example illustrated here is the ability to remove variation in the dividend–gain mix and thereby control for systematic tax effects without estimating the marginal tax rate for either dividends or gains. The benefits of the matched control methodology are illustrated for an eight-variable return forecast model that provides apparent performance benefits but whose performance assessment is complicated by measurement error, by extreme multicollinearity, and by possible specification and completeness issues including a possible yield tilt (tax tilt) from the high correlation of the forecast model variables with dividend yield. For the illustrative forecast, the cross sections of realized risky return, realized standard deviations, and skewness coefficients are nonlinear. Moreover, the risky return cross-sections change significantly as different combinations of control variables are imposed on the uncontrolled rank-ordered cross sections, e.g., the slope of the regression of realized returns on forecast score changes from 8.xx with no controls to 16zz with just the three tax controls. In addition to mitigating estimation bias, the primary statistical benefit is greatly increased statistical confidence and power relative to using multivariate regression to try to isolate forecast performance potential.
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Notes
- 1.
There is a large literature on statistical designs for the empirical estimation of multivariate functional dependencies but primarily focused on controlled or partially controlled studies rather than the observational samples that typically arise in epidemiology, demographics, the social sciences including especially economics and business, medicine, and many physical sciences including astronomy. Treatment response studies were an early statistical design focus and continue to be an ongoing design estimation concern. Because of the early and ongoing concern for treatment response studies, it is common to use the term response surface methods to refer to empirical methods for estimating functional dependencies. Most of the statistical literature pertains to controlled or partially controlled experiments in process industries, medicine, and advertising. For background readers are referred to the foundation works by Box (1954) and Box and Draper (1959); to the review article by Myers, Khuri, and Cornell (1989); and to the books on response surface methods: Khuri and Cornell (1996), Box and Draper (1987), and Box, Hunter, and Hunter (2005).
- 2.
Most of the statistical design literature cited in the previous footnote focuses on controlled and especially partially controlled studies. The ability to adapt response surface methods to observational studies is developed in Stone, Adolphson, and Miller (1993). They extend the use of response surface methods to observational data for those estimation situations in which it is pertinent to group data, e.g., to group observations on individual stocks into portfolios or households into income percentiles. The use of control variables to assess a conditional dependency (response subsurface) is a basic technique in controlled experiments that we adapt in this study to obtain a well-isolated portfolio-level dependency of realized risky return on a return forecast. Fortunately for compatibility with other finance panel studies using rank-ordered grouping, the optimal control matching can be structured as a power/efficiency improvement to the widely used relative rank-ordering used in many return dependency studies.
- 3.
The mathematics for correlation magnification is straightforward. The formula for the correlation coefficient between variables X and Y is covariance(X, Y)/[SD(X)SD(Y)], where SD stands for standard deviation. Ranking on variable X and grouping into fractile portfolios preserve a wide range of values for variable X. However, in each of the portfolios, the individual values of variable Y tend to average out to a value close to the sample average value. Thus, the portfolio-level standard deviation of Y is reduced, and for a very small number of portfolios (e.g., quintiles or deciles), SD(Y) tends to approach zero while the covariance in the numerator declines relatively slowly because of the wide range of portfolio-level values for the ranking variable X.
- 4.
Brennen (1970) shows that dividend yield is a significant omitted variable from the CAPM. Rosenberg (1974), Rosenberg and Rudd (1977), Rosenberg and Marathe (1979), Blume (1980), Rosenberg and Rudd (1982) and many subsequent researchers have empirically established the so-called dividend yield tilt. More recent studies include Peterson, Peterson, and Ang (1985), Fama and French (1988) and Pilotte (2003). For an extensive review of both dividend valuation and dividend policy and extensive references in this area, see Lease, Kose, Kalay, Loewenstein, and Sarig (2000).
- 5.
Other dispersion measures could be used, e.g., mean absolute deviation or interquartile range. Relative to these measures, within-portfolio variance gives greater weight to extreme departures from the portfolio average.
- 6.
Since beta measurement errors are known to regress toward the mean, the assumption of uncorrelated measurement errors used in the discussion here is almost certainly too strong for correctly assessing efficiency gains from reducing measurement error by grouping into rank-ordered fractile portfolios. In particular, beta values that are correlated with the forecast are very likely to have systematic variation in beta change values over the cross section. The control matching methodology developed in Sects. 8.4 and 8.5 will mitigate systematic changes in measurement error such as the well-known regression of betas toward the mean. When every portfolio in the cross section has the same beta value, each portfolio will have essentially the same ex post beta change. Hence, control matching provides the benefit of no systematic distortion from beta regression toward the mean.
- 7.
There is a large literature on statistical designs for the empirical estimation of multivariate functional dependencies but primarily focused on controlled or partially controlled studies rather than the observational samples that typically arise in epidemiology, demographics, the social sciences including especially economics and business, medicine, and many physical sciences including astronomy. Treatment response studies were an early statistical design focus and continue to be an ongoing design estimation concern. Because of the early and ongoing concern for treatment response studies, it is common to use the term response surface methods to refer to empirical methods for estimating functional dependencies. Most of the statistical literature pertains to controlled or partially controlled experiments in process industries, medicine, and advertising. For background readers are referred to the foundation works by Box (1954) and Box and Draper (1959); to the review article by Myers, Khuri, and Cornell (1989) and to the books on response surface methods: Khuri and Cornell (1996), Box and Draper (1987), and Box et~al. (2005).
- 8.
Most of the statistical design literature cited in the previous footnote focuses on controlled and especially partially controlled studies. The ability to adapt response surface methods to observational studies is developed in Stone et~al. (1993). They extend the use of response surface methods to observational data for those estimation situations in which it is pertinent to group data, e.g., to group observations on individual stocks into portfolios or households into income percentiles. The use of control variables to assess a conditional dependency (response subsurface) is a basic technique in controlled experiments that we adapt in this study to obtain a well-isolated portfolio-level dependency of realized risky return on a return forecast.
- 9.
Candidates for control variables are any variable believed to have a significant impact for explaining or predicting the cross-section of realized returns. Classes of return impacts include risk measures, e.g., beta, the book-price ratio, and firm size; tax valuation impacts, e.g., dividend yield, the dividend payout ratio, and possibly the debt tax shield as measured by the percentage of financing that is debt; and attractiveness measures that are indicative of future cashflow generation potential and asset usage efficiency, e.g., growth, return on investment, or sales intensity (sales per dollar of investment). Beta is the standard measure of volatility risk established as a return explanatory variable in formulations of the capital asset pricing model, for instance Sharpe (1964). It is also included in multifactor return modeling, as indicated by the Fama-French series, e.g., Fama and French (1992, 1996 2008a, 2008b).The variables EP and BP are the reciprocals of the price-earnings ratio and the price-book ratio, respectively. Their use as valuation and/or risk variables has been researched extensively beginning with Basu (1977), viewing dependency on the earnings-price ratio primarily as a valuation anomaly but recognizing the possibility that the earnings-price ratio could also be a risk instrument.The tax effect associated with the differential taxation of dividends and capital gains and the debt tax shield are discussed extensively in Sect. 8.7.3.
- 10.
Trying to put before-tax returns on an after-tax basis is fraught with problems. To put the dividend component of return on an after-tax basis requires an estimate of a time-varying marginal tax rate for ordinary income. To put the gain component of return on an after-tax basis requires the determination of the time-varying effective tax rate on capital gains.
- 11.
In a stock return forecast designed to find misvalued stocks, extreme values of some return variables are very likely the observations of greatest performance potential.
- 12.
In this study, the target average value is always the ex ante sample average value.
- 13.
See, for instance, Sharpe (1963, 1967, 1971) and Stone (1973).
- 14.
It is intuitive that minimizing the amount and distance of cross-portfolio shifting tends to preserve the original within-portfolio forecast distribution including within-portfolio variances. The substance of this approximation is to use portfolio rank-order distance as a substitute for actual return forecast differences. Since we map each return forecast into a near uniform distribution on the (0, 1) interval, we tend to ensure the validity of this approximation.
- 15.
The changed difference in changed rank is actually a stronger restriction on changing portfolio membership than the quadratic variance change it is approximating. Because the LP shifting measure penalizes very large rank shifts even more than the quadratic, the LP approximation tends to preclude large shifts in rank order even more than the quadratic. However, comparison of the LP and quadratic solutions showed that the LP and quadratic solutions were generally close.
- 16.
For details on the mean–variance optimization used, see Markowitz (1959, 1987).
- 17.
Markowitz and Xu (1994) later published the data mining test for backtest bias. Their test allows assessment of the expected difference between the best test model and an average of simulated policies.
- 18.
BGMTX is a one-step direct forecast of stock returns. The more common return forecast framework is a two-step return forecast in which an analyst predicts both a future value of a variable such as earnings and an associated future value multiple for that variable such as a future price–earnings ratio. These two predictions imply a prediction of future value. Under the assumption that the current price will converge toward this predicted future value, there is an implied prediction of a gain return. Given a prediction of future dividends, there is an implied stock return forecast. For a thorough treatment of the two-step framework and extensive references to the two-step return prediction literature, readers are referred to the CFA study guide by Stowe et~al. (2007). Because BGMTX is a direct one-step return prediction following a step-by-step determination of a normalized weighting of current and relative value ratios, it is amenable to a repeatable backtest.
- 19.
The ratio BP is of course the book-to-market ratio. BP is defined here as the ratio of book value per share to price per share. However, multiplying both numerator and denominator by the number of outstanding shares gives the ratio of book value to market value.
- 20.
Security analysis, Graham and Dodd (1934), is generally credited with establishing the idea of value investing. Graham and Dodd influenced Williams (1938), who made particular reference to their low P/E and net current approaches in The Theory of Investment Value. In turn, Williams (1938) influenced Markowitz’s thoughts on return and risk as noted in Markowitz (1991). Over the past 25 years, value-focused fundamental analysts and portfolio managers have expanded their value measures from primarily price relative to earnings and price relative to book value to include also price relative to cash flow and even price relative to sales. The choice of the eight fundamental variables in BGMTX reflects this expansion in focus, especially the expansion to include cash and sales ratios.
- 21.
When regression coefficients with t-values ≤ 1.96 are made equal to zero, there are no negative coefficients regardless of significance.
- 22.
Michaud (1989, 1998) recognizes that uncertainty about both the return and risk forecasts and other portfolio selection parameters is a source of risk/uncertainty in addition to the inherent uncertainty risk of investment and has formalized a very sophisticated resampling simulation to structure very thorough sensitivity analysis.
- 23.
See Grinold and Kahn (2000) for a thorough description of the MCI-Barra active tilt frameworks. See Menchero, Korozov, and Shepard (2010) for an updated version of the MCI-Barra equity risk modeling that includes both industry and country factors, a global equity risk factor, and additional style factors for value, size, momentum, etc.
- 24.
The magnitude of a tilt is defined as the absolute value of the difference in the relative weighting of stocks in the benchmark and the tilt portfolio. For instance, if a stock with a weight of 0.5 % is increased to 0.7 %, the tilt change is 0.2 %. If a stock with a relative weight of 0.15 % is excluded completely, the tilt change is 0.15 % to the tilt. The overall tilt percentage is the sum of all the tilt change percentages.
- 25.
- 26.
See Appendix 8.1 for the data to compute the ranges of realized risky return and standard deviations. Appendices 8.2 to 8.6 provide pertinent data for each of the 30 fractile portfolios for the cross sections with other control variables.
- 27.
Relative to the CRSP index, stocks in the backtest sample had a lower beta, generally about 10 % lower. While the backtest sample excluded generally low beta financial stocks, it also tilted toward larger, more mature companies because of the requirement of inclusion in both Compustat and CRSP for at least five years. This maturity tilt is the reason for the somewhat lower beta values for the backtest sample than for the overall CRSP sample.
- 28.
It is easy to check how well DP alone and DP and β in combination actually control for cross-sectional variation in the dividend–gain mix. For the forecast rank ordering in this study, in all time periods of 5 years or longer after 1972, the DP control alone does a good job of controlling for cross-sectional variation in the ex post dividend–gain mix. The term “good job” means that the average ex post dividend–gain ratio in each portfolio is very close to the sample average with no systematic variation over the cross section. Controlling for DP and EP together improves the control for variation in the dividend–gain mix by eliminating portfolio-to-portfolio variation and making most portfolios very close to the average. Controlling for DP and β in combination improves the control since high beta tends to be lower dividend payout. Likewise controlling for DP and size in combination improves the control for the dividend–gain mix for similar reasons, namely, the fact that small size tends to be higher beta and often zero or token dividend payout. Thus, the combination of tax controls and risk controls together improves on the tax controls alone in terms of ensuring very little portfolio-to-portfolio variation in the dividend–gain mix, especially for all time periods after 1972. The main caveat is slighter greater variation about the average dividend–gain mix for the three lowest-ranked and the three highest-ranked portfolios. This greater variation about the mean for the low-ranked and high-ranked portfolios is consistent with the much greater realized standard deviation for these portfolios as shown in Exhibit 8.5 as well as the much greater positive skewness for the three highest-ranked portfolios.
- 29.
A plot of realized semi-standard deviation for the case of all risk and all tax controls together is flat for the top 25 portfolios in the cross section, strong support for the assertion that the greater standard deviation for the top four portfolios is primarily a positive skewness effect and not downside uncertainty.
- 30.
Forecast value for mean–variance portfolio selection was established in BGMTX for 1978–1990. Guerard, Gultekin, and Stone (1997) added to the evidence of forecast value for the return forecast itself by using an endogenous APT to remove all explainable systematic return. Others have added both growth and momentum to show performance value well after the 1993 publication time.
- 31.
With 30 fractile portfolios in the cross sections of conditional return response observations, the difference for the top three returns combined and the bottom three portfolios combined represents a high-minus-low return for the top and bottom deciles of the cross section.
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Appendices
Appendices
8.1.1 Appendix 8.1. Rank-ordered portfolio data: no controls
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 2.98 | 10.29 | 8.96 | 0.802 | 0.096 |
2 | 3.90 | 6.22 | 9.11 | 0.279 | 0.057 |
3 | 6.99 | 8.84 | 7.96 | 0.065 | 0.093 |
4 | 11.70 | 7.72 | 7.01 | −0.231 | 0.092 |
5 | 14.38 | 6.10 | 6.81 | −0.300 | 0.075 |
6 | 18.02 | 8.89 | 6.41 | −0.543 | 0.116 |
7 | 21.51 | 7.94 | 6.13 | −0.509 | 0.108 |
8 | 24.98 | 7.75 | 6.09 | −0.519 | 0.106 |
9 | 28.05 | 9.18 | 5.76 | −0.682 | 0.133 |
10 | 31.50 | 7.85 | 5.68 | −0.593 | 0.115 |
11 | 34.84 | 6.48 | 5.54 | −0.714 | 0.097 |
12 | 37.98 | 7.96 | 5.56 | −0.606 | 0.119 |
13 | 41.04 | 9.92 | 5.48 | −0.591 | 0.151 |
14 | 44.48 | 9.20 | 5.43 | −0.528 | 0.141 |
15 | 47.98 | 9.39 | 5.39 | −0.503 | 0.145 |
16 | 51.03 | 9.42 | 5.35 | −0.384 | 0.147 |
17 | 54.45 | 8.79 | 5.35 | −0.417 | 0.137 |
18 | 57.89 | 9.93 | 5.54 | −0.213 | 0.149 |
19 | 60.96 | 9.48 | 5.50 | −0.209 | 0.144 |
20 | 64.44 | 11.74 | 5.56 | −0.241 | 0.176 |
21 | 67.87 | 9.97 | 5.69 | −0.077 | 0.146 |
22 | 70.89 | 10.04 | 5.78 | 0.034 | 0.145 |
23 | 73.96 | 10.36 | 6.08 | 0.462 | 0.142 |
24 | 77.43 | 12.39 | 6.35 | 0.287 | 0.163 |
25 | 80.87 | 11.95 | 6.49 | 0.530 | 0.154 |
26 | 84.41 | 12.93 | 6.95 | 0.421 | 0.155 |
27 | 87.06 | 14.29 | 7.35 | 1.061 | 0.162 |
28 | 91.08 | 13.03 | 8.26 | 0.993 | 0.131 |
29 | 94.29 | 15.50 | 9.57 | 1.271 | 0.135 |
30 | 96.59 | 19.12 | 10.62 | 2.308 | 0.150 |
8.1.2 Appendix 8.2. Rank-ordered portfolio data: only a beta control
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 4.21 | 7.77 | 7.36 | −0.248 | 0.088 |
2 | 5.20 | 5.74 | 7.59 | 0.080 | 0.063 |
3 | 8.54 | 9.37 | 7.14 | −0.447 | 0.109 |
4 | 13.52 | 7.72 | 6.22 | −0.617 | 0.103 |
5 | 16.19 | 7.64 | 6.40 | −0.636 | 0.099 |
6 | 19.69 | 9.61 | 5.83 | −0.565 | 0.137 |
7 | 23.12 | 7.73 | 5.98 | −0.612 | 0.108 |
8 | 26.27 | 7.60 | 5.81 | −0.477 | 0.109 |
9 | 29.14 | 7.56 | 5.78 | −0.507 | 0.109 |
10 | 32.34 | 9.83 | 5.68 | −0.629 | 0.144 |
11 | 35.49 | 7.94 | 5.56 | −0.400 | 0.119 |
12 | 38.51 | 7.73 | 5.77 | −0.541 | 0.112 |
13 | 41.44 | 7.68 | 5.61 | −0.567 | 0.114 |
14 | 44.72 | 8.70 | 5.64 | −0.402 | 0.128 |
15 | 48.02 | 8.08 | 5.54 | −0.484 | 0.121 |
16 | 50.97 | 9.39 | 5.65 | −0.404 | 0.139 |
17 | 54.24 | 8.85 | 5.59 | −0.290 | 0.132 |
18 | 57.51 | 9.04 | 5.76 | −0.034 | 0.131 |
19 | 60.47 | 9.17 | 5.75 | 0.197 | 0.133 |
20 | 63.79 | 11.05 | 5.84 | −0.012 | 0.158 |
21 | 67.07 | 10.47 | 5.78 | −0.180 | 0.151 |
22 | 69.96 | 10.37 | 5.92 | 0.149 | 0.146 |
23 | 72.92 | 10.89 | 5.85 | −0.010 | 0.155 |
24 | 76.15 | 12.23 | 6.09 | 0.042 | 0.167 |
25 | 79.65 | 12.76 | 6.07 | 0.287 | 0.175 |
26 | 83.18 | 11.10 | 6.50 | 0.632 | 0.142 |
27 | 85.85 | 13.27 | 6.78 | 0.247 | 0.163 |
28 | 89.97 | 15.10 | 7.65 | 1.263 | 0.164 |
29 | 93.48 | 15.11 | 8.68 | 1.483 | 0.145 |
30 | 95.94 | 19.89 | 9.55 | 1.585 | 0.173 |
8.1.3 Appendix 8.3. Rank-ordered portfolio data: only a size control
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 3.28 | 10.31 | 8.90 | 0.914 | 0.097 |
2 | 4.22 | 6.77 | 9.08 | 0.159 | 0.062 |
3 | 7.31 | 7.73 | 7.93 | 0.137 | 0.081 |
4 | 12.00 | 7.91 | 7.03 | −0.313 | 0.094 |
5 | 14.65 | 6.35 | 6.85 | −0.200 | 0.077 |
6 | 18.28 | 8.70 | 6.33 | −0.556 | 0.115 |
7 | 21.86 | 8.06 | 6.06 | −0.621 | 0.111 |
8 | 25.20 | 7.79 | 6.10 | −0.538 | 0.106 |
9 | 28.29 | 9.07 | 5.84 | −0.602 | 0.130 |
10 | 31.67 | 7.09 | 5.61 | −0.704 | 0.105 |
11 | 34.97 | 6.80 | 5.54 | −0.640 | 0.102 |
12 | 38.11 | 8.49 | 5.54 | −0.537 | 0.128 |
13 | 41.16 | 9.49 | 5.47 | −0.581 | 0.145 |
14 | 44.51 | 9.05 | 5.35 | −0.604 | 0.141 |
15 | 47.99 | 9.34 | 5.44 | −0.413 | 0.143 |
16 | 51.09 | 9.30 | 5.33 | −0.375 | 0.145 |
17 | 54.40 | 8.84 | 5.36 | −0.478 | 0.137 |
18 | 57.82 | 10.07 | 5.58 | −0.279 | 0.150 |
19 | 60.90 | 9.79 | 5.42 | −0.198 | 0.150 |
20 | 64.26 | 11.35 | 5.51 | −0.196 | 0.172 |
21 | 67.67 | 10.17 | 5.62 | −0.035 | 0.151 |
22 | 70.68 | 10.91 | 5.95 | 0.266 | 0.153 |
23 | 73.75 | 10.31 | 5.96 | 0.189 | 0.144 |
24 | 77.07 | 12.86 | 6.31 | 0.340 | 0.170 |
25 | 80.63 | 11.73 | 6.52 | 0.322 | 0.150 |
26 | 84.15 | 12.78 | 6.96 | 0.514 | 0.153 |
27 | 86.80 | 14.43 | 7.29 | 1.045 | 0.165 |
28 | 90.80 | 13.24 | 8.21 | 1.017 | 0.134 |
29 | 94.01 | 15.06 | 9.58 | 1.261 | 0.131 |
30 | 96.28 | 18.87 | 10.55 | 2.329 | 0.149 |
8.1.4 Appendix 8.4. Rank-ordered portfolio data: only a BP control
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 4.03 | 10.27 | 8.83 | 0.805 | 0.097 |
2 | 5.47 | 7.47 | 8.89 | 0.276 | 0.070 |
3 | 9.10 | 7.66 | 7.64 | −0.064 | 0.083 |
4 | 14.47 | 7.46 | 6.83 | −0.460 | 0.091 |
5 | 17.26 | 8.51 | 6.71 | −0.495 | 0.106 |
6 | 21.01 | 8.60 | 6.45 | −0.292 | 0.111 |
7 | 24.59 | 9.23 | 5.94 | −0.657 | 0.129 |
8 | 27.75 | 7.76 | 5.88 | −0.745 | 0.110 |
9 | 30.61 | 8.45 | 5.87 | −0.772 | 0.120 |
10 | 33.74 | 8.09 | 5.67 | −0.515 | 0.119 |
11 | 36.75 | 7.70 | 5.38 | −0.824 | 0.119 |
12 | 39.56 | 10.29 | 5.55 | −0.547 | 0.154 |
13 | 42.27 | 9.68 | 5.51 | −0.690 | 0.146 |
14 | 45.27 | 8.96 | 5.38 | −0.569 | 0.139 |
15 | 48.26 | 8.93 | 5.25 | −0.594 | 0.142 |
16 | 50.89 | 9.74 | 5.35 | −0.425 | 0.152 |
17 | 53.84 | 9.24 | 5.33 | −0.381 | 0.144 |
18 | 56.82 | 10.67 | 5.37 | −0.310 | 0.166 |
19 | 59.50 | 10.67 | 5.51 | −0.077 | 0.161 |
20 | 62.58 | 10.70 | 5.49 | −0.173 | 0.162 |
21 | 65.68 | 9.64 | 5.72 | −0.105 | 0.140 |
22 | 68.44 | 10.50 | 5.83 | −0.038 | 0.150 |
23 | 71.36 | 11.25 | 6.02 | 0.177 | 0.156 |
24 | 74.60 | 10.88 | 6.30 | 0.220 | 0.144 |
25 | 78.16 | 11.75 | 6.70 | 0.565 | 0.146 |
26 | 81.86 | 12.03 | 6.92 | 0.373 | 0.145 |
27 | 84.69 | 13.33 | 7.15 | 0.707 | 0.155 |
28 | 89.21 | 12.84 | 8.44 | 1.469 | 0.127 |
29 | 92.95 | 15.15 | 9.62 | 1.342 | 0.131 |
30 | 95.83 | 17.85 | 10.65 | 2.186 | 0.140 |
8.1.5 Appendix 8.5. Rank-ordered portfolio data: risk controls only
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 4.21 | 7.77 | 7.36 | −0.248 | 0.088 |
2 | 5.20 | 5.74 | 7.59 | 0.080 | 0.063 |
3 | 8.54 | 9.37 | 7.14 | −0.447 | 0.109 |
4 | 13.52 | 7.72 | 6.22 | −0.617 | 0.103 |
5 | 16.19 | 7.64 | 6.40 | −0.636 | 0.099 |
6 | 19.69 | 9.61 | 5.83 | −0.565 | 0.137 |
7 | 23.12 | 7.73 | 5.98 | −0.612 | 0.108 |
8 | 26.27 | 7.60 | 5.81 | −0.477 | 0.109 |
9 | 29.14 | 7.56 | 5.78 | −0.507 | 0.109 |
10 | 32.34 | 9.83 | 5.68 | −0.629 | 0.144 |
11 | 35.49 | 7.94 | 5.56 | −0.400 | 0.119 |
12 | 38.51 | 7.73 | 5.77 | −0.541 | 0.112 |
13 | 41.44 | 7.68 | 5.61 | −0.567 | 0.114 |
14 | 44.72 | 8.70 | 5.64 | −0.402 | 0.128 |
15 | 48.02 | 8.08 | 5.54 | −0.484 | 0.121 |
16 | 50.97 | 9.39 | 5.65 | −0.404 | 0.139 |
17 | 54.24 | 8.85 | 5.59 | −0.290 | 0.132 |
18 | 57.51 | 9.04 | 5.76 | −0.034 | 0.131 |
19 | 60.47 | 9.17 | 5.75 | 0.197 | 0.133 |
20 | 63.79 | 11.05 | 5.84 | −0.012 | 0.158 |
21 | 67.07 | 10.47 | 5.78 | −0.180 | 0.151 |
22 | 69.96 | 10.37 | 5.92 | 0.149 | 0.146 |
23 | 72.92 | 10.89 | 5.85 | −0.010 | 0.155 |
24 | 76.15 | 12.23 | 6.09 | 0.042 | 0.167 |
25 | 79.65 | 12.76 | 6.07 | 0.287 | 0.175 |
26 | 83.18 | 11.10 | 6.50 | 0.632 | 0.142 |
27 | 85.85 | 13.27 | 6.78 | 0.247 | 0.163 |
28 | 89.97 | 15.10 | 7.65 | 1.263 | 0.164 |
29 | 93.48 | 15.11 | 8.68 | 1.483 | 0.145 |
30 | 95.94 | 19.89 | 9.55 | 1.585 | 0.173 |
8.1.6 Appendix 8.6. Rank-ordered portfolio data: tax controls only
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 10.42 | 0.30 | 6.50 | −0.420 | 0.004 |
2 | 11.42 | 0.15 | 6.45 | −0.393 | 0.002 |
3 | 15.46 | 4.50 | 6.35 | −0.600 | 0.059 |
4 | 20.52 | 6.09 | 5.89 | −0.737 | 0.086 |
5 | 22.84 | 4.93 | 5.98 | −0.495 | 0.069 |
6 | 25.76 | 4.61 | 5.87 | −0.728 | 0.065 |
7 | 28.62 | 4.90 | 5.67 | −0.642 | 0.072 |
8 | 31.02 | 6.47 | 5.57 | −0.729 | 0.097 |
9 | 33.27 | 6.75 | 5.59 | −0.490 | 0.100 |
10 | 35.77 | 8.13 | 5.60 | −0.638 | 0.121 |
11 | 38.14 | 7.96 | 5.47 | −0.645 | 0.121 |
12 | 40.46 | 8.12 | 5.49 | −0.587 | 0.123 |
13 | 42.74 | 7.33 | 5.47 | −0.607 | 0.112 |
14 | 45.30 | 8.78 | 5.59 | −0.522 | 0.131 |
15 | 47.90 | 8.26 | 5.70 | −0.377 | 0.121 |
16 | 50.26 | 10.91 | 5.51 | −0.407 | 0.165 |
17 | 52.94 | 9.86 | 5.60 | −0.277 | 0.147 |
18 | 55.63 | 11.84 | 5.77 | −0.077 | 0.171 |
19 | 58.13 | 12.02 | 5.72 | −0.269 | 0.175 |
20 | 60.96 | 11.56 | 5.83 | −0.010 | 0.165 |
21 | 63.80 | 11.84 | 5.92 | 0.047 | 0.167 |
22 | 66.32 | 10.84 | 6.02 | 0.019 | 0.150 |
23 | 68.98 | 11.79 | 6.07 | 0.147 | 0.162 |
24 | 71.98 | 14.23 | 6.05 | 0.069 | 0.196 |
25 | 75.37 | 12.79 | 6.38 | 0.144 | 0.167 |
26 | 78.92 | 13.99 | 6.39 | 0.121 | 0.182 |
27 | 81.66 | 16.00 | 6.90 | 0.604 | 0.193 |
28 | 86.26 | 14.75 | 7.38 | 0.515 | 0.166 |
29 | 90.47 | 18.25 | 8.38 | 0.987 | 0.181 |
30 | 93.65 | 21.40 | 8.92 | 1.789 | 0.200 |
8.1.7 Appendix 8.7. Rank-ordered portfolio data: risk and tax controls
P# | FS | Rtn% | SD% | Skew | S Ratio |
|---|---|---|---|---|---|
1 | 10.42 | 0.30 | 6.50 | −0.420 | 0.004 |
2 | 11.42 | 0.15 | 6.45 | −0.393 | 0.002 |
3 | 15.46 | 4.50 | 6.35 | −0.600 | 0.059 |
4 | 20.52 | 6.09 | 5.89 | −0.737 | 0.086 |
5 | 22.84 | 4.93 | 5.98 | −0.495 | 0.069 |
6 | 25.76 | 4.61 | 5.87 | −0.728 | 0.065 |
7 | 28.62 | 4.90 | 5.67 | −0.642 | 0.072 |
8 | 31.02 | 6.47 | 5.57 | −0.729 | 0.097 |
9 | 33.27 | 6.75 | 5.59 | −0.490 | 0.100 |
10 | 35.77 | 8.13 | 5.60 | −0.638 | 0.121 |
11 | 38.14 | 7.96 | 5.47 | −0.645 | 0.121 |
12 | 40.46 | 8.12 | 5.49 | −0.587 | 0.123 |
13 | 42.74 | 7.33 | 5.47 | −0.607 | 0.112 |
14 | 45.30 | 8.78 | 5.59 | −0.522 | 0.131 |
15 | 47.90 | 8.26 | 5.70 | −0.377 | 0.121 |
16 | 50.26 | 10.91 | 5.51 | −0.407 | 0.165 |
17 | 52.94 | 9.86 | 5.60 | −0.277 | 0.147 |
18 | 55.63 | 11.84 | 5.77 | −0.077 | 0.171 |
19 | 58.13 | 12.02 | 5.72 | −0.269 | 0.175 |
20 | 60.96 | 11.56 | 5.83 | −0.010 | 0.165 |
21 | 63.80 | 11.84 | 5.92 | 0.047 | 0.167 |
22 | 66.32 | 10.84 | 6.02 | 0.019 | 0.150 |
23 | 68.98 | 11.79 | 6.07 | 0.147 | 0.162 |
24 | 71.98 | 14.23 | 6.05 | 0.069 | 0.196 |
25 | 75.37 | 12.79 | 6.38 | 0.144 | 0.167 |
26 | 78.92 | 13.99 | 6.39 | 0.121 | 0.182 |
27 | 81.66 | 16.00 | 6.90 | 0.604 | 0.193 |
28 | 86.26 | 14.75 | 7.38 | 0.515 | 0.166 |
29 | 90.47 | 18.25 | 8.38 | 0.987 | 0.181 |
30 | 93.65 | 21.40 | 8.92 | 1.789 | 0.200 |
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Stone, B.K. (2017). Portfolio Performance Assessment: Statistical Issues and Methods for Improvement. In: Guerard, Jr., J. (eds) Portfolio Construction, Measurement, and Efficiency. Springer, Cham. https://doi.org/10.1007/978-3-319-33976-4_8
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