View
11
Download
0
Category
Preview:
Citation preview
IMPROVING PORTFOLIO CONSTRUCTION THROUGH ADJUSTMENT FOR PARAMETER ESTIMATION ERROR:
AN EMPIRICAL COMPARISON OF ESTIMATION ERROR ADJUSTMENT APPROACHES IN THREE MAJOR ASIAN EQUITY MARKETS
James WilliamsNorthfield Asia Research Seminar Fall 2010
PRESENTATION ABSTRACT
The first part of the presentation briefly summarizes academic research findings regarding methods to improve mean variance optimization (MVO) results from reducing parameter estimation uncertainty.
The second part of the presentation focuses on the portfolio performance improvement from using parameter estimation adjustments in several major Asian markets during the recent global financial crisis.
Our goal is to determine whether estimation error adjustments would have led to better investment performance of optimized portfolios over the period, and whether any of the approaches significantly outperforms the others.
2
MARKET PERFORMANCE: JAN. 2007 – JULY 2010
-20.5
-25.7
-12.9
-16.9
-30-28-25-23-20-18-15-13-10
-8-5-30358
101315182023252830
TOPIX 1000 FTSE Xinhua 600 S&P ASX 300 S&P500
3
CUMULATIVE RETURN: JAN. 2007 – JULY 2010
50
145
79
78
405060708090
100110120130140150160170180190200210220230240250260270280
TOPIX 1000 FTSE Xinhua 600 AU Model 300 S&P500
4
MEAN VARIANCE OPTIMIZATION OVERVIEW
Mean variance optimization traces it roots back to Markowitz (1952)An optimal portfolio can be created that provides the greatest return for a given level of risk or the lowest level of risk for a given level of return.Need to know with complete certainty “certainty equivalence “ input parameters:
Mean of asset returnsStandard deviation of asset returnsCorrelation between assets
Optimizer as “error maximizers”The optimal mean variance portfolio is not necessary “optimal” since we the actual expected returns are not known. End up with suboptimal portfolio.Optimal portfolio ends up holding large positions with high expected returns and low standard deviation (and vice versa). Large estimation errors in inputs introduce errors in optimized portfolio weights.
5
ESTIMATION ERROR LITERATURE REVIEW
A number of academics and investment professionals have written about estimation error issues in MVO:
Stein (1955) shows that traditional sample statistics are not appropriate for multivariate problemsJobson and Korkie (1981), Michaud (1989, 1998), Chopra and Ziemba (1993)
Example: small changes in expected returns can lead to extreme weightings in optimized portfolio
Empirical tests by Chopra and Ziemba (1993) show that errors in return estimates are more important that errors in risk estimatesJorion (1992) and Broadie (1993) use Monte Carlo simulations to estimate the magnitude of the problemBlack and Litterman, (1991, 1992) develop bayesian process to reduce error in return estimatesLee (2000])and Satchell and Scowcroft (2000) discuss the Black-Litterman theory, He and Litterman (1999]), Jones, Lim, and Zangari(2007) focused on implementation.
6
METHODS TO IMPROVE INPUT PARAMETER ESTIMATION
Portfolio ConstraintsResampling
Michaud & Michaud methodology, (1998)
Bayesian Approaches Black-Litterman model
Uses market implied asset weights from equilibrium portfolio into the CAPM and solves for the optimal portfolio from these inputs.Model grounded in modern portfolio theory. Inputs can be placed into traditional Markowitz optimizer or a resampled type of optimizer.
Bayes-Stein estimator (Jorion 1986)Blend covariance matrix, (Ledoit and Wolf, 2004)
7
PORTFOLIO CONSTRAINTS
Implement portfolio constraints to create a more diversified portfolioConstrain by position sizeLimit by market cap size, sector weights, fundamentals such as dividend yield, or P/E rangesNo short-sellingTries to get a portfolio that is just right (see Goldilocks…)Too tight constraints will drive position size and portfolio construction, not the forecastsAd hoc
Other features in typical optimizer program include quadratic penaltiesallow the user to force the optimizer to create a portfolio whose average value of some variable (such as yield, P/E ratio, or relative strength) Pushes an optimizer in one or more directions (high market cap penalty vs. low dividend yield penalty)Tradeoffs: the more one pushes the portfolio toward a chosen value of one variable, the harder it becomes to push it toward the chosen value of another variable
8
RESAMPLING APPROACH
Meant to address weakness in MVO due to estimation error uncertainty.Approach used by Richard and Robert Michaud of New Frontier Advisors (NFA) uses monte carlo simulation methodMarkowitz tested resampling method versus a bayesianapproach and to his surprise, on average the resampling technique produces more diverse portfolios given the level of risk (Markowitz and Usmen 2003).
Later results contradict Markowitz study and show different Bayesian approaches would be better (Harvey, Liechty and Liechty, 2008)
9
RESAMPLED APPROACH (2)
Tends to outperform MVO only when long-only constraints are in place. (Scherer and Martin (2005)No theoretical basis to use average asset weights from hundreds of constructed optimized portfolios.The number of simulated observations are not set and is up to the manager to decide how many simulations to run based on level of manager confidence in risk return estimatesA resampling method was licensed by Northfield but we choose to use bayesian approaches.
10
BAYESIAN APPROACH
Traditional method of estimating returns considers that the actual returns and covariances of returns can be estimated using historical data.
Actual expected returns are unknown and fixed, single value. Bayesian approach assumes that the actual expected returns are unknown and random.
The goal in Bayesian statistics is to make as accurate an estimate as possible given the data and prior information. Bayes theorem named after Thomas Bayes, an English mathematician and minister.
11
BAYESIAN APPROACH (2)Based on subjective interpretation of probability. Combines forecasts from historical data with a prior distribution (prior belief). Use in investment field
Black-Litterman modelUses market implied asset weights from equilibrium portfolio in the CAPM model and solves for the optimal portfolio from these inputs.Model grounded in modern portfolio theory. Inputs can be placed into traditional Markowitz optimizer or a resampled type of optimizer.
Shrinkage estimatorsBayes-Stein estimator (Jorion 1986)Blend covariance matrix, (Ledoit and Wolf, 2004)
12
BLACK-LITTERMAN APPROACH
Idea is described in paper by Black & Litterman (1991, 1992)Uses Bayesian approach to combine investor’s views of expected return with the market equilibrium portfolio implied expected returns.
Uses weighted average of investor’s views and the market equilibrium –leads intuitive portfolios with sensible position weights. Equilibrium returns determined by using reverse optimization method where the vector of implied expected excess returns is calculated from known inputs (Idzorek, 2003)
Risk aversion coefficientCovariance matrix of excess returnsMarket capitalization weight of the assets
13
BLACK-LITTERMAN: IN PRACTICE
Expected security return should be consistent with the market equilibrium unless the investor has a specific view on the security.
Without any views of expected return, investor will end up holding the market portfolio (CAPM).Equilibrium returns used to “center” the optimal portfolio around the market portfolio.If investor has low confidence in her views, the resulting expected returns will be close to the market equilibrium implied returns.The higher the confidence in the investor’s views, the more the optimal portfolio will differ from the market equilibrium implied returns.
• Asset expected return given with a degree of confidenceEx. Singapore Telecom expected alpha of 5% +/- 2%
14
BLACK-LITTERMAN: NORTHFIELD APPROACH (BAYES ADJUST)
Squeezes forecasts for securities whose benchmark excess returns are highly correlated towards each other
A multivariate normally distributed prior is placed on benchmark relative returns.The center is zero or the benchmark relative implied alpha of a user-supplied equilibrium portfolio.The covariance equals the covariance of the returns themselves divide by the intensity of the prior.
Users can supply forecasted alphas with an associated confidence interval. (e.g. Telstra forecasted alpha is 3% +/-1%).
Alpha grouping by industry, sector, country also possible.
The procedure returns the most likely alphas given the prior, the forecasts, and the errors.
15
BAYES-STEIN ESTIMATOR
Derived from paper, by Jorion (1986). Unlike Black-Litterman which assumes market equilibrium, this process assumes that the true expected returns for an asset lies between the investor’s expectation, and a common value.This central, or prior value is usually the user’s expectation for the global minimum variance portfolio.
The rationale for this prior belief is that if an investor had no information about expected returns, but did have a valid understanding of risk they would prefer to hold the least risky portfolio available. All user supplied expected returns are squeezed toward the prior central value (often zero when benchmark relative), and the magnitude of adjustment for each asset is derived from the covariance among the assets.
16
BAYES-STEIN ESTIMATOR: NORTHFIELD APPROACH
Users set a value for the shrinkage estimatorThis method has been a part of our asset allocation (ART) product for a number of years.Assumes a prior on the mean returns, centered at the return of the minimum variance portfolio and having covariance proportional to the covariance of returns. Alphas are squeezed toward the alpha of the minimum variance portfolio. Small adjustment to the covariance matrix.
17
BLEND COVARIANCE APPROACH
Idea comes from a series of papers by Ledoit & Wolf (2003)Estimating the covariance matrix of past stock returns when universe of stocks is large compared to number of historical return observations, the sample covariance matrix is estimated with a lot of error. (Jobson and Korkie, 1980). Ledoit & Wolf developed shrinkage method that pulls the outlying high and low estimated coefficients towards the center.
18
BLEND COVARIANCE: NORTHFIELD APPROACH• Users can blend their chosen weighting of three alternative assumptions to
the full covariance matrix implied from the factor model in use. o The first alternative is that the covariance among securities can be described as a single-index
model (market beta only) as described in Sharpe (1964). o Single Index
• Correlates stocks only through the market portfolio. • Stock’s total variance is unchanged.• Reduce the multifactor risk model to CAPM
• The second alternative is that the correlation among securities is constant and equal across all pairwise relationships.
o Constant Correlation• Assigns all stocks the same pairwise correlation but different variances. • Correlation is backed out from the market portfolio. A stock’s total variance is unchanged. • More restrictive than CAPM ‐ all stocks have the same pairwise correlation, ρ, but
different variances
• The third alternative is that the covariance among securities is constant and equal across all pairwise relationships.
o Constant Covariance• Dulls the information in the covariance matrix. • All stocks have same variance and correlation with each other. • Correlation and variance are implied by the estimated variance of the market portfolio
(cap-weighted portfolio of all stocks in portfolio, benchmark, buy universe)• Least differentiated of the three – all stocks have the same pairwise correlation, ρ, and
the same variance, σ2
19
SIMPLE EMPIRICAL STUDY
Three Northfield country risk modelsAustraliaChina
Test period: December 2006 – June 2010Initial portfolio: 100% cash weightBenchmarks
Australia: Top 300 securities in monthly Australia model by market capitalizationChina: FTSE Xinhua 600
Optimization parametersTurnover: 1st month 200% turnover, 20% afterwardsTransaction costs: 30 bpsBuy Universe: benchmark as noted aboveRandomly generated alphas with alpha confidence level of 25% for each period.Maximum asset size: 10% of portfolio weightMaximum # of assets: 75 (Australia, China) 100 (Japan)
20
STUDY RESULTS - OVERVIEW
21
China Model Base Bayes Adjustment (Black‐Litterman) Bayes‐Stein Blend Covariance
25 50 75 40 60 80 Single IndexConstant
Cor. Constant Cov.Cumulative Return 129.57 82.63 72.80 69.12 117.67 108.13 96.00 136.71 140.95 145.43Total Active Return 52.64 19.42 12.03 9.75 44.21 37.34 28.72 58.03 60.50 63.80Avg. Active Return 1.00 0.42 0.27 0.22 0.87 0.75 0.59 1.09 1.13 1.17
Model Tracking Error 6.90 2.81 2.52 2.36 5.54 4.71 3.62 7.33 7.62 7.68
Portfolio Model SD 11.03 11.04 11.04 11.04 11.02 11.02 11.02 11.08 10.99 11.04Portfolio Real SD 13.44 12.81 12.66 12.68 13.28 13.15 12.99 13.55 13.55 13.57
Sharpe Ratio 1.05 0.83 0.78 0.77 1.00 0.96 0.90 1.07 1.10 1.10Information Ratio 1.68 1.78 1.26 1.10 1.83 1.87 1.94 1.70 1.69 1.74
Australia Model Base Bayes Adjustment (Black‐Litterman) Bayes‐Stein Blend Covariance
25 50 75 40 60 80 Single IndexConstant
Cor. Constant Cov.Cumulative Return ‐7.92 3.31 4.68 2.63 ‐8.94 ‐9.23 ‐7.09 ‐7.34 ‐7.16 ‐7.23Total Active Return 0.17 12.55 14.00 11.74 ‐1.06 ‐1.52 0.58 1.06 0.99 1.23Avg. Active Return 0.00 0.29 0.32 0.27 ‐0.02 ‐0.03 0.02 0.03 0.03 0.04
Model Tracking Error 5.62 4.99 5.11 5.08 4.91 4.37 3.99 5.63 5.71 5.85
Portfolio Model SD 4.34 4.57 4.57 4.55 4.31 4.30 4.28 4.38 4.23 4.41Portfolio Real SD 6.19 6.05 6.01 6.00 6.03 5.89 5.70 6.30 6.21 6.32
Sharpe Ratio ‐0.41 ‐0.17 ‐0.17 ‐0.22 ‐0.42 ‐0.42 ‐0.40 ‐0.39 ‐0.42 ‐0.38Information Ratio ‐0.02 0.70 0.79 0.71 ‐0.03 ‐0.03 0.01 0.06 0.05 0.06
Study Results: Black-Litterman
22
‐7.0
‐6.5
‐6.0
‐5.5
‐5.0
‐4.5
‐4.0
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Active Return AU
Base BayesAdjust_25 BayesAdjust_50 BayesAdjust_75
Study Results: Bayes-Stein
23
‐7.0
‐6.5
‐6.0
‐5.5
‐5.0
‐4.5
‐4.0
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Active Return AUBase Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
24
‐7.0
‐6.5
‐6.0
‐5.5
‐5.0
‐4.5
‐4.0
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Active Return AUBase Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
Study Results: Black-Litterman
25
3.5
3.7
3.9
4.1
4.3
4.5
4.7
4.9
5.1
5.3
5.5
5.7
5.9
6.1
Tracking Error AUBase BayesAdjust_25 BayesAdjust_50 BayesAdjust_75
Study Results: Bayes Stein
26
3.5
3.7
3.9
4.1
4.3
4.5
4.7
4.9
5.1
5.3
5.5
5.7
5.9
6.1
Tracking Error AU
Base Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
27
3.5
3.7
3.9
4.1
4.3
4.5
4.7
4.9
5.1
5.3
5.5
5.7
5.9
6.1
6.3
Tracking Error AU
Base Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
Study Results: Black-Litterman
28
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
Cumulative Return AU
Base BayesAdjust_25 BayesAdjust_50 BayesAjust_75
Study Results: Bayes Stein
29
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
Cumulative Return AUBase Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
30
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
Cumulative Return AUBase Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
STUDY RESULTS - OVERVIEW
31
China Model Base Bayes Adjustment (Black‐Litterman) Bayes‐Stein Blend Covariance
25 50 75 40 60 80 Single IndexConstant
Cor.Constant Cov.
Cumulative Return 129.57 82.63 72.80 69.12 117.67 108.13 96.00 136.71 140.95 145.43Total Active Return 52.64 19.42 12.03 9.75 44.21 37.34 28.72 58.03 60.50 63.80Avg. Active Return 1.00 0.42 0.27 0.22 0.87 0.75 0.59 1.09 1.13 1.17
Model Tracking Error 6.90 2.81 2.52 2.36 5.54 4.71 3.62 7.33 7.62 7.68
Portfolio Model SD 11.03 11.04 11.04 11.04 11.02 11.02 11.02 11.08 10.99 11.04Portfolio Real SD 13.44 12.81 12.66 12.68 13.28 13.15 12.99 13.55 13.55 13.57
Sharpe Ratio 1.05 0.83 0.78 0.77 1.00 0.96 0.90 1.07 1.10 1.10Information Ratio 1.68 1.78 1.26 1.10 1.83 1.87 1.94 1.70 1.69 1.74
Australia Model Base Bayes Adjustment (Black‐Litterman) Bayes‐Stein Blend Covariance
25 50 75 40 60 80 Single IndexConstant
Cor. Constant Cov.Cumulative Return ‐7.92 3.31 4.68 2.63 ‐8.94 ‐9.23 ‐7.09 ‐7.34 ‐7.16 ‐7.23Total Active Return 0.17 12.55 14.00 11.74 ‐1.06 ‐1.52 0.58 1.06 0.99 1.23Avg. Active Return 0.00 0.29 0.32 0.27 ‐0.02 ‐0.03 0.02 0.03 0.03 0.04
Model Tracking Error 5.62 4.99 5.11 5.08 4.91 4.37 3.99 5.63 5.71 5.85
Portfolio Model SD 4.34 4.57 4.57 4.55 4.31 4.30 4.28 4.38 4.23 4.41Portfolio Real SD 6.19 6.05 6.01 6.00 6.03 5.89 5.70 6.30 6.21 6.32
Sharpe Ratio ‐0.41 ‐0.17 ‐0.17 ‐0.22 ‐0.42 ‐0.42 ‐0.40 ‐0.39 ‐0.42 ‐0.38Information Ratio ‐0.02 0.70 0.79 0.71 ‐0.03 ‐0.03 0.01 0.06 0.05 0.06
Study Results: Black-Litterman
32
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Active Return CNBase BayesAdjust_25 BayesAdjust_50 BayesAdjust_75
Study Results: Bayes-Stein
33
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Active Return CNBase Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
34
‐3.5
‐3.0
‐2.5
‐2.0
‐1.5
‐1.0
‐0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Active Return CNBase Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
Study Results: Black-Litterman
35
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Tracking Error CNBase BayesAdjust_25 BayesAdjust_50 BayesAdjust_75
Study Results: Bayes Stein
36
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Tracking Error CNBase Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
37
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Tracking Error CNBase Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
Study Results: Black-Litterman
38
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
Cumulative Return CNBase BayesAdjust_25 BayesAdjust_50 BayesAjust_75
Study Results: Bayes Stein
39
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
Cumulative Return CNBase Bayes_Stein_40 Bayes_Stein_60 Bayes_Stein_80
Study Results: Blend Covariance
40
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
Cumulative Return CNBase Blend_SingleIndex Blend_Constant_Corr Blend_Constant_Cov
CONCLUSION
MVO attempts to form optimized portfolios that maximize portfolio expected return for a given amount of portfolio risk, or to minimize risk for a given level of expected return.However, limitation of working with uncertain estimates leads to estimation error problem and resulting portfolios that tend to overweight assets with higher returns and lower risks and underweight assets with lower returns and higher risks.There are several different methods available that will improve estimation error results Simple test over several markets during recent market past shows improved results from reducing errors in the parameter estimation process.Actions taken to reduce errors in estimated returns and risk inputs lead to “better” results – on average higher return, less risk than base case scenario.
41
REFERENCES
Black, Fischer and Robert Litterman. "Asset Allocation: Combining Investor Views With Market Equilibrium," Journal of Fixed Income, 1991, v1(2), 7-18.
——— (1992) "Global Portfolio Optimization," Financial Analysts Journal, v48(5,Sept/Oct).
Bulsing, Scowcroft, & Sefton. "Understanding Forecasting: A Unified Framework for Combining Both Analyst And Strategy Forecasts," UBS report, 2003.
Chopra, V. and W. Ziemba. "The Effect Of Errors In Means, Variances, and Covariances on Optimal Portfolio Choice,” Journal of Portfolio Management, 1993, v19(2), 6-12.
DiBartolomeo, Dan. “Parameter Estimation Error in Portfolio Optimization”, Northfield Research Seminar, 2006. http://www.northinfo.com/documents/204.pdf
Harvey, C., J Liechty and M. Liechty. 2008. “Bayes vs. Resampling: A Rematch.” Journal of Investment Management, Vo. 6, No. 1 (2008), pp. 1-17
He, G., and R. Litterman. “The Intuition Behind Black-Litterman Model Portfolios.” Working paper, Goldman Sachs Quantitative Resources Group, 1999.
Idzorek, Thomas. “A Step-by-Step Guide to the Black-Litterman Model”, Zephyr Associates Working Paper, 2003.
Jobson, J.D., and B. Korkie. “Putting Markowitz Theory to Work”. Journal of Portfolio Management, Vol. 7, No. 4 (1981), pp. 70-74.
42
REFERENCES
Jorion, Philippe. "Portfolio Optimization In Practice," Financial Analyst Journal, 1992, v48(1), 68-74.
——— (1985) "International Portfolio Diversification With Estimation Risk," Journal of Business, v58(3), 259-278.
——— (1986) "Bayes-Stein Estimation For Portfolio Analysis,“ Journal of Financial and Quantitative Analysis, v21(3), 279-292.
Ledoit, Olivier and Michael Wolf, “Honey, I Shrunk the Sample Covariance Matrix”, Journal of Portfolio Management, 2004.
• ———“Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection,” Journal of Empirical Finance, Dec v10(5):603–621
Markowitz, H., (1952a), “Portfolio Selection,” Journal of Finance, 7, 77–91.
——— (1952b), “The Utility of Wealth,” Journal of Political Economy 152–158.
——— (1959), “Portfolio Selection: Efficient Diversification of Investments,” Second Edition.
Markowitz, H. and N. Usmen. 2003. “Resampled Frontiers Versus Diffuse Bayes: An Experiment.” Journal of Investment Management, 1(4): 9-25.
Michaud, R. and R. Michaud. 2008 “Estimation Error and Portfolio Optimization: A Resampling Approach.” Journal of Investment Management, Vol. 6, No.1 pp. 8-28.
Shah, Anish. “Mitigating Estimation Error in Optimization”, Northfield Newport Research Seminar, 2010. http://www.northinfo.com/documents/367.pdf
——— (2007) “Alpha Scaling Revisited”, 2007. http://www.northinfo.com/documents/283.pdf
43
Recommended