A. PORTFOLIO CONSTRAINTS
1. Minimum weight constraint. Weights smaller than 0.05% are forced to zero.
2. Maximum weight constraint. Individual stock weights are capped at 5%.
3. Capacity constraint. The weight of a stock is capped at the lower of 1.5% or 20 times its weight in the corresponding cap-weighted portfolio. Note that this constraint dominates the maximum weight constraint.
4. Sector concentration constraint. Sector weights are not allowed to deviate more than ±5% from the corresponding cap-weighted sector weights.
5. Region concentration constraint. If the cap-weighted region weights are less than 2.5%, the minimum-variance region weights are capped at three times their weight in the cap-weighted portfolio. Otherwise, they are not allowed to deviate more than ±5% from the corresponding cap-weighted region weights.
6. Turnover constraint. The maximum allowable one-way index turnover is 20%.
B. MARKET AND REGION DEFINITIONS
Region 1 = DevEME, which includes Austria, Belgium, Denmark, Finland, Greece, Ireland, Israel, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden, and Switzerland
Region 2 = DevAPAC, which includes Australia, Hong Kong, New Zealand, and Singapore
Region 3 = France
Region 4 = Germany
Region 5 = United Kingdom
Region 6 = Japan
Region 7 = Canada
Region 8 = United States
Region 1 = EMEMEA, which includes Czech Republic, Egypt, Hungary, Morocco, Poland, and Turkey
Region 2 = EMAPAC, which includes Indonesia, Malaysia, Philippines, and Thailand
Region 3 = EMAME, which includes Chile, Colombia, Mexico, and Peru
Region 4 = South Africa
Region 5 = Russian Federation
Region 6 = India
Region 7 = China
Region 8 = Taiwan
Region 9 = South Korea
Region 10 = Brazil
C. EFFECTIVE NUMBER OF STOCKS
This is the reciprocal of the Herfindahl ratio, which was developed to gauge monopoly concentration in industry, repurposed for investment management. Hypothetically a portfolio of 100% weight in 1 stock has an Effective N of 1; a portfolio of equal weight to 1,000 stocks has an Effective N of 1,000. In another words, these minimum variance portfolios are as diversified as equally weighting only 30–40 stocks.
1. The S&P 500 Index closing price level was 676.53 on March 9, 2009, and 2103.84 on July 31, 2015, a change of 211%.
2. See Chow, Hsu, Kuo, and Li (2014); Soe (2012); Blitz, Pang, and van Vliet (2012).
3. The minimum-variance method is offered by several influential market providers, such as MSCI.
4. See Behr, Guettler, and Miebs (2008).
5. See Jagannathan and Ma (2003); Kempf and Memmel (2003); AGIC Systematic Investment Team (2012).
6. See Chow, Hsu, Kuo, and Li (2014), and Arnott (2006).
7. Methods available to mitigate the estimation errors inherent in sample covariance matrices include the Sharpe (1964) factor-based approach, the Elton and Gruber (1973) constant correlation approach, and the Ledoit and Wolf (2004) statistical shrinkage approach.
8. In brief, we employed an optimization routine to find a numerical solution of portfolio weights that minimizes portfolio variance under constraints. To ensure that the covariance structure inputs were positive definite, we applied principal component analysis to the covariance matrix, which was estimated using up to five years of monthly excess returns.
9. See the appendix for the mathematical definition of Effective N (here, the effective number of stocks).
AGIC Systematic Investment Team. 2012 “Specification of Constraints in Managed Volatility Strategies.” Allianz Global Investors Capital (September).
Arnott, Robert D. 2006. “Implementation Shortfall.” Editor’s Corner, Financial Analysts Journal, vol. 62, no. 3 (May/June):6–8.
Behr, Patrick, Andre Guettler, and Felix Miebs. 2008. “Is Minimum-Variance Investing Really Worth the While? An Analysis with Robust Performance Inference.”
Blitz, David, Juan Pang, and Pim van Vliet. 2012. “The Volatility Effect in Emerging Markets.” Robeco Research Paper (March).
Chow, Tzee-Man, Jason C. Hsu, Li-Lan Kuo, and Feifei Li. 2014. “A Study of Low Volatility Portfolio Construction Methods.” Journal of Portfolio Management, vol. 40, no. 4 (Summer):89–105.
Elton, Edwin J., and Martin J. Gruber. 1973. “Estimating the Dependence Structure of Share Prices—Implications for Portfolio Selection.” Journal of Finance, vol. 8, no. 5 (December):1203–1232.
Jagannathan, Ravi, and Tongshu Ma. 2003. “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps.” Journal of Finance, vol. 58, no. 4 (August):1651–1684.
Kempf, Alexander, and Christoph Memmel. 2003. “On the Estimation of the Global Minimum Variance Portfolio.”
Ledoit, Olivier, and Michael Wolf. 2004. “Honey, I Shrunk the Sample Covariance Matrix.” Journal of Portfolio Management, vol. 30, no. 4 (Summer):110–119.
Sharpe, William F. 1964. “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.” Journal of Finance, vol. 19, no. 3 (September):425–442.
Soe, Aye M. 2012. “The Low Volatility Effect: A Comprehensive Look.” (August 1).