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ON PORTFOLIO OPTIMIZATION: ARE THERE FINANCIAL GAINS OF USING ALTERNATIVE COVARIANCE METHODS?
Corresponding Author(s) : Ahsen Saghir
Humanities & Social Sciences Reviews,
Vol. 9 No. 3 (2021): May
Abstract
Purpose: The study evaluates the performance of alternative variance-covariance estimators as a fundamental ingredient to portfolio optimization.
Methodology: The study estimates eleven covariance matrices on the data of Pakistan stock exchange's non-financial sector firms covering the period from July 2006 to June 2020. The accuracy and efficiency of covariance estimators are assessed through two evaluation parameters: root mean square error and minimum variance portfolios (risk behavior).
Main findings: Empirical findings based on evaluation parameters suggest that more complex covariance estimators in the equity market of Pakistan yield no additional financial gains than the equally weighted portfolio of estimators.
Application of the study: As the estimation of the variance-covariance matrix is one of the essential elements of portfolio construction, this study guides investor(s) on selecting an appropriate covariance estimator among eleven estimators endorsed by literature.
Novelty/ originality of the study: Based on detailed analysis, the study documents that investor(s) of the Pakistan stock exchange cannot gain any additional benefit from more complex and tricky methods of variance-covariance estimators compared to a portfolio of estimators for the non-financial sector. Investors are advised to consider the equally weighted portfolio of estimators when formulating their investment strategy.
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- Bengtsson, C., & Holst, J. (2002). On portfolio selection: Improved covariance matrix estimation for Swedish asset returns. Department of Economics, Luud University.
- Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. The Review of Financial Studies, 4(2), 315-342. https://doi.org/10.1093/rfs/4.2.315 DOI: https://doi.org/10.1093/rfs/4.2.315
- Chan, L. K., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: Forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5), 937-974. DOI: https://doi.org/10.1093/rfs/12.5.937
- Chopra, V., & Ziemba, W. (1993). The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice. The Journal of Portfolio Management, 19(2), 6. https://doi.org/10.3905/jpm.1993.409440 DOI: https://doi.org/10.3905/jpm.1993.409440
- Chow, P. S., Cioffi, J. M., & Bingham, J. A. (1995). A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels. IEEE Transactions on communications, 43(2/3/4), 773-775. https://doi.org/10.1109/26.380108 DOI: https://doi.org/10.1109/26.380108
- DeMiguel, V., Garlappi, L., & Uppal, R. (2007). Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? The Review of Financial Studies, 22(5), 1915-1953. https://doi.org/10.1093/rfs/hhm075 DOI: https://doi.org/10.1093/rfs/hhm075
- Disatnik, D. J., & Benninga, S. (2007). Shrinking the covariance matrix. The Journal of Portfolio Management, 33(4), 55-63. DOI: https://doi.org/10.3905/jpm.2007.690606
- Elton, E. J., & Gruber, M. J. (1973). Estimating the dependence structure of share prices--implications for portfolio selection. The Journal of Finance, 28(5), 1203-1232. DOI: https://doi.org/10.1111/j.1540-6261.1973.tb01451.x
- Husnain, M., Hassan, A., & Lamarque, E. (2016). A Framework for Asset Allocation in Pakistani Equity Market: Simpler is Better. Pakistan Journal of Social Sciences (PJSS), 36(2), 881-893. https://bit.ly/3rcqd01
- Hwang, I., Xu, S., & In, F. (2018). Naive versus optimal diversification: Tail risk and performance. European Journal of Operational Research, 265(1), 372-388. https://doi.org/10.1016/j.ejor.2017.07.066 DOI: https://doi.org/10.1016/j.ejor.2017.07.066
- Jagannathan, R., & Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. The Journal of Finance, 58(4), 1651-1683. https://doi.org/10.3386/w8922 DOI: https://doi.org/10.1111/1540-6261.00580
- Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(3), 279-292. https://doi.org/10.2307/2331042 DOI: https://doi.org/10.2307/2331042
- King, B. F. (1966). Market and industry factors in stock price behavior. the Journal of Business, 39(1), 139-190. DOI: https://doi.org/10.1086/294847
- Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519-531. DOI: https://doi.org/10.1287/mnsc.37.5.519
- Kwan, C. C. (2011). An introduction to shrinkage estimation of the covariance matrix: A pedagogic illustration. Spreadsheets in Education (eJSiE), 4(3), 6.
- Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603-621. https://doi.org/10.1016/s0927-5398(03)00007-0 DOI: https://doi.org/10.1016/S0927-5398(03)00007-0
- Ledoit, O., & Wolf, M. (2004). Honey, I shrunk the sample covariance matrix. The Journal of Portfolio Management, 30(4), 110-119. https://doi.org/10.2139/ssrn.433840 DOI: https://doi.org/10.3905/jpm.2004.110
- Levy, H., & Markowitz, H. M. (1979). Approximating expected utility by a function of mean and variance. The American Economic Review, 69(3), 308-317.
- Liu, L., & Lin, H. (2010). Covariance estimation: do new methods outperform old ones? Journal of Economics and Finance, 34(2), 187-195. DOI: https://doi.org/10.1007/s12197-009-9104-4
- Ly, W. (2019). Optimized Portfolios vs. the Naive-Diversification Strategy Universitetet i Agder; University of Agder].
- Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91. https://doi.org/10.2307 /2975974 DOI: https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
- Michaud, R. O. (1989). The Markowitz optimization enigma: Is ‘optimized’optimal? Financial Analysts Journal, 45(1), 31-42. https://doi.org/10.2469/faj.v45.n1.31 DOI: https://doi.org/10.2469/faj.v45.n1.31
- Nguyen, H. L. (2018). Out-of-sample testing on portfolio performance in the Asian equity market: Can optimized portfolio outperformed simpler strategy? [Bachelor's Thesis, Aalto University, School of Business, Mikkeli Campus]. http://urn.fi/URN:NBN:fi:aalto-201809105086
- Pafka, S., & Kondor, I. (2004). Estimated Correlation Matrices and Portfolio Optimization. Physica A: Statistical Mechanics and its Applications, 343, 623-634. https://doi.org/10.1016/j.physa.2004.05.079 DOI: https://doi.org/10.1016/j.physa.2004.05.079
- Saghir, A., & Tirmizi, M. A. (2020). An Empirical Assessment of Alternative Methods of Variance-Covariance Matrix. International Review of Management and Business Research, 09(04). https://doi.org/10.30543/9-4(2020)-33 DOI: https://doi.org/10.30543/9-4(2020)-33
- Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management Science, 9(2), 277-293. DOI: https://doi.org/10.1287/mnsc.9.2.277
- Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. DOI: https://doi.org/10.1525/9780520313880-018
- Tamiz, M., & Jones, D. (1996). Goal programming and Pareto efficiency. Journal of Information and Optimization Sciences, 17(2), 291-307. https://doi.org/10.1080/02522667.1996.10699283 DOI: https://doi.org/10.1080/02522667.1996.10699283
- Tu, J., & Zhou, G. (2011). Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics, 99(1), 204-215. https://doi.org/10.1016/j.jfineco.2010.08.013 DOI: https://doi.org/10.1016/j.jfineco.2010.08.013
- Vasicek, O. A. (1973). A note on using cross-sectional information in Bayesian estimation of security betas. The Journal of Finance, 28(5), 1233-1239. DOI: https://doi.org/10.1111/j.1540-6261.1973.tb01452.x
- Zakamulin, V. (2017). Superiority of optimized portfolios to naive diversification: Fact or fiction? Finance Research Letters, 22, 122-128. https://doi.org/10.1016/j.frl.2016.12.007 DOI: https://doi.org/10.1016/j.frl.2016.12.007
References
Bengtsson, C., & Holst, J. (2002). On portfolio selection: Improved covariance matrix estimation for Swedish asset returns. Department of Economics, Luud University.
Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. The Review of Financial Studies, 4(2), 315-342. https://doi.org/10.1093/rfs/4.2.315 DOI: https://doi.org/10.1093/rfs/4.2.315
Chan, L. K., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: Forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5), 937-974. DOI: https://doi.org/10.1093/rfs/12.5.937
Chopra, V., & Ziemba, W. (1993). The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice. The Journal of Portfolio Management, 19(2), 6. https://doi.org/10.3905/jpm.1993.409440 DOI: https://doi.org/10.3905/jpm.1993.409440
Chow, P. S., Cioffi, J. M., & Bingham, J. A. (1995). A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels. IEEE Transactions on communications, 43(2/3/4), 773-775. https://doi.org/10.1109/26.380108 DOI: https://doi.org/10.1109/26.380108
DeMiguel, V., Garlappi, L., & Uppal, R. (2007). Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? The Review of Financial Studies, 22(5), 1915-1953. https://doi.org/10.1093/rfs/hhm075 DOI: https://doi.org/10.1093/rfs/hhm075
Disatnik, D. J., & Benninga, S. (2007). Shrinking the covariance matrix. The Journal of Portfolio Management, 33(4), 55-63. DOI: https://doi.org/10.3905/jpm.2007.690606
Elton, E. J., & Gruber, M. J. (1973). Estimating the dependence structure of share prices--implications for portfolio selection. The Journal of Finance, 28(5), 1203-1232. DOI: https://doi.org/10.1111/j.1540-6261.1973.tb01451.x
Husnain, M., Hassan, A., & Lamarque, E. (2016). A Framework for Asset Allocation in Pakistani Equity Market: Simpler is Better. Pakistan Journal of Social Sciences (PJSS), 36(2), 881-893. https://bit.ly/3rcqd01
Hwang, I., Xu, S., & In, F. (2018). Naive versus optimal diversification: Tail risk and performance. European Journal of Operational Research, 265(1), 372-388. https://doi.org/10.1016/j.ejor.2017.07.066 DOI: https://doi.org/10.1016/j.ejor.2017.07.066
Jagannathan, R., & Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. The Journal of Finance, 58(4), 1651-1683. https://doi.org/10.3386/w8922 DOI: https://doi.org/10.1111/1540-6261.00580
Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(3), 279-292. https://doi.org/10.2307/2331042 DOI: https://doi.org/10.2307/2331042
King, B. F. (1966). Market and industry factors in stock price behavior. the Journal of Business, 39(1), 139-190. DOI: https://doi.org/10.1086/294847
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519-531. DOI: https://doi.org/10.1287/mnsc.37.5.519
Kwan, C. C. (2011). An introduction to shrinkage estimation of the covariance matrix: A pedagogic illustration. Spreadsheets in Education (eJSiE), 4(3), 6.
Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603-621. https://doi.org/10.1016/s0927-5398(03)00007-0 DOI: https://doi.org/10.1016/S0927-5398(03)00007-0
Ledoit, O., & Wolf, M. (2004). Honey, I shrunk the sample covariance matrix. The Journal of Portfolio Management, 30(4), 110-119. https://doi.org/10.2139/ssrn.433840 DOI: https://doi.org/10.3905/jpm.2004.110
Levy, H., & Markowitz, H. M. (1979). Approximating expected utility by a function of mean and variance. The American Economic Review, 69(3), 308-317.
Liu, L., & Lin, H. (2010). Covariance estimation: do new methods outperform old ones? Journal of Economics and Finance, 34(2), 187-195. DOI: https://doi.org/10.1007/s12197-009-9104-4
Ly, W. (2019). Optimized Portfolios vs. the Naive-Diversification Strategy Universitetet i Agder; University of Agder].
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91. https://doi.org/10.2307 /2975974 DOI: https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Michaud, R. O. (1989). The Markowitz optimization enigma: Is ‘optimized’optimal? Financial Analysts Journal, 45(1), 31-42. https://doi.org/10.2469/faj.v45.n1.31 DOI: https://doi.org/10.2469/faj.v45.n1.31
Nguyen, H. L. (2018). Out-of-sample testing on portfolio performance in the Asian equity market: Can optimized portfolio outperformed simpler strategy? [Bachelor's Thesis, Aalto University, School of Business, Mikkeli Campus]. http://urn.fi/URN:NBN:fi:aalto-201809105086
Pafka, S., & Kondor, I. (2004). Estimated Correlation Matrices and Portfolio Optimization. Physica A: Statistical Mechanics and its Applications, 343, 623-634. https://doi.org/10.1016/j.physa.2004.05.079 DOI: https://doi.org/10.1016/j.physa.2004.05.079
Saghir, A., & Tirmizi, M. A. (2020). An Empirical Assessment of Alternative Methods of Variance-Covariance Matrix. International Review of Management and Business Research, 09(04). https://doi.org/10.30543/9-4(2020)-33 DOI: https://doi.org/10.30543/9-4(2020)-33
Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management Science, 9(2), 277-293. DOI: https://doi.org/10.1287/mnsc.9.2.277
Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. DOI: https://doi.org/10.1525/9780520313880-018
Tamiz, M., & Jones, D. (1996). Goal programming and Pareto efficiency. Journal of Information and Optimization Sciences, 17(2), 291-307. https://doi.org/10.1080/02522667.1996.10699283 DOI: https://doi.org/10.1080/02522667.1996.10699283
Tu, J., & Zhou, G. (2011). Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics, 99(1), 204-215. https://doi.org/10.1016/j.jfineco.2010.08.013 DOI: https://doi.org/10.1016/j.jfineco.2010.08.013
Vasicek, O. A. (1973). A note on using cross-sectional information in Bayesian estimation of security betas. The Journal of Finance, 28(5), 1233-1239. DOI: https://doi.org/10.1111/j.1540-6261.1973.tb01452.x
Zakamulin, V. (2017). Superiority of optimized portfolios to naive diversification: Fact or fiction? Finance Research Letters, 22, 122-128. https://doi.org/10.1016/j.frl.2016.12.007 DOI: https://doi.org/10.1016/j.frl.2016.12.007