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Modern Stochastics: Theory and Applications


A test on the location of tangency portfolio for small sample size and singular covariance matrix
Volume 12, Issue 1 (2025), pp. 43–59
Pub. online: 11 July 2024      Type: Research Article      Open accessOpen Access
Received
20 February 2024
Revised
20 June 2024
Accepted
21 June 2024
Published
11 July 2024

Abstract

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that $rank(\boldsymbol{\Sigma })=r\lt n\lt k+1$. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that $r/n\to c\in (0,1)$. Theoretical findings are completed by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

References

[1] 
Alfelt, G., Mazur, S.: On the mean and variance of the estimated tangency portfolio weights for small samples. Mod. Stoch. Theory Appl. 9(4), 453–482 (2022) MR4510383. https://doi.org/10.15559/22-vmsta212
[2] 
Bai, Z., Liu, H., Wong, W.-K.: Enhancement of the applicability of Markowitz’s portfolio optimization by utilizing random matrix theory. Math. Finance 19(4), 639–667 (2009) MR2583523. https://doi.org/10.1111/j.1467-9965.2009.00383.x
[3] 
Bauder, D., Bodnar, T., Mazur, S., Okhrin, Y.: Bayesian inference for the tangent portfolio. Int. J. Theor. Appl. Finance 21(08), 1850054 (2018) MR3897158. https://doi.org/10.1142/S0219024918500541
[4] 
Bodnar, T., Mazur, S., Podgórski, K.: A test for the global minimum variance portfolio for small sample and singular covariance. AStA Adv. Stat. Anal. 101(3), 253–265 (2017) MR3679345. https://doi.org/10.1007/s10182-016-0282-z
[5] 
Bodnar, T., Okhrin, Y.: On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory. Scand. J. Stat. 38(2), 311–331 (2011) MR2829602. https://doi.org/10.1111/j.1467-9469.2011.00729.x
[6] 
Bodnar, T., Mazur, S., Nguyen, H.: Estimation of optimal portfolio compositions for small sample and singular covariance matrix. Working paper 15, School of Business, Örebro University, Sweden (2022). https://www.oru.se/globalassets/oru-sv/institutioner/hh/workingpapers/workingpapers2022/wp-15-2022.pdf
[7] 
Bodnar, T., Mazur, S., Podgórski, K.: Singular inverse Wishart distribution and its application to portfolio theory. J. Multivar. Anal. 143, 314–326 (2016) MR3431434. https://doi.org/10.1016/j.jmva.2015.09.021
[8] 
Bodnar, T., Mazur, S., Podgórski, K., Tyrcha, J.: Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory. J. Stat. Plan. Inference 201, 40–57 (2019) MR3913439. https://doi.org/10.1016/j.jspi.2018.11.003
[9] 
Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31(3), 307–327 (1986) MR0853051. https://doi.org/10.1016/0304-4076(86)90063-1
[10] 
Britten-Jones, M.: The sampling error in estimates of mean-variance efficient portfolio weights. J. Finance 54(2), 655–671 (1999)
[11] 
Díaz-García, J.A., Gutiérrez-Jáimez, R., Mardia, K.V.: Wishart and pseudo-Wishart distributions and some applications to shape theory. J. Multivar. Anal. 63, 73–87 (1997) MR1491567. https://doi.org/10.1006/jmva.1997.1689
[12] 
Fama, E.F.: Foundations of finance: portfolio decisions and securities prices. (No Title) (1976)
[13] 
Fujikoshi, Y., Sakurai, T.: Consistency of test-based method for selection of variables in high-dimensional two-group discriminant analysis. Jpn. J. Stat. Data Sci. 2(1), 155–171 (2019) MR3969143. https://doi.org/10.1007/s42081-019-00032-4
[14] 
Gulliksson, M., Oleynik, A., Mazur, S.: Portfolio selection with a rank-deficient covariance matrix. Comput. Econ. (2023). doi:https://doi.org/10.1007/s10614-023-10404-4
[15] 
Ingersoll, J.E.: Theory of Financial Decision Making vol. 3. Rowman & Littlefield (1987)
[16] 
Javed, F., Mazur, S., Thorsén, E.: Tangency portfolio weights under a skew-normal model in small and large dimensions. J. Oper. Res. Soc. 75(7), 1395–1406 (2024). doi:https://doi.org/10.1080/01605682.2023.2249935
[17] 
Jobson, J.D., Korkie, B.: Estimation for Markowitz efficient portfolios. J. Am. Stat. Assoc. 75(371), 544–554 (1980) MR0590686
[18] 
Jorion, P.: Bayes-Stein estimation for portfolio analysis. J. Financ. Quant. Anal. 21(3), 279–292 (1986)
[19] 
Kan, R., Lassance, N.: Optimal portfolio choice with fat tails and parameter uncertainty (2024). https://ssrn.com/abstract=4652814
[20] 
Kan, R., Smith, D.R.: The distribution of the sample minimum-variance frontier. Manag. Sci. 54(7), 1364–1380 (2008)
[21] 
Kan, R., Zhou, G.: Optimal portfolio choice with parameter uncertainty. J. Financ. Quant. Anal. 42(3), 621–656 (2007)
[22] 
Kan, R., Wang, X., Zhou, G.: Optimal portfolio choice with estimation risk: no risk-free asset case. Manag. Sci. 68(3), 2047–2068 (2021)
[23] 
Karlsson, S., Mazur, S., Muhinyuza, S.: Statistical inference for the tangency portfolio in high dimension. Statistics 55(3), 532–560 (2021) MR4313438. https://doi.org/10.1080/02331888.2021.1951730
[24] 
Klein, R.W., Bawa, V.S.: The effect of estimation risk on optimal portfolio choice. J. Financ. Econ. 3(3), 215–231 (1976)
[25] 
Ledoit, O., Wolf, M.: Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks. Rev. Financ. Stud. 30(12), 4349–4388 (2017)
[26] 
Lee, M., Kim, D.: On the use of the Moore–Penrose generalized inverse in the portfolio optimization problem. Finance Res. Lett. 22, 259–267 (2017)
[27] 
Li, H., Bai, Z., Wong, W.-K., McAleer, M.: Spectrally-corrected estimation for high-dimensional Markowitz mean-variance optimization. Econom. Stat. 24, 133–150 (2022) MR4487968. https://doi.org/10.1016/j.ecosta.2021.10.005
[28] 
Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952) MR0103768
[29] 
Markowitz, H.M.: Foundations of portfolio theory. J. Finance 46(2), 469–477 (1991)
[30] 
Merton, R.C.: An analytic derivation of the efficient portfolio frontier. J. Financ. Quant. Anal. 7(4), 1851–1872 (1972)
[31] 
Muhinyuza, S.: A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting. Theory Probab. Math. Stat. 103, 103–119 (2020) MR4421345. https://doi.org/10.1090/tpms
[32] 
Muhinyuza, S., Bodnar, T., Lindholm, M.: A test on the location of the tangency portfolio on the set of feasible portfolios. Appl. Math. Comput. 386, 125519 (2020) MR4126729. https://doi.org/10.1016/j.amc.2020.125519
[33] 
Nadakuditi, R.R., Eldeman, A.: Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples. IEEE Trans. Signal Process. 56, 2625–2638 (2008) MR1500236. https://doi.org/10.1109/TSP.2008.917356
[34] 
Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59(2), 347–370 (1991) MR1097532. https://doi.org/10.2307/2938260
[35] 
Okhrin, Y., Schmid, W.: Distributional properties of portfolio weights. J. Econom. 134(1), 235–256 (2006) MR2328322. https://doi.org/10.1016/j.jeconom.2005.06.022
[36] 
Osborne, M.F.M.: Brownian motion in the stock market. Oper. Res. 7(2), 145–173 (1959) MR0104513. https://doi.org/10.1287/opre.7.2.145
[37] 
Palczewski, A., Palczewski, J.: Theoretical and empirical estimates of mean–variance portfolio sensitivity. Eur. J. Oper. Res. 234(2), 402–410 (2014) MR3144729. https://doi.org/10.1016/j.ejor.2013.04.018
[38] 
Pappas, D., Kiriakopoulos, K., Kaimakamis, G.: Optimal portfolio selection with singular covariance matrix. In: International Mathematical Forum, vol. 5, pp. 2305–2318 (2010) MR2727027
[39] 
Rachev, S.T., Mittnik, S.: Stable Paretian Models in Finance. Wiley, New York (2000)
[40] 
Rencher, A.C.: Methods of Multivariate Analysis. John Wiley & Sons, Inc. (2002) MR1885894. https://doi.org/10.1002/0471271357
[41] 
Srivastava, M.S.: Multivariate theory for analyzing high dimensional data. J. Japan Statist. Soc. 37(1), 53–86 (2007) MR2392485. https://doi.org/10.14490/jjss.37.53
[42] 
Tu, J., Zhou, G.: Data-generating process uncertainty: what difference does it make in portfolio decisions? J. Financ. Econ. 72(2), 385–421 (2004)
[43] 
Van der Vaart, A.W.: Asymptotic Statistics vol. 3. Cambridge university press (2000) MR1652247. https://doi.org/10.1017/CBO9780511802256
[44] 
Winkler, R.L.: Bayesian models for forecasting future security prices. J. Financ. Quant. Anal. 8(3), 387–405 (1973)
[45] 
Yonenaga, K., Suzukawa, A.: Distribution of the product of a Wishart matrix and a normal vector. Theory Probab. Math. Stat. 108, 209–224 (2023) MR4588246. https://doi.org/10.1090/tpms/1193

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Copyright
© 2025 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Tangency portfolio Hypothesis testing Singular Wishart distribution Singular covariance matrix Moore–Penrose inverse High-dimensional asymptotics

MSC2010
62P05 62P99 91B06 91G10 62F03 62F05

Funding
Svitlana Drin acknowledges financial support from the Knowledge Foundation Grant “Forecasting for Supply Chain Management” (Dnr: 20220115). Stepan Mazur acknowledges financial support from the project “Improved Economic Policy and Forecasting with High-Frequency Data” (Dnr: E47/22) funded by the Torsten Söderbergs Foundation and the internal research grants at Örebro University.

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