In this paper, we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately, we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed.

Keywords:

References

D. Avramov & G. Zhou (2010) Bayesian portfolio analysis, Annual Review of Financial Economics 2, 25–47. Web of Science, Google Scholar
N. Barberis (2000) Investing for the long run when returns are predictable, Journal of Finance 55, 225–264. Web of Science, Google Scholar
C. Barry (1974) Portfolio analysis under uncertain means, variances, and covariances, Journal of Finance 29, 515–522. Web of Science, Google Scholar
V. S. Bawa, S. Brown & R. Klein (1979) Estimation Risk and Optimal Portfolio Choice. Amsterdam: North Holland. Google Scholar
T. Bodnar, S. Mazur & Y. Okhrin (2017) Bayesian estimation of the global minimum variance portfolio, European Journal of Operational Research 256, 292–307. Web of Science, Google Scholar
T. Bodnar, S. Mazur, K. Podgórski & J. Tyrcha (2018a) Tangency portfolio weights for small sample and singular covariance matrix: Estimation and test theory, Journal of Statistical Planning and Inference, forthcoming. Web of Science, Google Scholar
T. Bodnar & Y. Okhrin (2011) On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory, Scandinavian Jornal of Statistics 38, 311–331. Web of Science, Google Scholar
T. Bodnar, N. Parolya & W. Schmid (2018b) Estimation of the global minimum variance portfolio in high dimensions, European Journal of Operational Research 266, 371–390. Web of Science, Google Scholar
M. Brandt (2010) Handbook of Financial Econometrics, 269–336. Amsterdam: North Holland. Google Scholar
S. Brown (1976) Optimal Portfolio Choice Under Uncertainty: A Bayesian Approach. Ph.D. thesis, Graduate School of Business, University of Chicago. Google Scholar
A. DasGupta (2008) Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer. Google Scholar
V. DeMiguel, L. Garlappi & R. Uppal (2009) Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? The Review of Financial Studies 22, 1915–1953. Web of Science, Google Scholar
P. Frost & J. Savarino (1986) An empirical Bayes approach to efficient portfolio selection, Journal of Financial and Quantitative Analysis 21, 293–305. Web of Science, Google Scholar
V. Golosnoy & Y. Okhrin (2007) Multivariate shrinkage for optimal portfolio weights, European Journal of Finance 13, 441–458. Web of Science, Google Scholar
V. Golosnoy & Y. Okhrin (2008) General uncertainty in portfolio selection: A case-based decision approach, Journal of Economic Behavior and Organization 67, 718–734. Web of Science, Google Scholar
A. Gupta & D. Nagar (2000) Matrix Variate Distributions. Boca Raton: Chapman and Hall/CRC. Google Scholar
A. Gupta, T. Varga & T. Bodnar (2013) Elliptically Contoured Models in Statistics and Portfolio Theory. Second edition. Springer. Google Scholar
L. Isserlis (1918) On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12, 134–139. Google Scholar
P. Jorion (1986) Bayes-stein estimation for portfolio analysis, Journal of Financial and Quantitative Analysis 21, 293–305. Web of Science, Google Scholar
R. Kan & G. Zhou (2007) Optimal portfolio choice with parameter uncertainty, Journal of Financial and Quantitative Analysis 42, 621–656. Web of Science, Google Scholar
S. Kandel & R. F. Stambaugh (1996) On the predictability of stock returns: An asset-allocation perspective, Journal of Finance 51, 385–424. Web of Science, Google Scholar
R. Klein & V. Bawa (1976) The effect of estimation risk on optimal portfolio choice, Journal of Financial Economics 3, 215–231. Web of Science, Google Scholar
S. Kotz & S. Nadarajah (2004) Multivariate t Distributions and Their Applications. Cambridge, UK: Cambridge University Press. Google Scholar
H. M. Markowitz (1952) Mean-variance analysis in portfolio choice and capital markets, Journal of Finance 7, 77–91. Web of Science, Google Scholar
A. M. Mathai & S. B. Provost (1992) Quadratic Forms an Random Variables. New York: Marcel Dekker. Google Scholar
C. D. Meyer (2000) Matrix Analysis and Applied Linear Algebra. SIAM. Google Scholar
R. J. Muirhead (1982) Aspects of Multivariate Statistical Theory. New York: Wiley. Google Scholar
L. Pastor (2000) Portfolio selection and asset pricing models, Journal of Finance 55, 179–223. Web of Science, Google Scholar
S. T. Rachev, J. S. J. Hsu, B. S. Bagasheva & F. J. Fabozzi (2008) Bayesian Methods in Finance. New Jersey: Wiley. Google Scholar
M. Sekerke (2015) Bayesian Risk Management: A Guide to Model Risk and Sequential Learning in Financial Markets. New Jersey: Wiley. Google Scholar
Z. Wang (2005) A shrinkage approach to model uncertainty and asset allocation, Review of Financial Studies 18, 673–705. Web of Science, Google Scholar
R. Winkler (1973) Bayesian models for forecasting future security prices, Journal of Financial and Quantitative Analysis 8, 387–405. Web of Science, Google Scholar
R. L. Winkler & C. B. Barry (1975) A Bayesian model for portfolio selection and revision, Journal of Finance 30, 179–192. Web of Science, Google Scholar