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Statistical Inference of Tangency Portfolio in Small and Large Dimension
Stockholm University, Faculty of Science, Department of Mathematics.
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers statistical test theory in portfolio theory. It analyses the asymptotic behavior of the considered tests in the high-dimensional setting, meaning k/n c ∈ (0, ∞) as n → ∞, where k and n are portfolio size and sample size, respectively. It also considers the high-dimensional asymptotic of the product of components involved in the computation of the optimal portfolio. The thesis comprises four manuscripts:

Paper I is concerned with the test on the location of the tangency portfolio on the set of feasible portfolios. Considering the independent and normally multivariate asset returns, we propose a finite-sample test on the mean-variance efficiency of the tangency portfolio (TP). We derive the distribution of the proposed test statistic under both the null and alternative hypotheses, using which we assess the power of the test and construct a confidence interval. The out-of-sample performance of the portfolio determined by the proposed test is conducted and through an extensive simulation study, we show the robustness of the developed test towards the violation of the normality assumptions. We also apply the developed test to real data in the empirical study.

Paper II extends the results of paper I. It is concerned with the study of the asymptotic distributions of the test on the existence of efficient frontier (EF) and the efficiency of the tangency portfolio in the mean-variance space in the high-dimension setting under both the null and alternative hypotheses. Finite-sample performance and robustness of the proposed tests are studied through an extensive simulation study.

In paper III, we study the distributional properties of the TP weights under the assumption of normally distributed logarithmic returns. The distribution of the weights of the TP is given under the form of a stochastic representation (SR). Using the derived SR we deliver the asymptotic distribution of the TP weights under a high-dimensional asymptotic regime. Besides, we consider tests about the elements of the TP weights and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the power function of the high-dimensional asymptotic and the exact tests. Moreover, in an empirical study, we apply the developed theory in analysing the TP weights in a portfolio made of stocks from the S&P 500 index.

In paper IV, we derive a stochastic representation of the product of a singular Wishart matrix and a singular Gaussian vector. We then use the derived SR in the obtention of the characteristic function of that product and in proving the asymptotic normality under the double asymptotic regime. The performance of the obtained asymptotic is shown in the simulation study.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2020. , p. 31
Keywords [en]
Tangency portfolio, Mean-variance portfolio, High-dimensional asymptotics, Test theory
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-180914ISBN: 978-91-7911-112-0 (print)ISBN: 978-91-7911-113-7 (electronic)OAI: oai:DiVA.org:su-180914DiVA, id: diva2:1426116
Public defence
2020-06-10, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Accepted. Paper 3: Manuscript.

Available from: 2020-05-18 Created: 2020-04-23 Last updated: 2020-05-26Bibliographically approved
List of papers
1. A test on the location of the tangency portfolio on the set of feasible portfolios
Open this publication in new window or tab >>A test on the location of the tangency portfolio on the set of feasible portfolios
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Due to the problem of parameter uncertainty, specifying the location of the tangency portfolio (TP) on the set of feasible portfolios becomes a challenging task. The set of feasible portfolios is a parabola in the mean-variance space with optimal portfolios lying on its upper part. Using statistical test theory, we want to decide if the tangency portfolio is mean-variance efficient, i.e. if it belongs to the upper limb of the efficient frontier. In the opposite case, the investor would prefer to invest into the risk-free asset or into the global minimum variance portfolio which lies in the vertex of the set of feasible portfolios. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we suggest a test on the location of the tangency portfolio on the set of feasible portfolios. The distribution of the test statistic is derived under both hypotheses, which we use to assess the power of the test and construct a confidence interval. Moreover, out-of-sample performance of the test is evaluated based on real data. The robustness to the assumption of normality is investigated via an extensive simulation study where we show that the new test is robust to the violation of the normality assumption and can also be used for heavy-tailed stochastic models. Moreover, in an empirical study we apply the developed theory to real data. We find that when the sample size is relatively large and a stable period is present on the market, then the mean-variance efficiency of the tangency portfolio can be statistically justified.

Keywords
Tangency portfolio, feasible portfolios, test theory, power function, out-of-sample performance
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-180906 (URN)
Available from: 2020-04-22 Created: 2020-04-22 Last updated: 2020-04-23Bibliographically approved
2. A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting
Open this publication in new window or tab >>A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting
(English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000Article in journal (Refereed) Accepted
Abstract [en]

In this paper we derive the asymptotic distribution of the test of the efficiency of the tangency portfolio in high-dimensional settings, namely when both the portfolio dimension and the sample size grow to infinity. Moreover, we propose a new test based on the estimator for the slope parameter of the efficient frontier in the mean-variance space when there is a possibility in investing into the riskless asset, and derive the asymptotic distribution of that test statistic under both the null and alternative hypotheses. Additionally, we study the finite sample performance of the derived theoretical results via simulations.

Keywords
Tangency Portfolio, Mean-variance portfolio, High-dimensional settings
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-180904 (URN)
Available from: 2020-04-22 Created: 2020-04-22 Last updated: 2020-04-23
3. Statistical Inference for the Tangency Portfolio in High Dimension
Open this publication in new window or tab >>Statistical Inference for the Tangency Portfolio in High Dimension
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study the distributional properties of the tangency portfolio (TP) weights assuming a normal distribution of the logarithmic returns. We derive a stochastic representation of the TP weights that fully describes their distribution. Under a high-dimensional asymptotic regime, i.e. the dimension of the portfolio, , and the sample size, , approach infinity such that , we deliver the asymptotic distribution of the TP weights. Moreover, we consider test about the elements of the TP and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the asymptotic distribution of the TP weights with the exact finite sample density. We also compare the high-dimensional asymptotic test with exact one. We document a good performance of the asymptotic approximations except for small sample sizes combined with  close to one. In an empirical study, we analyze the TP weights in portfolios containing stocks from the S&P 500 index.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-180992 (URN)
Available from: 2020-04-23 Created: 2020-04-23 Last updated: 2020-04-27Bibliographically approved
4. On the Product of a Singular Wishart Matrix and a Singular Gaussian Vector in High Dimension
Open this publication in new window or tab >>On the Product of a Singular Wishart Matrix and a Singular Gaussian Vector in High Dimension
2018 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 99, p. 37-50Article in journal (Refereed) Published
Abstract [en]

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation of this product is derived, using which its characteristic function and asymptotic distribution under the double asymptotic regime are established. We further document a good finite sample performance of the obtained high-dimensional asymptotic distribution via an extensive Monte Carlo study.

Keywords
Singular Wishart distribution, singular normal distribution, stochastic representation, high-dimensional asymptotics
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-164884 (URN)000493467200004 ()
Available from: 2019-01-20 Created: 2019-01-20 Last updated: 2020-04-27Bibliographically approved

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