Alternate Mathematical Approaches to Estimating Portfolio Efficiency: Incorporating a Multi-Asset Framework

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: 15 November 2024 | Viewed by 11167

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Interests: discrete and continuous models; weighted distributions; reliability and survival analysis; characterization problems in mathematical statistics; statistical inference; frailty models and association measures

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to exploring alternative approaches to measuring portfolio efficiency. While the distinction between an optimal and efficient portfolio is clear, it is not yet well understood how various tests perform under a multi-asset framework. The distributions of equities, bonds, corporate bonds, REITs, commodities, and currencies are often different, yet most tests assume the standard Gaussian distribution while evaluating portfolio efficiency and optimization. The issue will look at papers that discuss/utilize any or all of the following in their portfolio test designs:

  1. A multi-asset framework;
  2. Non-normal distributions underlying the data generating processes for asset prices: Non-normal return distributions, such as Poisson distribution, Merton's jump-diffusion model, and asymptotic Chi-Square return distribution, among others;
  3. Statistical tests such as tests in mean-variance space that explore the trigonometric properties (Gustafson, 2010) of the location of Markowitz-style efficient portfolios, tests utilizing GMM processes, or the likelihood ratio test (LRT) ((Zhou (1991), Gibbons, Ross, Shanken (1989)) and others;
  4. Efficiency tests could then be factored into additional tests related to portfolio performance (Sharpe, Treynor, Jensen tests, and other more recent methods).

Prof. Dr. Pankaj Agrrawal
Dr. Doureige Jurdi
Prof. Dr. Ramesh Gupta
Guest Editors

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Published Papers (4 papers)

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Research

19 pages, 4727 KiB  
Article
The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties
by Pankaj Agrrawal
Mathematics 2023, 11(9), 2198; https://doi.org/10.3390/math11092198 - 06 May 2023
Cited by 2 | Viewed by 3872
Abstract
This study is intended as a note and provides an extension to a much-used and established test for portfolio efficiency, the Gibbons, Ross, and Shanken GRS-Wald test. Tests devised to measure portfolio efficiency are crucial to the theoretical issues related to CAPM (Capital [...] Read more.
This study is intended as a note and provides an extension to a much-used and established test for portfolio efficiency, the Gibbons, Ross, and Shanken GRS-Wald test. Tests devised to measure portfolio efficiency are crucial to the theoretical issues related to CAPM (Capital Asset Pricing Model) testing and have applications for the fund manager who seeks to rank portfolio performance. This study looks at the GRS-Wald test for portfolio efficiency and extends it to make it visually more interpretive without any loss of generality in its structure. The geometrically recast statistic draws upon the trigonometric properties of a portfolio in the mean-variance space and a mathematical proof of the equivalence of the two statistics is provided. The GRS-Wald test is a widely used statistic in studies addressing the issue of portfolio efficiency and CAPM deviations. A simulation demonstrates the use of the recast GRS-Wald test in testing for the mean-variance efficiency of a test portfolio. The study also provides a table of the GRS-Wald test, based on a range of mean-variance locations (cosine of portfolio angles) at which the test portfolio and the efficient market portfolio can be placed. Full article
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19 pages, 5255 KiB  
Article
Portfolio Evaluation with the Vector Distance Based on Portfolio Composition
by Heonbae Jeon, Soonbong Lee, Hongseon Kim, Seung Bum Soh and Seongmoon Kim
Mathematics 2023, 11(1), 221; https://doi.org/10.3390/math11010221 - 01 Jan 2023
Viewed by 2188
Abstract
We propose a novel portfolio evaluation method, a distance-based approach, which directly evaluates the portfolio composition rather than portfolio returns. In this approach, we consider a portfolio as an estimator for an in-sample tangency portfolio, which we define as the optimal reference portfolio. [...] Read more.
We propose a novel portfolio evaluation method, a distance-based approach, which directly evaluates the portfolio composition rather than portfolio returns. In this approach, we consider a portfolio as an estimator for an in-sample tangency portfolio, which we define as the optimal reference portfolio. We then evaluate the portfolio by computing its vector distance to the optimal reference portfolio. In search of the proper distance-based performance measure, we choose four representative vector distances and compare their suitability as a new portfolio performance measure. Through extensive statistical analysis, we find that the Euclidean distance is the most proper distance-based performance measure of the four representative vector distances. We further verify that a portfolio with a large Euclidean distance is not desirable because not only does it provide a low utility implied by the first four moments of portfolio returns, but also it is not likely to maintain its long-term performance. Hence, the Euclidean distance can complement the return-based performance measures by confirming the reliability of a portfolio in its investment performance. Full article
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20 pages, 380 KiB  
Article
Comparing SSD-Efficient Portfolios with a Skewed Reference Distribution
by Francesco Cesarone, Raffaello Cesetti, Giuseppe Orlando, Manuel Luis Martino and Jacopo Maria Ricci
Mathematics 2023, 11(1), 50; https://doi.org/10.3390/math11010050 - 23 Dec 2022
Cited by 4 | Viewed by 2018
Abstract
Portfolio selection models based on second-order stochastic dominance (SSD) have the advantage of providing portfolios that reflect the behavior of risk-averse investors without the need to specify the utility function. Several scholars apply SSD conditions with respect to a reference distribution, typically that [...] Read more.
Portfolio selection models based on second-order stochastic dominance (SSD) have the advantage of providing portfolios that reflect the behavior of risk-averse investors without the need to specify the utility function. Several scholars apply SSD conditions with respect to a reference distribution, typically that of the market index, to find its dominant SSD portfolio. However, since the reference distribution could strongly influence asset allocation, in this article, we compare two SSD-based portfolio selection strategies with a reshaping of the reference distribution in terms of its skewness and, consequently, its variance. Through an extensive empirical analysis based on multiasset investment universes, we empirically show that the SSD portfolios dominating the new skewed benchmark index generally perform better. Full article
56 pages, 4583 KiB  
Article
The Impact of Options on Investment Portfolios in the Short-Run and the Long-Run, with a Focus on Downside Protection and Call Overwriting
by David Buckle
Mathematics 2022, 10(9), 1563; https://doi.org/10.3390/math10091563 - 06 May 2022
Cited by 3 | Viewed by 1962
Abstract
In this article, we analyse the impact of the introduction of options on an investment portfolio. Our first objective is to derive closed-form formulae for the standard measures of portfolio efficiency: risk premium, risk, Sharpe ratio, and beta, of any portfolio containing any [...] Read more.
In this article, we analyse the impact of the introduction of options on an investment portfolio. Our first objective is to derive closed-form formulae for the standard measures of portfolio efficiency: risk premium, risk, Sharpe ratio, and beta, of any portfolio containing any combination of options. Using these formulae on three examples of simple option strategies (call overwriting, put protection, and collars), we show how these statistics are altered by the inclusion of an option overlay in a portfolio. Our second objective is to show that if an option strategy is repeated over multiple investment time periods, the long-run return becomes normally distributed. Our motivation is to provide investors with the mathematics to measure the impact of the introduction of options on portfolio efficiency and encourage a potential portfolio rebalance to account for this impact. Then, we highlight that whilst options can create asymmetric non-normal outcomes, their repeated use may not alter the long-run portfolio return in the desired way and thus to encourage investors to assess if an option overlay will deliver the desired long-run outcome. Full article
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