Abstract
This paper originally proposes two unique closed-form solutions, respectively to risky assets only and a risk-free asset existing situations, of the mean-variance-skewness (MVS) optimization model subject to mean-sknewness-normalization constraints for portfolio selection. The efficient frontier and capital allocation surface (CAS) respectively derived from the two solutions are two hyperboloids, and tangent to each other at one hyperbola referred to as the market portfolio curve. Moreover, this curve intersects the mean-skewness plane of the portfolio return wtih zero-variance (zero-risk) at a line. Calculating the distance between a point on the coincident curve with the vertex of the CAS, we present a novel ratio to measure the performance of the risk-adjusted returns of market portfolio. The ratio is similar to the Sharpe ratio, moreover, under the more realistic assumption that portfolio returns follow a skew-normal distribution, the novel ratio can quantify the degree (or absence) of market portfolio exuberance.
Highlights
In the modern portfolio theory proposed by Markowitz [18], obtaining a mean-variance (MV) efficient portfolio denotes finding a weight vector of assets in a portfolio, which makes the minimal variance of the portfolio return, and is subject to the two constraints: a certain and pre-specified mean of the portfolio return has to be achieved; the weights have to be summed up to 1
It has been noticed that if the random vector of the asset returns is characterized by another special distribution beyond the normal one, the solution of the MVS optimization model may be unique
The novel ratio can measure the performance of the risk-adjusted returns of market portfolio versus their risks, and the degree of their exuberance that expresses the skewnesses of the risk-adjusted returns
Summary
In the modern portfolio theory proposed by Markowitz [18], obtaining a mean-variance (MV) efficient portfolio denotes finding a weight vector (proportions) of assets in a portfolio, which makes the minimal variance of the portfolio return (gross-rate), and is subject to the two constraints: a certain and pre-specified mean of the portfolio return has to be achieved; the weights have to be summed up to 1. It has been noticed that if the random vector of the asset returns is characterized by another special distribution beyond the normal one, the solution of the MVS optimization model may be unique. During this period, Azzalini and Valle [3] introduced the multivariate skew-normal distribution, which includes the normal distribution, but skews it with one simple parameter. Our purpose of this paper is to derive the unique closedform solution of the MVS optimization model (6) based on the skewnormal distribution and subject to mean-sknewness-normalization constraints. The novel ratio can measure the performance of the risk-adjusted returns of market portfolio versus their risks, and the degree (or absence) of their exuberance that expresses the skewnesses of the risk-adjusted returns
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