Journal of Risk and Financial Management Review Improved Covariance Matrix Estimation for Portfolio Risk Measurement: A Review Ruili Sun 1, Tiefeng Ma 2, Shuangzhe Liu 3 and Milind Sathye 4,* 1 Zhongyuan Bank Postdoctoral Programme, Zhongyuan Bank, Zhengzhou 450000, China; [email protected] 2 School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China; [email protected] 3 Faculty of Science and Technology, University of Canberra, Canberra 2601, Australia; [email protected] 4 Faculty of Business, Government and Law, University of Canberra, Canberra 2601, Australia * Correspondence: [email protected] Received: 17 January 2019; Accepted: 15 March 2019; Published: 24 March 2019 Abstract: The literature on portfolio selection and risk measurement has considerably advanced in recent years. The aim of the present paper is to trace the development of the literature and identify areas that require further research. This paper provides a literature review of the characteristics of financial data, commonly used models of portfolio selection, and portfolio risk measurement. In the summary of the characteristics of financial data, we summarize the literature on fat tail and dependence characteristic of financial data. In the portfolio selection model part, we cover three models: mean-variance model, global minimum variance (GMV) model and factor model. In the portfolio risk measurement part, we first classify risk measurement methods into two categories: moment-based risk measurement and moment-based and quantile-based risk measurement. Moment-based risk measurement includes time-varying covariance matrix and shrinkage estimation, while moment-based and quantile-based risk measurement includes semi-variance, VaR and CVaR. Keywords: portfolio selection; risk measure; fat tail; Copula; shrinkage; semi-variance; CVaR 1. Introduction This paper is motivated by three stylized facts about the operation of real-world financial markets. First, as real-world financial data are asymmetric and fat-tailed, the return series cannot be approximated by normal distribution. Second, financial time series are marked by volatility clustering and last, dependence structure of multivariate distribution is required to model such data and the model needs to be flexible enough to accommodate different types of financial data. Given these characteristics of financial data, many portfolio selection and risk measurement models have been developed to account for such data. Interestingly, we have not come across literature that reviews these developments in recent years. The present research would help fills this important gap. Accordingly, the aim of this paper is to review of development of the literature in the above areas and identify the directions for future research. As already stated, financial data are known to exhibit some unique characteristics such as fat tails (leptokurtosis), volatility clustering and possible asymmetry. When the tails of distribution have a higher density than that expected under conditions of normality, it is known as fat tailed data distribution. It is ‘a distribution that has an exponential decay (as in the normal) or a finite endpoint is considered thin tailed, while a power decay of the density function in the tails is considered a fat tailed J. Risk Financial Manag. 2019, 12, 48; doi:10.3390/jrfm12010048 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2019, 12, 48 2 of 34 J. Risk Financial Manag. 2019, 12, 48 2 of 34 distribution’ (LeBaron and Samanta 2004, p. 1). As financial data typically exhibit asymmetry and fat tails,and fat the tails, Gaussian the Gaussian distribution distribution cannot adequately cannot ad representequately it.represent Consequently, it. 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We proceedproceed asas follows:follows: Section2 2 provides provides anan overviewoverview ofof thethe aboveabove characteristicscharacteristics ofof financialfinancial data, Section 33 reviews the literature literature on on portfolio portfolio selection selection models, models, Section Section 44 reviews reviews the the literature literature on onfactor factor models, models, Section Section 5 5is is devoted devoted to to portfolio portfolio risk risk measurement measurement literature literature review andand SectionSection6 6 provides directionsdirections forfor futurefuture researchresearch andand conclusionsconclusions ofof thethe study.study. 2. Fat tail of Financial Data and Data Dependence 2. Fat tail of Financial Data and Data Dependence In this section, we review the literature on fat tails and data dependence. In this section, we review the literature on fat tails and data dependence. 2.1. The Concept of Fat Tails 2.1. The Concept of Fat Tails Mandelbrot(1963) first introduced the concept of fat tails in mathematical finance to describe cottonMandelbrot price changes. (1963) It wasfirst followedintroduced by the many concept econometric of fat tails studies in mathematical devoted to the finance quest forto describe suitable classescotton price of models changes. that It capturewas followed the essential by many statistical econometric properties studies of devoted stock and to the stock quest index for suitable returns, forclasses example, of models Heyde that et capture al.(2001 the), McAleeressential( statistica2005), Liul properties and Heyde of( stock2008) and and stockTsay (index2010), returns, among for manyexample, others. Heyde et al. (2001), McAleer (2005), Liu and Heyde (2008) and Tsay (2010), among many others.The fat tail distribution may have more than one definition, as there is no universal definition for The fat tail distribution may have more than one definition, as there is no universal definition the term tail in the first place. It generally refers to a probability distribution with a tail that looks fatter for the term tail in the first place. It generally refers to a probability distribution with a tail that looks than usual or the normal distribution. A good example may be the Student t distribution which is a fat fatter than usual or the normal distribution. A good example may be the Student t distribution which tailed distribution and exhibits tails that are fatter than the normal. It is also a leptokurtic distribution is a fat tailed distribution and exhibits tails that are fatter than the normal. It is also a leptokurtic which has excess positive kurtosis as illustrated in Figure1. distribution which has excess positive kurtosis as illustrated in Figure 1. Figure 1. Student t distribution is leptokurtic and has a fatter tail when compared to a standard Figure 1. Student t distribution is leptokurtic and has a fatter tail when compared to a standard normal normal distribution. distribution. Some researchers consider that a fat tail distribution refers to a subclass of heavy tailed distributions that exhibit power law decay behavior as well as infinite variance. One example may be a distribution X defined with a fat right tail by P(X > x) ∼ x−α as x → ∞, where P is the probability for the cumulative distribution, α > 0 is a (small) constant and referred to as the tail index, and the tilde J. Risk Financial Manag. 2019, 12, 48 3 of 34 Some researchers consider that a fat tail distribution refers to a subclass of heavy tailed distributions that exhibit power law decay behavior as well as infinite variance. One example −α mayJ. Risk beFinancial a distribution Manag. 2019, X 12 defined, 48 with a fat right tail by P(X > x) ∼ x as x ! ¥, where P is3 of the 34 probability for the cumulative distribution, α > 0 is a (small) constant and referred to as the tail index, andnotation the