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Understand your moments

Albeit the sophistication of modern finance, mean returns and variance are in the lead for the title of the most (over)used tools in the industry. They appear in assessments of past performances, day-to-day management of portfolios, and in fundamental formulas, like in the Capital Asset Pricing Model (CAPM). In seeking to explain the investment expectations, mean return and variance can be referred to as the first and second moments, respectively. Naturally, higher moments exist*. They could be employed to provide an additional explanation as to expected returns, and we shall examine what, why, when and whether to use them.


The third moment - coskewness (or systematic skewness) measures the excess return on the market excess return. The fourth - cokurtosis (or systematic kurtosis) measures whether extreme returns of an asset and its market occur at the same time. These higher moments came into prominence in many forms in the past few decades. For one, the distribution of returns is often not normally distributed. The rise of behavioural economics signalled that investors tend to be (irrationally) risk-averse, thus, willing to opt for positive coskewness and small cokurtosis, while forgoing higher returns and lower volatility 1). The introductions of the three- and four-moments CAPMs, which incorporated further descriptors to adjust for the likes of the size of the company, the growth potential, etc. These can be used to explain exotic investment instruments and strategies adopted by hedge funds 2). And there is empirical data to prove that the inclusion of higher moments substantially improves portfolio performance 3).


Nevertheless, higher moments are not a panacea, and there are valid drawbacks in considering them. For one, the complexity of computation is drastically increased with the inclusion of higher moments. For example, ten assets would require estimating 220 coskewness and 715 cokurtosis parameters. Doubling the number of assets would demand 1540 and 8855 parameters, respectively**. Moreover, the model becomes sensitive as it includes higher-order terms; and there are cases when four-moment models are bad at explaining the performance 4). This added difficulty of deciding when the higher moments are applicable or not, might not be practical for everyone.


Understanding the (higher) moments proved to radically improve the performance of the portfolios. These naturally present opportunities, but also can be viewed as an added cost/time spent on unproven models. Still, currently, we have better than ever access to (vast amounts of) data, while the cost of CPU has reduced drastically over the past decade. This suggests that the incorporation (or consideration) of higher moments in models is inevitable.


*For brave and inquisitive readers, unafraid of technicalities, moments correspond to the coefficients of the Taylor expansion of the expected utility. Thus, the first and second moments are the coefficients corresponding to the linear and quadratic factors.


**The complexity is cubic and quartic, respectively.



1Todd Mitton & Keith Vorkink, “Equilibrium Underdiversification and the Preference for Skewness” https://www.jstor.org/stable/4494802

2Angelo Ranaldo & Laurent Favre, “Hedge Fund Performance and Higher-Moment Market Models” https://doi.org/10.3905/jai.2005.608031

3Mike Buckle, Jing Chen & Julian M. Williams, “Realised higher moments: theory and practice” https://doi.org/10.1080/1351847X.2014.885456

4Daniel Chi-Hsiou Hung, Mark Shackleton & Xinzhong Xu, “CAPM, Higher Co-moment and Factor Models of UK Stock Returns”



N.B. This article does not constitute any professional investment advice or recommendations to buy, sell, or hold any investments or investment products of any kind, and should be treated as more of an illustrative piece for educational purposes.


To trial a truly powerful and comprehensive analytic software for investment decisions, fund allocation, and our new, innovative digital due diligence visit alternativesoft.com , call us on +44 20 7510 2003, or email us [email protected]



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