The 4 - Moment CAPM
Introduction
The 4-Moment Capital Asset Pricing Model is based on two academic papers (Jurcenzko and Maillet, The Four Moment CAPM: Some Basic Results , working paper, 2002, and Hwang and Satchell, Modeling Emerging Market Risk Premia Using Higher Moments , working paper, 1999).
 

When financial assets are normally distributed, the historical asset return, the asset standard deviation and its covariance with the market are enough to estimate the asset expected return. This model is the 2-Moment CAPM developed by Sharpe (1964), Lintner (1965) and Mossin (1966). We claim that the risk in not only in volatility and linear correlation, but in skewness, kurtosis, systematic skewness, and systematic kurtosis. The model developed below and applied in AlternativeSoft's platform is the Four-Moment CAPM, which accounts for the beta, the co-skewness, the co-kurtosis dependencies between the assets and the market portfolio.

 
The Model
The 2-Moment CAPM has the following form:
 
The Four-Moment CAPM has the following form:
 
with
 
The risk premium of a non-normally distributed asset is equal to:
• its market risk multiplied the market premium b1 plus
• its systematic skewness risk multiplied with the systematic skewness market premium b2 plus
• its systematic kurtosis risk multiplied with the systematic kurtosis market premium b3
 
If we assume that it is possible to construct a portfolio with zero beta, zero systematic kurtosis, and unitary systematic skewness, then the market premium of this portfolio will be b2. If we assume that it is possible to construct a portfolio with zero beta, zero systematic skewness, and unitary systematic kurtosis, then the market premium of this portfolio will be b3. Given these two assumptions, the market premium, b1, b2, b3, can be computed in a security market hyperplane as:
 
where E(R m ) is the world market expected annual return and R f is the expected annual risk free rate of return. S Z1 , is the systematic skewness between the portfolio Z1 and the market.  S Z2,m is the systematic kurtosis between the portfolio  Z2 and the market. By substituting b1, b2, b3, S Z1,m , and S Z2,m , in the Four-Moment CAPM equation and developing the terms, we finally obtain the asset i Four-Moment CAPM required rate of return:
 
 
We see that an asset required rate of return is composed of the risk free rate plus three premiums:
• The first premium is the reward of having in the portfolio an asset which is contributing positively to the world market beta
• The second premium is the reward of having in the portfolio an asset which is contributing negatively to the world market skewness
• The fourth premium is the reward of having in the portfolio an asset which is contributing positively to the world market kurtosis.
 
To determine the expected returns for hedge funds, fund of funds or mutual funds using the Four-Moment CAPM, AlternativeSoft's software platform is available.
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