# Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?

The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio and its first two moments. How can we measure how much of an asset's individual moments contribute to its portfolio weight?

For example, if the 2nd asset in a portfolio is given a weight of $$w_2=0.4$$, how much of this value is contributed by the following unknowns:

• Contribution of asset 2's mean to $$w_2$$ = ?
• Contribution of asset 2's variance to $$w_2$$ = ?
• Contribution of asset 2's skewness to $$w_2$$ = ?
• Contribution of asset 2's kurtosis to $$w_2$$ = ?

I can think of two possible approaches for deriving the contributions of asset $$n$$'s moments to its portfolio weight that I hope someone could derive a solution from:

1. moment $$m$$'s contribution could be derived from $$w_n$$ w.r.t. the analytical solution of the portfolio weight vector $$\boldsymbol{w}$$
2. moment $$m$$'s contribution could be derived from the portfolio version of moment $$m$$. i.e. derive the contributions of variance$$_n$$ from the portfolio variance formula, skew$$_n$$ from portfolio skewness, kurt$$_n$$ from portfolio kurtosis
• As you said, the MV method only incorporates M1 and M2 - moments of higher order have no influence on the optimal weights. Sep 30 '20 at 18:36
• Although skewness or kurtosis aren't actively used as inputs in the traditional model, a portfolio's returns will have a calculable skewness and kurtosis nonetheless, regardless if they are negligible values Sep 30 '20 at 19:48

It is not clear that this allocation would be useful or even possible.

Suppose you had a portfolio of two assets and that the optimal weights you had derived, based on a mean-variance approach were 0.4, 0.6.

These are independent of the 3rd and 4th moments, suggesting that whatever the 3rd and 4th moments were in these assets the 0.4/0.6 weights would be unaffected. By extension this must imply that the 'allocation' of the 3rd and 4th to the construction of weights is zero.

Indeed how would I allocate the proponents of the 1st and 2nd moments to the weight? Probably via analysis of the following derivatives:

$$\frac{\partial{W}}{\partial m_1}, \frac{\partial{W}}{\partial m_2}$$

where you actually have 8 quantities to consider here in some fashion, since $$W, m_1, m_2$$ are arrays of 2 instruments.