Interpreting Moment Contributions to Mean-Variance Portfolio Weights
Summary
The document asks how an asset’s mean, variance, skewness, and kurtosis contribute to its weight in a portfolio. It contrasts two possible approaches: studying how the optimal weight vector changes as moments change, or attributing weight through portfolio-level moment formulas. The motivating premise is that mean–variance optimization responds to expected returns and covariance risk, while higher moments are not part of that objective.
The response points out that if weights were chosen solely by a mean–variance model, skewness and kurtosis would not affect them, so their contribution under that model would be zero. It suggests examining derivatives of weights with respect to the first two moments to study sensitivity, while noting that multiple assets and moment inputs make attribution more involved. The exchange does not derive a general decomposition or settle how to allocate a weight among moments. Such contributions depend on the optimization objective and attribution convention; sensitivity analysis is not automatically a unique additive breakdown.
Key ideas
- Mean–variance optimization uses expected returns and covariance risk to determine portfolio weights.
- Skewness and kurtosis do not affect weights when they are absent from the objective.
- Weight sensitivities to the first two moments can be studied with derivatives.
- The document does not establish a unique additive allocation of a weight to individual moments.
- Moment attribution depends on the chosen portfolio objective and method.
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Full text
# Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?
# 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:
- moment $m$'s contribution could be derived from $w_n$ w.r.t. the analytical solution of the portfolio weight vector $\boldsymbol{w}$
- 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
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/58552
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.