Portfolio Skewness and Kurtosis from Asset Return Moments
Summary
The note explains how to estimate skewness and kurtosis for a two-asset portfolio from return observations. It first frames the inputs as aligned return time series combined using portfolio weights, with the resulting portfolio returns serving as the distribution whose higher moments are measured. The basic definitions use centered third and fourth moments, while the portfolio approach expands those calculations through co-skewness and co-kurtosis tensors.
For two assets, symmetry among tensor indices reduces the number of distinct co-moment terms that must be calculated. Portfolio kurtosis is then constructed from the weighted fourth raw moment and lower moments, normalized by portfolio volatility to the fourth power. The discussion uses empirical averages over a chosen sample, but leaves choices such as return frequency, lookback period, and finite-sample estimation conventions unspecified. Those choices can materially affect estimated higher moments.
Key ideas
- Portfolio skewness and kurtosis describe asymmetry and tail or peak behavior in portfolio returns.
- Construct portfolio returns from synchronized asset returns and their portfolio weights.
- Co-skewness and co-kurtosis tensors organize the joint higher moments of multiple assets.
- For two assets, index symmetry reduces the distinct tensor components that need to be estimated.
- Kurtosis is based on centered fourth moments and normalized by portfolio variance squared.
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# Calculating Portfolio Skewness & Kurtosis
# Calculating Portfolio Skewness & Kurtosis
I need to calculate the skewness and kurtosis of 2 asset portfolio, can someone please help me with the formulas and definition of terms? Thank you.
I have been using the matrices method and I am not sure if that is correct.
## Answer by Richard H (score 7)
https://quant.stackexchange.com/a/2275
"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is.
Skewness is defined as:
$E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$
and Kurtosis as:
$E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$
where X is your distro values (x_1, x_2, ... x_N), mean is the average of your distro values X (x_mean, a constant) and E[f(X)] is the Expectation of f(X) - i.e the mean of f(X).
So now you need to define your distributions. To be honest I don't know what the standards are for a given asset, but I imagine that if your asset price movements are ~ lognormal then you'll be wanting the daily (or whatever) percentage change in the value of the portfolio. These daily %age changes define your distribution X. Of course you'll need to consider how far back in time you go: 1 month data? 1 year?. So each daily %age change is your x_i. Calc the mean (probably close to zero), then your Skewness and Kurtosis per the formulas above.
## Answer by Derek Ploor (score 4)
https://quant.stackexchange.com/a/2370
Assuming you have return time series $$ r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T) $$ for the 2 assets and asset weights $w_1$ and $w_2$, we can follow the calculation of the $N$-asset portfolio skewness laid out in another answer for a similar question.
To extend it to include portfolio kurtosis, we need the co-kurtosis tensor $$ K_{ijkl} = E \left[ r_i \times r_j \times r_k \times r_l \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) \times r_k(t) \times r_l(t) $$ and moment $$ m_4 = \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N w_i w_j w_k w_l K_{ijkl} \quad, $$ then we can calculate portfolio kurtosis as $$ K_p = \frac{1}{\sigma_p^4} \left[ m_4 - 4 m_3 m_1 +6 m_2 m_1^2 - 3m_1^4 \right] \quad. $$
In the 2 asset portfolio case, the calculations of the higher-order tensors are not so daunting, as for the $2 \times 2 \times 2$ co-skewness tensor we only need to calculate $$\begin{split} S_{111} & \\ S_{112} &= S_{121} = S_{211} \\ S_{122} &= S_{212} = S_{221} \\ S_{222} & \end{split}$$ and for the $2 \times 2 \times 2 \times 2$ co-kurtosis tensor we only need to calculate $$\begin{split} K_{1111} & \\ K_{1112} &= K_{1121} = K_{1211} = K_{2111} \\ K_{1122} &= K_{1212} = K_{1221} = K_{2112} = K_{2211} \\ K_{1222} &= K_{2122} = K_{2221} = K_{2212} \\ K_{2222} & \end{split}$$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.