Calculating Portfolio Skewness & Kurtosis

Question

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 1

"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 2

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 $$\begin{split} K_{1111} & \\ K_{1112} &= K_{1121} = K_{1211} = K_{2111} \\ K_{1122} &= K_{1212} = K_{1221} = K_{2112} = K_{2211} \\ K_{1222} &= K_{2122} = K_{2221} = K_{2212} \\ K_{2222} & \end{split}$$