Term Glossary

Metrics, Models & Concepts

Central Moment


Central moment of return distribution is a moment of a probability distribution of returns about expected return: \begin{equation} \mu_n = E[(r - E[r])^n] \end{equation} For discrete return distributions: \begin{equation} Var(r)=\sum\limits _{i=1}^{\infty}(r_{i}-E(r))^{n}P(r=r_{i}) \end{equation} \begin{equation} \sum\limits _{i=1}^{\infty}P(r=r_{i})=1 \end{equation}

\(r\)
asset return
\(E(r)\)
expected return
\(P(r=r_i)\)
probability of i-th return realization
\(n\)
moment order, non-negative integer
For continous return distributions: \begin{equation} \mu_n=\intop_{-\infty}^{+\infty}(r-E(r))^{n}f(r)dr \end{equation}
\(r\)
asset return
\(E(r)\)
expected return
\(f(r)\)
probability density function of return distribution
\(n\)
moment order, non-negative integer
The first central moment \(\mu_1\) is 0. The second central moment \(\mu_2\) is called the variance. The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.


Function Reference
portfolio_moment, position_moment