Term Glossary

Metrics, Models & Concepts

Variance


Return variance is a quantity that expresses how much the rate of return deviates from the expected return: \begin{equation} \sigma^{2}=Var(r)=E(r−E(r))^{2} \end{equation} For discrete return distributions: \begin{equation} Var(r)=\sum\limits _{i=1}^{\infty}(r_{i}-E(r))^{2}P(r=r_{i})=\sum\limits _{i=1}^{\infty}r_{i}^{2}P(r=r_{i})-(\sum\limits _{i=1}^{\infty}r_{i}P(r=r_{i}))^{2} \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
For continous return distributions: \begin{equation} Var(r)=\intop_{-\infty}^{+\infty}(r-E(r))^{2}f(r)dr=\intop_{-\infty}^{+\infty}r^{2}f(r)dr-E^{2}(r)=\intop_{-\infty}^{+\infty}r^{2}f(r)dr-(\int\limits _{-\infty}^{\infty}\! r\, f_{r}(r)dr)^{2} \end{equation}
\(r\)
asset return
\(E(r)\)
expected return
\(f(r)\)
probability density function of return distribution


Function Reference
portfolio_variance, position_variance