### Covariance

Covariance provides a measure of the degree of the co-movement between two asset returns. \begin{equation} \sigma(r_i,r_j)=E[(r_i-E(r_i))(r_j-E(r_j))] \end{equation}

- \(r_i\)
- i-th asset return

Covariances play a key role in financial economics, especially in portfolio theory and in the Single-Index Model (SIM). Covariances among various assets' returns are used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.

Assuming co-movement of all assets is due to one common factor, as in SIM, excess returns can be decomposed to: \begin{equation} \sigma_{\text{ij}}=cov(r_{i},r_{j})=cov(\alpha_{i}+\beta_{i}r_{M}+\varepsilon_{i},\alpha_{j}+\beta_{j}r_{M}+\varepsilon_{j})=\beta_{j}\beta_{i}cov(r_{M},R_{M})=\beta_{j}\beta_{i}\sigma_{M}^{2} \end{equation}

- \(r_i\)
- i-th asset return
- \(\beta_i\)
- i-th asset market beta
- \(r_{M}\)
- market index (benchmark) return

This last equation greatly reduces the computations required to determine covariance because the covariance of the securities within a portfolio must be calculated using historical returns, and the covariance of each possible pair of securities in the portfolio must be calculated independently. With this equation, only the betas of the individual securities and the market variance need to be estimated to calculate covariance. The SIM greatly reduces the number of calculations that would otherwise have to be made for a large portfolio of thousands of securities.