Term Glossary

Metrics, Models & Concepts

Single Index Model

The Sharpe Single-Index Model (SIM) was proposed by William Sharpe to reduce the number of inputs (estimates) needed for portfolio optimization. The SIM starts with the observation that the returns on most assets co-move with the overall market, as measured by the “market portfolio”. Market portfolio is a value-weighted portfolio (index) of all assets in the economy. In practice, any of several broad-based market indices are used. The SIM expression for the random return of asset $A_i$ \begin{equation} r_i= \alpha_i + \beta_i r_I +\varepsilon_i, \end{equation} where $r_i$ is return of asset $A_i$, $r_I$ is return of market index, and $\varepsilon_i$ is the random component of $A_i$ asset's return that is unique to this asset, that is $Cov(\varepsilon_i,\varepsilon_j)=0$ and $Cov(\varepsilon_i,r_I)=0$. $\alpha_i$ and $\beta_i$ may be obtained using the following formulas \begin{equation} \beta_i=\frac{Cov(r_i, r_I)}{Var(r_I)}, \end{equation} \begin{equation} \alpha_i= E(r_i) -\beta_i E(r_I). \end{equation} From this expression, we can obtain the following relationship for the variance of the return of $A_i$: \begin{equation} \sigma^2_i= \beta_i^2\sigma^2_I +\sigma_{\varepsilon, i}^2, \end{equation} where \begin{equation} Var(r_I)=\sigma^2_I, \end{equation} \begin{equation} Var(\varepsilon_i) =\sigma_{\varepsilon, i}. \end{equation} Equations says the variance of an asset can be partitioned into systematic risk (a measure of how the asset covariates with the index) -- $\beta_i^2\sigma^2_I$ , and unsystematic variance (independent of economic), $\sigma_{\varepsilon, i}^2$. Therefore covariance of $r_i$ and $r_j$ takes the form \begin{equation} Cov(r_i, r_j)=\sigma_{i,j}=\beta_i*\beta_j*\sigma^2_I. \end{equation} So that the calculation of portfolio variation simplifies to \begin{equation} \sigma^2=\sum_{i=1}^n\sum_{j=1}^n \beta_i\beta_j \sigma^2_I\omega_i\omega_j + \sum_{i=1}^n \omega^2_i\sigma^2_{\varepsilon,i}. \end{equation} It requires n+1 calculation of variance and n covariance. With the SIM model, we have two new parameters portfolio, which also may be involved in the formulation of strategies for portfolio optimization. It's portfolio alpha and beta \begin{equation} \alpha=\sum_{i=1}^n\omega_i\alpha_i, \end{equation} \begin{equation} \beta=\sum_{i=1}^n\omega_i\beta_i. \end{equation} \begin{equation} \alpha= E(r) - \beta E(r_I). \end{equation} This leads to \begin{equation} \sigma^2=\beta^2\sigma^2_I+\sigma^2_\epsilon, \end{equation} where $\sigma^2_\epsilon$ is the portfolio’s unique component of risk.