Single-index model

The single-index model (SIM) is a simple asset pricing model to measure both the risk and the return of a stock. The model has been developed by William Sharpe in 1963 and is commonly used in the finance industry. Mathematically the SIM is expressed as:

${\displaystyle r_{it}-r_{f}=\alpha _{i}+\beta _{i}(r_{mt}-r_{f})+\epsilon _{it}\,}$

${\displaystyle \epsilon _{it}\sim N(0,\sigma _{i}^{2})\,}$

where:

r_it is return to stock i in period t

r_f is the risk free rate (i.e. the interest rate on treasury bills)

r_mt is the return to the market portfolio in period t

${\displaystyle \alpha _{i}}$ is the stock's alpha, or abnormal return

${\displaystyle \beta _{i}}$ is the stock's beta, or responsiveness to the market return

Note that ${\displaystyle r_{it}-r_{f}}$ is called the excess return on the stock, ${\displaystyle r_{mt}-r_{f}}$ the excess return on the market

${\displaystyle \epsilon _{it}}$ are the residual (random) returns, which are assumed independent normally distributed with mean zero and standard deviation ${\displaystyle \sigma _{i}}$

These equations show that the stock return is influenced by the market (beta), has a firm specific expected value (alpha) and firm-specific unexpected component (residual). Each stock's performance is in relation to the performance of a market index (such as the All Ordinaries). Security analysts often use the SIM for such functions as computing stock betas, evaluating stock selection skills, and conducting event studies.