Single-index model

This article is about the asset pricing model in economics. For a description of its more general application in semiparametric regression, see Semiparametric regression.

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:

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

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

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

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

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

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

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

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.

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See also

• Capital asset pricing model
• Multiple factor models

Further reading

• Sharpe, William F. (1963). "A Simplified Model for Portfolio Analysis". Management Science. 9 (2): 277–93. doi:10.1287/mnsc.9.2.277. S2CID 55778045.
• P. Diksha. "Sharpe Theory of Portfolio Management". Economics Discussion.
• Yip, Henry (2005). Spreadsheet Applications to securities valuation and investment theories. John Wiley and Sons Australia Ltd. ISBN 0470807962.