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Concept
Single-index model
Summary
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:
where:
rit is return to stock i in period t
rf is the risk free rate (i.e. the interest rate on treasury bills)
rmt is the return to the market portfolio in period t
is the stock's alpha, or abnormal return
is the stock's beta, or responsiveness to the market return
Note that is called the excess return on the stock, the excess return on the market
are the residual (random) returns, which are assumed independent normally distributed with mean zero and standard deviation
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.
To simplify analysis, the single-index model assumes that there is only 1 macroeconomic factor that causes the systematic risk affecting all stock returns and this factor can be represented by the rate of return on a market index, such as the S&P 500.
According to this model, the return of any stock can be decomposed into the expected excess return of the individual stock due to firm-specific factors, commonly denoted by its alpha coefficient (α), the return due to macroeconomic events that affect the market, and the unexpected microeconomic events that affect only the firm.
The term represents the movement of the market modified by the stock's beta, while represents the unsystematic risk of the security due to firm-specific factors.
Official source
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