Single-index model

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The single-index model (SIM) is an asset pricing model commonly used in the finance industry to measure risk and return of a stock. Mathematically the SIM is expressed as:

where:

rit − rf is the excess return on the stock

ai is the company's alpha

Bi is the company's beta

rmt − rf is the excess return on the market index

Eit is the residual return

The accuracy of the model is enhanced by the stock return's influence by market (beta) and firm-specific risk factors (alpha), unexpected returns (residual) and the 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.

Single-Index Model 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. Specifically, the return of stock i is: ri = αi + βirm + ei The term βirm represents the stock's return due to the movement of the market modified by the stock's beta, while ei represents the unsystematic risk of the security due to firm-specific factors. Macroeconomic events, such as interest rates or the cost of labor, causes the systematic risk that affects the returns of all stocks, and the firm-specific events are the unexpected microeconomic events that affect the returns of specific firms, such as the death of key people or the lowering of the firm's credit rating, that would affect the firm, but would have a negligible effect on the economy. The unsystematic risk due to firm-specific factors of a portfolio can be reduced to zero by diversification. The index model is based on the following: • Most stocks have a positive covariance because they all respond similarly to macroeconomic factors. • However, some firms are more sensitive to these factors than others, and this firm-specific variance is typically denoted by its beta (β), which measures its variance compared to the market for one or more economic factors. • Covariances among securities result from differing responses to macroeconomic factors. Hence, the covariance of each stock can be found by multiplying their betas and the market variance: • Cov(Ri, Rk) = βiβkσ2. 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. Hence, the index model greatly reduces the number of calculations that would otherwise have to be made for a large portfolio of thousands of securities.

[edit] Assumptions of single-index model

1.the return On the security besides the fundamental fact is guided by index can also be calculated by the associated with the index.

• Yip, Henry 2005: Spreadsheet Applications to securities valuation and investment theories, John Wiley and Sons Australia Ltd