Single Index Model
Encyclopedia
The single-index model is a simple asset pricing model commonly used in the finance
industry to measure risk and return of a stock
. Mathematically the SIM is expressed as:
where:
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.
The term
represents 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 changes in 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. In a portfolio, the unsystematic risk due to firm-specific factors 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(
) = 
.
This last equation greatly reduces the computations required to determine covariance because otherwise 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 to model a large portfolio of thousands of securities.
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
industry to measure risk and return of a stock
Stock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
. 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 alphaAlpha (investment)Alpha is a risk-adjusted measure of the so-called active return on an investment. It is the return in excess of the compensation for the risk borne, and thus commonly used to assess active managers' performances...
, or abnormal return -
is the stocks'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.
Assumptions of the 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.The term
represents 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 changes in 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. In a portfolio, the unsystematic risk due to firm-specific factors 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(
) = .This last equation greatly reduces the computations required to determine covariance because otherwise 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 to model a large portfolio of thousands of securities.