856:. This would require more computation, but still less than computing the covariance of each possible pair of securities in the portfolio. 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.
606:
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.
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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
518:
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
486:
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
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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
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Covariance 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:
913:
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491:). 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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This article is about the asset pricing model in economics. For a description of its more general application in semiparametric regression, see
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are the residual (random) returns, which are assumed independent normally distributed with mean zero and standard deviation
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The single-index model assumes that once the market return is subtracted out the remaining returns are uncorrelated:
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This is not really true, but it provides a simple model. A more detailed model would have multiple
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Most stocks have a positive covariance because they all respond similarly to macroeconomic factors.
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affecting all stock returns and this factor can be represented by the rate of return on a
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160:{\displaystyle r_{it}-r_{f}=\alpha _{i}+\beta _{i}(r_{mt}-r_{f})+\epsilon _{it}\,}
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represents the movement of the market modified by the stock's beta, while
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To simplify analysis, the single-index model assumes that there is only 1
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Spreadsheet
Applications to securities valuation and investment theories
883:
Sharpe, William F. (1963). "A Simplified Model for
Portfolio Analysis".
745:{\displaystyle E((R_{i,t}-\beta _{i}m_{t})(R_{k,t}-\beta _{k}m_{t}))=0,}
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842:{\displaystyle Cov(R_{i},R_{k})=\beta _{i}\beta _{k}\sigma ^{2}.}
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is the risk free rate (i.e. the interest rate on treasury bills)
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is the stock's beta, or responsiveness to the market return
225:{\displaystyle \epsilon _{it}\sim N(0,\sigma _{i}^{2})\,}
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34:model to measure both the risk and the return of a
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46:industry. Mathematically the SIM is expressed as:
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270:is the return to the market portfolio in period
609:The index model is based on the following:
381:is called the excess return on the stock,
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914:"Sharpe Theory of Portfolio Management"
571:{\displaystyle \beta _{i}(r_{m}-r_{f})}
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925:. John Wiley and Sons Australia Ltd.
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495:Assumptions of the single-index model
42:in 1963 and is commonly used in the
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519:events that affect only the firm.
38:. The model has been developed by
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449:{\displaystyle \epsilon _{it}}
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598:{\displaystyle \epsilon _{i}}
417:{\displaystyle r_{mt}-r_{f}}
374:{\displaystyle r_{it}-r_{f}}
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866:Capital asset pricing model
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476:{\displaystyle \sigma _{i}}
295:{\displaystyle \alpha _{i}}
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328:{\displaystyle \beta _{i}}
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21:Semiparametric regression
28:single-index model (SIM)
916:. Economics Discussion.
871:Multiple factor models
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921:Yip, Henry (2005).
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891:(2): 277–93.
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854:risk factors
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755:which gives
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509:market index
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912:P. Diksha.
513:S&P 500
944:Categories
932:0470807962
338:Note that
249:in period
828:σ
818:β
808:β
709:β
705:−
664:β
660:−
587:ϵ
553:−
531:β
522:The term
465:σ
435:ϵ
402:−
359:−
317:β
284:α
205:σ
189:∼
177:ϵ
145:ϵ
125:−
100:β
87:α
70:−
905:55778045
860:See also
617:factors.
235:where:
44:finance
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901:S2CID
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