Index Models

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Chapter 10

• Index Models

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Advantages of the Single Index Model

• Reduces the number of inputs for diversification.
• Easier for security analysts to specialize.

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Single Factor Model

• ri E(Ri) ßiF e
• ßi index of a securities particular return to
  the factor
• F some macro factor in this case F is
  unanticipated movement F is commonly related to
  security returns
• Assumption a broad market index like the SP500
  is the common factor.

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Single Index Model
Risk Prem
Market Risk Prem
or Index Risk Prem
the stocks expected return if the markets
excess return is zero
?a
i
(rm - rf) 0
ßi(rm - rf) the component of return due to
movements in the market index
ei firm specific component, not due to market
movements
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Risk Premium Format
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Security Characteristic Line
Excess Returns (i)
SCL
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Excess returns on market index
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Ri ? i ßiRm ei
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Using the Text Example from Table 10-1
Excess Mkt. Ret.
Excess GM Ret.
Jan. Feb. . . Dec Mean Std Dev
5.41 -3.44 . . 2.43 -.60 4.97
7.24 .93 . . 3.90 1.75 3.32
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Regression Results
?
rGM - rf ß(rm - rf)
?
ß
Estimated coefficient Std error of
estimate Variance of residuals 12.601 Std dev
of residuals 3.550 R-SQR 0.575
-2.590 (1.547)
1.1357 (0.309)
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Components of Risk

• Market or systematic risk risk related to the
  macro economic factor or market index.
• Unsystematic or firm specific risk risk not
  related to the macro factor or market index.
• Total risk Systematic Unsystematic

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Measuring Components of Risk

• ?i2 ?i2 ?m2 ?2(ei)
• where
• ?i2 total variance
• ?i2 ?m2 systematic variance
• ?2(ei) unsystematic variance

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Examining Percentage of Variance

• Total Risk Systematic Risk Unsystematic Risk
• Systematic Risk/Total Risk ?2
• ßi2 ? m2 / ?2 ?2
• ?i2 ?m2 / ?i2 ?m2 ?2(ei) ?2

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Index Model and Diversification
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Risk Reduction with Diversification
St. Deviation
Unique Risk s2(eP)s2(e) / n
bP2sM2
Market Risk
Number of Securities
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Industry Prediction of Beta

• Merrill Lynch Example
• Use returns not risk premiums
• a has a different interpretation
• a a rf (1-b)
• Forecasting beta as a function of past beta
• Forecasting beta as a function of firm size,
  growth, leverage etc.

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