3. Index Models
Chapter 10-11
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Factor Models
Reduces the number of inputs for diversification.
Easier to map risk sources
Easier for security analysts to specialize
Single factor model
ri = E(Ri) + ßiF + e
ßi = sensitivity of a securities’ return to
the factor
F= unanticipated factor movement (factor is
commonly related to security returns)
Assumption: a broad market index return is the
common factor.
10-2
Single Index Model
(ri - rf) =
Risk Prem
a i + ßi(rm - rf) + ei
Market Risk Prem
or Index Risk Prem
ai = the stock’s expected return if the
market’s excess return is zero
(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
10-3
Risk Premium Format
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premium
format
Ri = ai + ßi(Rm) + ei
10-4
Security Characteristic Line
Excess Returns (i)
SCL
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Excess returns
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on market index
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Ri = a i + ßiRm + ei
10-5
Using the Text Example from Table 10-1
Jan.
Feb.
.
.
Dec
Mean
Std Dev
Excess
GM Ret.
5.41
-3.44
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2.43
-.60
4.97
Excess
Mkt. Ret.
7.24
.93
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3.90
1.75
3.32
10-6
Regression Results
rGM - rf = a + ß(rm - rf)
a
Estimated coefficient
-2.590
Std error of estimate
(1.547)
Variance of residuals = 12.601
Std dev of residuals = 3.550
R-SQR = 0.575
ß
1.1357
(0.309)
10-7
Components of Risk
Market or systematic risk
risk related to the factor (market index)
Unsystematic or firm specific risk
risk not related to the factor
Total risk = Systematic + Unsystematic
i2 = i2 m2 + 2(ei)
where;
i2 = total variance
i2 m2 = systematic variance
2(ei) = unsystematic variance
10-8
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
10-9
Index Model and Diversification
RP  a P   P  eP
N
P  1N  P
i 1
N
a P  1 N a P
i 1
eP  1

2
p
N
N
e
i 1
P
2
  P2 M
  2 (e P )
10-10
Risk Reduction with Diversification
St. Deviation
Unique Risk
2(eP)=2(e) / n
P2M2
Market Risk
Number of
Securities
10-11
Industry Prediction of Beta
Merrill Lynch Example
Use returns not risk premiums
a has a different interpretation
a = a + rf (1-)
Forecasting beta as a function of past
beta
Forecasting beta as a function of firm size,
growth, leverage etc.
10-12
Multi Factor Model
Returns on a security come from two sources
Common macro-economic factor
Firm specific events
Possible common macro-economic factors
Gross Domestic Product Growth
Interest Rates
Use more than one factor in addition to market
return
Examples include gross domestic product, expected
inflation, interest rates etc.
Estimate a beta or factor loading for each factor
using multiple regression.
10-13
Multifactor Model Equation
Macro factors
Ri = αi + BetaGDP (GDP) + BetaIR (IR) + ei
Ri = Return for security i
BetaGDP= Factor sensitivity for GDP
BetaIR = Factor sensitivity for Interest Rate
ei = Firm specific events
Portfolio return factors
(Ri – Rf) = αi + BetaMKT (RM-Rf) + BetaHML(RHML) +
BetaSML (RSML) + ei
10-14
Multifactor SML Models
E(r) = rf + BGDPRPGDP + BIRRPIR
BGDP = Factor sensitivity for GDP
RPGDP = Risk premium for GDP
BIR = Factor sensitivity for Interest Rate
RPIR = Risk premium for GDP
10-15