bodie kane marcus perrakis ryaninvestments, fourth canadian edition copyright © mcgraw-hill ryerson...
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Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-1Slide 8-1
Chapter 8
Index Models Index Models and the and the Arbitrage Arbitrage Pricing TheoryPricing Theory
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-2Slide 8-2
Chapter Summary
Objective: To discuss the nature and illustrate the use of arbitrage. To introduce the index model and the APT.
The Single Index Model The Arbitrage Pricing Theory
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-3Slide 8-3
Advantages: Reduces the number of inputs for
diversification Easier for security analysts to specialize
Drawback: the simple dichotomy rules out important
risk sources (such as industry events)
The Single Index Model
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-4Slide 8-4
ßi = index of a security’s particular return to the factor
F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns
Single Factor Model
iiii eF)R(Er
Assumption: a broad market index like the S&P500 is the common factor
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-5Slide 8-5
Single Index Model
ifMiifi e)rr()rr(
i = stock’s expected return if market’s excess return is zero
i(rM-ri) = the component of return due to market movements
ei = the component of return due to unexpected firm-specific events
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-6Slide 8-6
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premiumformat
Ri = i + ßiRm + ei
Risk Premium Format
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-7Slide 8-7
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
Components of Risk
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-8Slide 8-8
i2 = total variance
i2 m
2 = systematic variance
2(ei) = unsystematic variance
Measuring Components of Risk
)e( i22
M2i
2i
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-9Slide 8-9
Total Risk = Systematic +Unsystematic
Examining Percentage of Variance
2
2M
2i2 squareR
2i
22 )e(
1
)e(22M
2
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-10Slide 8-10
Security Characteristic Line
Excess Returns (i)SCL
..
..
.... ..
..
.. ..
.. .. ..
.. ..
..
.. ..
......
..
..
..
....
......
....
..
....
....
..
.. ..
..
.. ..
..
.. ...... ..
.. .... ..Excess returnson market index
Ri = i + ßiRm + ei
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-11Slide 8-11
Using the Text Example from Table 8-1
Excess X Returns
Excess Mkt Returns
January 5.41 7.24
February 3.44 0.93
. . .
December 2.43 3.90
Mean -0.60 1.75
Std Deviation 4.97 3.32
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-12Slide 8-12
Regression Results
)rr(rr fMfXYZ
Estimated coefficient -2.590 1.1357
Std error of estimate (1.547) (0.309)
Variance of residuals = 12.601
Std dev of residuals = 3.550
R-SQR = 0.575
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-13Slide 8-13
Index Model and Diversification
n
1iPP
n
1iPP
n
1iPP en
1e;n1;n
1
iMiii eRR
)e( P2
M2
P2
P2
PMPPP eRR
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-14Slide 8-14
Risk Reduction with Diversification
Number of Securities
St. Deviation
Market Risk
Unique Risk
2(eP)=2(e) / n
P2M
2
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-15Slide 8-15
Industry Prediction of Beta
BMO Nesbitt Burns and Merrill Lynch examples BMO NB uses returns not risk premiums has a different interpretation: + rf (1-) Merill Lynch’s ‘adjusted ’
Forecasting beta as a function of past beta Forecasting beta as a function of firm size,
growth, leverage etc.
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-16Slide 8-16
Multifactor Models
Use factors in addition to market return Examples include industrial production, expected
inflation etc. Estimate a beta for each factor using multiple
regression Chen, Roll and Ross
Returns a function of several macroeconomic and bond market variables instead of market returns
Fama and French Returns a function of size and book-to-market value
as well as market returns
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-17Slide 8-17
Summary Reminder
Objective: To discuss the nature and illustrate the use of arbitrage. To introduce the index model and the APT.
The Single Index Model The Arbitrage Pricing Theory
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-18Slide 8-18
Arbitrage Pricing Theory
Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit
Since no investment is required, an investor can create large positions to secure large levels of profit
In efficient markets, profitable arbitrage opportunities will quickly disappear
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-19Slide 8-19
Arbitrage Example (pp. 293-295)
Stock Current Price
($)
Expected Return
(%)
Standard Deviation (%)
A 10 25.0 29.58
B 10 20.0 33.91
C 10 32.5 48.15
D 10 22.5 8.58
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-20Slide 8-20
Arbitrage Portfolio
Mean Standard Deviation
Correlation
Portfolio of A, B & C
25.83 6.400.94
D stock 22.25 8.58
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-21Slide 8-21
Arbitrage Action and Returns
Action: Short 3 shares of D and buy 1 of A, B & C to form portfolio PReturns: You earn a higher rate on the investment than you pay on the short sale
E(r)
P D
25.8322.25
6.40 8.58
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-22Slide 8-22
APT & Well-Diversified Portfolios
F is some macroeconomic factor For a well-diversified portfolio eP
approaches zero The result is similar to CAPM
PPPP eF)r(Er
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-23Slide 8-23
F
E(r)(%)
Portfolio
F
E(r)(%)
Individual Security
Portfolio & Individual Security Comparison
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-24Slide 8-24
E(r)%
Beta for F
10
76
Risk Free = 4
AD
C
.5 1.0
Disequilibrium Example
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-25Slide 8-25
Disequilibrium Example
Short Portfolio C Use funds to construct an equivalent risk
higher return Portfolio D D is comprised of A & Risk-Free Asset
Arbitrage profit of 1%
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-26Slide 8-26
M
Beta (Market Index)
Risk Free
1.0
[E(rM) - rf]
Market Risk Premium
E(r)
APT with Market Index Portfolio
Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition
Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 8-27Slide 8-27
APT applies to well diversified portfolios and not necessarily to individual stocks
With APT it is possible for some individual stocks to be mispriced - not lie on the SML
APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio
APT can be extended to multifactor models
APT and CAPM Compared