Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson...

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



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



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



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



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



Let: Ri = (ri - rf)

Rm = (rm - rf)

Risk premiumformat

Ri = i + ßiRm + ei

Risk Premium Format



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



i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

Measuring Components of Risk

)e( i22

M2i

2i



Total Risk = Systematic +Unsystematic

Examining Percentage of Variance

2

2M

2i2 squareR

2i

22 )e(

1

)e(22M

2



Security Characteristic Line

Excess Returns (i)SCL

..

..

.... ..

..

.. ..

.. .. ..

.. ..

..

.. ..

......

..

..

..

....

......

....

..

....

....

..

.. ..

..

.. ..

..

.. ...... ..

.. .... ..Excess returnson market index

Ri = i + ßiRm + ei



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



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



Index Model and Diversification

n

1iPP

n

1iPP

n

1iPP en

1e;n1;n

1

iMiii eRR

)e( P2

M2

P2

P2

PMPPP eRR



Risk Reduction with Diversification

Number of Securities

St. Deviation

Market Risk

Unique Risk

2(eP)=2(e) / n

P2M

2



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.



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



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



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



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



Arbitrage Portfolio

Mean Standard Deviation

Correlation

Portfolio of A, B & C

25.83 6.400.94

D stock 22.25 8.58



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



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



F

E(r)(%)

Portfolio

F

E(r)(%)

Individual Security

Portfolio & Individual Security Comparison



E(r)%

Beta for F

10

76

Risk Free = 4

AD

C

.5 1.0

Disequilibrium Example



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%



M

Beta (Market Index)

Risk Free

1.0

[E(rM) - rf]

Market Risk Premium

E(r)

APT with Market Index Portfolio



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

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson...

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