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Index Models and the Arbitrage Pricing Theory

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BMO Nesbitt Burns and Merrill Lynch examples. BMO NB uses returns not risk premiums ... M. Lynch's adjusted b':2/3(sample b) 1/3(1) Forecasting beta as a ... – PowerPoint PPT presentation

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Title: Index Models and the Arbitrage Pricing Theory


1
Chapter 8
Index Models and the Arbitrage Pricing Theory
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

3
The Single Index Model
  • 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)

4
Single Factor Model
  • ßi index of a securitys 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

5
Single Index Model
ai stocks expected return if markets excess
return is zero bi(rM-ri) the component of
return due to market movements ei the component
of return due to unexpected firm-specific events
6
Risk Premium Format
7
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

8
Measuring Components of Risk
  • ?i2 total variance
  • ?i2 ?m2 systematic variance
  • ?2(ei) unsystematic variance

9
Examining Percentage of Variance
  • Total Risk Systematic Unsystematic

10
Security Characteristic Line
11
Using the Text Example from Table 8-1
12
Regression Results
13
Index Model and Diversification
Portfolio
14
Risk Reduction with Diversification
15
Industry Prediction of Beta
  • BMO Nesbitt Burns and Merrill Lynch examples
  • BMO NB uses returns not risk premiums
  • a has a different interpretation a rf (1-b)
  • M. Lynchs adjusted b2/3(sample b)1/3(1)
  • Forecasting beta as a function of past beta
  • Forecasting beta as a function of firm size,
    growth, leverage etc.

16
Multifactor Models
  • Use other factors in addition to market return
  • Examples include industrial production, expected
    inflation etc.
  • Estimate a beta or factor loading for each factor
    using multiple regression

17
Multifactor Model Equation
  • Ri E(ri) 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

18
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 IR

19
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

20
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

21
APT Well-Diversified Portfolios
  • F is some macroeconomic factor
  • For a well-diversified portfolio eP approaches
    zero
  • The result is similar to CAPM

22
Portfolio Individual Security Comparison
23
Disequilibrium Example
24
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

25
APT with Market Index Portfolio
26
APT and CAPM Compared
  • 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

27
A Multifactor APT
  • A factor portfolio is a portfolio constructed so
    that it would have a beta equal to one on a given
    factor and zero on any other factor
  • These factor portfolios are the benchmark
    portfolios for a multifactor security market line
    for an economy with multiple sources of risk

28
Where Should we Look for Factors?
  • The multifactor APT gives no guidance on where to
    look for factors
  • 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

29
Arbitrage Example

30
Arbitrage Portfolio

31
Arbitrage Action and Returns
Action Short 3 shares of D and buy 1 of A, B
C to form portfolio P Returns You earn a higher
rate on the investment than you pay on the short
sale
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