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Chapter 8

Chapter 8. Index Models and the Arbitrage Pricing 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. The Single Index Model. Advantages :

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Chapter 8

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  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 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 Assumption: a broad market index like the S&P/TSX 60 is the common factor

  5. Single Index Model ai = stock’s expected return if market’s 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. Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Ri = i + ßiRm + ei 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. Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . . Ri = i + ßiRm + ei Security Characteristic Line

  11. Using the Text Example from Table 8-1

  12. Regression Results

  13. Index Model and Diversification

  14. St. Deviation Unique Risk s2(eP)=s2(e) / n bP2sM2 Market Risk Number of Securities 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) • Merill Lynch’s ‘adjusted b’ • 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 GDP

  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. E(r)(%) E(r)(%) F F Individual Security Portfolio Portfolio & Individual Security Comparison

  23. E(r)% 10 A D 7 6 C Risk Free = 4 .5 1.0 Beta for F 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. E(r) M [E(rM) - rf] Market Risk Premium Risk Free 1.0 Beta (Market Index) 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

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