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Chapter 8 Index Models and the Arbitrage Pricing Theory

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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

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**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)

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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&P500 is the common factor

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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

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**Risk Premium Format Let: Ri = (ri - rf) Risk premium format**

Rm = (rm - rf) Risk premium format Ri = i + ßiRm + ei

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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

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**Measuring Components of Risk**

i2 = total variance i2 m2 = systematic variance 2(ei) = unsystematic variance

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**Examining Percentage of Variance**

Total Risk = Systematic +Unsystematic

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**Security Characteristic Line**

Excess Returns (i) SCL . Excess returns on market index Ri = i + ßiRm + ei

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**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

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**Regression Results a b Estimated coefficient -2.590 1.1357**

Std error of estimate (1.547) (0.309) Variance of residuals = Std dev of residuals = 3.550 R-SQR = 0.575

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**Index Model and Diversification**

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**Risk Reduction with Diversification**

Number of Securities St. Deviation Market Risk Unique Risk s2(eP)=s2(e) / n bP2sM2

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**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.

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**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

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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

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**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

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**Arbitrage Example (pp. 293-295)**

Stock Current Price ($) Expected Return (%) Standard Deviation (%) A 10 25.0 29.58 B 20.0 33.91 C 32.5 48.15 D 22.5 8.58

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**Arbitrage Portfolio Mean Standard Deviation Correlation Portfolio**

of A, B & C 25.83 6.40 0.94 D stock 22.25 8.58

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**Arbitrage Action and Returns**

E(r) s P D 25.83 22.25 6.40 8.58 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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**APT & Well-Diversified Portfolios**

F is some macroeconomic factor For a well-diversified portfolio eP approaches zero The result is similar to CAPM

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**Portfolio & Individual Security Comparison**

E(r)(%) Portfolio F E(r)(%) Individual Security

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**Disequilibrium Example**

E(r)% Beta for F 10 7 6 Risk Free = 4 A D C .5 1.0

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**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%

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**APT with Market Index Portfolio M Beta (Market Index) Risk Free 1.0**

[E(rM) - rf] Market Risk Premium E(r)

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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

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