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&P500 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. Risk Premium Format Let: Ri = (ri - rf) Risk premium format
Rm = (rm - rf)
Risk premium
format
Ri = i + ßiRm + ei

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

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

12. 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

13. Index Model and Diversification

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

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 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

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

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

19. 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

20. Arbitrage Portfolio Mean Standard Deviation Correlation Portfolio
of A, B & C
25.83
6.40
0.94
D stock
22.25
8.58

21. 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

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

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

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

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%

26. APT with Market Index Portfolio M Beta (Market Index) Risk Free 1.0
[E(rM) - rf]
Market Risk Premium
E(r)

27. 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