1. LECTURE 8 : FACTOR MODELS
(Asset Pricing and Portfolio Theory)

2. Contents The CAPM Single index model Arbitrage portfolioS
Which factors explain asset prices ?
Empirical results

3. Introduction CAPM : Equilibrium model
One factor, where the factor is the excess return on the market.
Based on mean-variance analysis
Stephen Ross (1976) developed alternative model  Arbitrage Pricing Theory (APT)

4. Single Index Model

5. Single Index Model Alternative approach to portfolio theory.
Market return is the single index.
Return on a stock can be written as :
Ri = ai + biRm
ai = ai + ei
Hence Ri = ai + biRm + ei Equation (1)
Assume : Cov(ei, Rm) = 0
E(eiej) = 0 for all i and j (i ≠ j)

6. Single Index Model (Cont.)
Obtain OLS estimates of ai, bi and sei (using OLS)
Mean return :
ERi = ai + biERm
Variance of security return :
s2i = b2is2m + s2ei
Covariance of returns between securities :
sij = bibjs2m

7. Portfolio Theory and the Market Model
Suppose we have a 5 Stock Portfolio
Estimates required
Traditional MV-approach
5 Expected returns
5 Variances of returns
10 Covariances
Using the Single Index Model
5 OLS regressions
5 alphas and 5 betas
5 Variances of error term
1 Expected return of the market portfolio
1 Variance of market return

8. Factor Models

9. Single Factor Model ER
Slope = b a
Factor

10. Factor Model : Example Ri = ai + biF1 + ei Example :
Factor-1 is predicted rate of growth in industrial production
i mean Ri bi
Stock 1 15% 0.9
Stock 2 21% 3.0
Stock 3 12% 1.8

11. The APT : Some Thoughts The Arbitrage Pricing Theory
New and different approach to determine asset prices.
Based on the law of one price : two items that are the same cannot sell at different prices.
Requires fewer assumptions than CAPM
Assumption : each investor, when given the opportunity to increase the return of his portfolio without increasing risk, will do so.
Mechanism for doing so : arbitrage portfolio

12. An Arbitrage Portfolio

13. Arbitrage Portfolio Arbitrage portfolio requires no ‘own funds’
Assume there are 3 stocks : 1, 2 and 3
Xi denotes the change in the investors holding (proportion) of security i, then X1 + X2 + X3 = 0
No sensitivity to any factor, so that b1X1 + b2X2 + b3X3 = 0
Example : 0.9 X X X3 = 0
(assumes zero non factor risk)

14. Arbitrage Portfolio (Cont.)
Let X1 be 0.1.
Then
0.1 + X2 + X3 = 0
X X3 = 0
2 equations, 2 unknowns.
Solving this system gives
X2 = 0.075
X3 =

15. Arbitrage Portfolio (Cont.)
Expected return
X1 ER1 + X2 ER2 + X3 ER3 > 0
Here 15 X X X3 > 0 (= 0.975%)
Arbitrage portfolio is attractive to investors who
Wants higher expected returns
Is not concerned with risk due to factors other than F1

16. Portfolio Stats / Portfolio Weights (Example)
Old Portf.
Arbitr. Portf.
New Portf. X1
1/3
0.1
0.433 X2
0.075
0.408 X3
-0.175
0.158
Properties
ERp
16%
0.975%
16.975% bp
1.9
0.00 sp
11%
small
approx 11%

17. Pricing Effects Stock 1 and 2 Stock 3
Buying stock 1 and 2 will push prices up
Hence expected returns falls
Stock 3
Selling stock 3 will push price down
Hence expected return will increase
Buying/selling stops if all arbitrage possibilities are eliminated.
Linear relationship between expected return and sensitivities
ERi = l0 + l1bi
where bi is the security’s sensitivity to the factor.

18. Interpreting the APT ERi = l0 + l1bi l0 = rf
l1 = pure factor portfolio, p* that has unit sensitivity to the factor
For bi = 1
ERp* = rf + l1
or l1 = ERp* - rf (= factor risk premium)

19. Two Factor Model : Example
Ri = ai + bi1F1 + bi2F2 + ei
i ERi bi1 bi2
Stock 1 15%
Stock 2 21%
Stock 3 12%
Stock 4 8%

20. Multi Factor Models Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + ei
ERi = l0 + l1 bi1 + l2 bi2 + … + lkbik

21. Identifying the Factors
Unanswered questions :
How many factors ?
Identity of factors (i.e. values for lamba)
Possible factors (literature suggests : 3 – 5)
Chen, Roll and Ross (1986)
Growth rate in industrial production
Rate of inflation (both expected and unexpected)
Spread between long-term and short-term interest rates
Spread between low-grade and high-grade bonds

22. Testing the APT

23. Testing the Theory
Proof of any economic theory is how well it describes reality.
Testing the APT is not straight forward
theory specifies a structure for asset pricing
theory does not say anything about the economic or firm characteristics that should affect returns.
Multifactor return-generating process
Ri = ai + S bijFj + ei
APT model can be written as
ERi = rf + S bijlj

24. Testing the Theory (Cont.)
bij : are unique to each security and represent an attribute of the security
Fj : any I affects more than 1 security (if not all).
lj : the extra return required because of a security’s sensitivity to the jth attribute of the security

25. Testing the Theory (Cont.)
Obtaining the bij’s
First method is to specify a set of attributes (firm characteristics) : bij are directly specified
Second method is to estimate the bij’s and then the lj using the equation shown earlier.

26. Principal Component Analysis (PCA)
Technique to reduce the number of variables being studied without losing too much information in the covariance matrix.
Objective : to reduce the dimension from N assets to k factors
Principal components (PC) serve as factors
First PC : (normalised) linear combination of asset returns with maximum variance
Second PC : (normalised) linear combination of asset returns with maximum variance of all combinations orthogonal to the first component

27. Pro and Cons of Principal Component Analysis
Advantage :
Allows for time-varying factor risk premium
Easy to compute
Disadvantage :
interpretation of the principal components, statistical approach

28. Summary APT alternative approach to explain asset pricing
Factor model requiring fewer assumptions than CAPM
Based on concept of arbitrage portfolio
Interpretation : lamba’s are difficult to interpret, no economics about the factors and factor weightings.

29. References
Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 7
Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.5 (The Arbitrage Pricing Theory)

30. END OF LECTURE