1. Index Models
Chapter 8

2. Advantages of the Single Index Model
Reduces the number of inputs for diversification.
Can be used to estimate covariance estimates for MPT-type analysis
Easier for security analysts to specialize.
Provides “easy-to-understand” expected return estimates if factor is the VW market excess return.
CAPM based

3. Single Factor Model ri = E(Ri) + ßiF + e Where
E(Ri) = return to the asset that is not due to risk (i.e., the risk-free rate component).
ßi = index of a securities’ 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.

4. a Single Index Model (ri - rf) = i + ßi(rm - rf) + ei Risk Prem
Market Risk Prem
or Index Risk Prem a
= the stock’s expected return if the
market’s excess return is zero: i
(rm - rf) = 0
ßi(rm - rf) = the component of return due to
movements in the market index
ei = firm specific component, not due to market
movements

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

6. Example: advantages of using factor model to estimate covariances

7. Study: Article on Modern Portfolio Theory and how well it really works
See class web page, handouts, chapter 8, “Article on Modern Portfolio Theory and how well it really works.”
Motivation
How good of a job do single factor and multifactor models do at estimating covariances in a MPT setting?
In other words, can we form a higher Sharpe ratio portfolio using factor model estimated covariances instead of a covariance matrix based off historical time-series of returns?

9. Study: Article on Modern Portfolio Theory and how well it really works
Results
The SR of the minimum variance portfolio is:
0.69 using a single factor model (with VW as the factor)
0.64 using historical return data
0.62 using a nine factor model
Conclusions
The single factor model does a better job than just using historical data.
No improvement in the SR using a much larger number of factors.

10. Security Characteristic Line
The slope = beta
Excess Returns (i)
SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . . . . . . . . . . .
Ri =  i + ßiRm + ei

11. Example
Excess
GM Ret.
Excess
Mkt. Ret.
Jan.
Feb. .
Dec
Mean
Std Dev
5.41
-3.44 .
2.43
-.60
4.97
7.24
.93 .
3.90
1.75
3.32

13.   Regression Results rGM - rf = + ß(rm - rf) ß Estimated coefficient
Std error of estimate
Variance of residuals =
Std dev of residuals = 3.550
R-SQR = = (correlation coefficient)2
-2.590
(1.547)
1.1357
(0.309)

15. 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

16. The Components of Total Risk
i2 = i2 m2 + 2(ei)
The first term is market risk, and the second term is firm specific risk
where;
i2 = total variance (or total risk)
i2 m2 = systematic variance
2(ei) = unsystematic variance

17. Simple: Just take the variance of the security characteristic line.
I know what you are thinking! You are wondering how they derived the expression for total risk, right?
Simple: Just take the variance of the security characteristic line.

18. Examining Percentage of Variance
Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk/Total Risk = 2 = R-SQR
ßi2  m2 / 2 = 2
i2 m2 /[i2 m2 + 2(ei)] = 2
Lets do some examples!

19. Index Model and Diversification

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

21. Oh no! Some more examples
Her stock could be a real “steal” right now…hahaha

22. Industry Prediction of Beta
Merrill Lynch Example
Use returns not risk premiums
a has a different interpretation
a = a + rf (1-b)
Forecasting beta as a function of past beta
Forecasting beta as a function of firm size, growth, leverage etc.

25. Up next: the case for multi-factor models and a study
Should the “correct” asset pricing model have multiple factors?
Study on how well multifactor models predict the cross-section of future stock returns.

33. Example
See external class web page, Excel handouts, chapter 8, “The use of macroeconomic variables to predict the S&P500”

34. Study: Economic Forces and the Stock Market
See class web page, handouts, chapter 8, “Economic Forces and the Stock Market”
Motivation
Are betas from other factors (besides the market beta) priced in the cross-section of expected returns?
Method
Perform “two-pass” regressions. First, estimate factor loadings from time series regressions to obtain firm betas for multiple factors, then use cross-sectional regressions to see if the factors are priced.

35. Study: Economic Forces and the Stock Market
Macroeconomic variables used in the study:
MP = monthly growth in industrial production
DEI = change in expected inflation
UI = unexpected inflation
UPR = risk premium (junk spread)
UTS = term structure (slope of the yield curve)

36. Study: Economic Forces and the Stock Market

37. Study: Economic Forces and the Stock Market
Conclusions:
MP (industrial productions), UI (inflation), UPR (junk spread) appear to be priced.
CAPM beta is not priced.

38. Study: Multifactor explanations of asset pricing anomalies
See class web page, handouts, chapter 8, “Multifactor explanations of asset pricing anomalies ”
Motivation
Can a multifactor model, using returns from well diversified portfolios as the factors, explain most or all of the common return anomalies?
Aside: size, E/P, C/P, B/M, lagged returns, and other firm-specific variables (which all appear to predict stock returns) are called anomalies because the CAPM cannot explain their ability to forecast future stock returns.

39. Study: Multifactor explanations of asset pricing anomalies
The three-factor model
Ri-rf = ai + bi(Rm-rf) + si(SMB) + hi(HML) + ei
Where
Ri = return to stock i (or portfolio)
SMB = small minus big; the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks.
HML= high minus low; the difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks.
bi, si, and hi = factor loadings
See also on external web page:
Excel Handouts, chapter 8, “Example performance calculations” for an example of estimating a 3-factor model alpha on IBM monthly stock returns.
Handouts, chapter 4, “Factor models used in performance calculations.”

40. Study: Multifactor explanations of asset pricing anomalies
Testing methodology:
Create a time-series of returns to a given anomaly.
Example: cash-to-price. Sort all firms each year into deciles on lagged annual c/p. Hold portfolios for one year, then rebalance.
Excess return spread:
Decile 1 (low c/p) = 0.43%/month
Decile 10 (high c/p) = 1.20%/month
Spread = 0.77% month.
Can time series variations in the market return, SMB, and HML explain this spread?
Estimate the time series model:
R(cp10-cp1)-rf = ai + bi(Rm-rf) + si(SMB) + hi(HML) + ei
Null: a=0.
If we reject the null, then the 3-factor model can’t explain or “price” the anomaly.

45. Study: Multifactor explanations of asset pricing anomalies
Conclusions
The 3-factor model can explain most accounting ratio anomalies.
The 3-factor model can’t explain momentum.