0. Fin 525 Week 8
Asset Pricing Models

1. Financial Analysis Is More Art Than Science
There are no “Generally Accepted Financial Principles”
Important (and often arbitrary) choices
Frequency of data (daily, weekly, monthly, quarterly, annual)
When to start and end time periods
How far back in time to go
How many data points to use
You are stuck using the data that is available to you
Professor Ross Miller • Fall 2007

2. The Usefulness of Historical Data
Not very useful for computing the absolute level of returns unless the data goes back through several business cycles (20 or more years)
Do not expect GMCR to go up by over 80% in price every year
Do expect the S&P 500 to generate a total return of about 10% (or at least 5% above the risk-free rate) every year (at least until baby boomers begin to retire in large numbers)
Historical data is more useful for determining volatilities, correlations, and betas (which are a kind of correlation)
Professor Ross Miller • Fall 2007

3. The Standard Deviation of Returns
This is also known as the stock’s volatility; however, there are other methods of measuring volatility
Volatility is a standard measure of the stock’s risk—higher volatility means more risk
Annualizing volatility is somewhat tricky:
For weekly returns, we multiply by 52 to annualize
Professor Ross Miller • Fall 2007

4. Some Numbers for the 2 Years from September 2005 to September 2007
Professor Ross Miller • Fall 2007

5. Correlation
A function that connects two parallel series of numbers, such as stock returns (Excel uses CORREL)
Varies from -1 to 1
1 is perfectly correlated
0 is uncorrelated (as if the two series were chosen independently of one another
-1 is perfectly negatively correlated (when one series zigs, the other series zags)
Professor Ross Miller • Fall 2007

6. Correlation Among the Stocks in the Project (From the “Correlations” Worksheet)
Professor Ross Miller • Fall 2007

7. Two Ways to Reduce Portfolio Volatility
Add more assets
Even though most assets are to some degree correlated with one another though common factors, as long as that correlation is less than 1, added assets tends to reduce portfolio volatility
Sell the appropriate assets short as a hedge
Related stocks—for example, those in the same industry
Professor Ross Miller • Fall 2007

8. A Simple Experiment
Go to the Portfolio Volatility What-If worksheet in Fin525Fall2007StockProject.xls spreadsheet
Add 1,000 shares of any single stock to the 10,000 shares of GMCR
Notice that the portfolio volatility (standard deviation) goes down in every case
Professor Ross Miller • Fall 2007

9. Why Does This Work?
While the additional stocks may amplify risk factors they have in common with GMCR, their “specific risks” promote diversity, which lowers portfolio volatility
Because this risk is easy to diversify away, the market does not reward anyone for bearing it
The observation that only holding undiversifiable risk can increase one’s expected returns is at the heart of the Capital Asset Pricing Model (CAPM)
Professor Ross Miller • Fall 2007

10. The Financial World Consists of “Risk Factors”
The standard form of CAPM assumes that there is a single risk factor that is commonly represented by the S&P 500 Index (SPY works just fine as a substitute)
More complex factor models include factors for things like growth vs. value and size (large cap vs. small cap)
GMCR contains substantial risk not captured by any factor
Professor Ross Miller • Fall 2007

11. How Do We Find Common Risk Factors?
Regression analysis and related statistical tools (discriminant analysis, factor analysis, neural networks, etc.)
These tools perform what is known as variance decomposition
The variance of the stock or portfolio that we are interested in is decomposed into two parts
Specific (or idiosyncratic) variance/risk/volatility
Market (or systematic) variance/risk/volatility
Professor Ross Miller • Fall 2007

12. The CAPM Equation (again)
Expected return = Risk-free return + Premium for risk
Where
E(ri) is the expected return for stock i rf is the risk-free rate of return i is the beta for stock i E(rM) is the expected market rate of return
Professor Ross Miller • Fall 2007

13. CAPM as a Regression Equation
The independent variable (x in most textbooks) is the excess return on the market index
The dependent variable (y in most textbooks) is the excess return on the asset/portfolio
Beta (the slope of the regression line) is the amount of market risk in the asset/portfolio
Alpha (the intercept of the regression line) is the risk-adjusted performance of the asset/portfolio
Professor Ross Miller • Fall 2007

14. Beta, Beta, Oh Which Beta to Use
Betas are available in abundance on the Internet
For many stocks, different sources have very different value of beta depending on how they compute it
Frequency of data used
Amount of history
“Adjustments” to beta’s value, some valid, some not
How you use beta can effect which one to use
In general, if you know what you are doing, you are better off computing your own betas
Professor Ross Miller • Fall 2007

15. What About the Risk-Free Rate and the Market Premium?
The yield of 3-month T-bills makes a good risk-free rate
You can find it here
At the close of trading on 10/11/2007 it was 4.11%
A reasonable number for the risk premium, E(rM)-rf , under current conditions is 6%
For a weekly history of risk-free rates, the Fed’s Eurodollar time series used to work well, more recently, their 3-month T-bill rate works better
Professor Ross Miller • Fall 2007

16. The CAPM Regression in Graphic Form
Asset Excess Returns . . . . . . . . . . . . . . . . . . .
Market Excess Returns . . . . . . . . . . . . . . . . . . . . . . .
Professor Ross Miller • Fall 2007

17. A Useful Equation from BKM (page 260)
i can be anything (stock or portfolio) and this equation separates the variance of that stock (or portfolio) into a systematic piece and a specific piece
Linear regression or handy functions within Excel can be used to find beta
Professor Ross Miller • Fall 2007

18. Why is This Equation Useful?
The amount of variance captured by the market is known as R2 (often written R-squared)
You can (in a statistical sense) get rid of all systematic risk by selling short the market index in an amount indicated by beta
You are left with the specific risk, which you can then do what you want with (leverage, diversify, etc.)
Professor Ross Miller • Fall 2007

19. How Things Tie Together
When you have a single market factor (BKM refer to it as an “index”), then the R in R2 is the same as the correlation between the portfolio/asset and the “market” (usually written as r, just to confuse you)
You do not have to use any form of regression analysis to get beta in Excel, you can use the CORREL function on excess returns to get r, and then use the formula on the previous slide to solve for beta:
Professor Ross Miller • Fall 2007

20. Regression Analysis in Excel
In Fin525Fall2007StockProject.xls look at the worksheet called “GMCR Regression for CAPM beta”
Excel has built-in function for single variable regression
Excel has an Analysis ToolPak for doing all kinds of regression (single and multiple variable)
One can also “hard-wire” regressions into Excel using the matrix math and summation functions
Professor Ross Miller • Fall 2007

21. Regressing GMCR Excess Weekly Returns Against SPY Excess Weekly Returns
Professor Ross Miller • Fall 2007

22. The Regression Equation
Here is the regression equation for GMCR relative to SPY:
So, alpha = or 86 b.p. (per week) and beta = 1.22
Professor Ross Miller • Fall 2007

23. Things to Make You Happy
Beta and R2 do not depend on the frequency (daily, weekly, monthly, or whatever) of the data you use
All that matters is that the same time period is used consistently
Professor Ross Miller • Fall 2007

24. Things to Make You Unhappy
Alpha, sigma, and anything involving returns does depend on the frequency of the data used in the regression
To convert a weekly alpha into an annual alpha (approximately), use the same compounding conversion that we used to returns earlier
Professor Ross Miller • Fall 2007

25. One Way to Minimize Portfolio Volatility
Excel’s Analysis ToolPak comes with a Solver tool.
Fin525Fall2007StockProject.xls comes with the Solver already set up to minimize the portfolio volatility over the historical 104-week time period
Your problem is how to adapt this spreadsheet to come up with a forward-looking portfolio
Note that the prices of the stocks can be quite different now than they were two years ago
Professor Ross Miller • Fall 2007

26. A Final Word to Preserve Your Professor’s Sanity
There is no “good” or “bad” level of volatility— just because a certain number could have been achieved looking backward does not mean that it can be achieved going forwards
Certain of my students in the past have developed an unhealthy obsession with certain levels of volatility, please do not repeat their mistake
Professor Ross Miller • Fall 2007

27. For Week 9
Look at Fin525Fall2007StockProject.xls and figure out how to modify it to come up with a hedging portfolio
You can also look at Fin525Spring2007GMCRTrack.xls to see what happened this past spring
Professor Ross Miller • Fall 2007