1. Capital Asset Pricing and Arbitrage Pricing Theory
Chapter 7
Capital Asset Pricing and Arbitrage Pricing Theory
Describes the financial instruments traded in primary and secondary markets.
Discusses Market indexes. Discusses options and futures.
McGraw-Hill/Irwin
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. 1

2. 7.1 The Capital Asset Pricing Model

3. Capital Asset Pricing Model (CAPM)
Equilibrium model that underlies all modern financial theory
Derived using principles of diversification, but with other simplifying assumptions
Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development
This was Nobel Prize winning work first proposed by William Sharpe.
This model captures all of the risk/return tradeoff we have been discussing up until now.
Derived with very strong limiting assumptions (discuss on next slides)

4. Simplifying Assumptions
Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
No taxes and no transaction costs
Think of a world where individuals are all very similar except their initial wealth and their level of risk aversion. ( I was born poor and chicken)
“Price Takers” means that individuals do not effect prices. (Big assumption! An individual’s behavior does not effect price.)

5. Simplifying Assumptions (cont.)
Information is costless and available to all investors
Investors are rational mean-variance optimizers
Homogeneous expectations
Rational mean-variance optimizers means that all investors attempt to construct efficient frontier portfolios like we discussed in chapter 7.
Homogeneous expectations means that if two investors examine the same investment opportunity they will have identical beliefs about the expected returns, variance of returns and correlations with other investments. Isn’t that a heroic assumption!

6. Resulting Equilibrium Conditions
All investors will hold the same portfolio for risky assets; the “market portfolio”
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
Market price of risk or return per unit of risk depends on the average risk aversion of all market participants
The result will be that all investors will find that the Market portfolio is on the efficient frontier. In fact, it will be the Tangent portfolio. So all investors will choose to hold a portion of their wealth in the risk free and a portion in the risky Market portfolio.

7. Capital Market Line E(r) CML M E(rM) rf s sm
M = The value weighted “Market” Portfolio of all risky assets.
E(r)
CML
E(rM) M
Efficient Frontier rf
The Capital Market Line should look very familiar. It is basically the Capital allocation line with the Market portfolio, M, as the tangent portfolio.
All investors should now choose to invest somewhere along the Capital Market Line.
Remember, any other portfolio on the efficient frontier will be dominated by the tangent portfolio or any other point on the Capital market line. s sm

8. Known Tangency Portfolio of CML
Equilibrium conditions: All investors will hold the __________________________________________
same portfolio for risky assets; the “market portfolio”
Pricing of individual securities is therefore related to the risk that individual securities have when they are included in the market portfolio.
The equilibrium conditions we just described suggest that all investors will hold the same portfolio of risky investments.
Pricing of individual securities is therefore related to the risk that individual securities have when they are included in the market portfolio.

9. Slope and Market Risk Premium
M = rf = E(rM) - rf =
Market portfolio Risk free rate
Excess return on the market portfolio {
= Optimal Market price of risk
= Slope of the CML
For one standard deviation of the Market we expect to get the reward of the risk premium.
Stress that this ratio should be highest for market. →

10. Expected Return and Risk on Individual Securities
The risk premium on individual securities is a function of the individual security’s __________________________________________
What type of individual security risk will matter, systematic or unsystematic risk?
An individual security’s total risk (2i) can be partitioned into systematic and unsystematic risk:
contribution to the risk of THE market portfolio
The equilibrium conditions we just described suggest that all investors will hold the same portfolio of risky investments.
Pricing of individual securities is therefore related to the risk that individual securities have when they are included in the market portfolio.
s2i = bi2 sM2 + s2(ei)
M = market portfolio of all risky securities 10

11. Expected Return and Risk on Individual Securities
Individual security’s contribution to the risk of the market portfolio is a function of the __________ of the stock’s returns with the market portfolio’s returns and is measured by BETA
covariance
With respect to an individual security, systematic risk can be measured by bi =
[COV(ri,rM)] / s2M
The equilibrium conditions we just described suggest that all investors will hold the same portfolio of risky investments.
Pricing of individual securities is therefore related to the risk that individual securities have when they are included in the market portfolio. 11

12. Individual Stocks: Security Market Line
Slope SML = =
Equation of the SML (CAPM)
E(ri) = rf + bi[E(rM) - rf]
(E(rM) – rf )/ βM
price of risk for market
E(r)
SML
E(rM) rf
The security market line now changes the x axis to represent the risk associated with a given individual investment. Beta is the relevant measure of risk of a particular asset with respect to the market portfolio.
Here as Beta, the measure of risk of an asset increases, the expected return increases.
Note that Beta M = 1. ß ß
= 1.0 M

13. Sample Calculations for SML
Equation of the SML
E(ri) = rf + bi[E(rM) - rf]
E(rm) - rf = rf =
bx = 1.25
E(rx) =
by = .6
E(ry) =
.08
.03
Return per unit of systematic risk = 8% & the return due to the TVM = 3%
(.08) = .13 or 13%
Any Beta above one should result in a return above the market return.
What is the return of a stock with a beta of 1?
(0.08) = or 7.8%
If b = 1?
If b = 0?

14. Graph of Sample Calculations
E(r)
SML
Rx=13% ßx
1.25
.08
RM=11%
If the CAPM is correct, only β risk matters in determining the risk premium for a given slope of the SML.
Ry=7.8% ßy .6
Here we can look at the returns with respect to the asset betas in a graphical format. 3% ß
1.0 ßM

15. According to the SML, the E(r) should be _____ 1.25 13% 15%
Suppose a security with a b of ____ is offering an expected return of ____
According to the SML, the E(r) should be _____
1.25
13%
15%
13%
E(r) = (.08) = 13%
Is the security under or overpriced?
Underpriced: It is offering too high of a rate of return for its level of risk
The difference between the return required for the risk level as measured by the CAPM in this case and the actual return is called the stock’s _____ denoted by __
What is the __ in this case?
alpha  
 = +2% Positive  is good, negative  is bad
+  gives the buyer a + abnormal return

16. More on Alpha and Beta E(rM) = 14% βS = 1.5 rf = 5%
Required return = rf + β S [E(rM) – rf] =
If you believe the stock will actually provide a return of ____, what is the implied alpha?
 =
14%
1.5 5%
[14 – 5] = 18.5%
17%
Get Beta of portfolio by the sum of each weight times the beta of the asset.
Get the risk premium of the portfolio by the sum of each weight times the risk premium of the asset.
Or, by the portfolio Beta times the Market risk premium.
17% % = -1.5%
A stock with a negative alpha plots below the SML & gives the buyer a negative abnormal return 16

17. Portfolio Betas βP = Wi βi
If you put half your money in a stock with a beta of ___ and ____ of your money in a stock with a beta of ___and the rest in T-bills, what is the portfolio beta?
Wi βi
1.5
30%
0.9
0.50(1.5) (0.9) (0) = 1.02
All portfolio beta expected return combinations should also fall on the SML.
All (E(ri) – rf) / βi should be the same for all stocks.
Get Beta of portfolio by the sum of each weight times the beta of the asset.
Get the risk premium of the portfolio by the sum of each weight times the risk premium of the asset.
Or, by the portfolio Beta times the Market risk premium. 17

18. Measuring Beta Concept:
Method
We need to estimate the relationship between the security and the “Market” portfolio.
Can calculate the Security Characteristic Line or SCL using historical time series excess returns of the security, and unfortunately, a proxy for the Market portfolio.

19. 7.2 The CAPM and Index Models

20. Security Characteristic Line (SCL)
Excess Returns (i)
Dispersion of the points around the line measures ______________.
The statistic is called e
SCL . . . .
Slope =  . . .
unsystematic risk . . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . . . .
= 
What should  equal? .
The security characteristic line is basically the historical values of the excess returns of an individual asset versus those of the market portfolio. The CAPM model suggests that alpha should equal zero since all returns are based on the constant rf, the stock beta and the market risk. . . . . . . . . . . . .
Ri = a i + ßiRM + ei

21. GM Excess Returns May 00 to April 05
“True”  is between 0.81 and 1.74!
If rf = 5% and rm – rf = 6%, then we would predict GM’s return (rGM) to be
5% (6%) = 12.66%
Note I used FF Rm-Rf from the same time period. Tell them this is not the same as the Excess Returns in the text.
(Note there is a conceptual error here, because I used book’s ER for GM and FF data for RM-Rf, however it works for illustrative purposes)
0.5858
(Adjusted) = 33.18%
8.57%
7-21

22. Adjusted Betas Adjusted β = 2/3 (Calculated β) + 1/3 (1) =
Calculated betas are adjusted to account for the empirical finding that betas different from _ tend to move toward _ over time. 1
A firm with a beta __ will tend to have a ___________________ in the future. A firm with a beta ___ will tend to have a ____________________ in the future.
>1
lower beta (closer to 1)
< 1
higher beta (closer to 1)
Adjusted β = =
2/3 (Calculated β) + 1/3 (1)
2/3 (1.276) + 1/3 (1)
1.184
The adjusted beta forecast is adjusted to account for the empirical finding that betas tend to have a regression toward the mean.
A firm with a high beta (Beta >1) will tend to have a beta closer to 1 in the future, and vice versa. 22

23. 7.3 The CAPM and the Real World

24. Evaluating the CAPM
The CAPM is “false” based on the ____________________________.
validity of its assumptions
The CAPM could still be a useful predictor of expected returns. That is an empirical question.
Huge measurability problems because the market portfolio is unobservable.
Conclusion: As a theory the CAPM is untestable.

25. Evaluating the CAPM practicality
However, the __________ of the CAPM is testable.
Betas are ___________ at predicting returns as other measurable factors may be.
More advanced versions of the CAPM that do a better job at ___________________________ are useful at predicting stock returns.
not as useful
estimating the market portfolio
Still widely used and well understood.

26. Evaluating the CAPM
The _________ we learn from the CAPM are still entirely valid.
principles
Investors should diversify.
Systematic risk is the risk that matters.
A well diversified risky portfolio can be suitable for a wide range of investors.
The risky portfolio would have to be adjusted for tax and liquidity differences.
Differences in risk tolerances can be handled by changing the asset allocation decisions in the complete portfolio.
Diversification predates CAPM anyway, but it is implied by the CAPM.
Even if the CAPM is “false,” the markets can still be “efficient.”

27. 7.4 Multifactor Models and the CAPM

28. Fama-French (FF) 3 Factor Model
Fama and French noted that stocks of ____________ and stocks of firms with a _________________ have had higher stock returns than predicted by single factor models.
smaller firms
high book to market
Problem: Empirical model without a theory
Will the variables continue to have predictive power?

29. Fama-French (FF) 3 factor Model
FF proposed a 3 factor model of stock returns as follows:
rM – rf = Market index excess return
Ratio of ______________________________________ measured with a variable called ____:
HML: High minus low or difference in returns between firms with a high versus a low book to market ratio.
_______________ measured by the ____ variable
SMB: Small minus big or the difference in returns between small and large firms.
book value of equity to market value of equity
HML
Firm size variable
SMB

30. Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM

31. Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
If rf = 5%, rm – rf = 6%, & return on HML portfolio will be 5%, then we would predict GM’s return (rGM) to be
5% % (6%) (5%) = 13.06%
Note with negative alpha you don’t want to recommend this stock.
* * *
0.6454
(Adjusted) = 38.52%
8.22%
Compared to single factor model:
Better Adjusted R2; lower βM higher E(r), but negative alpha.

32. 7.5 Factor Models and the Arbitrage Pricing Theory

33. Arbitrage Pricing Theory (APT)
Zero investment:
Efficient markets:
Arises if an investor can construct a zero investment portfolio with a sure profit
Since no net investment outlay is required, an investor can create arbitrarily large positions to secure large levels of profit
The arbitrage pricing theory was created by Stephen Ross and it is similar to the CAPM but it requires far fewer limiting assumptions.
Arbitrage example Apples cost 1 per pound
Pie crust cost 1 dollar.
Pie cost 3 dollars and a pie is simply the apples thrown into the crust.
With efficient markets, profitable arbitrage opportunities will quickly disappear

34. Simple Arbitrage Example
If all of these stocks cost ___ today are there any arbitrage opportunities? $8
Portfolio Cost Final Outcome C
(A+B) /
Short
Buy
The A&B combo dominates portfolio C, but costs the same.
Arbitrage opportunity: Buy A&B combo and short C, $0 net investment, sure gain of $1
The opportunity should not persist in competitive capital markets.
rf = 1/8 = 12.5%!
If we buy a portfolio of half of A and half of B. This portfolio will strictly dominate C. We would want to do an infinite arbitrage.
Buy as much as possible of A+B/2 and sell the same amount of C short. The arbitrage opportunity would disappear when the prices adjusted so no arbitrage was remaining.

35. Arbitrage Pricing Example
Suppose Rf = ___ and a well diversified portfolio P has a beta of ___ and an alpha of ___ when regressed against a systematic factor S. Another well diversified portfolio Q has a beta of ___ and an alpha of ___.
If we construct a portfolio of P and Q with the following weights:
What should αp = ___ 6%
1.3 2%
0.9 1%
WP = and WQ = ;
Then βp =
αp =
-2.25
3.25
Note: Σ W = 1
(-2.25 x 1.3) + (3.25 x 0.9) = 0
(-2.25 x 2%) + (3.25 x 1%) = %
αp = -1.25% means an investor will earn rf – 1.25% or 4.75% on portfolio PQ.
In theory one could short this portfolio and pay 4.75%, and invest in the riskless asset and earn 6%, netting the 1.25% difference.
Arbitrage should eliminate the negative portfolio alpha quickly.
Formulas for the weights are in the text. 35

36. Arbitrage Pricing Model
The result: For a well diversified portfolio
Rp = βpRS (Excess returns)
(rp,i – rf) = βp(rS,i – rf)
and for an individual security
(rp,i – rf) = βp(rS,i – rf) + ei
Advantage of the APT over the CAPM:
RS is the excess return on a portfolio with a beta of 1 relative to systematic factor “S”
No particular role for the “Market Portfolio,” which can’t be measured anyway
Easily extended to multiple systematic factors, for example
(rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + ei
RS is the excess return on a portfolio with a beta of 1 relative to systematic factor “S” AND has a beta of zero to all other systematic factors. These are called factor portfolios. 36

37. APT and CAPM Cont.
APT applies to well diversified portfolios but not necessarily to all individual stocks
APT employs fewer restrictive assumptions
APT does NOT specify the systematic factors
Chen, Roll and Ross (1986) suggest:
 Industrial production
 Yield curve
 Default spreads
 Inflation
In order to actually use this you would need to find 4 well diversified portfolios:
Portfolio 1 would have to have a beta of 1 wrt Δ Industrial Production and betas of zero on the other 3 factors.
Portfolio 2 would have to have a beta of 1 wrt Δ Yield curve and betas of zeros wrt the other 3 factors. …
(rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + βp,4(r4,i – rf) + ei

38. Selected Problems

39. Problem 1 a. E(rX) = 5% + 0.8(14% – 5%) = 12.2% X =
CAPM: E(ri) = rf + β(E(rM)-rf) a.
CAPM: E(ri) = 5% + β(14% -5%)
E(rX) =
X =
E(rY) =
Y =
5% + 0.8(14% – 5%) = 12.2%
14% – 12.2% = 1.8%
5% + 1.5(14% – 5%) = 18.5%
17% – 18.5% = –1.5% 39

40. Problem 1
X = 1.8%
Y = -1.5%
Which stock?
Well diversified: Relevant Risk Measure? Best Choice?
Which stock?
Held alone: Relevant Risk Measure? Best Choice? 
β: CAPM Model
Calculate Sharpe ratios
Stock X with the positive alpha 40

41. Problem 1 (continued) Sharpe Ratios Held Alone: Sharpe Ratio X =
Sharpe Ratio Y =
Sharpe Ratio Index =
(0.14 – 0.05)/0.36 = 0.25
(0.17 – 0.05)/0.25 = 0.48
Better
(0.14 – 0.05)/0.15 = 0.60 41

42. Problem 2 E(rP) = rf + b[E(rM) – rf] 20% = 5% + b(15% – 5%)42

43. Problem 3 E(rP) = rf + b[E(rM) – rf] E(rp) when double the beta:
If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity:
Price = Dividend / E(r)
$40 = Dividend / 0.13
At the new discount rate of 19%, the stock would be worth:
$5.20 / 0.19 = $27.37
13% = 7% + β(8%) or
β = 0.75
E(rP) = 7% + 1.5(8%) or
E(rP) = 19%
so the Dividend = $40 x 0.13 = $5.20
The increase in stock risk has reduced the value of the stock by
($ $40) / $40 = %.

44. Problem 4 False. b = 0 implies E(r) = rf , not zero.
Depends on what one means by ‘volatility.’ If one means the  then this statement is false. Investors require a risk premium for bearing systematic (i.e., market or undiversifiable) risk.
False. You should invest 0.75 of your portfolio in the market portfolio, which has β = 1, and the remainder in T-bills. Then:
bP = (0.75 x 1) + (0.25 x 0) = 0.75 44

45. Problems 5 & 6 9.
10.
Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower.
Possible. Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.
If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk as measured by beta, rather than the standard deviation, which includes nonsystematic risk. Thus, Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.

46. Problem 7 Calculate Sharpe ratios for both portfolios:
Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the portfolio with the highest return per unit of risk. 46

47. Problem 8 8. Need to calculate Sharpe ratios?
Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard deviation with a higher expected return. 47

48. Problem 9 Given the data, the SML is: E(r) = 10% + b(18% – 10%)
A portfolio with beta of 1.5 should have an expected return of:
E(r) = 10% + 1.5(18% – 10%) = 22%
Not Possible: The expected return for Portfolio A is 16% so that Portfolio A plots below the SML (i.e., has an  = –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM. 48

49. Problem 10 E(r) = 10% + b(18% – 10%)
The SML is the same as in the prior problem. Here, the required expected return for Portfolio A is:
10% + (0.9  8%) = 17.2%
Not Possible: The required return is higher than 16%. Portfolio A is overpriced, with  = –1.2%. 49

50. Problem 11
Sharpe A =
Sharpe M =
Possible: Portfolio A's ratio of risk premium to standard deviation is less attractive than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio.
(16% - 10%) / 22% = .27
(18% - 10%) / 24% = .33 50

51. Problem 12
Since the stock's beta is equal to 1.0, its expected rate of return should be equal to ______________________.
E(r) =
0.18 =
the market return, or 18%
or P1 = $109 51

52. Problem 13 r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0
We can’t tell which adviser did the better job selecting stocks because we can’t calculate either the alpha or the return per unit of risk.
CAPM: ri = 6% + β(14%-6%)
r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0, rf = 6%; rM = 14%
a1 =
a2 =
The second adviser did the better job selecting stocks (bigger + alpha)
19% –
16% –
[6% + 1.5(14% – 6%)] =
19% – 18% = 1%
[6% + 1.0(14% – 6%)] =
16% – 14% = 2% 52

53. Problem 14
McKay should borrow funds and invest those funds proportionally in Murray’s existing portfolio (i.e., buy more risky assets on margin). In addition to increased expected return, the alternative portfolio on the capital market line (CML) will also have increased variability (risk), which is caused by the higher proportion of risky assets in the total portfolio.
McKay should substitute low beta stocks for high beta stocks in order to reduce the overall beta of York’s portfolio. Because York does not permit borrowing or lending, McKay cannot reduce risk by selling equities and using the proceeds to buy risk free assets (i.e., by lending part of the portfolio). 54

54. Problem 15
Since the beta for Portfolio F is zero, the expected return for Portfolio F equals the risk-free rate.
For Portfolio A, the ratio of risk premium to beta is:
The ratio for Portfolio E is:
(10% - 4%)/1 = 6%
(9% - 4%)/(2/3) = 7.5%
Create Portfolio P by buying Portfolio E and shorting F in the proportions to give βp = βA = 1, the same beta as A. βp =Wi βi
1 = WE(βE) + (1-WE)(βF);
E(rp) =
WE = 1 / (2/3) or
WE = 1.5 and
WF = (1-WE) = -.5
1.5(9) (4) = 11.5%,
Buying Portfolio P and shorting A creates an arbitrage opportunity since both have β = 1
p,-A =
11.5% - 10% = 1.5% 55

55. Problem 16 E(IP) = 4% & E(IR) = 6%; E(rstock) = 14%
βIP = 1.0 & βIR = 0.4
Actual IP = 5%, so unexpected ΔIP = 1%
Actual IR = 7%, so unexpected ΔIR = 1%
The revised estimate of the expected rate of return of the stock would be the old estimate plus the sum of the unexpected changes in the factors times the sensitivity coefficients, as follows:
Revised estimate =
Note IP is defined as the growth rate of Industrial Production and of course the inflation rate is a change variable
E(rstock) + Δ due to unexpected Δ Factors
14% +
[(1  1%) + (0.4  1%)] =
15.4%