Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin
Chapter 7
Capital Asset Pricing and Arbitrage
Pricing Theory
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
7-2
Capital Asset Pricing Model (CAPM)
•
•
•
Equilibrium model that underlies all modern financial theory: What should be the “appropriate” level of return commensurate with a given amount of “risk” for an individual security
Derived using principles of diversification, with other simplifying assumptions
7-3
Simplifying Assumptions•
•
•
•
Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
No taxes and no transaction costs
7-4
Simplifying Assumptions (cont.)
•
•
•
Information is costless and available to all investors
Investors are rational mean-variance optimizers
Homogeneous expectations
7-5
Resulting Equilibrium Conditions
•
All investors will hold the same portfolio for risky assets; the “market portfolio” and CML
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
Equilibrium security prices or “risk-appropriate” level of return is determined according to CAPM
•
•
7-6
E(r)
rf
E(rM)M
CML
m
Capital Market Line
M = The value weighted “Market” Portfolio of all risky assets. Allinvestors will hold the same portfoliofor risky securities
Efficient Frontier
7-7
M = rf =
E(rM) - rf =
Slope and Market Risk Premium
{Excess return on the
market portfolio
MME(rM) - rf = Optimal Market price of risk
= Slope of the CML
Market portfolioRisk free rate
E(rE(r))
E(rE(rMM))
rrff
MMCMLCML
mm
Capital Market Line
M = The value weighted M = The value weighted ““MarketMarket””Portfolio of all risky assets.Portfolio of all risky assets.
→
7-8
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:
2i =i
2 M2 + 2(ei)
M = market portfolio of all risky securities
contribution to the risk of THE market portfolio
7-9
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
With respect to an individual security, systematic risk can be measured byi= [COV(ri,rM)] / 2
M
covariance
7-10
E(r)E(r)
E(rE(rMM))
rrff
SMLSML
MMßßßß = 1.0= 1.0
Individual Stocks: Security Market LineSlope SML =
=
Equation of the SML (=CAPM)
E(ri) = rf + [E(rM) - rf]
[E(rM) – rf ]
price of risk for market
7-11
Sample Calculations for SML
E(rm) - rf = rf =
x = 1.25
E(rx) =
y = 0.6
E(ry) =
Equation of the SML
E(ri) = rf + [E(rM) - rf]i
0.03 + (0.08)*1.25 = 0.13 or 13%
0.03 + (0.08)*0.6 = 0.078 or 7.8%
If = 1?
If = 0?
0.08 0.03
Return per unit of systematic risk = 8% & the risk free return = 3%
7-12
E(r)E(r)SMLSML
ßß
ßßMM
1.01.0
RRMM=11%=11%
3%3%
RRxx=13%=13%
ßßxx
1.251.25
RRyy=7.8%=7.8%
ßßyy
0.60.6
0.080.08
Graph of Sample Calculations
If the CAPM is correct, only β risk matters in determining the risk premium for a given slope of the SML.
7-13
E(rE(r))
15%15%
SMLSML
ßß1.01.0
RRmm=11%=11%
rrff=3%=3%
1.251.25
Disequilibrium Example
Suppose a security with a of ____ is offering an expected return of ____
According to the SML, the E(r) should be _____
1.2515%
13%
Underpriced: It is offering a higher 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?
E(r) = 0.03 + 1.25(.08) = 13%
Is the security under or overpriced?
= +2% Positive is good, negative is bad
+ gives the buyer a + abnormal return
alpha
13%
7-14
More on Alpha and Beta
E(rM) =
βS =
rf =
Required return = rf + [E(rM) – rf] βS
=
If you project that the stock will actually provide a return of ____, what is the implied alpha?
=
5 + [14 – 5]*1.5 = 18.5%
17%
17% - 18.5% = -1.5%
14%
1.5
5%
A stock with a negative alpha plots below the SML & gives the buyer a negative abnormal return
7-15
Portfolio Betas
βP =
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?
βP = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02
1.530% 0.9
Wi βi
• All portfolio beta expected return combinations should also fall on the SML.
7-16
Measuring Beta
• Concept:
• Method
Can calculate the Security Characteristic Line or SCL using historical time series excess returns of the security, and a proxy for the Market portfolio (DJI, S&P, etc).
We need to estimate the relationship between the security and the “Market” portfolio.
7-17
Security Characteristic Line (SCL)Excess Returns (i)
..
..
........
.. ..
.. ....
.. ....
.. ..
.. ....
......
.. ..
.. ....
.. ....
.. ..
.. ....
.. ....
.. ..
..
.. ...... .... .... ..
Excess returnson market index
Ri = i + ßiRM + ei
Slope =
= What should equal?
SCLDispersion of the points around the line measures ______________.unsystematic risk
7-18
GM Excess Returns May 00 to April 05
0.5858(Adjusted) = 33.18%
8.57%
-0.0143 1.276
0.01108 0.2318
“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%)*1.276 = 12.66%
7-18
7-19
Adjusted Betas
Adjusted β =
=
=
2/3 (Calculated β) + 1/3 (1)
2/3 (1.276) + 1/3 (1)
1.184
Calculated betas are adjusted to account for the empirical finding that betas different from _ tend to move toward _ over time.
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 1
lower beta (closer to 1)>1< 1
higher beta (closer to 1)
7-21
Evaluating the CAPM
• The CAPM is “false” based on the ____________________________.
•
–
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.
validity of its assumptions
7-22
Evaluating the CAPM
• 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.
practicality
not as useful
estimating the market portfolio
Still widely used and well understood.
7-23
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.
7-25
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.
–
Problem: Empirical model without a theory
high book to marketsmaller firms
7-26
Fama-French (FF) 3 factor ModelFF 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 equityHML
Firm size variable SMB
7-28
Fama-French (FF) 3 factor ModelrGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
0.6454
(Adjusted) = 38.52%
8.22%
-0.0262* 1.2029* 0.6923* 0.3646
0.0116 0.2411 0.2749 0.3327
Compared to single factor model:
Better Adjusted R2; lower βM higher E(r), but negative alpha.
If rf = 5%, rm – rf = 6%, & return on HML portfolio will be 5%, then we would predict GM’s return (rGM) to be
5% + -2.62% + 1.2029(6%) + 0.6923(5%) = 13.06%
7-29
Arbitrage Pricing Theory (APT)• Arbitrage:
• 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
With efficient markets, profitable arbitrage opportunities will quickly disappear
7-30
Simple Arbitrage Example
Portfolio Cost Final Outcome
C 8 9(A+B) / 28 10
• •
•
If all of these stocks cost ___ today are there any arbitrage opportunities?
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.
$8
7-31
Arbitrage Pricing ExampleSuppose Rf = ___ and a well diversified portfolio P has a beta of ___ and an alpha of ___. 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%?
6% 1.31%
0.9 2%
WP = and WQ = ;
Then βp =
αp = 1.25% means an investor will earn rf 1.25% on portfolio PQ.
In theory one could short this portfolio and pay 1.25%, and invest in the riskless asset and earn 6%, netting the 4.75% difference.
Arbitrage should eliminate the portfolio alpha quickly.
(-2.25 x 1.3) + (3.25 x 0.9) = 0
(-2.25 x 1%) + (3.25 x 2%) = 1.25%
WP = - β Q / (β P - β Q)
WQ = β P / (β P - β Q)
WP = - β Q / (β P - β Q)
WQ = β P / (β P - β Q)
Note: Σ W = 1
-2.25 3.25
7-32
Arbitrage Pricing ModelThe result: For a well diversified portfolio
Rp = βpRS (Excess returns)
(rp – rf) = βp(rS – rf)
and for an individual security
(ri – rf) = βi(rS – rf) + ei
Advantage of the APT over the CAPM:•
• –
No particular role for the “Market Portfolio,” which can’t be measured anyway
Easily extended to multiple systematic factors, for example
=> (ri – 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”
7-33
APT and CAPM Cont.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
7-35
Problem 1
– 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%
a. CAPM: E(ri) = 5% + β(14% -5%)
CAPM: E(ri) = rf + β(E(rM)-rf)
7-36
Problem 1
b. Which stock?
i. Well diversified:Relevant Risk Measure?
Best Choice?
b. Which stock?
ii. Held alone:Relevant Risk Measure?
Best Choice?β: CAPM Model
Stock X with the positive alpha
Calculate Sharpe ratios
X = 1.8%
Y = -1.5%
7-37
Problem 1
b. (continued) Sharpe Ratios
ii. 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
(0.14 – 0.05)/0.15 = 0.60
Better
σ
rE(r)Ratio Sharpe f
7-39
Problem 3
E(rP) = rf + [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
7-40
Problem 4
a.
a.
b.
•
False. = 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:
P = (0.75 x 1) + (0.25 x 0) = 0.75
7-41
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.
7-42
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.
0.5.12
.10.16Sharpe A
0.33
.24
.10.18SharpeM
σ
rE(r)Ratio Sharpe f
7-43
Problem 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.
8.
7-44
Problem 9
9.
Given the data, the SML is:
E(r) = 10% + (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.
7-45
Problem 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%.
E(r) = 10% + (18% – 10%)
7-46
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
7-47
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 =
0
011
P
PPD
100
100P9 1 or P1 = $109
)r)β(E(rrP
PPD:mEquilibriu In fMf
0
011
the market return, or 18%
7-48
Problem 13
a.
b.
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 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.
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 1.0, rf = 6%; rM = 14%
1 =
2 =
The second adviser did the better job selecting stocks (bigger + alpha)
19% –
16% –
19% – 18% = 1%
16% – 14% = 2%
CAPM: ri = 6% + β(14%-6%)
[6% + 1.5(14% – 6%)] =
[6% + 1.0(14% – 6%)] =
7-50
Problem 14
a.
b.
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).
7-51
Problem 15
i.
ii.
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:
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) or1.5(9) + -0.5(4) = 11.5%,
11.5% - 10% = 1.5%
(10% - 4%)/1 = 6%
(9% - 4%)/(2/3) = 7.5%
WE = 1.5 and WF = (1-WE) = -.5
p,-A =
Buying Portfolio P and shorting A creates an arbitrage opportunity since both have β = 1
7-52
Problem 16
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 = 14% +
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%
E(rstock) + Δ due to unexpected Δ Factors[(1 1%) + (0.4 1%)] =15.4%