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Transcript of Capital Asset Pricing and Arbitrage Pricing Theory Department of Banking and Finance SPRING 200 7 -0...
Capital Asset Pricing and Arbitrage Pricing Theory
Department of Banking and Finance
SPRING 2007-08
by
Asst. Prof. Sami Fethi
2 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM)
The asset pricing models aim to use the concepts of portfolio valuation and market equilibrium in order to determine the market price for risk and appropriate measure of risk for a single asset.
Capital Asset Pricing Model (CAPM) has an observation that the returns on a financial asset increase with the risk. CAPM concerns two types of risk namely unsystematic and systematic risks. The central principle of the CAPM is that, systematic risk, as measured by beta, is the only factor affecting the level of return.
3 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM)
The Capital Asset Pricing Model (CAPM) was developed independently by Sharpe (1964), Lintner (1965) and Mossin (1966) as a financial model of the relation of risk to expected return for the practical world of finance.
CAPM is originally depending on the mean variance theory which was demonstrated by Markowitz’s portfolio selection model (1952) aiming higher average returns with lower risk.
4 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM)
Equilibrium model that underlies all modern financial theory
Derived using principles of diversification with simplified assumptions
Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development
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Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM) From the point that most of the investors think that the
variance or standard deviation of their portfolio’s return will enable them to quantify the risk, portfolio selection model is used in order to find the efficient portfolios and secondly to generate an equation that relates the risk of an asset to its expected return.
The reason for this is that portfolios are expected to have maximum return given the variance of future returns. Therefore, Mean Variance Analysis is one of the tools for achieving higher average returns with lower risk and as a second tool Capital Asset Pricing Model’s main parameters are depending on mean and variance of returns.
6 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM)
Moreover, CAPM requires that in the equilibrium the market portfolio must be an efficient portfolio. One way to establish its efficiency is to argue that if investors have homogenous expectations, the set of optimal portfolios they would face would be using the same values of expected returns, variances and co variances.
Therefore, the efficiency of the market portfolio and the CAPM are joint hypothesis and it is not possible to test the validity of one without the other (Roll, 1977).
7 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
CAPM Assumptions-SummaryCAPM Assumptions-Summary
Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs Information is costless and available to all
investors Investors are rational mean-variance optimizers Homogeneous expectations
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Ch 7: CAPM and APT
AssumptionsAssumptions
Asset markets are frictionless and information liquidity is high.
All investors are price takers; so that, they are not able to influence the market price by their actions.
All investors have homogenous expectations about asset returns and what the uncertain future holds for them.
All investors are risk averse and they operate in the market rationally and perceive utility in terms of expected return.
9 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Assumptions (cont.)Assumptions (cont.)
All investors are operating in perfect markets which enables them to operate without tax payments on returns and without cost of transactions entailed in trading securities.
All securities are highly divisible for instance they can be traded in small parcels (Elton and Gruber, 1995, p.294).
All investors can lend and borrow unlimited amount of funds at the risk-free rate of return.
All investors have single period investment time horizon in means of different expectations from their investments leads them to operate for short or long term returns from their investments.
10 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Resulting Equilibrium ConditionsResulting Equilibrium Conditions
All investors will hold the same portfolio for risky assets – market portfolio
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
Risk premium on the market depends on the average risk aversion of all market participants
Risk premium on an individual security is a function of its covariance with the market
11 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
E(r)E(r)
E(rE(rMM))
rrff
MMCMLCML
mm
Capital Market LineCapital Market Line
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Ch 7: CAPM and APT
Capital Market LineCapital Market Line
If a fully diversified investor is able to invest in the market portfolio and lend or borrow at the risk free rate of return, the alternative risk and return relationships can be generally placed around a market line which is called the Capital Market Line (CML).
13 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Capital Market LineCapital Market Line
CML: E(rp)= rF+ λσp
E(rp): Expected return on portfolio
rF : Return on the risk free assetλ : Market price riskσp : Market portfolio risk
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Ch 7: CAPM and APT
M = Market portfoliorf = Risk free rate
E(rM) - rf = Market risk premium
E(rM) - rf = Market price of risk
= Slope of the CAPM
Slope and Market Risk PremiumSlope and Market Risk Premium
MM
15 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Expected Return and Risk on Individual Expected Return and Risk on Individual SecuritiesSecurities
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio
Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio
16 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Security Market LineSecurity Market Line
The SML shows the relationship between risk measured by beta and expected return. The model states that the stock’s expected return is equal to the risk-free rate plus a risk premium obtained by the price of the risk multiplied by the quantity of the risk.
17 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
E(r)E(r)
E(rE(rMM))
rrff
SMLSML
MMßßßß = 1.0= 1.0
Security Market LineSecurity Market Line
18 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
SML RelationshipsSML Relationships
= [COV(ri,rm)] / m
2
Slope SML = E(rm) - rf
= market risk premium
SML = rf + [E(rm) - rf]
(σSpS,M) is the market price of risk
SML: E(rS)=rF+ λσSpS,M
19 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Sample Calculations for SMLSample Calculations for SML
E(rm) - rf = .08 rf = .03
x = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%
y = .6
e(ry) = .03 + .6(.08) = .078 or 7.8%
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Ch 7: CAPM and APT
E(r)E(r)
RRxx=13%=13%
SMLSML
mm
ßß
ßß1.01.0
RRmm=11%=11%RRyy=7.8%=7.8%
3%3%
xxßß1.251.25
yyßß.6.6
.08.08
Graph of Sample CalculationsGraph of Sample Calculations
21 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Disequilibrium ExampleDisequilibrium Example
Suppose a security with a beta of 1.25 is offering expected return of 15%
According to SML, it should be 13%Underpriced: offering too high of a rate of
return for its level of risk
22 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
E(r)E(r)
15%15%
SMLSML
ßß1.01.0
RRmm=11%=11%
rrff=3%=3%
1.251.25
Disequilibrium ExampleDisequilibrium Example
23 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Security Characteristic LineSecurity Characteristic LineExcessExcess Returns (i) Returns (i)
SCLSCL
..
..
........
.. ..
.. ....
.. ....
.. ..
.. ....
......
.. ..
.. ....
.. ....
.. ..
.. ....
.. ....
.. ..
..
.. ...... .... .... ..
ExcessExcess returns returnson market indexon market index
RRii = = ii + ß + ßiiRRmm + e + eii
24 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Using the Text Example p. 231, Table 7.5Using the Text Example p. 231, Table 7.5
Jan.Jan.Feb.Feb.....DecDecMeanMeanStd DevStd Dev
5.415.41-3.44-3.44
..
..2.432.43-.60-.604.974.97
7.247.24.93.93
..
..3.903.901.751.753.323.32
ExcessExcessMkt. Ret.Mkt. Ret.
ExcessExcessGM Ret.GM Ret.
25 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Estimated coefficientEstimated coefficientStd error of estimateStd error of estimateVariance of residuals = 12.601Variance of residuals = 12.601Std dev of residuals = 3.550Std dev of residuals = 3.550R-SQR = 0.575R-SQR = 0.575
ßß
-2.590-2.590(1.547)(1.547)
1.13571.1357(0.309)(0.309)
rrGMGM - r - rf f = + ß(r= + ß(rmm - r - rff))
Regression Results:Regression Results:
26 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Arbitrage Pricing TheoryArbitrage Pricing Theory
Arbitrage Pricing Theory was developed by Stephen Ross (1976). His theory begins with an analysis of how investors construct efficient portfolios and offers a new approach for explaining the asset prices and states that the return on any risky asset is a linear combination of various macroeconomic factors that are not explained by this theory namely.
Similar to CAPM it assumes that investors are fully diversified and the systematic risk is an influencing factor in the long run. However, unlike CAPM model APT specifies a simple linear relationship between asset returns and the associated factors because each share or portfolio may have a different set of risk factors and a different degree of sensitivity to each of them.
27 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
The Assumptions of APTThe Assumptions of APT Capital asset returns’ properties are consistent
with a linear structure of the factors. The returns can be described by a factor model.
Either there are no arbitrage opportunities in the capital markets or the markets have perfect competition.
The number of the economic securities are either inestimable or so large that the law of large number can be applied that makes it possible to form portfolios that diversify the firm specific risk of individual stocks.
Lastly, the number of the factors can be estimated by the investor or known in advance (K. C. John Wei, 1988)
28 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
The Model of APTThe Model of APT
k Ri= E( Ri )+ ∑ δj βij+ εi
j=1 where, R i : The single period expected rate on
security i , i =1,2….,n δj : The zero mean j factor common to the
all assets under consideration βij : The sensitivity of security i’s returns to
the fluctuations in the j th common factor portfolio
εi : A random of i th security that constructed to have a mean of zero.
29 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Arbitrage Pricing Theory-brieflyArbitrage Pricing Theory-briefly
• Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit
Since no investment is required, an investor can create large positions to secure large levels of profit
In efficient markets, profitable arbitrage opportunities will quickly disappear
30 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Arbitrage Example Arbitrage Example
Current Expected Standard
Stock Price$ Return% Dev.%
A 10 25.0 29.58
B 10 20.0 33.91
C 10 32.5 48.15
D 10 22.5 8.58
31 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Arbitrage PortfolioArbitrage Portfolio
Mean Stan. Correlation
Return Dev. Of Returns
Portfolio
A,B,C 25.83 6.40 0.94
D 22.25 8.58
32 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Arbitrage Action and ReturnsArbitrage Action and Returns
E. Ret.E. Ret.
St.Dev.St.Dev.
* * PP* * DD
Short 3 shares of D and buy 1 of A, B & C to form Short 3 shares of D and buy 1 of A, B & C to form PP
You earn a higher rate on the investment than You earn a higher rate on the investment than you pay on the short saleyou pay on the short sale
33 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
APT and CAPM ComparedAPT and CAPM Compared
APT applies to well diversified portfolios and not necessarily to individual stocks
With APT it is possible for some individual stocks to be mispriced - not lie on the SML
APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio
APT can be extended to multifactor models
34 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-market riskExample-market risk
Suppose the risk free rate is 5%, the average investor has a risk-aversion coefficient of A* is 2, and the st. dev. Of the market portfolio is 20%.
A) Calculate the market risk premium. B) Find the expected rate of return on the market. C) Calculate the market risk premium as the risk-
aversion coefficient of A* increases from 2 to 3. D) Find the expected rate of return on the market
referring to part c.
35 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-market riskAnswer-market risk
A) E(rm-rf)=A*σ2m
Market Risk Premium =2(0.20)2=0.08B) E(rm) = rf +Eq. Risk prem
= 0.05+0.08=0.13 or 13%
C) Market Risk Premium =3(0.20)2=0.12D) E(rm) = rf +Eq. Risk prem
= 0.05+0.12=0.17 or 17%
36 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-risk premiumExample-risk premium
Suppose an av. Excess return over Treasury bill of 8% with a st. dev. Of 20%.
A) Calculate coefficient of risk-aversion of the av. investor.
B) Calculate the market risk premium as the risk-aversion coefficient is 3.5
37 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-risk premiumAnswer-risk premium
A) A*= E(rm-rf)/ σ2m =0.085/0.202=2.1
B) E(rm)-rf =A*σ2m =3.5(0.20)2=0.14 or 14%
38 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-ERORExample-EROR
Suppose the risk premium of the market portfolio is 9%, and the estimated beta is 1.3. The risk premium for stock is 1.3 times the market risk premium.
A) Calculate expected ROR if T-bill rate is 5%. B) Calculate ROR and the risk premium if the
estimated of the beta is 1.2 for the company. C) Find the company’s risk premium if market
risk premium and beta are 8% and 1.3 respectively.
39 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-ERORAnswer-EROR
A) E(rc)=rf+βc[E(rm-rf)]
=5%+1.3(9%)=16.7%B) E(rc)=rf+βc[E(rm-rf)]
=5%+1.2(9%)=15.8% 1.2(9%)=10.8% risk premium C) 1.3(8%)=10.4%
40 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-Portfolio beta and risk premiumExample-Portfolio beta and risk premium
Consider the following portfolio:
A) Calculate the risk premium on each portfolio
B) Calculate the total portfolio if Market risk premium is 7.5%.
Asset BetaRisk
prem.Portfolio Weight
X 1.2 9% 0.5
Y 0.8 6 0.3
Z 0.0 0 0.2
Port. 0.84 1.0
41 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-Portfolio beta and risk premiumAnswer-Portfolio beta and risk premium
A) (9%) (0.5)=4.5 (6%) (0.3)=1.8 =6.3%B) 0.84(7.5)=6.3%
42 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-risk premiumExample-risk premium
Suppose the risk premium of the market portfolio is 8%, with a st. dev. Of 22%.
A) Calculate portfolio’s beta. B) Calculate the risk premium of the portfolio
referring to a portfolio invested 25% in x motor company with beta 0f 1.15 and 75% in y motor company with a beta of 1.25.
43 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-risk premiumAnswer-risk premium
A) βy= 1.25, βx= 1.15
βp=wy βy+ wx βx
=0.75(1.25)+0.25(1.15)=1.225
B) E(rp)-rf=βp[E(rm)-rf]
=1.225(8%)=9.8%
44 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-SMLExample-SML
Suppose the return on the market is expected to be 14%, a stock has a beta of 1.2, and the T-bill rate is 6%.
A) Calculate the expected return of the SML B) If the return is 17%, calculate alpha of the
stock
45 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-SMLAnswer-SML A) E(rp)=rf+β[E(rm)-rf]
=6+1.2(14-6)=15.6%E(r)E(r)
17%17% SMLSML
ßß1.01.0
15.6%15.6%
14%14%
6%6%
1.21.2
MM
StockStock
α=α= 17-15.6=1.417-15.6=1.4
46 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-SMLExample-SML Stock xyz has an expected return of 12% and risk of
beta is 1.5. Stock ABC is expected to return 13% with a beta of 1.5. The market expected return is 11% and rf=5%.
A) Based on CAPM, which stock is a better buy? B) What is the alpha of each stock? C) Plot the relevant SML of the two stocks D) rf is 8% ER on the market portfolio is 16%, and
estimated beta is 1.3, what is the required ROR on the project?
E) If the IRR of the project is 19%, what is the project alpha?
47 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-SMLAnswer-SML
A and B) α=E(r)-{rf+β[E(rm)-rf]}
αXYZ= 12-{5+1.0[11-5]}=1– UNDERVALUED
αABC= 13-{5+1.5[11-5]}= -1– OVERVALUED
48 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-SML-CAnswer-SML-C
E(r)E(r)
14%14% SMLSML
ßß1.01.0
13%13%
12%12%
5%5%
1.51.5
xyzxyz
StockStock
α=α= 13-14=-113-14=-1
ααABCABC
=13-12=1=13-12=1
49 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-SMLAnswer-SML
D) E(r)={rf+β[E(rm)-rf]}
= 8+1.3[16-8]=18.4%E) If the IRR of the project is 19%, it is
desirable. However, any project with an IRR by using similar beta is less than 18.4% should be rejected.
50 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-SMLExample-SML
Consider the following table:
Market Return
Aggressive stock
Defensive stock
5% 2% 3.5%
20 32 14
51 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Example-SML cont..Example-SML cont..
A) What are the betas of the two stock? B) What is the E(ROR) on each stock if Market
return is equally likely to be 5% or 20%? C) If T-bill rate is 8% and Market return is equally
likely to be 5% or 20%, draw SML for the economy?
D) Plot the two securities on the SML graph and show the alphas
52 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Answer-SMLAnswer-SML
A) βA=2-32/5-20=2 βB=3.5-14/5-20=0.7
B) E(rA)={rf+β[E(rm)-rf]}
=0.5(2%+32%)=17% =0.5(3.5%+14%)=8.75%
53 Investment Management © 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
The EndThe End
Thanks