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CAPM, BETA AND APT
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CAPITAL MARKET THEORY: AN
OVERVIEW
Capital market theory extends portfolio theory and
seeks to develops a model for pricing all risky assets
based on their relevant risks
Asset Pricing Models
Capital asset pricing model (CAPM) allows for the calculation
of the required rate of return for any risky asset based on
the securitys beta
Arbitrage Pricing Theory (APT) is a multi-factor model for
determining the required rate of return
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ASSUMPTIONS OF CAPITAL MARKET
THEORY
All investors are Markowitz efficient investors (rational)who invest on the efficient frontier.
Investors can borrow or lend any amount of money atthe risk-free rate of return (RFR).
Investors have homogeneous expectations
All investors are risk averse and efficiently diversified.Only the systematic risk is of concern in determiningestimated return.
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ASSUMPTIONS OF CAPITAL MARKET
THEORY All investments are infinitely divisible, which means
that it is possible to buy or sell fractional shares of any
asset or portfolio.
There are no taxes or transaction costs involved in
buying or selling assets.
Capital markets are efficient and no single investor
can influence price (Perfect competition).
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MAKING ASSUMPTIONS
Some of these assumptions are clearly unrealistic
Relaxing many of these assumptions would have only
minor influence on the model and would not change
its main implications or conclusions.
The primary way to judge a theory is on how well itexplains and helps predict behavior, not on its
assumptions.
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THE CAPITAL ASSET PRICING MODEL
An assets covariance with the market portfolio is the
relevant risk measure (helps in beta calculation)
This can be used to determine an appropriate requiredrate of return on a risky asset
CAPM indicates what should be the expected or
required rates of return on risky assets
This helps to value an asset by providing an appropriate
discount rate to use in dividend valuation models
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DETERMINING THE EXPECTED
RETURN
The expected rate of return of a risk asset isdetermined by the RFR plus a risk premium
for the individual asset
The risk premium is determined by the
systematic risk of the asset (beta) and the
prevailing market risk premium (RM-RFR)
RFR)-(RRFR)E(R Mi iF!
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WHAT IS THE MARKET RISK PREMIUM
[RM - RRF]
The additional return over the risk-free rate
needed to compensate investors for assuming an
average amount of risk.
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CAPM AND BETA
)Cov(RFR-R
RFR)E(R Mi,2M
Mi
W
!
RFR)-R(Cov
RFR M2M
Mi,
W!
2
M
Mi,CovW
We then define as beta
RFR)-(RRFR)E(R Mi iF!
)( iF
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CAPM AND REQUIRED RETURN FOR IBMS
COMMON STOCK
Let rRF=6%, rM=12%, and the Beta of IBM commonstock is IBM=1.3
rIBM = rRF + IBM[rM - rRF]
rIBM = 0.06 + 1.3[0.12 0.06]
rIBM = 0.06 + 1.3[0.06]
rIBM = 0.06 + 0.078 =0.138 or 13.8%
Note the following items:
The market risk premium is [12% - 6%] = 6%
IBMs risk premium is 1.3[12% 6%] = 7.8%.
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THE SECURITY MARKET LINE (SML)
The CAPM model is linear and when plotted ona graph paper gives a straight line called SML.
The graphical version of CAPM is SecurityMarket Line.
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GRAPH OF SML
)E(Ri
)Beta(Cov 2Mim/W
0.1
mR
SML
0
Negative Beta
RFR
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OVER AND UNDER PRICED
In equilibrium, all assets and all portfolios of assets
should plot on the SML
The SML gives the market going rate of return or what you
should earn as a return for a security
Any security with an expected return that plots above the
SML is underpriced
Any security with an expected return that plots below theSML is overpriced
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WHAT IF THE SML CHANGES?
Two possible SML changes can definitely occur. Eachhas important effects.
(1) Risk-free rates rRF can change. This can be due to theInflation Premium (IP) or the real rate of interest changing.Note that in this case, the RM must also increase by thesame amount so that the market risk premium [rM-rRF] isunchanged.
(2) The market risk premium [rM-rRF] can change. Thiswould typically be due to changing attitudes toward riskaversion by investors.
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WHAT IF THE SML CHANGES?
Following slides allow for two separate SML changes:
In the first case, we let the risk-free rate rRF increase by 2%
(from 5% to 7%). The market required return rM mustincrease by the same amount (from 10% to 12%), so thatthe market risk premium [rM-rRF] remains at 5%.
In the second case, we allow for a 2% increase in the
market risk premium (from 5% to 7%). The risk-free rateremains the same at 5%, however, rM must increase by 2%(from 10% to 12%) so that the market risk premium cannow be [rM-rRF] = 7%.
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THE SECURITY MARKET LINE
Risk-free rate increases from 5% to 7%. The
new SML is ri = 0.07 + 0.05i
SML, old
ri (%)
0 0.5 1.0 1.5
12
10
7
5
Risk, i
Mkt. risk prem. = 5%
rRF=7%
SML, new
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THE SECURITY MARKET LINE
The market risk premium increases from 5 to
7%. The new SML is ri = 0.05 + 0.07i
SML, old
ri (%)
0 0.5 1.0 1.5
12
10
5
Risk, i
Mkt. risk prem. = 7%
rRF=5%
SML, new
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HOW TO INTERPRET BETA ()
Beta measures a stocks degree of systematic ormarket risk. It can also be thought of as the stockscontribution to the risk of a well-diversified portfolio.
= 1: the stock has average market risk. The stockgenerally tends to go up (down) by the same percentageamount as the market.
= 1.5: The stock generally tends to go up (down) by 50%
(1.5x) more than the market.
= 0.5: The stock generally tends to go up (down) by half asmuch as the market.
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HOW TO INTERPRET BETAS (),
CONTINUED = 0: the stock has no correlation with movements in the
overall stock market. All of this firms risk would actually befirm-specific risk.
< 0: The stock generally tends move in a directionopposite that of the market (very rare).
Firms that supply basic consumer goods (Proctor &Gamble) and utilities (phone, cable, gas, or electric)
tend to have low Betas (lower than 1.0, often around0.4 to 0.6).
Firms that are in economically cyclical industrieswould have higher Betas (greater then 1.0).
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EXPECTED RETURN AND BETA OF A
PORTFOLIO OF STOCKS
Both portfolio expected returns and portfolio Betas are
always a weighted average of the stocks that comprise
the portfolio.
You invest $10,000: $6000 in Apple Computer and
$4000 in Proctor & Gamble (PG).
Let rRF=5% and rM=10%. Let APPLE=1.3 and PG=0.6
The investment weights are (6000/10000)=0.6 for Apple
and (4000/10000)=0.4 for PG. The weights must always
sum up to 1.
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EXPECTED RETURN AND BETA OF A
PORTFOLIO OF STOCKS, CONTINUED
The portfolio Beta is always the weighted
average of each of the stocks betas.
P
= wapple
apple
+ wpg
pg
P = 0.6(1.30) + 0.4(0.6)
P = 1.02
Now find the portfolio required return.
rP = 0.05 + 1.02[0.10 0.05] = 0.101 or 10.1%
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THE CONCEPT OF BETA
Beta(b) measures how the return of an individual asset (or evena portfolio) varies with the market.
b = 1.0 : same risk as the market
b < 1.0 : less risky than the market
b > 1.0 : more risky than the market
Beta is the slope of the regression line (y = a + bx) between astocks return(y) and the market return(x) over time, b fromsimple linear regression.
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CALCULATING BETA: THE
CHARACTERISTIC LINE
The systematic risk input of an individual asset is derived from a
regression model, referred to as the assets characteristic line
with the model portfolio:
IFE ! tM,iiti, RRwhere:Ri,t = the rate of return for asset i during period t
RM,t = the rate of return for the market portfolio M during t
miii R-R FE !
2M
Mi,CovW
F !i
error termrandomthe!I
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ESTIMATING THE BETA COEFFICIENT
Generally, these quantities are not known.
We usually rely on their historical values to provide uswith an estimate of beta.
If we know the securitys correlation with the market, its
standard deviation, and the standard deviation of the
market, we can use the definition of beta:
M
jMj
jW
WVF
,!
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INTERPRETING THE BETA
COEFFICIENTThe beta of the market portfolio is always equal to
1.0:
1,
!!
M
MMM
M
W
WVF
1, !MMVSince
The beta of the risk-free asset is always equal to 0:
0,
!!M
rMr
r
ff
fW
WV
F
0!frWSince
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PROJECT, UN-GEARED & GEARED
BETA Beta Geared: The Beta attaching to the ordinary
shares of a geared firm. These bear a risk higherthan the firms basic activity.
Beta Un-geared: The geared Beta stripped of theeffect of gearing. Corresponds to the activity Beta inan equivalent un-geared firm.
An indication of the systematic riskiness attachingto the returns on ordinary shares. It equates to theasset Beta for an un-geared firm, or is adjustedupwards to reflect the extra riskiness of shares in a
geared firm., i.e. the Geared Beta
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ARBITRAGE PRICING THEORY (APT)
CAPM is criticized because of the difficulties in
selecting a proxy for the market portfolio as a
benchmark
An alternative pricing theory with fewer
assumptions was developed:
Arbitrage Pricing Theory
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ASSUMPTIONS OF ARBITRAGE PRICING
THEORY
Capital markets are perfectly competitive
Investors always prefer more wealth to lesswealth with certainty
The stochastic process generating assetreturns can be presented as a k-factor model
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ARBITRAGE PRICING THEORY
Multiple factors expected to have an impact on allassets:
Inflation
Growth in GNP Major political upheavals
Changes in interest rates
And many more.
Contrast with CAPM assumption that only beta isrelevant
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ARBITRAGE PRICING THEORY (APT)
where:
= the expected return on an asset with zerosystematic risk where
ikkiii bbbE PPPP ! ...22110
0P
1P = the risk premium related to each of the common factors - for
example the risk premium related to interest rate risk
bik = the pricing relationship between the risk premium and asset i - that is how
responsive asset i is to this common factorK
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EXAMPLE OF TWO STOCKS
AND A TWO-FACTOR MODEL
= changes in the rate of inflation. The risk premium
related to this factor is 1 percent for every 1 percent
change in the rate
1P
)01.(1
!P
= percent growth in real GNP. The average risk premium related to this
factor is 2 percent for every 1 percent change in the rate
= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3
percent
2P
)02.( 2 !P
)03.(3
!P3P
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EXAMPLE OF TWO STOCKS
AND A TWO-FACTOR MODEL
= the response of asset Xto changes in the rate of
inflation is 0.501x
b)50.(
1!
xb
= the response of asset Y to changes in the rate of inflation is 2.00
1yb
= the response of asset X to changes in the growth rate of real GNP is
1.50
= the response of asset Y to changes in the growth rate of real GNP is
1.75
2xb
2yb
)50.1( 2 !xb
)75.1( 2 !yb
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EXAMPLE OF TWO STOCKS
AND A TWO-FACTOR MODEL
= .03 + (.01)bi1 + (.02)bi2
Ex = .03 + (.01)(0.50) + (.02)(1.50)
= .065 = 6.5%
Ey = .03 + (.01)(2.00) + (.02)(1.75)
= .085 = 8.5%
22110 iii bbE PPP !
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EMPIRICAL TESTS OF THE APT
Studies by Roll and Ross and by Chen support APT by
explaining different rates of return with some better
results than CAPM
Dhrymes and Shanken question the usefulness of APT
because it was not possible to identify the factors and
therefore may not be testable