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LECTURE 7 :
THE CAPM
(Asset Pricing and Portfolio Theory)
THE CAPM
Quantitative Asset Pricing
Dirk Nitzsche (E-mail : [email protected])
Contents
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The market portfolio
Factor models : The CAPM
Equilibrium model for asset pricing
(SML)
Performance measures / Risk adjusted
rate of return
Using the CAPM to appraise projects
The UK Stock Market (FT
All Share Index)
FT All Share Price Index
3500
3000
2500
2000
1500
1000
500
0
01/12/1996
28/08/1999
24/05/2002
17/02/2005
Daily Share price between 1st April 1997 and 12th Nov. 2004
FT All Share Index
(Return)
FT All Share Pricve Index (Returns)
0.06
0.04
0.02
0
01/12/1996
-0.02
-0.04
-0.06
28/08/1999
24/05/2002
17/02/2005
BA and Newcastle Utd.
Share Price
BA Share Price
Newcastle United Share Price
800
160
700
140
600
120
500
100
400
80
300
60
200
40
100
20
0
28/10/1995
24/07/1998
19/04/2001
14/01/2004
10/10/2006
0
28/10/1995
24/07/1998
19/04/2001
14/01/2004
10/10/2006
Tesco Share Price
Tesco Share Price
350
300
250
200
150
100
50
0
28/10/1995
24/07/1998
19/04/2001
14/01/2004
10/10/2006
Assumptions to Derive
the CAPM
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Assumption 1 :
– Investors agree in their forecasts of expected
returns, standard deviation and correlations
– Therefore all investors optimally hold risky assets
in the same relative proportions
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Assumption 2 :
– Investors generally behave optimally. In
equilibrium prices of securities adjust so that
when investors are holding their optimal
portfolio, aggregate demand equals its supply.
The Capital Market Line
ERi = rf + [(ERm – rf)/sm] si
ER
Portfolio M
Slope of the CML
CML
ERm - rf
rf
sm
Standard deviation
Risk Premium, Beta and
Market Portfolio
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
Suppose risk premium on the market is a function of
its variance.
The market : ERm – rf = As2m
ERi = rf + [(ERm-rf)/s2m] s2i
bi  sim / s2m
Rem. : Portfolio risk is covariance
ERi = rf + bi[ERm – rf]
or ERi –rf = bi[ERm – rf]
(SML)
s2i = b2i s2m + var(ei)
total risk = systematic risk + nonsystematic risk
Systematic and Non
Systematic Risk
ER
CML
Asset with systematic risk
ONLY
Portfolio M
rf
Assets with non systematic risk
Standard deviation
The Security Market Line
Expected return
and
actual return
SML
Q (buy)
M
P
expected
return
T (sell)
r
actual
return
S (sell)
0.5
1
1.2
The larger is bi, the larger is ERi
Beta, bi
Properties of Betas
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Betas represent an asset’s systematic
(market or non-diversifiable) risk
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Beta of the market portfolio : bm = 1
Beta of the risk free asset : b = 0
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Beta of a portfolio : bp = Swibi
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Applications of Beta
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Market timing
… if you expect the market to go up you want to move into
higher beta stocks to get more exposure to the ‘bull market’.
… vice versa if market goes down you want less exposure to the
stock market and hence should buy stocks with lower betas
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Portfolio construction
… to construct a customised portfolio. (Rem : bp = S bi)
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Performance measures
Risk Management
Calculating the WACC
… to use for the DPV for assessing the viability of a project or
the value of a company.
US Companies : Betas and
Sigma (7th December 1979)
Company
Name
Coca Cola
Beta
Volatility
1.19
18%
Exxon Corp
General Electric
General Motor
Gillette
Lockheed
0.67
1.26
0.81
1.09
3.02
18%
15%
19%
21%
43%
Performance Measures / Risk
Adjusted Rate of Return
Sharpe Ratio, Treynor
Ratio
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Shape ratio (from the CML)
SRi = (ERi –rf)/si
Risk is measured by the standard deviation (total
risk of security)
Aim : to maximise the Sharpe ratio
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Treynor ratio (from the SML)
TRi = (ERi – rf)/bi
Risk is measured by beta (market risk only)
Aim : to maximise the Treynor ratio
Performance Evaluation
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CAPM can be used to evaluate the performance
of an investment portfolio (i.e. mutual fund)
Step 1 :
Calculate the summary stats of the investment
portfolio (e.g. average rate of return, variance)
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Step 2 :
Calculate the covariance between the investment
portfolio and the market portfolio and the variance
of the market portfolio : b = Cov(Ri,Rm)/Var(Rm)
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Step 3 :
Calculate Jensen’s alpha performance measure :
(Ri – rf) = a + b(Rm - rf)
Obtaining Beta
Estimating the Betas
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Time series regression :
(Ri-rf )t = ai + bi (Rm-rf )t + eit
(alternative models also available)
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Decision :
– How much historical data to use (i.e. 1 years, 2
years, 5 years, 10 years or what) ?
– What data frequency to use (i.e. daily data,
weekly data, monthly data or what) ?
– Should I use the model above or an alternative
model ?
Adjusted Beta
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To price securities, need to obtain forecast of betas
Estimating betas, can use historic data
– Assume betas are ‘mean reverting’ (to mean of market
beta)
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High betas (b > 1)  lower beta in future
Low betas (b < 1)  higher betas in future
Adjusted beta = (w) estimated beta + (1-w) 1.0
(i.e. w = 2/3 and (1-w) = 1/3)
(Can test whether betas are ‘mean’ reverting and then
estimate w.)
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Q.: Is beta constant over time ?
CAPM and Investment
Appraisal
CAPM and Investment
Appraisal (All Equity Firm)
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Expected return ERi on equity (calculated from the CAPM
formula) is (often) used as the discount rate in a DPV calculation
to assess a physical investment project for an all equity financed
firm
We use ERi because it reflects the riskiness of the firm’s new
investment project – provided the ‘new’ investment project has
the same ‘business risk’ characteristics as the firm’s existing
project.
This is because ERi reflects the return required by investors to
hold this share as part of their portfolio (of shares) to
compensate them, for the (beta-) risk of the firm (i.e. due to
covariance with the market return, over the past).
CAPM and Investment
Appraisal (Levered Firm)
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What if the new project is so large it will radically
alter the debt equity mix, in the future ?
How do we measure the equity return ER (then the
WACC) to be used as the discount rate ?
(MM result : Equity holder requires higher return
ERi as the debt to equity ratio increases.)
We calculate this ‘new’ equity return by using the
‘levered beta’ in the CAPM equation as :
bL(new) = bU (1 + (1-t))(B/S)new)
How Does beta (L) Vary
with Debt-equity Ratio ?
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B/(B+S)
0%
50%
70%
90%
(B/S)
new
0%
100%
233%
900%
beta (L) Leverage effect
1.28 (= bU)
2.1
3.2
8.7
0
0.82
1.92
7.4
Above uses bL(new) = bU (1 + (1-t))(B/S)new) with t = 0.36
How Does beta (L) Vary with
Debt-equity Ratio ? (Cont.)
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The ‘leverage-beta’ increases with leverage (B/S)
and hence so does the required return on equity ERi
given by the CAPM and hence the discount rate for
cash flows
ERi can then be used with the bond yield to
calculate WACC, if debt and equity finance is being
used for the ‘new’ project.
See Cuthbertson and Nitzsche (2001) ‘Investments
: Spot and Derivatives Markets’
Variants of the CAPM
Zero Beta CAPM
ER
ER
M
M
ERz
ERz
beta
Z
sigma
Portfolio Z is not the minimum
variance portfolio.
Zero Beta CAPM (Cont.)
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Two factor model :
ERi = ERZ + (ERm – ERZ) bi
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Portfolio s has smallest variance :
ss2 = Xz2sz2 + (1 – Xz)2sm2
∂ss2/∂Xz = 2Xzsz2 – 2sm2 + 2Xzsm2 = 0
Solving for Xz : Xz = sm2 / (sm2+sz2)
Zero Beta CAPM (Cont.)
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Since sm2 and sz2 must be positive
– Positive weights on both assets (M and Z)
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Since ERz < ERm
– Portfolio ‘S’ (Z and M) must have higher
expected return than Z
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Since the minimum variance portfolio has
higher ER and lower sigma than Z, Z cannot
be on the efficient portion of the efficient
frontier.
Zero Beta CAPM (Cont.)
ER
M
S
Z
S : minimum variance portfolio
sigma
The Consumption CAPM
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Different definition of equilibrium in the
capital market
Key assumption :
– Investors maximise a multiperiod utility
function over lifetime consumption
– Homogeneous beliefs about asset
characteristics
– Infinitely lived population, one consumption
good
The Consumption CAPM
(Cont.)
Et(rt – rf ) + ½(s2t(ri*) = -covt(m,ri*)
where
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ri* = ln(Ri*)
M = ln(M)
M = q(Ct+1/Ct)-g
Excess return on asset-i depends on the
covariance between ri* and consumption.
The higher is the ‘covariance’ with
consumption growth, the higher the ‘risk’ and
the higher the average return.
References
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Cuthbertson, K. and Nitzsche, D. (2004)
‘Quantitative Financial Economics’,
Chapters 7
Cuthbertson, K. and Nitzsche, D. (2001)
‘Investments : Spot and Derivatives
Markets’, Chapter 10.3
END OF LECTURE