Investment and Portfolio
Management
Lecture 1: CAPM Review and Testing,
Security Selection, CAPM Applications
Professor Robert Kosowski
© Imperial College Business School
My Brief Bio
• Professor in the Finance Group of Imperial College Business School, Imperial
College London.
• Prior to joining Imperial College London, Assistant Professor of Finance at
INSEAD where he taught in the MBA, Executive Education and PhD programs.
• Advisor to private and public organizations (Specialist Advisor (UK House of
Lords, 2009-2010), Expert Technical Consultant (IMF, 2008, 2016)). Worked for
Goldman Sachs in London, the Boston Consulting Group in Germany and
Deutsche Bank in New York City. Consultant for IMF, SWFs and other financial
services firms. Head of Quantitative Research (Part-time) Unigestion.
• Research interests include asset pricing and financial econometrics with a
focus on the performance of hedge funds, business cycles, analyst
recommendations and derivatives trading. Awarded European Finance
Association 2007 and several INQUIRE UK Best Paper Awards
• Research published in top peer-reviewed finance journals such as The Journal
of Finance, Review of Financial Studies and The Journal of Financial
Economics and has been featured in publications such as The Financial Times
and The Wall Street Journal.
• BA and MA in Economics from Trinity College, Cambridge University, and a
MSc in Economics and PhD from the London School of Economics.
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Investment and Portfolio Management (IPM) Course
Overview
Asset
Allocation
1. MV-Portfolio
Choice and CAPM
8. Options and
Performance
Evaluation
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Expected
Return Models
2. APT/CCAPM
3. Market Eff. &
Behavioural Fin.
7. Commodities
Note: Numbers indicate the lecture in IPM
Predictability
in Returns
4. Term Structure
Of Interest Rates
5. IR Risk
6. FX
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Lecture Overview
• 1. Motivation, Course Introduction, Readings
• 2. Index Model
• 3. CAPM
• 4. Examples of Use: Applying the CAPM to Corporate Finance
and Portfolio Management
• 5. Testing the CAPM
Appendix:
– New Evidence on CAPM Based on Announcement Days
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Review of Important Concepts
• What is systematic risk
• What is idiosyncratic risk?
• What is the relationship between expected
return and risk
• How are total risk and systematic risk
measured?
• What is the difference
– between expected and realized return?
– between ex post and ex ante return
– between the windscreen and the rear view
mirror
• Is the CAPM about the cross-section or the
time-series of returns?
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1. Motivating Example (1), CAPM Application:
America’s Hottest Investor*
Never mind the rocky market. After a string of supersmart
calls, mutual fund manager Ken Heebner is putting up the
best numbers of his sterling career.
Since May 1998, Focus has an average annualized return
of 24%, the best ten-year record of any U.S. mutual fund,
compared with only 4% for Standard & Poor's 500.
Focus, which has $7.4 billion in assets, is already up 15%
in 2008 (as of May 19), but it is 2007 that will be
remembered as Heebner's pièce de résistance.
Fueled by big bets on energy, fertilizer, and metals, Focus
soared 80% last year, vs. 5% for the S&P 500. "I told Ken it
was like he was walking between the raindrops," says CGM
president Bob Kemp, who oversees sales and marketing at
the firm, of the year Heebner had in 2007.
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Source: America’s Hottest
Investor* , Fortune Magazine,
May 27, 2008,
http://money.cnn.com/2008/0
5/23/magazines/fortune/birge
r_americas_hottest_investor.f
ortune/index.htm?postversion
=2008052706
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Motivating Example (2): Did Heebner’s fund beat the
market on a risk adjusted basis over recent period?
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Source: http://finance.yahoo.com/q?s=CGMFX
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From Mean-Variance to the CML: 2 Risky Assets (Debt and Equity)
• As an illustration, see the Mean-Standard Deviation Animation that
we created in https://www.edx.org/course/finance-essentials-mba-success-imperialx-icbs002
• P has highest Sharpe Ratio
– SR=(Excess Return/ St. Dev)
• Below you can see BKM Table 7.1
MVP
• What position would an investor with
Risk Aversion A=4 take in portfolio p?
• Assume that portfolio P consists of
40% debt and 60% equity. Appendix
A shows how to calculate these
percentages.
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What position would an investor with Risk Aversion A=4 take in
portfolio p?
y=
E ( rp ) − rf
A
2
p
0.11 − 0.05
=
= 74.39%
2
4  0.142
ywD = 0.4  0.7439 = 29.76%
ywE = 0.6  0.7439 = 44.63%
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Why do we need a factor model?
• A variance covariance matrix is needed in one of the simplest examples
of a risk managed portfolio, the minimum variance portfolio (see graph)
• For this purpose we need to first calculate the sample covariance matrix
• But this becomes problematic when the number of assets (N) is of the
same order of magnitude or larger than the number of observations (T).
• Example:
• In typical applications, there can be over a thousand stocks to choose
from, but rarely more than ten years of monthly data (i.e. N = 1,000 and
T = 120).
• For background and references see
– https://systematicinvestor.wordpress.com/2012/02/26/portfolio-optimization-why-do-we-need-a-risk-model/
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Single Factor Model
• Consider a single factor model that decomposes security return into
expected and unexpected component
ri = E (ri ) + ei ,
(
ei ~ N 0,  (ei )
2
)
• Consider a common (macro) factor
ri = E (ri ) + m + ei , where  i2 =  m2 +  2 (ei )
• Incorporate exposures to this macro factor
ri = E (ri ) +  i m + ei
 i2 =  i2 m2 +  2 (ei )
Cov(ri , rj ) = Cov( i m + ei ,  j m + e j ) =  i  j
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m
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2. Single-index approach
• Pros/Cons of Single-factor approach
– Purely statistical and simple, but what is the factor?
• Single-index model
– Assume factor is equity market or S&P500
– Use regression to estimate single index model (for FORD, for example)
Ri (t ) = ri − rf =  i +  i RM (t ) + ei (t )
• Expected Return-Beta Relationship
E (Ri ) =  i + i E (RM )
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Index Model and Diversification
• Portfolio’s variance:
 =   +  ( eP )
2
P
2
P
2
M
2
• Variance of the equally weighted
portfolio of firm-specific
components:
2
1 2
1 2
 ( eP ) =     ( ei ) =  ( e )
n
i =1  n 
n
2
• When n gets large,
becomes negligible
 (eP )
2
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Estimating Beta: SCL of Ford
𝑅2
2
2
𝛽𝐹𝑜𝑟𝑑
𝜎𝑀𝑎𝑟𝑘𝑒𝑡
= 2
2
𝛽𝐹𝑜𝑟𝑑 𝜎𝑀𝑎𝑟𝑘𝑒𝑡
+ 𝜎 2 (𝑒𝐹𝑜𝑟𝑑 )
=
0.3943
Source: BKM (11th ed), Chapter 8, observations 2011- Dec 2015
Consider more recent Ford valuation example (based on Dividend Discount Model in
2018, for illustration).
Dividends (2019)= $0.6 (forward dividend on 5/10/2018)
rf=3.4% based on US 30 year rate on 5/10/2018
Beta (2018)=0.72, on 5/10/2018
MRP=5%,
g=0.5% (assumed)
P2018=Div2019/(E(r)-g)= .... versus actual price of ....
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Betas by Sector
Industry Name N
Natural Gas Utility
Water Utility
Tobacco
Average Beta Market D/E Ratio
Unlevered Beta
0.65
62.04%
0.7
77.89%
0.73
21.57%
0.44
0.47
0.63
Educational Services
Food Processing
Telecom. Utility
0.79
0.87
1.03
8.89%
28.98%
84.06%
0.75
0.71
0.63
Aerospace/Defense
Precious Metals
1.15
1.18
23.64%
6.76%
0.97
1.12
Financial Svcs. (Div.)
Automotive
Entertainment
Steel (Integrated)
Advertising
1.37
1.5
1.72
1.72
1.79
135.83%
108.58%
37.99%
36.84%
36.55%
0.65
0.8
1.3
1.32
1.36
Semiconductor Equip
1.79
5.84%
1.7
Heavy Truck/Equip Makers
1.94
46.41%
1.42
Public/Private Equity
2.18
104.42%
1.07
Source: http://people.stern.nyu.edu/adamodar/New_Home_Page/datafile/Betas.html
Data Used: Value Line database; Date of Analysis: Data used is as of 2011; see above link for updates
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3. CAPM: Model of Expected Returns
• So far we assumed that the market index is used in the
single-index model
– Is there a model that tells us what to use as an index or
benchmark?
• CAPM
– It is the 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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Equilibrium – Capital Market Line (CML) and Security
Market Line (SML)
Mean - Std. Dev. Diagram (Assumptions)
Expected Return
Mean - Beta Diagram (Predictio n)
Expected Return
Tangency (Market) Portfolio ' m'
Tangency (Market) Portfolio
Security 1
Security 1
Security 2
Security 2
Security 3
Security 3
rf
rf
Std. Dev.
of Return
Capital market line
Beta
 =1
Security market line
Important understanding check: Why can securities lie below CML, but cannot
below/above SML
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CAPM Assumptions
1. Individual investors are price takers
2. Single-period investment horizon
3. Investments are limited to traded financial assets
4. No taxes and transaction costs
5. Information is costless and available to all investors
6. Investors are rational mean-variance optimizers
7. There are homogeneous expectations
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CAPM Derivation: Using GE Example
• Covariance of GE return with the market portfolio:
n

 n
Cov(rGE , rM ) = Cov rGE ,  wk rk  =  wk Cov(rGE , rk )
k =1

 k =1
• Therefore, the reward-to-risk ratio for investments in GE would
be:


E (rGE ) − rf
GE' s contributi on to risk premium wGE E (rGE ) − rf
=
=
GE' s contributi on to variance
wGECov(rGE , rM ) Cov(rGE , rM )
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Using GE Example Continued
• Reward-to-risk ratio for investment in market portfolio:
Market risk premium E (rM ) − rf
=
Market var iance
 M2
• Reward-to-risk ratios of GE and the market portfolio:
E (rGE ) − rf
Cov(r GE , rM )
=
E (rM ) − rf

2
M
• And the risk premium for GE:
E (rGE ) − rf =
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Cov(r GE , rM )

2
M
E (r ) − r 
M
f
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The Index Model and Realized Returns
• To move from expected to realized returns—use the index
model in excess return form and regression:
Ri ,t =  i +  i RM ,t + ei ,t
• The index model beta coefficient turns out to be the same
beta as that of the CAPM expected return-beta
relationship
• Why do investors care about systematic and not
idiosyncratic risk?
• What should be R-squared of this regression?
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Applying the CAPM in practice
• Estimating the CAPM and Beta
• Tests of the CAPM
– Why test it ?
• ‘Roll’ critique of the CAPM
• New Evidence Based on Announcement Days
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Bloomberg - Beta estimation
Example
of using
adjusted
beta in
long-short
portfolio in
practice
(see also
AJO case)
Bloomberg commands:
∙ BP/ LN Equity <GO> (type “bp/ ln” – press F8 – press Enter)
∙ BETA <GO> (type “beta” – press Enter)
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Example of exam style question
The correlation between the ABC corporation and the S&P 500
Index is 0.65. The standard deviation of the market is 18% and that
of ABC is 27%. The implied beta of ABC is closest to:
a. 1.06
b. 0.93
c. 1.20
d. 0.98
e. 1.23
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How the CAPM can be used to measure investment
performance – Alpha and Beta
Portfolio Return (r)
Fund B
α
Security
Market Line
Fund A
M
β * (rM-rf)
Risk
Free
Rate
Beta
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4. CAPM Application to Motivating
Example: CGM Focus
Source: http://finance.yahoo.com/q/rk?s=CGMFX
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Exam style question
You are a portfolio analyst and in charge of evaluating portfolio A,
which consists entirely of UK common stocks. The table below shows
data on that portfolio and on a diversified portfolio of UK stocks (the
FTSE100), which you could think of as “the market”.
Average
Annual Rate
of Return
Portfolio A
9
FTSE100
7
UK risk-free 2
rate
Standard
Deviation
Return
21
11
N/A
Beta
of
0.4
1.00
N/A
Which of the following is correct
a)
b)
c)
d)
e)
The FTSE 100 is cheap
Portfolio A's Sharpe ratio is close to 0.50
Portfolio A's Sharpe ratio is close to 0.33
Portfolio A's return adjusted Sharpe ratio is negative
Portfolio A's Sharpe ratio is inconsistent with its beta
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5. CAPM Tests: What the CAPM Predicts And How to Test It
E (Ri ) =  i + i E (RM )
Essence of CAPM: expected return on any asset is a positive
linear function of its beta and that beta is the only measure
of risk needed to explain the cross-section of expected
returns.
How to test it:
The basic idea of the approach is the use of a time series
(first pass) regression to estimate betas and the use of a
cross–sectional (second pass) regression to test the
hypothesis derived from the CAPM.
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CAPM Test Methodology: Two-Pass Regression
• Step 1 (First-Pass Regression):
For each of the N securities included in the sample, we
first run the following regression over time to estimate beta:
Rit = rit − rft = ai + bi Rmt + eit
(1)
• Step 2 (Second-Pass Regression)
We run the following cross-section regression over the
sample period over the N securities:
Ri = ri − rf =  0 +  1bi
for i=1,...,N(say 100 stocks)
Ri = ri − rf =  0 +  1bi +  2 2 ( ei )
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(2)
(2 extended)
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Test Result: SML slope is “too flat” and intercept is “too
high”.
Return %
CAPM
Estimated
SML
 0 = 0.127(0.006)
 1 = 0.042(0.006)
 2 = 0.310(0.026)
rM − rf = 0.165
Beta
Source: original Lintner (1965) coefficient estimates (standard errors), see BKM Ch.13.1
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Lecture Summary
1. Motivation
2. Index Model
3. CAPM
4. Examples of Use: Applying the CAPM to Corporate
Finance and Portfolio Management
5. Testing the CAPM
Appendix:
New Evidence on CAPM Based on Announcement
Days
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Appendix A: What is the portfolio with the Steepest Capital
Market Line?
Max SR
p
=
E (rp ) − rf
wi
p
wD =
E ( RD )  E2 − E ( RE ) Cov ( RD , RE )
E ( RD )  E2 + E ( RE )  D2 −  E ( RD ) + E ( RE )  Cov ( RD , RE )
8 − 5 ) 400 − (13 − 5 ) 72
(
=
= 0.4
(8 − 5) 400 + (13 − 5 )144 − (8 − 5 + 13 − 5 ) 72
E ( rp ) = (.4  8 ) + ( 0.6 13) = 11%
SRP =
11 − 5
14.2
Portfolio holds 40%
in bonds!
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Appendix B: Recent resurrection for the CAPM (1): CAPM and
Return %
Announcement Days
CAPM
Estimate
d SML
• Earlier tests over last 40 years: Market beta does not
help explain the cross-section of stock returns
• Savor and Wilson (2013):
– Beta/returns (cross-section) and volatility/returns (time-series)
relations look very differently depending on type of trading day
– Market beta is positively related to returns on announcement days
Beta
Certain important information is released in the form of prescheduled
announcements; If asset prices respond to this news, the risk of holding affected
assets will be higher around announcements
For individual stocks; Beta-sorted portfolios; Industry portfolios
Even for other assets (bonds, currencies, etc.)
– Beta is after all an important measure of systematic risk
What are scheduled macroeconomic news announcements?
• Employment: Jan 1964 – Dec 2011 (573 observations)
• CPI: Jan 1964 – Jan 1971 (85 observations)
• PPI: Feb 1971 – Dec 2011 (491 observations)
• PPI comes out before CPI every month
• FOMC interest rate decisions: Jan 1978 – Dec 2011 (294 observations)
• Unscheduled announcements are excluded
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Appendix B: Recent resurrection for the CAPM (2)
Savor and Wilson (2013) Figure 1: Average Excess Returns for 10
Beta-Sorted Portfolios
18.0
16.0
A
v
.
14.0
12.0
E
x
c
10.0
e
s
s
RP = 10.3 (t-stat = 13.7)
Adj. R2 = 95.9%
Ann. Day
8.0
R
e
t
u
r
n
6.0
(
4.0
Non Ann. Day
RP = -1.5 (t-stat = -3.7)
Adj. R2 = 63.1%
b
p
s
)
2.0
0.0
0.4
-2.0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
CAPM Beta
Source: Savor, Pavel G. and Wilson, Mungo Ivor, Asset Pricing: A Tale of Two Days (August 2013). AFA 2013
San Diego Meetings Paper. Available at SSRN: http://ssrn.com/abstract=2024422
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Appendix B: Recent resurrection for the CAPM (3)
Table 1: Daily Excess Returns for 10 Beta-sorted Portfolios
Panel A: Fama-MacBeth regressions
Value-weighted
A-Day
N-Day
A-Day N-day
Equal-weighted
Beta
0.00092
[2.81]
Av. R2
0.514
0.00020 -0.00010
[3.64]
[-0.89]
0.492
Intercept
0.00013
[0.90]
-0.00007
[-0.44]
0.00103
[2.89]
Beta
0.00094
[2.96]
Av. R2
0.574
0.00069 -0.00031
[16.60]
[-2.80]
0.564
Intercept
0.00089
[8.59]
0.00020
[1.99]
0.00126
[3.57]
Panel B: Pooled regression
Value-weighted
Intercept
Beta
0.00024 -0.00015
[3.26]
[-1.18]
Ann.
0.00016
[0.81]
Equal-weighted
Ann. *
Beta
0.00084
[2.74]
R2
0.001
Intercept
Beta
0.00079 -0.00039
[10.55]
[-2.85]
Ann.
0.00061
[3.01]
Ann. *
Beta
0.00119
[3.62]
R2
0.001
Source: Savor, Pavel G. and Wilson, Mungo Ivor, Asset Pricing: A Tale of Two Days (August 2013). AFA 2013
San Diego Meetings Paper. Available at SSRN: http://ssrn.com/abstract=2024422
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