Portfolio Managment
3-228-07
Albert Lee Chun
Proof of the Capital Asset
Pricing Model
Lecture 6
0
Course Outline
Sessions 1 and 2 : The Institutional Environment
 Sessions 3, 4 and 5: Construction of Portfolios
 Sessions 6 and 7: Capital Asset Pricing Model
 Session 8: Market Efficiency
 Session 9: Active Portfolio Management
 Session 10: Management of Bond Portfolios
 Session 11: Performance Measurement of Managed
Portfolios

1
Plan for Today
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Fun Proof of the CAPM
Zero-Beta CAPM (not on the syllabus)
A few examples
Revision for the mid-term
Albert Lee Chun
Portfolio Management
2
A Fun Proof of the CAPM
Albert Lee Chun
Portfolio Management
3
CAPM Says that
E(R port )
for any security i that we pick,
the expected return of that
security is given by
Capital
Market
Line
M
security i
 port
Albert Lee Chun
Portfolio Management
4
Why does CAPM work?
E(R port )
Green line traces out the
set of possible portfolios P
using security i and M by
varying w,
M
P
Rf
Capital
Market
Line
where w is the
weight on
security i in
portfolio P
security i
 port
Albert Lee Chun
Portfolio Management
5
Why does CAPM work?
E(R port )
Note that w=1 corresponds
to security i and w=0
gives us the market
portfolio M,
w=0
M
P
Rf
Capital
Market
Line
where w is the
weight on
security i in
portfolio P
security i
w=1
 port
Albert Lee Chun
Portfolio Management
6
Why does CAPM work?
E(R port )
For any weight w, we can
easily compute the expected
return and the variance of
portfolio P,
w=0
M
P
Rf
Capital
Market
Line
where w is the
weight on
security i in
portfolio P
security i
w=1
 port
Albert Lee Chun
Portfolio Management
7
Why does CAPM work?
E(R port )
Note that the CML (orange
line) is tangent to both the
risky efficient frontier (blue
line) and the green line at M.
w=0
M
P
Rf
Capital
Market
Line
security i
w=1
Intuition: The orange
line, the blue line and
the green line all touch
at only 1 point M.
Why?
 port
Albert Lee Chun
Portfolio Management
8
Why does CAPM work?
E(R port )
Slope of the green line at
M, is equal to the slope of
the blue line at M which is
equal to the slope of the
CML(orange line)!
w=0
Rf
Capital
Market
Line
M
security i
Intuition: The orange
line, the blue line and
the green line all touch
at only 1 point M.
Why?
 port
Albert Lee Chun
Portfolio Management
9
Why does CAPM work?
E(R port )
Slope of the green line at M, is
equal to the slope of the blue line
at M which is equal to the slope of
the CML(orange line)!
w=0
Capital
Market
Line
M
The slope of the CML
Rf
security i
 port
Albert Lee Chun
Portfolio Management
10
Why does CAPM work?
(slope = slope = slope)
E(R port )
Capital
Market
Line
w=0
M
Therefore, the slope of
all 3 lines at M is
Rf
Albert Lee Chun
security i
Portfolio Management
11
Why does CAPM work?
E(R port )
Mathematically the slope of the green
line at M is:
Capital
Market
Line
w=0
M
The slope of all 3 lines
at M is
Rf
security i
 port
Albert Lee Chun
Portfolio Management
12
Why does CAPM work?
Note that we can also express the slope of the green line as as:
E(R port )
w=0
Rf
This slope has to
equal the slope of
the CML at M!
M
security i
=
 port
Albert Lee Chun
Portfolio Management
13
We want to
find the slope
of the green
line
Proof of CAPM
by
differentiating
these at w = 0
and using this
relation
to set the slope
at (w = 0)
equal to the
slope of the
CML
Albert Lee Chun
=
Portfolio Management
14
Proof of CAPM
E(R port )
=
w=0
Rf
M
security i
To prove CAPM we use the fact that the green slope has to
equal the slope of the CML at M.
 port
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Portfolio Management
15
Let’s Take a Few Derivatives
Derivative of expected
return w.r.t w.
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Portfolio Management
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Let’s Take a Few Derivatives
Derivative of standard deviation w.r.t. w
Evaluate the derivative at w = 0, which is at the market portfolio!
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Portfolio Management
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Equate the Slopes
=
=
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Portfolio Management
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Equating the Slopes
Capital
Market
Line
w=0
Rf
M
security i
 port
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Portfolio Management
19
Now Solve for E(Ri)
Voila! We just proved the CAPM!!
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Portfolio Management
20
We just showed that
E(R port )
for any security i that we pick,
the expected return of that
security is given by
M
Rf
security i
So we just won the
Nobel Prize!
 port
Albert Lee Chun
Portfolio Management
21
Zero-Beta Capital Asset Pricing Model
(Not on the Syllabus: However, understanding this might be
useful for solving other problems on the exam.)
Albert Lee Chun
Portfolio Management
22
Suppose There is No Risk Free Asset
E(R port )
Can we say something about the expected
return of a particular asset in this
economy?
Efficient
frontier
 port
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Portfolio Management
23
Zero Beta CAPM
Fisher Black (1972)
There exists an efficient portfolio that is uncorrelated
with the market portfolio, hence it has zero beta.
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Portfolio Management
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Zero-Beta CAPM World
E(R i )
Efficient
frontier
E(R ZB )
Albert Lee Chun
Zero-Beta
Portfolio
Portfolio Management

25
Zero-Beta SML
E(R i )
SML
E(R M )
E(R ZB )
0
Albert Lee Chun
1.0
Portfolio Management
Beta
26
Example CAPM
Suppose there are 2 efficient risky securities:
Security
E(r)
Beta
Egg
0.07
0.50
Bert
0.10
0.80
You do not know E(Rm) or Rf.
Suppose that Karina is thinking about buying the following:
Security
E(r)
Beta
Karina
0.16
1.30
Should she buy the security?
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Portfolio Management
27
Under Valued or Overvalued
Undervalued
E(ri )
Buy!
SML
Market
E(rm )
Bert
Egg
Overvalued
rf
Don`t Buy!
0
Albert Lee Chun
1.0
Portfolio Management
Beta
28
Example CAPM
We know that for the two efficient securities:
E(REgg) = rf + BEgg(E(Rm)- Rf)
E(RBert) = rf + BBert(E(Rm)- Rf)
And if Karina is an efficient security we would have:
E(RKarina) = rf + BKarina(E(Rm) - Rf)
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Portfolio Management
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Example CAPM
First find the expected return on the market and the risk-free
retrun by solving 2 equations in 2 unknowns:
E(REgg) = (1- BEgg) Rf + BEgg E(Rm)
E(RBert) = (1- BBert) Rf + BBert E(Rm)
Some algebra:
(E(REgg) - (1- BEgg) Rf )/ BEgg = (E(RBert) - (1- BBert) Rf )/ BBert
Rf = [BBert E(REgg) - BEgg E(RBert)]/ [BEgg(1-BBert ) + BBert (1- BEgg) ]
E(Rm)= (E(REgg) - (1- BEgg) Rf )/ BEgg
Albert Lee Chun
Portfolio Management
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Example CAPM
Security
Egg
Bert
Karina
E(r)
.07
.1
.16
Beta
.5
.8
1.3
Rf = [BBert E(REgg) - BEgg E(RBert)]/ [-BEgg(1-BBert ) + BBert (1- BEgg)
]
= .02
E(Rm)= (E(REgg) - (1- BEgg) Rf )/ BEgg
= .12
E(RKarina) = rf + BKarina(E(Rm) - Rf)
=.02 + 1.3*(.12 - .02) = .15 < .16
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Portfolio Management
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Stock is Under Valued
Undervalued
E(ri )
Buy!
Karina
16%
SML
Market
E(rm )
15%
Bert
Egg
rf
0
Albert Lee Chun
1.0
Portfolio Management
Béta
32
Another Example
State of the
Economy
Probability
Return
Eggbert
Rerurn
Dingo
Risk-Free Rate
Bad
0.20
0.04
0.07
0.03
Good
0.45
0.10
0.10
0.03
Great
0.35
0.22
0.19
0.03
Expected
Return
?
?
Variance
?
?
Coefficient of
Correlation
with the
market
0.712
0.842
Covariance with
the Market
0.0015
?
Albert Lee Chun
Portfolio Management
Example
The expected return on the market portfolio is 9%.
A) Determine the covariance between the return on
Dingo and the return on the market portfolio.
B) Determine the rate of return on Dingo using CAPM.
Would you recommend that investors buy shares of
Dingo? (Justify your answer)
Albert Lee Chun
Portfolio Management
Solution :
E(re) = 13,00%
E(rd) = 12,55%
Var(re) = 0,004860
Var(rd) = 0,002365
STD(re) = 0,069714
STD(rd) = 0,048629
STD Market= 0,030220
Var Market = 0,000913
Covariance of Dingo with the market = 0,001237
Beta of Dingo = 1,35
Expected Reeturn of the Market = 9%
Expect Return of Dingo according to CAPM :
E(rd) = Rf + BetaDingo (E(Rm) - Rf) = 11,13%
12,55% > 11,13% - Buy! Lies above the SML.
Albert Lee Chun
Portfolio Management
35
Midterm
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Focus on solving examples that I gave you to do at
home and what we did in class.
Do the math as well as know the intuition.
The lecture notes are more important than the book,
although the book is important too.
Focus on Lectures 3 – 6
Albert Lee Chun
Portfolio Management