Portfolio Managment 3-228-07 Albert Lee Chun

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Portfolio Managment3-228-07 Albert Lee Chun
Proof of the Capital Asset Pricing Model
Lecture 6
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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

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Plan for Today

• Fun Proof of the CAPM
• Zero-Beta CAPM (not on the syllabus)
• A few examples
• Revision for the mid-term

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A Fun Proof of the CAPM
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CAPM Says that
for any security i that we pick, the expected
return of that security is given by
Capital Market Line
M
security i
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Why does CAPM work?
Green line traces out the set of possible
portfolios P using security i and M by varying w,
Capital Market Line
where w is the weight on security i in portfolio
P
M
P
security i
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Why does CAPM work?
Note that w1 corresponds to security i and w0
gives us the market portfolio M,
Capital Market Line
where w is the weight on security i in portfolio
P
w 0
M
P
security i
w 1
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Why does CAPM work?
For any weight w, we can easily compute the
expected return and the variance of portfolio P,
Capital Market Line
where w is the weight on security i in portfolio
P
w 0
M
P
security i
w 1
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Why does CAPM work?
Note that the CML (orange line) is tangent to
both the risky efficient frontier (blue line) and
the green line at M.
Capital Market Line
w 0
M
P
Intuition The orange line, the blue line and the
green line all touch at only 1 point M. Why?
security i
w 1
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Why does CAPM work?
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)!
Capital Market Line
w 0
M
Intuition The orange line, the blue line and the
green line all touch at only 1 point M. Why?
security i
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Why does CAPM work?
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)!
Capital Market Line
w 0
M
The slope of the CML
security i
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Why does CAPM work?
(slope slope slope)
Capital Market Line
w 0
M
Therefore, the slope of all 3 lines at M is
security i
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Why does CAPM work?
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
security i
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Why does CAPM work?
Note that we can also express the slope of the
green line as as
This slope has to equal the slope of the CML at M!
w 0
M


security i
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Proof of CAPM
We want to find the slope of the green line by
differentiating these at w 0 and using this
relation to set the slope at (w 0) equal to
the slope of the CML

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Proof of CAPM

w 0
M

security i
To prove CAPM we use the fact that the green
slope has to equal the slope of the CML at M.
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Lets Take a Few Derivatives
Derivative of expected return w.r.t w.
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Lets 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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Equate the Slopes


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Equating the Slopes
Capital Market Line
w 0
M
security i
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Now Solve for E(Ri)
Voila! We just proved the CAPM!!
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We just showed that
for any security i that we pick, the expected
return of that security is given by
M
security i
So we just won the Nobel Prize!
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Zero-Beta Capital Asset Pricing Model(Not on the
Syllabus However, understanding this might be
useful for solving other problems on the exam.)
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Suppose There is No Risk Free Asset
Can we say something about the expected return of
a particular asset in this economy?
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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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Zero-Beta CAPM World
Zero-Beta Portfolio
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Zero-Beta SML
SML
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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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Under Valued or Overvalued
Undervalued Buy!
SML
Market
Bert
Egg
Overvalued Dont Buy!
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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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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

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Example CAPM
Security E(r) Beta Egg
.07 .5 Bert .1 .8 Karina .16 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 lt .16
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Stock is Under Valued
Undervalued Buy!
Karina
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SML
Market
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Bert
Egg
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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 ?
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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)

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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 gt 11,13 -
Buy! Lies above the SML.
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Midterm

• 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