Download - Portfolio Managment 3-228-07 Albert Lee Chun
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Portfolio ManagmentPortfolio Managment3-228-073-228-07
Albert Lee ChunAlbert Lee Chun
Proof of the Capital Asset Proof of the Capital Asset Pricing Model Pricing Model
Lecture 6
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Course Outline Course Outline
Sessions 1 and 2 : The Institutional Environment Sessions 1 and 2 : The Institutional Environment Sessions 3, 4 and 5: Construction of PortfoliosSessions 3, 4 and 5: Construction of Portfolios Sessions Sessions 66 and 7: and 7: Capital Asset Pricing ModelCapital Asset Pricing Model Session 8: Market EfficiencySession 8: Market Efficiency Session 9: Active Portfolio ManagementSession 9: Active Portfolio Management Session 10: Management of Bond PortfoliosSession 10: Management of Bond Portfolios Session 11: Performance Measurement of Managed Session 11: Performance Measurement of Managed PortfoliosPortfolios
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Albert Lee Chun Portfolio Management 3
Plan for TodayPlan for Today
Fun Proof of the CAPMFun Proof of the CAPM Zero-Beta CAPM (not on the syllabus)Zero-Beta CAPM (not on the syllabus) A few examplesA few examples Revision for the mid-termRevision for the mid-term
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Albert Lee Chun Portfolio Management 4
A Fun Proof of the CAPMA Fun Proof of the CAPM
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Albert Lee Chun Portfolio Management 5
CAPM Says that CAPM Says that
security i
Capital Market
Line
for any security i that we pick, the expected return of that
security is given by
M
)E(R port
port
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Albert Lee Chun Portfolio Management 6
Why does CAPM work?Why does CAPM work?
)E(R port
security i
P
Capital Market
Line
Green line traces out the set of possible portfolios P using security i and M by
varying w,
port
M where w is the weight on
security i in portfolio P
fR
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Albert Lee Chun Portfolio Management 7
Why does CAPM work?Why does CAPM work?
)E(R port
security i
Capital Market
Line
port
w = 1
PMw = 0 where w is the
weight on security i in portfolio P
Note that w=1 corresponds to security i and w=0 gives us the market
portfolio M,
fR
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Albert Lee Chun Portfolio Management 8
Why does CAPM work?Why does CAPM work?
security i
Capital Market
Line
For any weight w, we can easily compute the expected
return and the variance of portfolio P,
port
w = 1
P
)E(R port
M where w is the weight on
security i in portfolio P
w = 0
fR
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Albert Lee Chun Portfolio Management 9
Why does CAPM work?Why does CAPM work?
)E(R port
security i
Capital Market
Line
port
Intuition: The orange line, the blue line and
the green line all touch at only 1 point M.
Why?w = 1
P
Note that the CML (orange line) is tangent to both the
risky efficient frontier (blue line) and the green line at M.
Mw = 0
fR
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Albert Lee Chun Portfolio Management 10
Why does CAPM work?Why does CAPM work?
)E(R port
security i
Capital Market
Line
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)!
port
Intuition: The orange line, the blue line and
the green line all touch at only 1 point M.
Why?
Mw = 0
fR
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Albert Lee Chun Portfolio Management 11
Why does CAPM work?Why does CAPM work?
)E(R port
security i
Capital Market
Line
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)!
port
The slope of the CML
Mw = 0
fR
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Albert Lee Chun Portfolio Management 12
Why does CAPM work?Why does CAPM work?
security i
Capital Market
Line
Therefore, the slope of all 3 lines at M is
Mw = 0
(slope = slope = slope))E(R port
fR
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Albert Lee Chun Portfolio Management 13
Why does CAPM work?Why does CAPM work?
)E(R port
security i
Capital Market
Line
Mathematically the slope of the green line at M is:
port
Mw = 0
The slope of all 3 lines at M is
fR
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Albert Lee Chun Portfolio Management 14
Why does CAPM work?Why does CAPM work?
security i
Note that we can also express the slope of the green line as as:
port
=
This slope has to equal the slope of
the CML at M!
Mw = 0
)E(R port
fR
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Albert Lee Chun Portfolio Management 15
Proof of CAPMProof 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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Albert Lee Chun Portfolio Management 16
Proof of CAPMProof of CAPM
security i
port
=
To prove CAPM we use the fact that the green slope has to equal the slope of the CML at M.
Mw = 0
)E(R port
fR
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Albert Lee Chun Portfolio Management 17
Let’s Take a Few DerivativesLet’s Take a Few Derivatives
Derivative of expected return w.r.t w.
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Albert Lee Chun Portfolio Management 18
Let’s Take a Few DerivativesLet’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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Albert Lee Chun Portfolio Management 19
Equate the SlopesEquate the Slopes
=
=
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Albert Lee Chun Portfolio Management 20
Equating the SlopesEquating the Slopes
security i
Capital Market
Line
port
Mw = 0
fR
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Albert Lee Chun Portfolio Management 21
Now Solve for E(RNow Solve for E(Rii))
Voila! We just proved the CAPM!!
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Albert Lee Chun Portfolio Management 22
We just showed that We just showed that
security i
for any security i that we pick, the expected return of that
security is given by
M
)E(R port
port
So we just won the Nobel Prize!
fR
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Albert Lee Chun Portfolio Management 23
Zero-Beta Capital Asset Pricing ModelZero-Beta Capital Asset Pricing Model(Not on the Syllabus: However, understanding this might be (Not on the Syllabus: However, understanding this might be
useful for solving other problems on the exam.)useful for solving other problems on the exam.)
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Albert Lee Chun Portfolio Management 24
Suppose There is No Risk Free Asset Suppose There is No Risk Free Asset
Can we say something about the expected return of a particular asset in this
economy?)E(R port
port
Efficientfrontier
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Albert Lee Chun Portfolio Management 25
Zero Beta CAPMZero Beta CAPM
Fisher Black (1972)Fisher Black (1972)
There exists an efficient portfolio that is uncorrelated There exists an efficient portfolio that is uncorrelated with the market portfolio, hence it has zero beta.with the market portfolio, hence it has zero beta.
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Albert Lee Chun Portfolio Management 26
Zero-Beta CAPM World Zero-Beta CAPM World
Efficientfrontier
)E(R i
)E(R ZB Zero-Beta Portfolio
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Albert Lee Chun Portfolio Management 27
Zero-Beta SMLZero-Beta SML
)E(R i
Beta0.1
)E(R ZB
SML
0
)E(R M
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Albert Lee Chun Portfolio Management
Example CAPMExample CAPM
Suppose there are 2 efficient risky securities:Suppose there are 2 efficient risky securities:
SecuritySecurity E(r) E(r) BetaBeta
EggEgg 0.070.07 0.500.50
BertBert 0.100.10 0.800.80
You do not know E(Rm) or Rf.You do not know E(Rm) or Rf.
Suppose that Karina is thinking about buying the following:Suppose that Karina is thinking about buying the following:
SecuritySecurity E(r) BetaE(r) Beta
KarinaKarina 0.160.16 1.301.30
Should she buy the security?Should she buy the security?
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Albert Lee Chun Portfolio Management 29
Under Valued or OvervaluedUnder Valued or Overvalued
Beta0.10
Undervalued
Buy!
Overvalued
Don`t Buy!
SML
fr
)E(ri
)E(rm
EggBert
Market
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Albert Lee Chun Portfolio Management
Example CAPMExample CAPM
We know that for the two efficient securities:We know that for the two efficient securities:
E(RE(REggEgg) = r) = rff + B + BEggEgg(E(R(E(Rm)m)- R- Rff))
E(RE(RBertBert) = rf + B) = rf + BBertBert(E(R(E(Rm)m)- R- Rff))
And if Karina is an efficient security we would have:And if Karina is an efficient security we would have:
E(RE(RKarinaKarina) = rf + B) = rf + BKarinaKarina(E(R(E(Rm) m) - R- Rff))
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Albert Lee Chun Portfolio Management
Example CAPMExample CAPM
First find the expected return on the market and the risk-free First find the expected return on the market and the risk-free retrun by solving 2 equations in 2 unknowns:retrun by solving 2 equations in 2 unknowns:
E(RE(REggEgg) = (1- B) = (1- BEggEgg)) RRff + B + BEgg Egg E(RE(Rm)m)
E(RE(RBertBert) = (1- B) = (1- BBertBert)) RRff + B + BBert Bert E(RE(Rm)m)
Some algebra:Some algebra:
(E(R(E(REggEgg) - (1- B) - (1- BEggEgg) ) RfRf )/ B )/ BEggEgg = = (E(R(E(RBertBert) - (1- B) - (1- BBertBert) ) RfRf )/ B )/ BBertBert
RRff = = [[BBBertBert E(RE(REggEgg) - B) - BEgg Egg E(RE(RBertBert)]/ [B)]/ [BEggEgg(1-B(1-BBert Bert ) + B) + BBert Bert (1- B(1- BEggEgg) ]) ]
E(RE(Rm)m)= = (E(R(E(REggEgg) - (1- B) - (1- BEggEgg) ) RfRf )/ B )/ BEggEgg
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Albert Lee Chun Portfolio Management
Example CAPMExample CAPM
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Security E(r) BetaEgg .07 .5Bert .1 .8
Karina .16 1.3
Rf = Rf = [B[BBertBert E(RE(REggEgg) - B) - BEggEgg E(R E(RBertBert)]/ [-B)]/ [-BEggEgg(1-B(1-BBertBert ) + B ) + BBertBert (1- B (1- BEggEgg) ]) ]
= .02= .02E(Rm)= E(Rm)= (E(R(E(REggEgg) - (1- B) - (1- BEggEgg) ) RfRf )/ B )/ BEggEgg
= .12= .12
E(RE(RKarinaKarina)) = rf + B = rf + BKarinaKarina(E(Rm) - Rf)(E(Rm) - Rf)
=.02 + 1.3*(.12 - .02) = =.02 + 1.3*(.12 - .02) = .15.15 < .16 < .16
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Albert Lee Chun Portfolio Management 33
Stock is Under Valued Stock is Under Valued
Béta0.10
Undervalued
Buy!
SML
fr
)E(ri
)E(rm
EggBert
Market
Karina
16%
15%
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Albert Lee Chun Portfolio Management
Another Example
State of the Economy
Probability Return Eggbert
RerurnDingo
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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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)
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Albert Lee Chun Portfolio Management
Solution : Solution :
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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.
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Albert Lee Chun Portfolio Management
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