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

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


A Fun Proof of the CAPMA Fun Proof of the CAPM


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


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



)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



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



)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



)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



)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



security i

Capital Market

Line

Therefore, the slope of all 3 lines at M is

Mw = 0

(slope = slope = slope))E(R port

fR



)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



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


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


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


Let’s Take a Few DerivativesLet’s Take a Few Derivatives

Derivative of expected return w.r.t w.


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!


Equate the SlopesEquate the Slopes

=

=


Equating the SlopesEquating the Slopes

security i

Capital Market

Line

port

Mw = 0

fR


Now Solve for E(RNow Solve for E(Rii))

Voila! We just proved the CAPM!!


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


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.)


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


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.


Zero-Beta CAPM World Zero-Beta CAPM World

Efficientfrontier

)E(R i

)E(R ZB Zero-Beta Portfolio


Zero-Beta SMLZero-Beta SML

)E(R i

Beta0.1

)E(R ZB

SML

0

)E(R M

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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Under Valued or OvervaluedUnder Valued or Overvalued

Beta0.10

Undervalued

Buy!

Overvalued

Don`t Buy!

SML

fr

)E(ri

)E(rm

EggBert

Market



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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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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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


Stock is Under Valued Stock is Under Valued

Béta0.10

Undervalued

Buy!

SML

fr

)E(ri

)E(rm

EggBert

Market

Karina

16%

15%


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 ?


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)


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.


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

portfolio managment 3-228-07 albert lee chun

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