lecture02 portfolio theory

7/30/2019 Lecture02 Portfolio Theory

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PORTFOLIO THEORY

Instructor: Dr. Kumail Rizvi

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RISK, RETURNAND PORTFOLIO

THEORY

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WHY TO INVEST?

To fund education plans

To fund retirement needs

To fund future liabilities in the form of insurance

claims To provide income to meet ongoing needs of a

university by endowment

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HOWTO INVEST?

Putting All Your Eggs in One Basket

Standalone Approach

Carrying Your Eggs in More Than One Basket

Portfolio Approach

A portfolio is a collection of different securities

such as stocks and bonds, that are combined

and considered a single asset

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PORTFOLIO APPROACH

Portfolio approach

provides the benefit of

diversification.

Diversification is logical

If you drop the basket,

all eggs break

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DIVERSIFICATION:

AVOID DISASTER & REDUCE RISK

Diversification has two faces:

1. Diversification helps to immunize from potentiallycatastrophic events such as the outright failure of one

of the constituent investments.(If only one investment is held, and the issuing firm goesbankrupt, the entire portfolio value and returns are lost. If aportfolio is made up of many different investments, theoutright failure of one is more than likely to be offset by gainson others, helping to make the portfolio immune to suchevents.)

2. Diversification results in an overall reduction inreturns volatility with little sacrifice in expectedreturns

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RETURNOFAPORTFOLIO

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EXPECTED RETURNOFAPORTFOLIO

The Expected Return on a Portfolio is simply theweighted average of the returns of the individual assets

that make up the portfolio:

The portfolio weight of a particular security is the

percentage of the portfolios total value that is invested

in that security.

)(n

1i

iip ERwER[8-9]

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EXPECTED RETURNOFAPORTFOLIOEXAMPLE

Portfolio value = $2,000 + $5,000 = $7,000rA = 14%, rB = 6%,

wA = weight of security A = $2,000 / $7,000 = 28.6%

wB = weight of security B = $5,000 / $7,000 = (1-28.6%)=71.4%

%288.8%284.4%004.4

)%6(.714)%14(.286)(n

1i

iip ERwER

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RANGEOF RETURNSINATWO ASSET

PORTFOLIO

In a two asset portfolio, simply by changing the weight of

the constituent assets, different portfolio returns can be

achieved.

Because the expected return on the portfolio is a simple

weighted average of the individual returns of the assets,

you can achieve portfolio returns bounded by the highest

and the lowest individual asset returns.

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RANGEOF RETURNSINATWO ASSET

PORTFOLIO

Example 1:

Assume ERA= 8% and ERB = 10%

(See the following 6 slides based on Figure 1)

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EXPECTED PORTFOLIO RETURNAFFECTON PORTFOLIO RETURNOF CHANGING RELATIVE WEIGHTSIN A

AND B

ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

FIGURE 1

ERA=8%

ERB= 10%



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AND B

FIGURE 1

ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

A portfolio manager can select the relative weights of the two

assets in the portfolio to get a desired return between 8%

(100% invested in A) and 10% (100% invested in B)



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AND B

ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

FIGURE 1

ERA

=8%

ERB= 10%

The potential returns of

the portfolio are

bounded by the highest

and lowest returns of

the individual assets

that make up the

portfolio.



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ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

FIGURE 1

ERA=8%

ERB= 10%

The expected return

on the portfolio if

100% is invested in

Asset A is 8%.

%8%)10)(0(%)8)(0.1( BBAAp ERwERwER


AND B



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AND B

FIGURE 1

ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The expected return

on the portfolio if

100% is invested in

Asset B is 10%.

%10%)10)(0.1(%)8)(0( BBAAp ERwERwER



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AND B

FIGURE 1

ExpectedReturn%

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The expected return

on the portfolio if 50%

is invested in Asset A

and 50% in B is 9%.

%9%5%4

%)10)(5.0(%)8)(5.0(

BBAAp ERwERwER



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RISKIN PORTFOLIOS

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MODERN PORTFOLIO THEORY- MPT

Prior to the establishment of Modern Portfolio Theory (MPT),

most people only focused upon investment returnsthey

ignored risk.

With MPT, investors had a tool that they could use to

dramatically reduce the risk of the portfolio without a significant

reduction in the expected return of the portfolio.

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EXPECTED RISK FOR PORTFOLIOSSTANDARD DEVIATIONOFATWO-ASSET PORTFOLIOUSING COVARIANCE

))()((2)()()()( ,2222 BABABBAAp COVwwww [8-11]

Risk of Asset A

adjusted forweight in the

portfolio

Risk of Asset B

adjusted for

weight in the

portfolio

Factor to take into

account co-movementof returns. This factor

can be negative.

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EXPECTED RISK FOR PORTFOLIOSSTANDARD DEVIATIONOFATWO-ASSET PORTFOLIOUSING

CORRELATION COEFFICIENT

))()()()((2)()()()( ,2222 BABABABBAAp wwww [8-15]

Factor that takes into

account the degree of

co-movement of

returns. It can have anegative value if

correlation is negative.

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GROUPING INDIVIDUAL ASSETSINTO

PORTFOLIOS

The riskiness of a portfolio that is made of different risky

assets is a function of three different factors:

the riskiness of the individual assets that make up the

portfolio

the relative weights of the assets in the portfolio

the degree of co-movement of returns of the assets making

up the portfolio

The standard deviation of a two-asset portfolio may be

measured using the Markowitz model:

BABABABBAAp wwww ,2222 2

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RISKOFATHREE-ASSET PORTFOLIO

The data requirements for a three-asset portfolio grows dramatically

if we are using Markowitz Portfolio selection formulae.

We need 3 (three) correlation coefficients between A and B; A and C;

and B and C.

A

B C

a,b

b,c

a,c

CACACACBCBCBBABABACCBBAAp wwwwwwwww ,,,222222 222

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RISKOFAFOUR-ASSET PORTFOLIO

The data requirements for a four-asset portfolio grows

dramatically if we are using Markowitz Portfolio selection

formulae.

We need 6 correlation coefficients between A and B; A and C; A

and D; B and C; C and D; and B and D.

A

C

B D

a,b a,d

b,c c,d

a,cb,d

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COVARIANCE

A statistical measure of the correlation of the

fluctuations of the annual rates of return of

different investments.

)-)((Prob_

,

1

_

,i BiB

n

i

AiAAB kkkkCOV

[8-12]

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CORRELATION

The degree to which the returns of two stocks co-

move is measured by the correlation coefficient ().

The correlation coefficient () between the returns

on two securities will lie in the range of +1 through

- 1.+1 is perfect positive correlation

-1 is perfect negative correlation

BA

ABAB

COV

[8-13]

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COVARIANCEAND CORRELATION

COEFFICIENT

Solving for covariance given the correlation

coefficient and standard deviation of the two

assets:

BAABABCOV [8-14]

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IMPORTANCEOF CORRELATION

Correlation is important because it affects the

degree to which diversification can be achieved

using various assets. Theoretically, if two assets returns are perfectly

negatively correlated, it is possible to build a

riskless portfolio with a return that is greater than

the risk-free rate.

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AFFECTOF PERFECTLY NEGATIVELY CORRELATED

RETURNSELIMINATIONOF PORTFOLIO RISK

Time 0 1 2

If returns of A and B are

perfectly negatively correlated,

a two-asset portfolio made up

of equal parts of Stock A and B

would be riskless. There would

be no variability

of the portfolios returns overtime.

Returns on Stock A

Returns on Stock B

Returns on Portfolio

Returns

%

10%

5%

15%

20%

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AFFECTOF PERFECTLY NEGATIVELY CORRELATED

RETURNSNUMERICAL EXAMPLE

Weight of Asset A = 50.0%

Weight of Asset B = 50.0%

Year Return onAsset AReturn on

Asset B

Expected

Return on thePortfolio

xx07 5.0% 15.0% 10.0%

xx08 10.0% 10.0% 10.0%

xx09 15.0% 5.0% 10.0%

Perfectly Negatively

Correlated Returns

over time

%10%5.7%5.2

)%15(.5)%5(.5)(n

1i

iip ERwER

%10%5.2%5.7

)%5(.5)%15(.5)(n

1i

iip ERwER

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DIVERSIFICATION POTENTIAL

The potential of an asset to diversify a portfolio is dependent

upon the degree of co-movement of returns of the asset with

those other assets that make up the portfolio.

In a simple, two-asset case, if the returns of the two assets are

perfectly negatively correlated it is possible (depending on therelative weighting) to eliminate all portfolio risk.

This is demonstrated through the following series of

spreadsheets, and then summarized in graph format.

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EXAMPLEOF PORTFOLIO COMBINATIONS

AND CORRELATION

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 1

B 14.0% 40.0%

Weight of A Weight of B

Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 17.5%

80.00% 20.00% 6.80% 20.0%

70.00% 30.00% 7.70% 22.5%

60.00% 40.00% 8.60% 25.0%50.00% 50.00% 9.50% 27.5%

40.00% 60.00% 10.40% 30.0%

30.00% 70.00% 11.30% 32.5%

20.00% 80.00% 12.20% 35.0%

10.00% 90.00% 13.10% 37.5%

0.00% 100.00% 14.00% 40.0%

Portfolio Components Portfolio Characteristics

Perfect

Positive

Correlation

no

diversificatio

n

Both

portfolio

returns and

risk are

bounded by

the range

set by theconstituent

assets when

=+1

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AND CORRELATION

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 15.9%

80.00% 20.00% 6.80% 17.4%

70.00% 30.00% 7.70% 19.5%

60.00% 40.00% 8.60% 21.9%50.00% 50.00% 9.50% 24.6%

40.00% 60.00% 10.40% 27.5%

30.00% 70.00% 11.30% 30.5%

20.00% 80.00% 12.20% 33.6%

10.00% 90.00% 13.10% 36.8%

0.00% 100.00% 14.00% 40.0%


Positive

Correlation

weak

diversification

potential

When =+0.5

these portfolio

combinations

have lower

risk

expected

portfolio returnis unaffected.

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AND CORRELATION

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 14.1%

80.00% 20.00% 6.80% 14.4%

70.00% 30.00% 7.70% 15.9%

60.00% 40.00% 8.60% 18.4%50.00% 50.00% 9.50% 21.4%

40.00% 60.00% 10.40% 24.7%

30.00% 70.00% 11.30% 28.4%

20.00% 80.00% 12.20% 32.1%

10.00% 90.00% 13.10% 36.0%

0.00% 100.00% 14.00% 40.0%


No

Correlation

some

diversification

potential

Portfolio

risk is

lower

than the

risk of

eitherasset A or

B.

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AND CORRELATION

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 12.0%

80.00% 20.00% 6.80% 10.6%

70.00% 30.00% 7.70% 11.3%

60.00% 40.00% 8.60% 13.9%50.00% 50.00% 9.50% 17.5%

40.00% 60.00% 10.40% 21.6%

30.00% 70.00% 11.30% 26.0%

20.00% 80.00% 12.20% 30.6%

10.00% 90.00% 13.10% 35.3%

0.00% 100.00% 14.00% 40.0%


Negative

Correlation

greater

diversification

potential

Portfolio risk

for more

combinations

is lower than

the risk of

either asset

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AND CORRELATION

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -1

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 9.5%

80.00% 20.00% 6.80% 4.0%

70.00% 30.00% 7.70% 1.5%

60.00% 40.00% 8.60% 7.0%50.00% 50.00% 9.50% 12.5%

40.00% 60.00% 10.40% 18.0%

30.00% 70.00% 11.30% 23.5%

20.00% 80.00% 12.20% 29.0%

10.00% 90.00% 13.10% 34.5%

0.00% 100.00% 14.00% 40.0%


Perfect

Negative

Correlation

greatest

diversification

potential

Risk of the

portfolio is

almost

eliminated at

70% investedin asset A

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Diversification of a Two Asset

Portfolio Demonstrated Graphically

The Effect of Correlation on Portfolio Risk:

The Two-Asset Case

Expected

Return

Standard Deviation

0%

0% 10%

4%

8%

20% 30% 40%

12%

B

AB= +1

A

AB = 0

AB = -0.5AB = -1

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IMPACTOFTHE CORRELATION COEFFICIENT

Figure 2 (see the next slide) illustrates the

relationship between portfolio risk () and the

correlation coefficient

The slope is not linear a significant amount ofdiversification is possible with assets with no correlation (it

is not necessary, nor is it possible to find, perfectly

negatively correlated securities in the real world)

With perfect negative correlation, the variability of portfolio

returns is reduced to nearly zero.

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EXPECTED PORTFOLIO RETURNIMPACTOFTHE CORRELATION COEFFICIENT

FIGURE 2

15

10

5

0

StandardDevia

tion

(%)ofPortfolio

Returns

Correlation Coefficient

()

-1 -0.5 0 0.5 1



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FOR N-ASSETS PORTFOLIO

Kumail Rizvi, PhD, CFA, FRM 8 - 40


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EVOLUTIONOF MODERN

PORTFOLIO THEORY

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EVOLUTIONOF MODERN PORTFOLIO

THEORY

Efficient Frontier

Capital Allocation Line (CAL)

Capital Market Line (CML)

Security Market Line (SML) Capital Asset Pricing Model (CAPM)

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THE EFFICIENT FRONTIERThe Capital Asset Pricing Model (CAPM)

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EFFICIENT FRONTIER

Markowitz efficient frontier contains allportfolios of risky assets that rational, risk

averse investors will choose.

Specifically, Harry Markowitzs efficient

portfolios are:

Those portfolios providing the minimum risk for a

certain level of return

Those portfolios providing the maximum return for

their level of risk

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A S


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ACHIEVABLE SETOF PORTFOLIO

COMBINATIONSGETTINGTOTHEN ASSET CASE

In a real world investment universe with all ofthe investment alternatives (stocks, bonds,money market securities, hybrid instruments,gold real estate, etc.) it is possible to constructmany different alternative portfolios out of riskysecurities.

Each portfolio will have its own unique expectedreturn and risk.

Whenever you construct a portfolio, you can

measure two fundamental characteristics of theportfolio: Portfolio expected return (ERp)

Portfolio risk (p)

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THE ACHIEVABLE SETOF PORTFOLIO

COMBINATIONS

You could start by randomly assembling ten risky

portfolios.

The results (in terms of ERp

and p

)might look like

the graph on the following page:

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ACHIEVABLE PORTFOLIO COMBINATIONSTHE FIRST TEN COMBINATIONS CREATED

Portfolio Risk (p)

10 Achievable

Risky PortfolioCombinations

ERp

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THE ACHIEVABLE SETOF PORTFOLIO

COMBINATIONS

You could continue randomly assembling more

portfolios.

Thirty risky portfolios might look like the graph on thefollowing slide:

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ACHIEVABLE PORTFOLIO COMBINATIONSTHIRTYCOMBINATIONS NAIVELYCREATED

Portfolio Risk (p)

30 Risky Portfolio

Combinations

ERp

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ACHIEVABLE SETOF PORTFOLIO

COMBINATIONSALL SECURITIESMANYHUNDREDSOF DIFFERENT COMBINATIONS

When you construct many hundreds of different

portfolios naively varying the weight of the individual

assets and the number of types of assets themselves,

you get a set of achievable portfolio combinations asindicated on the following slide:

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Portfolio Risk (p)

ERp

ACHIEVABLE PORTFOLIO COMBINATIONSMORE POSSIBLE COMBINATIONS CREATED

E

E is the

minimu

m

variance

portfolio

Achievable Set of

Risky Portfolio

Combinations

The highlighted

portfolios are

efficient in that

they offer the

highest rate of

return for a given

level of risk.Rationale

investors will

choose only from

this efficient set.

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Portfolio Risk (p)

Achievable Set of

Risky Portfolio

Combinations

ERp

ACHIEVABLE PORTFOLIO COMBINATIONSEFFICIENT FRONTIER (SET)

E

Efficient

frontier is

the set of

achievable

portfolio

combinationsthat offer the

highest rate

of return for

a given level

of risk.

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NOTTO FORGETTHE BULGE OF

DIVERSIFICATION12

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EFFICIENT FRONTIER12

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Global

minimum-

variance

Efficient Frontier


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EFFICIENT FRONTIER12

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CAPITAL MARKET LINE12

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CAPITAL MARKET LINE12

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SECURITY MARKET LINE12

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SECURITY MARKET LINE12

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CAPITAL ASSET PRICING MODEL12

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PORTFOLIO PERFORMANCE MEASURES12

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PORTFOLIO PERFORMANCE MEASURES12

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EXERCISE:12

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lecture02 portfolio theory

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