risk and diversification

8/6/2019 Risk and Diversification

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Risk and Return

What is Return?????

Quantification of Return????

What is the Measurement Unit????

What is Risk?????

Quantification of Risk??????

What is the Measurement Unit?????


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Risk and Return

Risk the chance that realized returns differsignificantly from expected returns

We assume investors like returns and dislike risk

Historically, there has been a tradeoff betweenrisk and return

To achieve a higher return, an investor should bewilling to accept more risk

Lower Risk is obtained only through lower returns

Can we measure Risk?


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Measures of Risk

Standard Deviation/Variance measures a portfolios total volatilityand its total risk (that is, systematic

and unsystematic risk) The total risk is divided into two parts.

Total Risk= Systematic Risk +Unsystematic Risk


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Variance and Standard

DeviationStandard Deviation:- Extent of deviation of returnsfrom the average value of returns. Square root of theaverage of square of deviation of the observedreturns from their expected value of returns.

Variance in the average value of the squares ofdeviation of the observed returns from the expectedvalue of return.


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Variance and StandardDeviation

The lower the variance and standarddeviation, the less risk is associated withthat asset.

The risk in question deals with theuncertainty that is present with possiblereturn scenarios.

For example, if there is no risk ( anexpected return with absolute certainty),then there is no variance or standarddeviation of the return.


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Investment Risks

Systematic Risks/Non Diversifiable/Uncontrollable

Market risk

Interest rate risk

Purchasing power risk Foreign currency (exchange rate) risk

Reinvestment risk

Unsystematic Risks/Diversifiable/Controllable Business risk

Financial risk

Default risk

Country (or regulation) risk


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Measure of Systematic Risk

Beta a commonly used measure ofsystematic risk that is derived fromregression analysis

Unsystematic risk is residual.


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MANAGING RISK

A person is said to be risk averse ifhe/she exhibits a dislike ofuncertainty.


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MANAGING RISK

Individuals can reduce risk, choosingany of the following:

Buy insurance

Diversify

Accept a lower return on theirinvestments


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The Markets for Insurance

One way to deal with risk is to buy

insuranceinsurance.

The general feature of insurancecontracts is that a person facing arisk pays a fee to an insurancecompany, which in return agrees to

accept all or part of the risk.


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Diversification of UnsystematicRisk

Diversification refers to the reductionof risk achieved by replacing a singlerisk with a large number of smaller

unrelated risks.


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Unsystematic risk is the risk thataffects only a single person. Theuncertainty associated with specific

companies.


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Aggregate risk is the risk that affectsall economic actors at once, the

uncertainty associated with theentire economy.

Diversification cannot removeaggregate risk.


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Diversification

49

20

IUnsystematic

risk

Aggregate

risk

Number OF STOCKS

Number of

isk (standard Deviation)

(Less risk)

0 1 4 6 810 20 30 40


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The Tradeoff between Risk and Return

3.1

8.3

Return

Risk0 5 10 15 20

Nostocks

25%

stocks

50%stocks

75%stocks

100%stocks

Need notbe a linearline


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Type of Calculations in Risk

and Return


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Ex Ante

"before the event

A term that refers to future events, such as future returnsor prospects of a company. Using ex-ante analysis helps to givean idea of future movements in price or the future impact ofa newly implemented policy.

An example of ex-ante analysis is when an investmentcompany values a stock ex-ante and then compares thepredicted results to the actual movement of the stock's price.

Limitation-Returns and risk can be calculated after-the-fact (ie. You use actual realized return data). This isknown as an ex postcalculation.

Or you can use forecast datathis is an ex antecalculation.


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Ex Post

Ex-post translated from Latin means "after the fact". The use of historical returns has traditionally beenthe most common way to predict the probability ofincurring a loss on any given day. Ex-post is theopposite of ex-ante, which means "before theevent".

Companies may try to obtain ex-post data toforecast future earnings. Another common use for

ex-post data is in studies such as value at risk (VaR),a probability study used to estimate the maximumamount of loss a portfolio could incur on any givenday.


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Ex Post Calculation of Risk

and return


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Risk- Standard Deviation (expost)

The formula for the standarddeviation when analyzing sampledata (realized returns or ex post) is:

1

)(1

2

=

=

n

kkn

i

ii

Where kis a realized return on thestock and n is the number of returnsused in the calculation of the mean.


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Holding Period Return(For investments that yield dividend cash flow

returns)

0

101

PriceBeginning

DividendPriceBeginning-PriceEnding

P

DPPHPR

HPR

+=

+=


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The Bias Inherent in theArithmetic Average

Arithmetic averages can yield incorrect results becauseof the problems of bias inherent in its calculation.

Consider an investment that was purchased for 10, rose to20 and then fell back to 10.

Let us calculate the HPR in both periods:

The arithmetic average return earned on this investmentwas:

%5020

10

20

2010

%10010

10

10

1020

2

1

=

=

=

==

=

HPR

HPR

%252

%50

2

%50%100==

=Average


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The Bias Inherent in theArithmetic Average

Example Continued ...

The answer is clearly incorrect since the investorstarted with 10 and ended with 10.

The correct answer may be obtained through the

use of the geometric average:

0111)1(

1)]5)(.2[(

1%))]50(1%)(1001[(

1)1(

2/1

2/1

2/1

1

===

=

++=

+= =

n i

n

i

rerageGeometricA


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Geometric Versus ArithmeticAverage Returns

Consider two investments with the following realized returns over thepast few years:

Holding Period Returns

Year

IBM

Stock

Government

Bonds

2000 12.0% 6.0%

2001 12.0% 6.0%

2002 12.0% 6.0%

2003 12.0% 6.0%

2004 12.0% 6.0%

2005 12.0% 6.0%

If the returns are equal over time, the arithmetic averagereturn will equal the geometric average return.


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%126

72

6

%12%12%12%12%12%12

:ReturnAverageArithmetic

654321_

==

+++++=

+++++=

N

HPRHPRHPRHPRHPRHPRR


Year

IBM

Stock

Government

Bonds

2000 12.0% 6.0%

2001 12.0% 6.0%

2002 12.0% 6.0%

2003 12.0% 6.0%

2004 12.0% 6.0%2005 12.0% 6.0%

%12

1973822685.1

1)12.1)(12.1)(12.1)(12.1)(12.1)(12.1(

1)]1)(1)(1)(1)(1)(1(

:ReturnAverageGeometric

16667.

6

1

6

1

654321

_

=

=

=

++++++= HPRHPRHPRHPRHPRHPRG

SAME

ANSWER !


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%126

72

6

%8%32%5%33%30%40

:ReturnAverageArithmetic

654321_

==

+++=

+++++=

N

HPRHPRHPRHPRHPRHPRR


Year

IBM

Stock

Government

Bonds

2000 40.0% 11. 0%

2001 -30. 0% 4. 0%

2002 33.0% 8.0%

2003 5.0% 3.0%

2004 32.0% 6.0%

2005 -8.0% 4.0%

A rit hmet ic A verage = 12. 0% 6. 0%Standard Deviat ion = 27.71% 3.03%

%84.8

1661991408.1

1)92)(.32.1)(05.1)(33.1)(70.0)(40.1(

1)]1)(1)(1)(1)(1)(1(

:ReturnAverageGeometric

16667.

6

1

6

1

654321

_

=

=

=

++++++= HPRHPRHPRHPRHPRHPRG

NOT THESAMEANSWER !

Now consider volatilereturns:

Volatility of returns over time eats awayat your realized returns!!!The greater the

volatility the greater the difference between the arithmeticand geometric average. Arithmetic average OVERSTATES the


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Measuring Returns

When you are trying to find average returns,especially when those returns rise and fall,always remember to use the geometric

average. The greater the volatility of returns over time,

the greater the difference you will observebetween the geometric and arithmetic

averages.


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Ex Ante Calculation of Risk

and Return


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Ex Ante -Expected Return

An investor might say that a given assetwill be expected to yield a 10% return.This is however apoint estimate.

Pressed further, the investor will admit

that the asset could possibly provide areturn of -10% under certain conditionsor as high as 25%.

The uncertainty in the actual range of

possible returns is indeed a form of risk.


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Expected Return of Securitywhen Probability is Given

Say that the investorbelieves that with a 30%probability, a given assetwill have a 10% return. A-10% return isdetermined to happenwith a 10% probability. The 25% return canoccur with a 60%

probability.

Probability ofReturn

PossibleReturn

30% 10%

10% -10%

60% 25%


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Expected Return, whenProbabilities are given

In order to find the expected return thefollowing formula is used:

For our example:

Expected Return = (0.30)(10%) + (0.10)(-10%) + (0.60)(25%) = 19%

Expected Return = (Probability of Return) X ( Possible

Return)


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So Why is Expected ReturnImportant?

Expected Return is the most basic form of riskanalysis.

An asset with perfect certainty of return will

have only one possible return. This is rare. The challenge is to determine proper probability

weights in order to calculate an expected returnvalue that accurately captures the risk

associated with an assets returns.


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Deviation, When Probability

is given Used to Quantify the risks associated with

possible returns.

For our previous example the variance was 130and the standard deviation was 11.4%

Variance = (Probability of Return) X ( Possible Return ExpectedReturn)^2

=

=n

i

iiiPkk

1

2)(

= P1 (X1-X)2 + P2 (X2-X)

2..


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Coefficient of Variation

Sometimes Variance and StandardDeviation can be misleading.

If conditions for two or more investmentalternatives are not similar then ameasure of relative variability is needed.

A widely used measure of relativevariability is the Coefficient of Variation(CV)


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Coefficient of Variation

The Coefficient of Variation is usuallycalculated with the following formula:

CV = Standard Deviation/ ExpectedRate of Return

- or

CV = Standard Deviation / Mean


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An Example of CV

Assume Stock A and StockB have widely differingrates of return andstandard deviations of

return.

Using standard deviationanalysis, Stock A seems tobe less risky than Stock B.

StockA

StockB

ExpectedReturn

7% 12%

Standard

Deviation

5% 7%


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However.

Using CV analysis, the results aredifferent.

CV of Stock A = 5% / 7% = 0.714

CV of Stock B = 7% / 12% = 0.583

The CV figure shows that Stock B hasless relative variability or lower riskper unit of expected return.


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Conclusion

Use Standard Deviation to compare differentassets and choose the one with the leastamount of risk and the highest possible return.

Historical Standard Deviation can be used tolook at the past performance of an asset. Canalso be used to compare two or more assets.

The Coefficient of Variation should be used tocompare assets in different industries or widely

differing expected returns.


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Calculations

Calculation of Beta

Calculation of total Risk= 2

Systematic Risk= B2. 2m

2= Systematic risk + unsystematic risk

B2. 2m + 2ex

=Cov12 /2m = dx dy / dx2


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Beta

A securitys beta is

41

2

2

( , )

where return on the market index

variance of the market returns

return on Security

i mi

m

m

m

i

COV R R

R

R i

=

=

=

=

% %

%

%


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Risk and Return for Portfolio


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Markowitz Portfolio Theory andNormality

If returns are normally distributedthey are completely described by

their mean and variance So investors can choseportfolios based

solely on the mean and variance

Investors will prefer portfolios withhigh means and low variances


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Markowitz Portfolio Theory

0

2

4

6

8

10

12

14

16

18

20

-50 0 50

% proba

bility

% return

0

2

4

6

8

10

12

14

16

18

20

-50 0 50

Investment C

Investmen

t D

C is preferred to D, standard deviations are the

same, but C hasa higher mean. If C & D had the same mean, theone with the lower variance would be preferred.


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Risk and Return - MPT

Prior to the establishment of Modern Portfolio Theory, most people only focused uponinvestment returnsthey ignored risk.

With MPT, investors had a tool that they could useto dramatically reduce the risk of the portfoliowithout a significant reduction in the expectedreturn of the portfolio.

Harry Markowitz: Founder of Portfolio Theory


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Harry Markowitzs Portfolio Selection Journal of Finance article (1952) set thestage for modern portfolio theory The first major publication indicating the

important of security return correlation in theconstruction of stock portfolios

Markowitz showed that for a given level of

expected return and for a given securityuniverse, knowledge of the covariance andcorrelation matrices are required

47


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Harry Markowitzs efficientportfolios:

Those portfolios providing the maximum

return for their level of risk

Those portfolios providing the minimum

risk for a certain level of return

48


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A portfolios performance is the resultof the performance of its components

The return realized on a portfolio is a linear

combination of the returns on theindividual investments

The variance of the portfolio is nota linear

combination of component variances

49


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The degree to which the returns oftwo stocks co-move is measured bythe correlation coefficient.

The correlation coefficient betweenthe returns on two securities will liein the range of +1 through - 1.

+1 is perfect positive correlation.

-1 is perfect negative correlation.

M k it P tf li Th


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Markowitz Portfolio Theory

A

B

StandardDeviation (%)

Expected Return(%)

40% A, 60% B

Expected Returns and Standard Deviations vary givendifferent weights for shares in the portfolio.


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Efficient Frontier

A

B

Return

Risk s

AB*

Combining A and B


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Efficient Frontier

Return

Risk

Low Risk

HighReturn

High Risk

HighReturn

Low Risk

Low

Return

High Risk

Low

Return


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Efficient Frontier

Return

Risk

Low Risk

HighReturn

High Risk

HighReturn

Low Risk

Low

Return

High Risk

Low

Return

Effi i t F ti


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Efficient Frontier

Standard

Deviation

Expected Return(%)

The jelly fish shape contains all possible combinations of risk and

return: The feasible set

The red line constitutes the efficient frontier: Highest return forgiven risk


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Portfolio Risk and Return- Ex

Ante


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Portfolio Return

The expected return of a portfolio isa weighted average of the expectedreturns of the components:

57

1

1

( ) ( )

where proportion of portfolio

invested in security and

1

n

p i i

i

i

n

i

i

E R x E R

x

i

x

=

=

=

=

=

% %


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Portfolio variance

Understanding portfolio variance isthe essence of understanding themathematics of diversification

The variance of a linear combination ofrandom variables is not a weightedaverage of the component variances

58

Grouping Individual Assets into


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Grouping Individual Assets intoPortfolios

The riskiness of a portfolio that is made ofdifferent risky assets is a function of threedifferent factors: the riskiness of the individual assets that make up the

portfolio the relative weights of the assets in the portfolio

the degree of comovement of returns of the assetsmaking up the portfolio

The standard deviation of a two-asset portfoliomay be measured using the Markowitz model:

BABABABBAAp wwww ,2222 2++=

Variance of A Linear


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Variance of A LinearCombination

Return variance is a securitys total risk

Most investors want portfolio variance to be as low as possible without having to give up any return

60

2 2 2 2 2 2 p A A B B A B AB A B

x x x x = + +

Total Risk Risk from A Risk from B Interactive Risk

Variance of A Linear


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Variance of A LinearCombination

If two securities have low correlation, the interactiverisk will be small

If two securities are uncorrelated, the interactive

risk drops out If two securities are negatively correlated,

interactive risk would be negative and would reducetotal risk

61

Efficient Frontier


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Efficient Frontier

Example Correlation Coefficient = .4Shares s % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Weighted Average Standard Deviation = 33.6

Standard Deviation Portfolio = 28.1

Return = weighted avg = Portfolio = 17.4%

Lets Add share New Co to the portfolio

Gain from

Diversification

Efficient Frontier


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Efficient Frontier

Example Correlation Coefficient = .3

Shares s % of Portfolio Avg Return

Portfolio 28.1 50% 17.4%

New CorpNew Corp 3030 50%50% 19%19%

NEW Standard Deviation = weighted avg = 31.80NEW Standard Deviation = Portfolio = 23.43

NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk


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Formulas

Correlation r= Cov12 / 1 2

Cov12 = dx dy /n-1 where dxis deviation from mean

r= dx dy dx2 dy2

Risk of a Three asset


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Risk of a Three-assetPortfolio

CACACACBCBCBBABABACCBBAAp wwwwwwwww ,,,222222

222 +++++=

We need 3 (three) correlation coefficientsbetween A and B; A and C; and B and C.

A

B C

a,b

b,c

a,c


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Risk of a Four-asset Portfolio

The data requirements for a four-asset portfoliogrows dramatically if we are using Markowitz Portfolioselection 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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The n-Security Case

A covariance matrix is a tabularpresentation of the pair wisecombinations of all portfolio

components The required number of covariance's to

compute a portfolio variance is (n2 n)/2

Any portfolio construction techniqueusing the full covariance matrix is calleda Markowitz model

67


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Diversification Potential

The potential of an asset to diversify a portfolio isdependent upon the degree of co-movement ofreturns of the asset with those other assets that makeup the portfolio.

In a simple, two-asset case, if the returns of the twoassets are perfectly negatively correlated it is possible(depending on the relative weighting) to eliminate allportfolio risk.

This is demonstrated through the following chart.

Example of Portfolio


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Example of PortfolioCombinations 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 BExpected

ReturnStandardDeviation

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 PositiveCorrelation no

diversification



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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0.5

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% 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



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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%

Portfolio Components Portfol io Characteristics

No Correlation some

diversification

potential

Lower

risk

than

assetA



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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%


NegativeCorrelation

greater

diversification

potential


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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 = -0.5

AB = -1

Diversification of a Two Asset Portfolio Demonstrated Graphically

An Exercise using T bills Stocks and


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An Exercise using T-bills, Stocks andBonds

Base Data: Stocks T-bills Bonds

Expected Return 12.73383 6.151702 7.007872

Standard Deviat ion 0.168 0.042 0.102

Correlation Coefficient Matrix:

Stocks 1 -0.216 0.048

T-bills -0.216 1.000 0.380

Bonds 0.048 0.380 1.000

Portfolio Combinations:

Combination Stocks T-bills Bonds

Expected

Return Variance

Standard

Deviation

1 80.0% 10.0% 10.0% 11.5 0.0181 13.5%

2 90.0% 10.0% 0.0% 12.1 0.0226 15.0%

3 80.0% 20.0% 0.0% 11.4 0.0177 13.3%

4 70.0% 20.0% 10.0% 10.8 0.0138 11.7%

5 60.0% 20.0% 20.0% 10.3 0.0106 10.3%

6 50.0% 25.0% 25.0% 9.7 0.0079 8.9%

7 40.0% 20.0% 40.0% 9.1 0.0065 8.1%8 30.0% 0.0% 70.0% 8.7 0.0080 8.9%

9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%

10 10.0% 70.0% 20.0% 7.0 0.0018 4.3%

11 0.0% 100.0% 0.0% 6.2 0.0017 4.2%

Weights Portfolio

Results Using only Three Asset


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g yClasses

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 5.0 10.0 15.0 20.0

Standard Deviation of the Portfolio (%)

PortfolioExpectedReturn(%) Efficient Set

MinimumVariance

Portfolio


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Plotting Achievable Portfolio Combinations

Expected Return on

the Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%


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The Efficient Frontier

Expected Return on

the Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%

i i i


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Data Limitations

Because of the need for so muchdata, MPT was a theoretical idea formany years.

Later, a student of Markowitz, namedWilliam Sharpe worked out a wayaround thatcreating the Beta

Coefficient as a measure of volatilityand then later developing the CAPM.

i


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Question

Assume the following statistics for Stocks A, B, and C:

80

Stock A Stock B Stock C

Expected return .20 .14 .10

Standard deviation .232 .136 .195

Q ti


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Question

The correlation coefficients between the threestocks are:

81

Stock A Stock B Stock C

Stock A 1.000

Stock B 0.286 1.000

Stock C 0.132 -0.605 1.000

Q ti


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Question

An investor seeks a portfolio return of 12%.

Which combinations of the stocks accomplish this objective? Which of those combinationsachieves the least amount of risk?

82

Q ti


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Question

83

Q ti


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Question

Calculate the variance of the B/C combination:

84

2 2 2 2 2

2 2

2

(.50) (.0185) (.50) (.0380)

2(.50)(.50)( .605)(.136)(.195)

.0046 .0095 .0080

.0061

p A A B B A B AB A B x x x x = + +

= +

+

= +

=

Q ti


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Question

Calculate the variance of the A/C combination:

85

2 2 2 2 2

2 22

(.20) (.0538) (.80) (.0380)

2(.20)(.80)(.132)(.232)(.195)

.0022 .0243 .0019

.0284

p A A B B A B AB A B x x x x = + +

= +

+

= + +

=

Q ti


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Question

Investing 50% in Stock B and 50% in Stock C achieves an expected return of12% with the lower portfolio variance. Thus, the investor will likely prefer thiscombination to the alternative of investing 20% in Stock A and 80% in Stock C.

86

C t f D i


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Concept of Dominance

A portfolio dominates all others if: For its level of expected return, there is

no other portfolio with less risk

For its level of risk, there is no otherportfolio with a higher expected return

87

C t f D i


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Concept of Dominance

In the previous example, the B/C combination dominates theA/C combination:

88

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.005 0.01 0.015 0.02 0.025 0.03

Risk

Expec

tedRe

turn

B/C combination

dominates A/C

Minimum Variance


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Portfolio

The minimum variance portfolio isthe particular combination of

securities that will result in the leastpossible variance

Solving for the minimum varianceportfolio requires basic calculus

89

Minimum Variance


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Portfolio For a two-security minimum variance

portfolio, the proportions invested instocks A and B are:

90

2

2 2 2

1

B A B ABA

A B A B AB

B A

x

x x

=

+

=


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Minimum Variance


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Portfolio

Solution: The weights of the minimum variance portfolios in thiscase are:

92

2

2 2

.06 (.224)(.245)(.5)59.07%

2 .05 .06 2(.224)(.245)(.5)

1 1 .5907 40.93%

B A B ABA

A B A B AB

B A

x

x x

= = =

+ +

= = =

Minimum Variance


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Portfolio

0

0.2

0.4

0.6

0.8

1

1.2

0 0.01 0.02 0.03 0.04 0.05 0.06

Weigh

tA

risk and diversification

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