chapter 2 - risk, return, and portfolio theory.ppt

Chapter 2Chapter 2

Risk, Return Risk, Return and Portfolio and Portfolio

TheoryTheory

Risk, Return Risk, Return and Portfolio and Portfolio

TheoryTheory

Learning ObjectivesLearning Objectives

• The difference among the most important types of The difference among the most important types of returnsreturns

• How to estimate expected returns and risk for How to estimate expected returns and risk for individual securitiesindividual securities

• What happens to risk and return when securities are What happens to risk and return when securities are combined in a portfoliocombined in a portfolio

• What is meant by an “efficient frontier”What is meant by an “efficient frontier”• Why diversification is so important to investorsWhy diversification is so important to investors

CHAPTER 2 – Risk, Return and Portfolio Theory

What are investment returns?

Investment returns measure the financial results of an investment.

Returns may be historical /realized or prospective/expected (anticipated).

Returns can be expressed in:

Dollar terms.

Percentage terms. CHAPTER 2 – Risk, Return and Portfolio Theory

What is the return on an investment that costs $1,000 and is sold after 1 year for $1,100?

Dollar return:

Percentage return:

$ Received - $ Invested $1,100 - $1,000 = $100.

$ Return/$ Invested $100/$1,000 = 0.10 = 10%.


What is investment risk?

Typically, investment returns are not known with certainty.

Investment risk pertains to the probability of earning a return less than that expected.

The greater the chance of a return far below the expected return, the greater the risk.


Measuring ReturnsMeasuring Returns

Measuring ReturnsMeasuring ReturnsIntroductionIntroduction

Ex Ante Returns/ Expected returnEx Ante Returns/ Expected return• Return calculations may be done Return calculations may be done ‘before-the-‘before-the-

fact,’ fact,’ in which case, assumptions must be in which case, assumptions must be made about the futuremade about the future

Ex Post Returns/realized Ex Post Returns/realized • Return calculations done Return calculations done ‘after-the-fact,’ ‘after-the-fact,’ in in

order to analyze what rate of return was order to analyze what rate of return was earned.earned.


Measuring Average ReturnsMeasuring Average ReturnsEx Post ReturnsEx Post Returns

• Measurement of historical rates of return that Measurement of historical rates of return that have been earned on a security or a class of have been earned on a security or a class of securities allows us to identify trends or securities allows us to identify trends or tendencies that may be useful in predicting tendencies that may be useful in predicting the future.the future.


Measuring Average ReturnsMeasuring Average ReturnsArithmetic AverageArithmetic Average

Where:Where:rrii = the individual returns = the individual returns

nn = the total number of observations = the total number of observations

• Most commonly used value in statisticsMost commonly used value in statistics• Sum of all returns divided by the total number of Sum of all returns divided by the total number of

observationsobservations


(AM) Average Arithmetic 1

n

rn

ii

Measuring Expected (Ex Ante) ReturnsMeasuring Expected (Ex Ante) Returns

• While past returns might be interesting, While past returns might be interesting, investor’s investor’s are most concerned with future are most concerned with future returns.returns.

• Sometimes, historical average returns will not Sometimes, historical average returns will not be realized in the future.be realized in the future.

• Developing an independent estimate of ex Developing an independent estimate of ex ante returns usually involves use of ante returns usually involves use of forecasting discrete scenarios with forecasting discrete scenarios with outcomesoutcomes and and probabilities of occurrenceprobabilities of occurrence..


Estimating Expected ReturnsEstimating Expected ReturnsEstimating Ex Ante (Forecast) ReturnsEstimating Ex Ante (Forecast) Returns

• The general formulaThe general formula

Where:Where:ERER = the expected return on an investment = the expected return on an investment

RRii = the estimated return in scenario = the estimated return in scenario ii

ProbProbi i = the probability of state = the probability of state i i occurring occurring


)Prob((ER)Return Expected 1

i

n

iir

Estimating Expected ReturnsEstimating Expected ReturnsEstimating Ex Ante (Forecast) ReturnsEstimating Ex Ante (Forecast) Returns

Example:Example:

This is type of forecast data that are required to This is type of forecast data that are required to make an ex ante estimate of expected return.make an ex ante estimate of expected return.

State of the EconomyProbability of Occurrence

Possible Returns on

Stock A in that State

Economic Expansion 25.0% 30%Normal Economy 50.0% 12%Recession 25.0% -25%


Estimating Expected ReturnsEstimating Expected ReturnsEstimating Ex Ante (Forecast) Returns Using a Spreadsheet ApproachEstimating Ex Ante (Forecast) Returns Using a Spreadsheet Approach

Example Solution:Example Solution:

Sum the products of the probabilities and possible Sum the products of the probabilities and possible returns in each state of the economy.returns in each state of the economy.

(1) (2) (3) (4)=(2)×(3)

State of the EconomyProbability of Occurrence

Possible Returns on

Stock A in that State

Weighted Possible

Returns on the Stock

Economic Expansion 25.0% 30% 7.50%Normal Economy 50.0% 12% 6.00%Recession 25.0% -25% -6.25%

Expected Return on the Stock = 7.25%


Estimating Expected ReturnsEstimating Expected ReturnsEstimating Ex Ante (Forecast) Returns Using a Formula ApproachEstimating Ex Ante (Forecast) Returns Using a Formula Approach

Example Solution:Example Solution:

Sum the products of the probabilities and possible Sum the products of the probabilities and possible returns in each state of the economy.returns in each state of the economy.

7.25%

)25.0(-25%0.5)(12% .25)0(30%

)Prob(r)Prob(r )Prob(r

)Prob((ER)Return Expected

332211

1i

n

iir


Measuring RiskMeasuring Risk

RiskRisk

• Probability of incurring harmProbability of incurring harm• For investors, risk For investors, risk is the probability of earning is the probability of earning

an inadequate return.an inadequate return.– If investors require a 10% rate of return on a given If investors require a 10% rate of return on a given

investment, then any return less than 10% is investment, then any return less than 10% is considered harmful.considered harmful.


RiskRiskIllustratedIllustrated


Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm

The range of total possible returns on the stock A runs from -30% to more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% represent risk to the investor.

A

RangeRange

• The difference between the maximum and The difference between the maximum and minimum values is called the rangeminimum values is called the range


Differences in Levels of RiskDifferences in Levels of RiskIllustratedIllustrated


Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm The wider the range of probable outcomes the greater the risk of the investment.

A is a much riskier investment than BB

A

Refining the Measurement of RiskRefining the Measurement of RiskStandard Deviation (Standard Deviation (σσ))

• Range measures risk based on only two Range measures risk based on only two observations (minimum and maximum value)observations (minimum and maximum value)

• Standard deviation uses all observations.Standard deviation uses all observations.– Standard deviation can be calculated on forecast or Standard deviation can be calculated on forecast or

possible returns as well as historical or ex post possible returns as well as historical or ex post returns.returns.

(The following two slides show the two different formula used for Standard (The following two slides show the two different formula used for Standard Deviation)Deviation)


Measuring RiskMeasuring RiskEx post Standard DeviationEx post Standard Deviation

nsobservatio ofnumber the

year in return the

return average the

deviation standard the

:

_

n

ir

r

Where

i


1

)(post Ex 1

2_

n

rrn

ii

Measuring RiskMeasuring RiskExample Using the Ex post Standard DeviationExample Using the Ex post Standard Deviation

ProblemProblem

Estimate the standard deviation of the historical returns on investment A Estimate the standard deviation of the historical returns on investment A that were: 10%, 24%, -12%, 8% and 10%.that were: 10%, 24%, -12%, 8% and 10%.

Step 1 – Calculate the Historical Average ReturnStep 1 – Calculate the Historical Average Return

Step 2 – Calculate the Standard DeviationStep 2 – Calculate the Standard Deviation

%0.85

40

5

10812-2410 (AM) Average Arithmetic 1

n

rn

ii


%88.121664

664

4

404002564

4

2020162

15

)814()88()812()824(8)-(10

1

)(post Ex

22222

222221

2_

n

rrn

ii

Measuring RiskMeasuring RiskEx ante Standard DeviationEx ante Standard Deviation

A Scenario-Based Estimate of RiskA Scenario-Based Estimate of Risk


)()(Prob anteEx 2

1i ii

n

i

ERr

Scenario-based Estimate of RiskScenario-based Estimate of RiskExample Using the Ex ante Standard Deviation – Raw DataExample Using the Ex ante Standard Deviation – Raw Data

State of the Economy Probability

Possible Returns on Security A

Recession 25.0% -22.0%Normal 50.0% 14.0%Economic Boom 25.0% 35.0%


GIVEN INFORMATION INCLUDES:

- Possible returns on the investment for different discrete states

- Associated probabilities for those possible returns

Scenario-based Estimate of RiskScenario-based Estimate of RiskEx ante Standard Deviation – Spreadsheet ApproachEx ante Standard Deviation – Spreadsheet Approach

• The following two slides illustrate an The following two slides illustrate an approach to solving for standard deviation approach to solving for standard deviation using a spreadsheet model.using a spreadsheet model.


Scenario-based Estimate of RiskScenario-based Estimate of RiskFirst Step – Calculate the Expected ReturnFirst Step – Calculate the Expected Return



Weighted Possible Returns

Recession 25.0% -22.0% -5.5%Normal 50.0% 14.0% 7.0%Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%


Determined by multiplying the probability times the

possible return.

Expected return equals the sum of the weighted possible returns.

Scenario-based Estimate of RiskScenario-based Estimate of RiskSecondSecond Step – Measure the Weighted and Squared Deviations Step – Measure the Weighted and Squared Deviations




Deviation of Possible

Return from Expected

Squared Deviations

Weighted and

Squared Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531

Expected Return = 10.3% Variance = 0.0420

Standard Deviation = 20.50%




Deviation of Possible

Return from Expected

Squared Deviations

Weighted and

Squared Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531

Expected Return = 10.3% Variance = 0.0420

Standard Deviation = 20.50%


Second, square those deviations from the mean.

The sum of the weighted and square deviations is the variance in percent squared terms.

The standard deviation is the square root of the variance (in percent terms).

First calculate the deviation of possible returns from the expected.

Now multiply the square deviations by their probability of occurrence.

Scenario-based Estimate of RiskScenario-based Estimate of RiskExample Using the Ex ante Standard Deviation FormulaExample Using the Ex ante Standard Deviation Formula

herein.herein.

%5.20205.

0420.

)06126(.25.)00141(.5.)10401(.25.

)8.24(25.)8.3(5.)3.32(25.

)3.1035(25.)3.1014(5.)3.1022(25.

)()()(

)()(Prob anteEx

222

222

2331

2222

2111

2

1i

ERrPERrPERrP

ERr ii

n

i




Recession 25.0% -22.0% -5.5%Normal 50.0% 14.0% 7.0%Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%


Modern Portfolio TheoryModern Portfolio Theory

PortfoliosPortfolios

• A portfolio is a collection of different securities such as A portfolio is a collection of different securities such as stocks and bonds, that are combined and considered a stocks and bonds, that are combined and considered a single assetsingle asset

• The risk-return characteristics of the portfolio is obviously The risk-return characteristics of the portfolio is obviously different than the characteristics of the assets that make different than the characteristics of the assets that make up that portfolio, especially with regard to risk.up that portfolio, especially with regard to risk.

• Combining different securities into portfolios is done to Combining different securities into portfolios is done to achieve achieve diversificationdiversification..


Expected Return of a PortfolioExpected Return of a PortfolioModern Portfolio TheoryModern Portfolio Theory

The Expected Return on a Portfolio is simply the weighted The Expected Return on a Portfolio is simply the weighted average of the returns of the individual assets that make up the average of the returns of the individual assets that make up the portfolio:portfolio:

The portfolio weight of a particular security is the percentage of The portfolio weight of a particular security is the percentage of the portfolio’s total value that is invested in that security.the portfolio’s total value that is invested in that security.


)( n

1i

iip ERwER

Expected Return of a PortfolioExpected Return of a PortfolioExampleExample

%288.8%284.4%004.4

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

1i

iip ERwER


Portfolio value = $2,000 + $5,000 = $7,000Portfolio value = $2,000 + $5,000 = $7,000

rrAA = 14%, r = 14%, rBB = 6%, = 6%,

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

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

Range of Returns in a Two Asset PortfolioRange of Returns in a Two Asset Portfolio

In a two asset portfolio, simply by changing the weight of the In a two asset portfolio, simply by changing the weight of the constituent assets, different portfolio returns can be achieved.constituent assets, different portfolio returns can be achieved.

Because the expected return on the portfolio is a simple Because the expected return on the portfolio is a simple weighted average of the individual returns of the assets, you can weighted average of the individual returns of the assets, you can achieve portfolio returns bounded by the highest and the lowest achieve portfolio returns bounded by the highest and the lowest individual asset returns.individual asset returns.



Example 1:Example 1:

Assume ERAssume ERA A = 8% and ER= 8% and ERBB = 10% = 10%

(See the following 6 slides based on Figure 8-4)(See the following 6 slides based on Figure 8-4)


Expected Portfolio ReturnExpected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and BAffect on Portfolio Return of Changing Relative Weights in A and B


Ex

pe

cte

d R

etu

rn %

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

2 - 4 FIGURE

ERA=8%

ERB= 10%



2 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

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)



Ex

pe

cte

d R

etu

rn %

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

2 - 4 FIGURE

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.



Ex

pe

cte

d R

etu

rn %

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

2 - 4 FIGURE

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



8 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

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



8 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

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


Example 1:Example 1:

Assume ERAssume ERA A = 14% and ER= 14% and ERBB = 6% = 6%

(See the following 2 slides )(See the following 2 slides )


Range of Returns in a Two Asset PortfolioRange of Returns in a Two Asset PortfolioE(r)E(r)AA= 14%, E(r)= 14%, E(r)BB= 6%= 6%


A graph of this relationship is found on the following slide.

Expected return on Asset A = 14.0%Expected return on Asset B = 6.0%

Weight of Asset A

Weight of Asset B

Expected Return on the

Portfolio0.0% 100.0% 6.0%

10.0% 90.0% 6.8%20.0% 80.0% 7.6%30.0% 70.0% 8.4%40.0% 60.0% 9.2%50.0% 50.0% 10.0%60.0% 40.0% 10.8%70.0% 30.0% 11.6%80.0% 20.0% 12.4%90.0% 10.0% 13.2%100.0% 0.0% 14.0%

Range of Returns in a Two Asset PortfolioRange of Returns in a Two Asset Portfolio E(r)E(r)AA= 14%, E(r)= 14%, E(r)BB= 6%= 6%

Range of Portfolio Returns

0.00%2.00%4.00%6.00%8.00%

10.00%12.00%14.00%16.00%

Weight Invested in Asset A

Exp

ecte

d R

etur

n on

Tw

o A

sset

Por

tfol

io


Expected Portfolio ReturnsExpected Portfolio ReturnsExample of a Three Asset PortfolioExample of a Three Asset Portfolio

CHAPTER 2 – Risk, Return and Portfolio TheoryK. Hartviksen

Relative Weight

Expected Return

Weighted Return

Stock X 0.400 8.0% 0.03Stock Y 0.350 15.0% 0.05Stock Z 0.250 25.0% 0.06 Expected Portfolio Return = 14.70%

Risk in PortfoliosRisk in Portfolios

Risk, Return and Portfolio TheoryRisk, Return and Portfolio Theory

Modern Portfolio Theory - MPTModern Portfolio Theory - MPT

• Prior to the establishment of Modern Portfolio Theory Prior to the establishment of Modern Portfolio Theory (MPT), most people only focused upon investment (MPT), most people only focused upon investment returns…they ignored risk.returns…they ignored risk.

• With MPT, investors had a tool that they could use to With MPT, investors had a tool that they could use to dramatically reduce the riskdramatically reduce the risk of the portfolio of the portfolio without a without a significant reductionsignificant reduction in the expected return of the in the expected return of the portfolio.portfolio.


Expected Return and Risk For PortfoliosExpected Return and Risk For PortfoliosStandard Deviation of a Two-Asset Portfolio using CovarianceStandard Deviation of a Two-Asset Portfolio using Covariance

))()((2)()()()( ,2222

BABABBAAp COVwwww


Risk of Asset A adjusted for weight

in the portfolio

Risk of Asset B adjusted for weight

in the portfolio

Factor to take into account comovement of returns. This factor

can be negative.

Expected Return and Risk For PortfoliosExpected Return and Risk For PortfoliosStandard Deviation of a Two-Asset Portfolio using Correlation CoefficientStandard Deviation of a Two-Asset Portfolio using Correlation Coefficient

))()()()((2)()()()( ,2222

BABABABBAAp wwww


Factor that takes into account the degree of

comovement of returns. It can have a negative value if correlation is

negative.

Grouping Individual Assets into PortfoliosGrouping Individual Assets into Portfolios

• The riskiness of a portfolio that is made of different risky The riskiness of a portfolio that is made of different risky assets is a function of three different factors:assets is a function of three different factors:– the riskiness of the individual assets that make up the portfoliothe riskiness of the individual assets that make up the portfolio

– the relative weights of the assets in the portfoliothe relative weights of the assets in the portfolio

– the degree of comovement of returns of the assets making up the the degree of comovement of returns of the assets making up the portfolioportfolio

• The standard deviation of a two-asset portfolio may be The standard deviation of a two-asset portfolio may be measured using the Markowitz model:measured using the Markowitz model:


BABABABBAAp wwww ,2222 2

Risk of a Three-Asset PortfolioRisk of a Three-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

ρa,b

ρb,c

ρa,c

CACACACBCBCBBABABACCBBAAp wwwwwwwww ,,,222222 222

Risk of a Four-asset PortfolioRisk of a Four-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

ρa,b ρa,d

ρb,c ρc,d

ρa,c

ρb,d

CovarianceCovariance

• A statistical measure of the correlation of A statistical measure of the correlation of the fluctuations of the annual rates of the fluctuations of the annual rates of return of different investments.return of different investments.


)-)((Prob _

,1

_

,i BiB

n

iiiAAB kkkkCOV

CorrelationCorrelation

• The degree to which the returns of two stocks The degree to which the returns of two stocks co-move is measured by the correlation co-move is measured by the correlation coefficient (coefficient (ρρ).).

• The correlation coefficient (The correlation coefficient (ρρ) between the ) between the returns on two securities will lie in the range of returns on two securities will lie in the range of +1 through - 1.+1 through - 1.

+1 is perfect positive correlation+1 is perfect positive correlation

-1 is perfect negative correlation-1 is perfect negative correlation


BA

ABAB

COV

Covariance and Correlation CoefficientCovariance and Correlation Coefficient

• Solving for covariance given the correlation Solving for covariance given the correlation coefficient and standard deviation of the two coefficient and standard deviation of the two assets:assets:


BAABABCOV

Importance of CorrelationImportance of Correlation

• Correlation is important because it affects the Correlation is important because it affects the degree to which diversification can be degree to which diversification can be achieved using various assets.achieved using various assets.

• Theoretically, if two assets returns are Theoretically, if two assets returns are perfectly positively correlated, it is possible to perfectly positively correlated, it is possible to build a riskless portfolio with a return that is build a riskless portfolio with a return that is greater than the risk-free rate.greater than the risk-free rate.


Affect of Perfectly Negatively Correlated ReturnsAffect of Perfectly Negatively Correlated ReturnsElimination of Portfolio RiskElimination of 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 variabilityof the portfolios returns over time.

Returns on Stock A

Returns on Stock B

Returns on Portfolio

Returns%

10%

5%

15%

20%

Example of Perfectly Positively Correlated ReturnsExample of Perfectly Positively Correlated ReturnsNo Diversification of Portfolio RiskNo Diversification of Portfolio Risk


Time 0 1 2

If returns of A and B are perfectly positively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be risky. There would be no diversification (reduction of portfolio risk).

Returns%

10%

5%

15%

20%

Returns on Stock A

Returns on Stock B

Returns on Portfolio

Affect of Perfectly Negatively Correlated ReturnsAffect of Perfectly Negatively Correlated ReturnsElimination of Portfolio RiskElimination of 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 variabilityof the portfolios returns over time.

Returns%

10%

Returns on Portfolio5%

15%

20%

Returns on Stock B

Returns on Stock A

Affect of Perfectly Negatively Correlated ReturnsAffect of Perfectly Negatively Correlated ReturnsNumerical ExampleNumerical Example

%10%5.2%5.7

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

1i

iip ERwER


Weight of Asset A = 50.0%Weight of Asset B = 50.0%

YearReturn on

Asset AReturn on

Asset B

Expected Return on the

Portfolioxx07 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

Diversification PotentialDiversification Potential

• The potential of an asset to diversify a portfolio is The potential of an asset to diversify a portfolio is dependent upon the degree of co-movement of returns of dependent upon the degree of co-movement of returns of the asset with those other assets that make up the the asset with those other assets that make up the portfolio.portfolio.

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

• This is demonstrated through the following series of This is demonstrated through the following series of spreadsheets, and then summarized in graph format.spreadsheets, and then summarized in graph format.


Example of Portfolio Combinations and Example of Portfolio Combinations and CorrelationCorrelation

AssetExpected

ReturnStandard Deviation

Correlation Coefficient

A 5.0% 15.0% 1B 14.0% 40.0%

Weight of A Weight of BExpected


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

diversification

Both portfolio returns and risk are bounded by the range set by the constituent assets when ρ=+1


AssetExpected



A 5.0% 15.0% 0.5B 14.0% 40.0%



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 ρ=+0.5 these portfolio combinations have lower risk – expected portfolio return is unaffected.


AssetExpected



A 5.0% 15.0% 0B 14.0% 40.0%



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 either asset A or B.


AssetExpected



A 5.0% 15.0% -0.5B 14.0% 40.0%



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


AssetExpected



A 5.0% 15.0% -1B 14.0% 40.0%



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% invested in asset A


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 = -0.5

AB = -1

Impact of the Correlation CoefficientImpact of the Correlation Coefficient

• Figure 2-7 (see the next slide) illustrates the Figure 2-7 (see the next slide) illustrates the relationship between portfolio risk (relationship between portfolio risk (σσ) ) and the and the correlation coefficientcorrelation coefficient– The slope is not linear a significant amount of The slope is not linear a significant amount of

diversification is possible with assets with no diversification is possible with assets with no correlation (it is not necessary, nor is it possible to correlation (it is not necessary, nor is it possible to find, perfectly negatively correlated securities in the find, perfectly negatively correlated securities in the real world)real world)

– With perfect negative correlation, the variability of With perfect negative correlation, the variability of portfolio returns is reduced to nearly zero.portfolio returns is reduced to nearly zero.


Expected Portfolio ReturnExpected Portfolio ReturnImpact of the Correlation CoefficientImpact of the Correlation Coefficient


2 - 7 FIGURE

15

10

5

0

Sta

nd

ard

De

via

tio

n (

%)

of

Po

rtfo

lio

Re

turn

s

Correlation Coefficient (ρ)

-1 -0.5 0 0.5 1

Zero Risk PortfolioZero Risk Portfolio

• We can calculate the portfolio that removes all risk.We can calculate the portfolio that removes all risk.

• When When ρρ = -1, then = -1, then

• Becomes:Becomes:


BAp ww )1(

))()()()((2)()()()( ,2222

BABABABBAAp wwww

An Exercise using T-bills, Stocks and BondsAn Exercise using T-bills, Stocks and Bonds

Base Data: Stocks T-bills BondsExpected Return(%) 12.73383 6.151702 7.0078723

Standard Deviation (%) 0.168 0.042 0.102

Correlation Coefficient Matrix:Stocks 1 -0.216 0.048T-bills -0.216 1 0.380Bonds 0.048 0.380 1

Portfolio Combinations:

Combination Stocks T-bills BondsExpected Return Variance

Standard Deviation

1 100.0% 0.0% 0.0% 12.7 0.0283 16.8%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% 30.0% 0.0% 10.8 0.0134 11.6%5 60.0% 40.0% 0.0% 10.1 0.0097 9.9%6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%7 40.0% 60.0% 0.0% 8.8 0.0044 6.6%8 30.0% 70.0% 0.0% 8.1 0.0028 5.3%9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%10 10.0% 90.0% 0.0% 6.8 0.0014 3.8%

Weights Portfolio


Historical averages for

returns and risk for three asset

classes

Historical correlation coefficients

between the asset classes

Portfolio characteristics for each combination of securities

Each achievable portfolio combination is plotted on expected return, risk (σ) space, found on the following slide.

Achievable PortfoliosAchievable PortfoliosResults Using only Three Asset ClassesResults Using only Three Asset Classes

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 (%)

Po

rtfo

lio E

xpec

ted

Ret

urn

(%

) Efficient Set

Minimum Variance

Portfolio


The plotted points are attainable portfolio

combinations.

The efficient set is that set of achievable portfolio

combinations that offer the highest rate of return for a

given level of risk. The solid blue line indicates the efficient

set.

Achievable Two-Security PortfoliosAchievable Two-Security PortfoliosModern Portfolio TheoryModern Portfolio Theory


2 - 9 FIGURE

Ex

pe

cte

d R

etu

rn %

Standard Deviation (%)

13

12

11

10

9

8

7

6

0 10 20 30 40 50 60

This line represents the set of portfolio combinations that are achievable by varying relative weights and using two non-correlated securities.

DominanceDominance

• It is assumed that investors are rational, It is assumed that investors are rational, wealth-maximizing and risk averse.wealth-maximizing and risk averse.

• If so, then some investment choices dominate If so, then some investment choices dominate others.others.


Investment ChoicesInvestment ChoicesThe Concept of Dominance IllustratedThe Concept of Dominance Illustrated


A B

C

Return%

Risk

10%

5%

To the risk-averse wealth maximizer, the choices are clear, A dominates B,A dominates C.

A dominates B because it offers the same return but for less risk.

A dominates C because it offers a higher return but for the same risk.

20%5%

Efficient FrontierEfficient FrontierThe Two-Asset Portfolio CombinationsThe Two-Asset Portfolio Combinations


A is not attainable

B,E lie on the efficient frontier and are attainable

E is the minimum variance portfolio (lowest risk combination)

C, D are attainable but are dominated by superior portfolios that line on the line

above E

2 - 10 FIGURE

Ex

pe

cte

d R

etu

rn %


A

E

B

C

D

Efficient FrontierEfficient FrontierThe Two-Asset Portfolio CombinationsThe Two-Asset Portfolio Combinations


2 - 10 FIGURE

Ex

pe

cte

d R

etu

rn %


A

E

B

C

D

Rational, risk averse investors will only want to hold portfolios such as B.

The actual choice will depend on her/his risk preferences.

DiversificationDiversification


DiversificationDiversification

• We have demonstrated that risk of a portfolio can be We have demonstrated that risk of a portfolio can be reduced by spreading the value of the portfolio across, reduced by spreading the value of the portfolio across, two, three, four or more assets.two, three, four or more assets.

• The key to efficient diversification is to choose assets The key to efficient diversification is to choose assets whose returns are less than perfectly positively whose returns are less than perfectly positively correlated.correlated.

• Even with random or naïve diversification, risk of the Even with random or naïve diversification, risk of the portfolio can be reduced.portfolio can be reduced.– This is illustrated in Figure 8 -11 and Table 8 -3 found on the This is illustrated in Figure 8 -11 and Table 8 -3 found on the

following slides.following slides.• As the portfolio is divided across more and more securities, the risk As the portfolio is divided across more and more securities, the risk

of the portfolio falls rapidly at first, until a point is reached where, of the portfolio falls rapidly at first, until a point is reached where, further division of the portfolio does not result in a reduction in risk.further division of the portfolio does not result in a reduction in risk.

• Going beyond this point is known as superfluous diversification.Going beyond this point is known as superfluous diversification.


DiversificationDiversificationDomestic DiversificationDomestic Diversification


2 - 11 FIGURE

14

12

10

8

6

4

2

0

Sta

nd

ard

Dev

iati

on

(%

)

Number of Stocks in Portfolio

0 50 100 150 200 250 300

Average Portfolio RiskJanuary 1985 to December 1997

DiversificationDiversificationDomestic DiversificationDomestic Diversification


Average Monthly Portfolio

Return (%)

Standard Deviation of Average

Monthly Portfolio Return (%)

Ratio of Portfolio Standard Deviation to

Standard Deviation of a Single Stock

Percentage of Total Achievable Risk Reduction

1 1.51 13.47 1.00 0.002 1.51 10.99 0.82 27.503 1.52 9.91 0.74 39.564 1.53 9.30 0.69 46.375 1.52 8.67 0.64 53.316 1.52 8.30 0.62 57.507 1.51 7.95 0.59 61.358 1.52 7.71 0.57 64.029 1.52 7.52 0.56 66.17

10 1.51 7.33 0.54 68.30

14 1.51 6.80 0.50 74.1940 1.52 5.62 0.42 87.2450 1.52 5.41 0.40 89.64100 1.51 4.86 0.36 95.70200 1.51 4.51 0.34 99.58222 1.51 4.48 0.33 100.00

Source: Cleary, S. and Copp D. "Diversification with Canadian Stocks: How Much is Enough?" Canadian Investment Review (Fall 1999), Table 1.

Table 8-3 Monthly Canadian Stock Portfolio Returns, January 1985 to December 1997


Total Risk of an Individual AssetTotal Risk of an Individual AssetEquals the Sum of Market and Unique RiskEquals the Sum of Market and Unique Risk

• This graph illustrates This graph illustrates that total risk of a that total risk of a stock is made up of stock is made up of market risk (that market risk (that cannot be diversified cannot be diversified away because it is a away because it is a function of the function of the economic ‘system’) economic ‘system’) and unique, company-and unique, company-specific risk that is specific risk that is eliminated from the eliminated from the portfolio through portfolio through diversification.diversification.


[8-19]

Sta

nd

ard

Dev

iati

on

(%

)


Average Portfolio Risk

Diversifiable (unique) risk

Nondiversifiable (systematic) risk

risk )systematic-(non Uniquerisk c)(systematiMarket risk Total

International DiversificationInternational Diversification

• Clearly, diversification adds value to a Clearly, diversification adds value to a portfolio by reducing risk while not reducing portfolio by reducing risk while not reducing the return on the portfolio significantly.the return on the portfolio significantly.

• Most of the benefits of diversification can be Most of the benefits of diversification can be achieved by investing in 40 – 50 different achieved by investing in 40 – 50 different ‘positions’ (investments)‘positions’ (investments)

• However, if the investment universe is However, if the investment universe is expanded to include investments beyond the expanded to include investments beyond the domestic capital markets, additional risk domestic capital markets, additional risk reduction is possible.reduction is possible.

(See Figure 8 -12 found on the following slide.)(See Figure 8 -12 found on the following slide.)


DiversificationDiversificationInternational DiversificationInternational Diversification


2 - 12 FIGURE

100

80

60

40

20

0

Pe

rce

nt

ris

k

Number of Stocks

0 10 20 30 40 50 60

International stocks

U.S. stocks

11.7

Summary and ConclusionsSummary and Conclusions

In this chapter you have learned:In this chapter you have learned:– How to measure different types of returnsHow to measure different types of returns– How to calculate the standard deviation and How to calculate the standard deviation and

interpret its meaninginterpret its meaning– How to measure returns and risk of portfolios How to measure returns and risk of portfolios

and the importance of correlation in the and the importance of correlation in the diversification process.diversification process.

– How the efficient frontier is that set of achievable How the efficient frontier is that set of achievable portfolios that offer the highest rate of return for a portfolios that offer the highest rate of return for a given level of risk.given level of risk.


Concept Review QuestionsConcept Review Questions


Concept Review Question 1Concept Review Question 1Ex Ante and Ex Post ReturnsEx Ante and Ex Post Returns

What is the difference between ex ante and What is the difference between ex ante and ex post returns?ex post returns?


chapter 2 - risk, return, and portfolio theory.ppt

Documents

perfectly positively correlated

perfectly negatively correlated

estimating expected returnsestimating

changing relative weights

perfect positive correlation

expected portfolio returnaffect

modern portfolio theory

perfect negative correlation