b)risk and return i

RiskandReturnIOutline1IndividualSecurities2ExpectedReturn,Variance,andCovariance3TheReturnandRiskforPortfolios4TheEfficientSetforTwoAssets5TheEfficientSetforManySecurities6RisklessBorrowingandLending7MarketEquilibrium8RelationshipbetweenRiskandExpectedReturn(CAPM)

1

1.IndividualSecurities Characteristicsofindividualsecuritiesthatareofinterest:

ExpectedReturn VarianceandStandardDeviation CovarianceandCorrelation

Rate of ReturnScenario Probability Stock fund Bond fund

Recession 33.3% -7% 17%Normal 33.3% 12% 7%

Boom 33.3% 28% -3%

Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund.

Note: These are future possible returns and their probabilities, not historical returns

2.ExpectedReturn,Variance,andCovariance

1

2 2

1

( )

( ( ))

n

i ii

n

i ii

E r p r

p r E r

This image cannot currently be displayed.Stock fund Bond Fund

Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 17%Normal 12% 7%Boom 28% -3%Expected returnVarianceStandard Deviation

BS

SBn

i BrE

iBr

srE

isr

ip

SB

1))())(((

Stock fund Bond FundRate of Squared Rate of Squared

Scenario Return Deviation Return Deviation Recession -7% 17%Normal 12% 7%Boom 28% -3%Expected return 11.00% 7.00%VarianceStandard Deviation

%11%)28(31%)12(3

1%)7(31)(

3

1

iiiS rprE

4


Scenario Return Deviation Return Deviation Recession -7% 324 sq% 17% 100 sq%Normal 12% 1 sq% 7% 0 sq%Boom 28% 289 sq% -3% 100 sq%Expected return 11.00% 7.00%Variance Standard Deviation

222 %324%)11%7()]([ ii rEr5


Scenario Return Deviation Return Deviation Recession -7% 324 sq% 17% 100 sq%Normal 12% 1 sq% 7% 0 sq%Boom 28% 289 sq% -3% 100 sq%Expected return 11.00% 7.00%Variance 205 sq% 66.67 sq%Standard Deviation

2223

1%205%)2891324(

31)]([

ii

ii rErp

6

Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 324 sq% 17% 100 sq%Normal 12% 1 sq% 7% 0 sq%Boom 28% 289 sq% -3% 100 sq%Expected return 11.00% 7.00%Variance 205 sq% 66.67 sq%Standard Deviation 14.3% 8.2%

%3.14%205 2

7

19949.0)2.8)(3.14(

)350(31

%)350(31%)7%3%)(11%28(3

1

%)7%7%)(11%12(31%)7%17%)(11%7(3

11

)()(

2

SB

n

iiBrEiBrsrEisripSB


3.TheReturnandRiskforPortfolios

Note that stocks have a higher expected return than bonds, and higher risk.

Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.


9

Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0%Normal 12% 7% 9.5%Boom 28% -3% 12.5%

Expected return 11.00% 7.00%Variance 205 sq% 66.67 sq%Standard Deviation 14.31% 8.16%

In each of the states, the rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

%5%)17%(50%)7%(50 SSBBP rwrwr

10

Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0%Normal 12% 7% 9.5%Boom 28% -3% 12.5%

Expected return 11.00% 7.00% 9.0%Variance 205 sq% 66.67 sq%Standard Deviation 14.31% 8.16%

The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

%9%)7%(50%)11%(50)()()( SSBBP rEwrEwrEAlso a weighted average of the returns in each of the scenario.

%9%)5.12(31%)5.9(3

1%)5(31)( prE

11

Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 16 sq%Normal 12% 7% 9.5% 0.25 sq%Boom 28% -3% 12.5% 12.25 sq%

Expected return 11.00% 7.00% 9.0%Variance 205 sq% 66.67 sq%Standard Deviation 14.31% 8.16%

222 %16%)9%5()]([ ipip rEr

12


Expected return 11.00% 7.00% 9.0%Variance 205 sq% 66.67 sq% 9.5 sq%Standard Deviation 14.31% 8.16% 3.11%

The variance of the portfolio is

2223

1%5.9)%25.1225.016(

31)]([

iip

iip rErp

13

Alternatively, the variance of the portfolio can be computed as follow:

BSSBSB2

S2

S2

B2

B

BSSSBB2

S2

S2

B2

B

BSSSBB2

SS2

BB2P

w2wwww2www

))(w2(w)(w)(w

where BS is the correlation coefficient and equals -1



14

Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.



15

Portfolo Risk and Return Combinations

5.0%6.0%7.0%8.0%9.0%

10.0%11.0%12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Por

tfolio

Ret

urn

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%

10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%

50.00% 3.08% 9.00%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

4.TheEfficientSetforTwoAssets

We can consider other portfolio weights besides 50% in stocks and 50% in bonds.

100% bonds

100% stocks

Find the weights of the min variance portfolio

Portfolo Risk and Return Combinations

5.0%6.0%7.0%8.0%9.0%

10.0%11.0%12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Por

tfol

io R

etur

n

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%

10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

100% stocks

100% bonds

Note that some portfolios are better than others. They have higher returns for the same level of risk or less. These comprise the efficient frontier.

TwoSecurityPortfolioswithVariousCorrelations

100% bonds

retu

rn

100% stocks

= 0.2 = 1.0

= -1.0

18

PortfolioRisk/ReturnofTwoSecurities:CorrelationEffects

Relationshipdependsoncorrelationcoefficient

1.0< < +1.0

Thesmallerthecorrelation,thegreatertheriskreductionpotential

If =+1.0,noriskreductionispossible

19

PortfolioRiskasaFunctionoftheNumberofStocksinthePortfolio

Nondiversifiable risk; Systematic Risk; Market Risk

Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk

n

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not.

Thus diversification can eliminate some, but not all of the risk of individual securities. Investors are only rewarded for bearing systematic risk.

Portfolio risk

5.TheEfficientSetforManySecurities

Consideraworldwithmanyriskyassets;wecanstillidentifytheopportunityset ofriskreturncombinationsofvariousportfolios.

retu

rn

P

Individual Assets

21

Giventheopportunitysetwecanidentifytheminimumvarianceportfolio.

retu

rn

P

minimum variance portfolio

Individual Assets

22

Thesectionoftheopportunitysetabove theminimumvarianceportfolioistheefficientfrontier.

retu

rn

P

minimum variance portfolio

Individual Assets

23

6.RisklessBorrowingandLending

Inadditiontoriskysecurities,consideraworldthatalsohasariskfreesecurity

rf

retu

rn

24

Nowaninvestorcanallocatehismoneybetweentheriskfreesecurityandaportfolioofriskysecurities

rfre

turn

25

Withariskfreesecurityandtheefficientfrontieridentified,theinvestorwillchoosethecapitalallocationlinewiththesteepestslope.

retu

rn

P

rf

26

Ifallinvestorshavethesameperceptionregardingtheprobabilitydistributionofriskysecurities(homogeneousexpectations)andiftheycanborrowandlendatthesamerate

theywillagreeonthesameriskyportfolioM.

27

7.MarketEquilibrium

Allinvestorshavethesamecapitalallocationline.ThislineistheCapitalMarketLine(CML).

rf

retu

rn

Optimal Risky Porfolio M

28

JustwheretheinvestorchoosestoinvestalongtheCMLdependsonhisrisktolerance.

rfre

turn

M

29

TheSeparationProperty

TheSeparationProperty statesthatthemarketportfolio,M,isthesameforallinvestors thatis,thechoiceofthemarketportfolioisseparate fromtheinvestorsrisktolerance.

100% bonds

100% stocks

rf

retu

rn

M

30

EquationofCML

( )( ) m fj f j

m

E r rE r r

CML applies only to 2 benchmark securities: the market portfolio and the risk free asset.

It does not apply to individual assets or just anyportfolio.

31

8.CAPM TheCMLprovidestheequilibriumrelationshipforcombinationsofM

andtheriskfreeassetbutdoesnotsayanythingabouttheexpectedreturnsonindividualsecuritiesorotherportfolios.

Wewillnowturntoarelationshipthatdoesso CAPM. TheCAPMismathematicallyderivedfromtheCML.

Fromourearlierdiscussions, Themarketrewardsinvestorsonlyforholdingsystematicrisk. Beta measuresthesystematicrisk.

32

DefinitionofRiskwhenInvestorsholdtheMarketPortfolio

Thebestmeasureoftheriskofasecurityinalargeportfolioisthebeta()ofthesecurity.

Betameasurestheresponsivenessofasecuritytomovementsinthemarketportfolio.

)()(

2,

M

Mii R

RRCov

Clearly, the estimate of beta will depend upon the choice of a proxy for the market portfolio.

33

CAPM

Inequilibrium,allassetsandportfoliosmusthavethesamerewardtoriskratioandtheyallmustequaltherewardtoriskratioforthemarket.

Asm =1bydefinition,rearrangingtheabovegivestheCAPMequation

E(Ri)=Rf +i (E(RM) Rf)

M

fM

i

fi RRERRE

)()(

34

Estimatingwithregression

Exce

ss S

ecur

ity R

etur

nsExcess Return

on market %

Slope = i

(Ri -Rf)= i + i (Rm -Rf )+ ei35

ExpectedReturnonanIndividualSecurity ThisformulaiscalledtheCapitalAssetPricingModel(CAPM)

)( FMiFi RRRR

Assume i = 0, then the expected return is RF. Assume i = 1, then Mi RR

Expected return on a security

= Riskfree rate +Beta of security

Market risk premium

36

b)risk and return i

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