chapter 08 risk and rates of return

8/2/2019 Chapter 08 Risk and Rates of Return

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CHAPTER8

RiskandRatesofReturnn Stand-aloneriskn Por3olioriskn Risk&return:CAPM/SML


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Investmentreturns

Therateofreturnonaninvestmentcanbecalculatedasfollows:

(AmountreceivedAmountinvested)

Return=________________________

Amountinvested

Forexample,if$1,000isinvestedand$1,100isreturnedaOeroneyear,therateofreturnforthisinvestmentis:

($1,100-$1,000)/$1,000=10%.


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Whatisinvestmentrisk?

Twotypesofinvestmentrisk Stand-alonerisk Por3oliorisk

InvestmentriskisrelatedtotheprobabilityofearningalowornegaVveactualreturn.

ThegreaterthechanceoflowerthanexpectedornegaVvereturns,theriskiertheinvestment.


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ProbabilitydistribuVons

AlisVngofallpossibleoutcomes,andtheprobabilityofeachoccurrence.

Canbeshowngraphically.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y


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SelectedRealizedReturns,

19262004

AverageStandard

ReturnDeviaVon

Small-companystocks 17.5%.1%

Large-companystocks 12.4 20. L-Tcorporatebonds 6.2 8.6

L-Tgovernmentbonds 5.8 9.

U.S.Treasurybills .8 .1

Source:BasedonStocks,Bonds,Bills,andInfla1on:(Valua1onEdi1on)2005Yearbook(Chicago:IbbotsonAssociates,2005),p28.


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InvestmentalternaVves

Economy Prob. T-Bill HT Coll USR MP

Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0%

Below avg 0.2 5.5% -7.0% 13.0% -14.0% -3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg 0.2 5.5% 30.0% -11.0% 41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0% 26.0% 38.0%


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WhyistheT-billreturnindependentofthe

economy?DoT-billspromiseacompletely

risk-freereturn?

n T-bills will return the promised 5.5%, regardlessof the economy.

n No, T-bills do not provide a completely risk-freereturn, as they are still exposed to inflation.

Although, very little unexpected inflation is likelyto occur over such a short period of time.

n T-bills are also risky in terms of reinvestment raterisk.

n T-bills are risk-free in the default sense of theword.


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HowdothereturnsofHTandColl.behavein

relaVontothemarket?

HTMoveswiththeeconomy,andhasaposiVvecorrelaVon.Thisistypical.

Coll.Iscountercyclicalwiththeeconomy,andhasanegaVvecorrelaVon.Thisisunusual.


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CalculaVngtheexpectedreturn

12.4%(0.1)(45%)(0.2)(30%)(0.4)(15%)

(0.2)(-7%)(0.1)(-27%)r

Prr

returnofrateexpectedr

HT

^

N

1iii

^

^

=+

++

+=

=

=

=


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Summaryofexpectedreturns

Expectedreturn

HT 12.4%

Market 10.5%

USR 9.8%T-bill 5.5%

Coll. 1.0%

HThasthehighestexpectedreturn,andappearstobethebestinvestmentalternaVve,butisitreally?Havewefailedtoaccountforrisk?


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CalculaVngstandarddeviaVon

deviationStandard=

2

Variance ==

i

2N

1ii P)r(r

=

=


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StandarddeviaVonforeachinvestment

15.2%

18.8%20.0%

13.2%0.0%

(0.1)5.5)-(5.5

(0.2)5.5)-(5.5(0.4)5.5)-(5.5(0.2)5.5)-(5.5(0.1)5.5)-(5.5

P)r(r

M

USRHT

CollbillsT

2

22

22

billsT

N

1ii

2^

i

=

==

==

+

++

+

=

=

=

21


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ComparingstandarddeviaVons

USR

Prob.T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)


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CommentsonstandarddeviaVon

asameasureofrisk

StandarddeviaVon(i)measurestotal,orstand-alone,risk.

Thelargeriis,thelowertheprobabilitythatactualreturnswillbeclosertoexpectedreturns.

LargeriisassociatedwithawiderprobabilitydistribuVonofreturns.


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Comparingriskandreturn

Security Expectedreturn, r

Risk,

T-bills 5.5% 0.0%HT 12.4% 20.0%

Coll* 1.0% 13.2%

USR* 9.8% 18.8%

Market 10.5% 15.2%

* Seem out of place.

^


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CoefficientofariaVon(C)

Astandardizedmeasureofdispersionaboutthe

expectedvalue,thatshowstheriskperunitof

return.

r

returnExpected

deviationStandardCV

==


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Riskrankings,

bycoefficientofvariaVon

C

T-bill 0.0

HT 1.6

Coll. 1.2

USR 1.9 Market 1.4

n Collections has the highest degree of risk per unitof return.

n HT, despite having the highest standard deviationof returns, has a relatively average CV.


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IllustraVngtheCasameasureof

relaVverisk

A=B,butAisriskierbecauseofalargerprobabilityoflosses.In

otherwords,thesameamountofrisk(asmeasuredby)forsmallerreturns.

0

A B

Rate of Return (%)

Prob.


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Investoratudetowardsrisk

Riskaversionassumesinvestorsdislikeriskandrequirehigherratesofreturnto

encouragethemtoholdriskiersecuriVes.

Riskpremiumthedifferencebetweenthereturnonariskyassetandarisklessasset,

whichservesascompensaVonforinvestorsto

holdriskiersecuriVes.


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Por3olioconstrucVon:

Riskandreturn

Assumeatwo-stockpor3olioiscreatedwith$50,000investedinbothHTandCollecVons.

Apor3oliosexpectedreturnisaweightedaverageofthereturnsofthepor3olioscomponentassets.

StandarddeviaVonisalilemoretrickyandrequiresthatanewprobabilitydistribuVonforthepor3olioreturnsbedevised.


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CalculaVngpor3olioexpectedreturn

6.7%(1.0%)0.5(12.4%)0.5r

rwr

:averageweightedaisr

p

^

N

1i

i

^

ip

^

p

^

=+=

==


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8-22

AnalternaVvemethodfordeterminingpor3olio

expectedreturn

Economy Prob. HT Coll Port.

Recession 0.1 -27.0% 27.0% 0.0%

Below avg 0.2 -7.0% 13.0% 3.0% Average 0.4 15.0% 0.0% 7.5%

Above avg 0.2 30.0% -11.0% 9.5%

Boom 0.1 45.0% -21.0% 12.0%

6.7%(12.0%)0.10(9.5%)0.20

(7.5%)0.40(3.0%)0.20(0.0%)0.10rp^

=++

++=


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CalculaVngpor3oliostandard

deviaVonandC

0.516.7%

3.4%CV

3.4%

6.7)-(12.00.10

6.7)-(9.50.206.7)-(7.50.40

6.7)-(3.00.20

6.7)-(0.00.10

p

21

2

2

2

2

2

p

==

=

+

+

+

+

=


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Commentsonpor3olioriskmeasures

p=.4%ismuchlowerthantheiofeitherstock(HT=20.0%;Coll.=1.2%).

p

=.4%islowerthantheweightedaverageof

HTandColl.s(16.6%).

Therefore,thepor3olioprovidestheaveragereturnofcomponentstocks,butlowerthanthe

averagerisk.

Why?NegaVvecorrelaVonbetweenstocks.


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Generalcommentsaboutrisk

5%foranaveragestock. MoststocksareposiVvely(thoughnot

perfectly)correlatedwiththemarket(i.e.,between0and1).

Combiningstocksinapor3oliogenerallylowersrisk.


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ReturnsdistribuVonfortwoperfectly

negaVvelycorrelatedstocks(=-1.0)

-10

15 15

25 2525

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM


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ReturnsdistribuVonfortwoperfectly

posiVvelycorrelatedstocks(=1.0)

Stock M

0

15

25

-10

Stock M

0

15

25

-10

Portfolio MM

0

15

25

-10


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CreaVngapor3olio:Beginningwithonestockandaddingrandomly

selectedstockstopor3olio

pdecreasesasstocksadded,becausetheywouldnotbeperfectlycorrelatedwiththe

exisVngpor3olio.

Expectedreturnofthepor3oliowouldremainrelaVvelyconstant.

EventuallythediversificaVonbenefitsofaddingmorestocksdissipates(aOerabout10stocks),

andforlargestockpor3olios,ptendstoconvergeto20%.


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IllustraVngdiversificaVoneffectsofa

stockpor3olio

# Stocks in Portfolio10 20 30 40 2,000+

Diversifiable Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)35


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Breakingdownsourcesofrisk

Stand-alonerisk=Marketrisk+Diversifiablerisk

MarketriskporVonofasecuritysstand-aloneriskthatcannotbeeliminatedthroughdiversificaVon.Measuredbybeta.

DiversifiableriskporVonofasecuritysstand-aloneriskthatcanbeeliminatedthroughproperdiversificaVon.


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Failuretodiversify

Ifaninvestorchoosestoholdaone-stockpor3olio(doesntdiversify),wouldtheinvestorbecompensatedfortheextrarisktheybear?

N!Stand-aloneriskisnotimportanttoawell-

diversifiedinvestor.

RaVonal,risk-averseinvestorsareconcernedwithp,whichisbaseduponmarketrisk.

Therecanbeonlyoneprice(themarketreturn)foragivensecurity.

NocompensaVonshouldbeearnedforholdingunnecessary,diversifiablerisk.


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CapitalAssetPricingModel(CAPM)

Modellinkingriskandrequiredreturns.CAPMsuggeststhatthereisaSecurityMarketLine(SML)thatstatesthatastocksrequiredreturnequalsthe

risk-freereturnplusariskpremiumthatreflectsthestocksriskaOerdiversificaVon.

ri=rRF+(rMrRF)bi

Primaryconclusion:TherelevantriskinessofastockisitscontribuVontotheriskinessofawell-diversifiedpor3olio.


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Beta

Measuresastocksmarketrisk,andshowsastocksvolaVlityrelaVvetothemarket.

Indicateshowriskyastockisifthestockisheldinawell-diversifiedpor3olio.


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Commentsonbeta

Ifbeta=1.0,thesecurityisjustasriskyastheaveragestock.

Ifbeta>1.0,thesecurityisriskierthanaverage(Aggressivestock)

Ifbeta


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CanthebetaofasecuritybenegaVve?

Yes,ifthecorrelaVonbetweenStockiandthemarketisnegaVve(i.e.,i,m


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CalculaVngbetas

Well-diversifiedinvestorsareprimarilyconcernedwithhowastockisexpectedtomoverelaVvetothemarketinthefuture.

Withoutacrystalballtopredictthefuture,analystsareforcedtorelyonhistoricaldata.AtypicalapproachtoesVmatebetaistorunaregressionofthesecurityspastreturnsagainstthepastreturnsofthemarket.

Theslopeoftheregressionlineisdefinedasthebetacoefficientforthesecurity.


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IllustraVngthecalculaVonofbeta

.

.

.

ri

_

rM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM^ ^

Year r M ri

1 15% 18%2 -5 -10

3 12 16


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Betacoefficientsfor

HT,Coll,andT-Bills

ri

_

kM

_

-20 0 20 40

40

20

-20

HT: b = 1.30

T-bills: b = 0

Coll: b = -0.87


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Comparingexpectedreturnsand

betacoefficients

Security ExpectedReturn Beta

HT 12.4% 1.2

Market 10.5 1.00

USR 9.8 0.88T-Bills 5.5 0.00

Coll. 1.0 -0.87

RiskiersecuriVeshavehigherreturns,sotherankorderisK.


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TheSecurityMarketLine(SML):

CalculaVngrequiredratesofreturn

SML:ri=rRF+(rMrRF)bi

ri=rRF+(RPM)bi

Assumetheyieldcurveisflatandthatr RF=5.5%andRP

M=5.0%.


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Whatisthemarketriskpremium?

AddiVonalreturnovertherisk-freerateneededtocompensateinvestorsforassuming

anaverageamountofrisk.

Itssizedependsontheperceivedriskofthestockmarketandinvestorsdegreeofrisk

aversion.

ariesfromyeartoyear,butmostesVmatessuggestthatitrangesbetween4%and8%peryear.


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CalculaVngrequiredratesofreturn

rHT =5.5%+(5.0%)(1.2) =5.5%+6.6% =12.10%

rM =5.5%+(5.0%)(1.00) =10.50%

rUSR =5.5%+(5.0%)(0.88) =9.90% rT-bill =5.5%+(5.0%)(0.00) =5.50% rColl =5.5%+(5.0%)(-0.87) =1.15%


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Expectedvs.Requiredreturns

r)r(Overvalued1.21.0Coll.

r)r(uedFairly val5.55.5bills-T

r)r(Overvalued9.99.8USR

r)r(uedFairly val10.510.5Market

r)r(dUndervalue12.1%12.4%HT

rr

^

^

^

^

^

^

chapter 08 risk and rates of return

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