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Chapter 7Risk and Rates of Return

Learning Objectives

After reading this chapter, students should be able to:

Explain the difference between stand-alone risk and risk in a portfolio context.

Explain how risk aversion affects a stocks required rate of return.

Discuss the difference between diversifiable risk and arket risk, and explain how each

t!pe of risk affects well-diversified investors.

Explain what the "A#$ is and how it can be used to estiate a stocks required rate ofreturn.

Discuss how changes in the general stock and the bond arkets could lead to changes inthe required rate of return on a firs stock.

Discuss how changes in a firs operations ight lead to changes in the required rate ofreturn on the firs stock.

Chapter 8: Risk and Rates of Return Learning Objectives 117

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Lecture Suggestions

%isk anal!sis is an iportant topic, but it is difficult to teach at the introductor! level. &e'ust tr! to give students an intuitive overview of how risk can be defined and easured, and

leave a technical treatent to advanced courses. (ur priar! goals are to be sure studentsunderstand )*+ that investent risk is the uncertaint! about returns on an asset, )+ theconcept of portfolio risk, and )+ the effects of risk on required rates of return.

&hat we cover, and the wa! we cover it, can be seen b! scanning the slides andntegrated "ase solution for "hapter /, which appears at the end of this chapter solution.0or other suggestions about the lecture, please see the 12ecture 3uggestions4 in "hapter 5,where we describe how we conduct our classes.

DAS O! C"A#$%R: & O' (8 DAS )(*+,inute periods-

118 Lecture Suggestions Chapter 7: Risk and Rates of Return

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Ans.ers to %nd+of+Chapter /uestions

7+1 a0 6o, it is not riskless. 7he portfolio would be free of default risk and liquidit! risk,but inflation could erode the portfolios purchasing power. f the actual inflation

rate is greater than that expected, interest rates in general will rise to incorporatea larger inflation preiu )#+ and8as we saw in "hapter 98the value of theportfolio would decline.

b0 6o, !ou would be sub'ect to reinvestent rate risk. ou ight expect to 1rollover4 the 7reasur! bills at a constant )or even increasing+ rate of interest, but ifinterest rates fall, !our investent incoe will decrease.

c0 A ;.3. governent-backed bond that provided interest with constant purchasingpower )that is, an indexed bond+ would be close to riskless. 7he ;.3. 7reasur!currentl! issues indexed bonds.

7+ a0 7he probabilit! distribution for coplete certaint! is a vertical line.

b0 7he probabilit! distribution for total uncertaint! is the

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be constructed, investors would probabl! be better off 'ust purchasing 7reasur! bills,or other Cero beta investents.

7+( 3ecurit! A is less risk! if held in a diversified portfolio because of its negativecorrelation with other stocks. n a single-asset portfolio, 3ecurit! A would be orerisk! because AF Gand "HAF "HG.

7+4 6o. 0or a stock to have a negative beta, its returns would have to logicall! beexpected to go up in the future when other stocks returns were falling. Iust becausein one !ear the stocks return increases when the arket declined doesnt ean thestock has a negative beta. A stock in a given !ear a! ove counter to the overallarket, even though the stocks beta is positive.

7+7 7he risk preiu on a high-beta stock would increase ore than that on a low-betastock.

%#' %isk #reiu for 3tock ' )r$J r%0+b'.

f risk aversion increases, the slope of the 3$2 will increase, and so will the arketrisk preiu )r$J r%0+. 7he product )r$J r%0+b'is the risk preiu of the 'thstock. f b'

is low )sa!, ?.5+, then the product will be sallK %#'will increase b! onl! half theincrease in %#$. Lowever, if b'is large )sa!, .?+, then its risk preiu will rise b!twice the increase in %#$.

7+8 According to the 3ecurit! $arket 2ine )3$2+ equation, an increase in beta willincrease a copan!s expected return b! an aount equal to the arket riskpreiu ties the change in beta. 0or exaple, assue that the risk-free rate is9@, and the arket risk preiu is 5@. f the copan!s beta doubles fro ?.B to*.9 its expected return increases fro *?@ to *M@. 7herefore, in general, acopan!s expected return will not double when its beta doubles.

7+3 a0 A decrease in risk aversion will decrease the return an investor will require onstocks. 7hus, prices on stocks will increase because the cost of equit! will decline.

b0 &ith a decline in risk aversion, the risk preiu will decline as copared to thehistorical difference between returns on stocks and bonds.

c0 7he iplication of using the 3$2 equation with historical risk preius )whichwould be higher than the 1current4 risk preiu+ is that the "A#$ estiatedrequired return would actuall! be higher than what would be reflected if the orecurrent risk preiu were used.

7+1* %#$ r$J r%0r$ %#$ = r%0

9@ = 5.@ **.@

%equired return on stock %isk-free return = )3tocks beta+)$arket risk preiu+ 5.@ = *. N 9@ *.5@.

1* Answers and Solutions Chapter 7: Risk and Rates of Return

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7+11( )

N2

t avg

t 1

r r

EstimatedN 1

=

=

rAvg )*?.5@ = B.99@ J /.M?@ J ./5@ = **.B@ = *.9@+O9 9.B@.

sqrtP)*?.5@ J 9.B@+ = )B.99@ J 408@+ = )J/.M?@ J 9.B@+ = )J./5@ J9.B@+

=)**.B@ J 9.B@+ = )*.9@ J 9.B@+O5+ *?.9@.

Chapter 7: Risk and Rates of Return Answers and Solutions 11

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So5utions to %nd+of+Chapter #rob5e,s

7+1 rQ )?.*+)-5?@+ = )?.+)-5@+ = )?.M+)*9@+ = )?.+)5@+ = )?.*+)9?@+

**.M?@.

)-5?@ J **.M?@+)?.*+ = )-5@ J **.M?@+)?.+ = )*9@ J **.M?@+)?.M+ = )5@ J **.M?@+)?.+ = )9?@ J **.M?@+)?.*+

/*.MMK 9.9@.

"H **.M?@

9.9A@ .M.

7+ nvestent Geta>5,??? ?.B M?,??? *.M

7otal >/5,???

bp )>5,???O>/5,???+)?.B+ = )>M?,???O>/5,???+)*.M+ *.*.

7+& r%0 B@K r$ *@K b ?.BK r R

r r%0= )r$J r%0+b B@ = )*@ J B@+?.B **.@.

7+2 r%0 5.5@K %#$ 9.5@K r$ R

r$ 5.5@ = )9.5@+* *@.

r when b *. R

r 5.5@ = 9.5@)*.+ *.@.

7+( a0 r **@K r%0 /@K %#$ M@.

r r%0= )r$J r%0+b**@ /@ = M@bM@ M@b

b *.

1 Answers and Solutions Chapter 7: Risk and Rates of Return

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b0 r%0 /@K %#$ 9@K b *.

r r%0= )r$J r%0+b /@ = )9@+* *@.

7+4 a0 =

=6

*i

iir#rQ .

:rQ ?.*)-5@+ = ?.)?@+ = ?.M)?@+ = ?.)5@+ = ?.*)M5@+

*M@ versus *@ for M,???,??

>9??,???)-?.5?+ =

>M,???,??

>*,???,??)*.5+ =

>M,???,??>,???,?? )?./5+

bp )?.*+)*.5+ = )?.*5+)-?.5?+ = )?.5+)*.5+ = )?.5+)?./5+ ?.*5 J ?.?/5 = ?.*5 = ?./5 ?./95.

rp r%0= )r$J r%0+)bp+ 9@ = )*M@ J 9@+)?./95+ *.*@.

Chapter 7: Risk and Rates of Return Answers and Solutions 1&

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Alternative solution: 0irst, calculate the return for each stock using the "A#$ equationPr%0= )r$J r%0+bS, and then calculate the weighted average of these returns.

r%0 9@ and )r$J r%0+ B@.

3tock nvestent Geta r r%0= )r$J r%0+b &eight

A > M??,??? *.5? *B@ ?.*?G 9??,??? )?.5?+ ?.*5" *,???,??? *.5 *9 ?.5D ,???,??? ?./5 * ? .5?

7otal >M,???,??? * .??

rp *B@)?.*?+ = @)?.*5+ = *9@)?.5+ = *@)?.5?+ *.*@.

7+3 n equilibriu:

rI IrQ *.5@.

rI r%0= )r$J r%0+b*.5@ M.5@ = )*?.5@ J M.5@+b

b *..

7+1* &e know that b% *.5?, b3 ?./5, r$ *@, r%0 /@.

ri r%0= )r$J r%0+bi /@ = )*@ J /@+bi.

r% /@ = 9@)*.5?+ *9.?@r3 /@ = 9@)?./5+ ** .5

M .5 @

7+11 An index fund will have a beta of *.?. f r$is *.?@ )given in the proble+ and the risk-free rate is 5@, !ou can calculate the arket risk preiu )%#$+ calculated as r$J r%0as follows:

r r%0= )%#$+b*.?@ 5@ = )%#$+*.?/.?@ %#$.

6ow, !ou can use the %#$, the r%0, and the two stocks betas to calculate their requiredreturns.

Gradford:

rG r%0= )%#$+b

5@ = )/.?@+*.M5 5@ = *?.*5@ *5.*5@.

0arle!:

r0 r%0= )%#$+b 5@ = )/.?@+?.B5


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7+1( a0 ;sing 3tock < )or an! stock+:

@ r%0= )r$J r%0+b*5?,??

>*M,5?)b+ = >*5?,??

>/,5??)*.??+

*.* ?.5b = ?.?5*.?/ ?.5b

*.*9 b.

6ew portfolio beta ?.5)*.*9+ = ?.?5)*./5+ *.*5/5 *.*9.

Alternative solutions:

*. (ld portfolio beta *.* )?.?5+b*= )?.?5+b= ... = )?.?5+b?

*.* +b) i )?.?5+

ib *.*O?.?5 .M.

6ew portfolio beta ).M J *.? = *./5+)?.?5+ *.*5/5 *.*9.

. ib excluding the stock with the beta equal to *.? is .M J *.? *.M, so thebeta of the portfolio excluding this stock is b *.MO* *.*9. 7he beta of thenew portfolio is:

*.*9)?.5+ = *./5)?.?5+ *.*5/5 *.*9.

7+17 bL% *.BK b2% ?.9. 6o changes occur.

r%0 9@. Decreases b! *.5@ to M.5@.

r$ *@. 0alls to *?.5@.

6ow 3$2: ri r%0= )r$J r%0+bi.

rL% M.5@ = )*?.5@ J M.5@+*.B M.5@ = 9@)*.B+ *5.@


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range of =?.5, investing in a portfolio of stocks would definitel! be aniproveent over investing in the single stock.

7+1

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??/ )?.5?+ )/.9?+ )M.?5+??B /.?? 9.? 9.95

$ean **.? **.? **.?3td. Dev. ?./ ?./B ?.*"oef. Har. *.BM *.BM *./B

e0 A risk-averse investor would choose the portfolio over either 3tock A or 3tock Galone, since the portfolio offers the sae expected return but with less risk. 7hisresult occurs because returns on A and G are not perfectl! positivel! correlated )rAG ?.BB+.

7+& a0 $rQ ?.*)-B@+ = ?.)?@+ = ?.M)*@+ = ?.)?@+ = ?.*)5?@+ *@.

r%0 9@. )given+

7herefore, the 3$2 equation is:

ri r%0= )r$J r%0+bi 9@ = )*@ J 9@+bi 9@ = )/@+bi.

b0 0irst, deterine the funds beta, b0. 7he weights are the percentage of fundsinvested in each stock:

A >*9?O>5?? ?..G >*?O>5?? ?.M." >B?O>5?? ?.*9.D >B?O>5?? ?.*9.E >9?O>5?? ?.*.

b0 ?.)?.5+ = ?.M)*.+ = ?.*9)*.B+ = ?.*9)*.?+ = ?.*)*.9+ ?.*9 = ?.BB = ?.BB = ?.*9 = ?.* *.?BB.

6ext, use b0 *.?BB in the 3$2 deterined in #art a:

0rQ 9@ = )*@ J 9@+*.?BB 9@ = /.9*9@ *.9*9@.

c0 r6 %equired rate of return on new stock 9@ = )/@+*.5 *9.5@.

An expected return of *5@ on the new stock is below the *9.5@ required rate ofreturn on an investent with a risk of b *.5. 3ince r6 *9.5@ F 6rQ *5@, thenew stock should not be purchased. 7he expected rate of return that would ake thefund indifferent to purchasing the stock is *9.5@.

Chapter 7: Risk and Rates of Return Answers and Solutions 13

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Co,prehensiveSpreadsheet #rob5e,

Note to Instructors:7he solution to this proble is not provided to students at the back of their text. nstructors

can access the Excelfile on the textbooks web site or the nstructors %esource "D.

7+2 a0

b0

(n a stand-alone basis, it would appear that Gartan is the ost risk!, %e!noldsthe least risk!.

c0

%e!nolds now looks ost risk!, because its risk )3D+ per unit of return is highest.

d0

t is clear that Gartan oves with the arket and %e!nolds oves counter to thearket. 3o, Gartan has a positive beta and %e!nolds a negative one.

e0 Gartans calculations:

1&* Comprehensive/Spreadsheet Problem Chapter 7: Risk and Rates of Return

Bartman Reynolds Index

Standard deviation of return 31.5% 9.7% 13.8%


Coefficient of ariation 1.!7 3."3 !."7


#!!8 #$.7% 1.1% 3#.8%

#!!7 $.#% 13.#% 1.#%

#!!" "#.8% 1!.!% 3$.9%

#!!5 #.9% !.$% 1$.8%

#!!$ "1.!% 11.7% 19.!%

&v' Returns #9.$% #.7% #!."%

-20%

-10%

0%

10%

20%

0%

40%

!0%

60%

"0%

0.0% 10.0% 20.0% 0.0% 40.0%

Stoc(s)

Returns

Index Returns

Stoc( Returns s. Index

#artma$

Re$o&ds

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9ntegrated Case

7+(

China Deve5op,ent 9ndustria5 6ankis! and eturn

Assue that !ou recentl! graduated with a a'or in finance. ou 'ust landed a

'ob as a financial planner with "hina Developent ndustrial Gank )"DG+, a large

financial services corporation. our first assignent is to invest >*??,??? for a

client. Gecause the funds are to be invested in a business at the end of * !ear,

!ou have been instructed to plan for a *-!ear holding period. 0urther, !our boss

has restricted !ou to the investent alternatives in the following table, shown

with their probabilities and associated outcoes. )0or now, disregard the ites

at the botto of the dataK !ou will fill in the blanks later.+

Returns on A5ternative 9nvest,ents

%sti,ated Rate of Return

State ofthe%cono,

#rob0 $+6i55s"igh$ech

Co55ec+tions

;0S0Rubber

arket#ortfo5io

+Stock#ortfo5io

%ecession ?.* 5.5@ -/.?@ /.?@ 9.?@a -*/.?@ ?.?@

Gelow Avg. ?. 5.5 -/.? *.? -*M.? -.?Average ?.M 5.5 *5.? ?.? .? *?.? /.5

Above Avg. ?. 5.5 ?.? -**.? M*.? 5.?

Goo ?.* 5.5 M5.? -*.? 9.? B.? *.?

r+hat )r< - *.?@ .B@ *?.5@

Std0 dev0

)-?.? *. *B.B *5. .M

Coeff0 of =ar0 )C=- *. *. *.M ?.5

beta )b- -?.B/ ?.BB

a6ote that the estiated returns of ;.3. %ubber do not alwa!s ove in the sae direction asthe overall econo!. 0or exaple, when the econo! is below average, consuerspurchase fewer tires than the! would if the econo! were stronger. Lowever, if the econo!is in a flat-out recession, a large nuber of consuers who were planning to purchase a newcar a! choose to wait and instead purchase new tires for the car the! currentl! own. ;nderthese circustances, we would expect ;.3. %ubbers stock price to be higher if there was arecession than if the econo! was 'ust below average.

Chapter 7: Risk and Rates of Return Integrated Case 1&&

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"DGs econoic forecasting staff has developed probabilit! estiates for the

state of the econo!K and its securit! anal!sts have developed a sophisticated

coputer progra, which was used to estiate the rate of return on each

alternative under each state of the econo!. Ligh 7ech nc. is an electronics fir,

"ollections nc. collects past-due debts, and ;.3. %ubber anufactures tires and

various other rubber and plastics products. "DG also aintains a 1arket

portfolio4 that owns a arket-weighted fraction of all publicl! traded stocksK !ou

can invest in that portfolio, and thus obtain average stock arket results. iven

the situation as described, answer the following questions.

A0 )1- >h is the $+bi55?s return independent of the state of the

econo,@ Do $+bi55s pro,ise a co,p5ete5 risk+free return@%p5ain0

Answer: P3how 3/-* through 3/-/ here.S 7he 5.5@ 7-bill return does not

depend on the state of the econo! because the 7reasur! ust )and

will+ redee the bills at par regardless of the state of the econo!.

7he 7-bills are risk-free in the default risk sense because the

5.5@ return will be realiCed in all possible econoic states. Lowever,

reeber that this return is coposed of the real risk-free rate, sa!

@, plus an inflation preiu, sa! .5@. 3ince there is uncertaint!

about inflation, it is unlikel! that the realiCed real rate of return would

equal the expected @. 0or exaple, if inflation averaged .5@ over

the !ear, then the realiCed real return would onl! be 5.5@ J .5@

@, not the expected @. 7hus, in ters of purchasing power, 7-bills

are not riskless.

Also, if !ou invested in a portfolio of 7-bills, and rates thendeclined, !our noinal incoe would fallK that is, 7-bills are exposed

to reinvestent rate risk. 3o, we conclude that there are no trul!

risk-free securities in the ;nited 3tates. f the 7reasur! sold inflation-

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indexed, tax-exept bonds, the! would be trul! riskless, but all actual

securities are exposed to soe t!pe of risk.

A0 )- >h are "igh $ech?s returns epected to ,ove .ith the

econo,B .hereas Co55ections? are epected to ,ove counter

to the econo,@

Answer: P3how 3/-B here.S Ligh 7echs returns ove with, hence are

positivel! correlated with, the econo!, because the firs sales, and

hence profits, will generall! experience the sae t!pe of ups and

downs as the econo!. f the econo! is booing, so will Ligh 7ech.

(n the other hand, "ollections is considered b! an! investors to be

a hedge against both bad ties and high inflation, so if the stock

arket crashes, investors in this stock should do relativel! well.

3tocks such as "ollections are thus negativel! correlated with )ove

counter to+ the econo!. )6ote: n actualit!, it is alost ipossible to

find stocks that are expected to ove counter to the econo!.+

60 Ca5cu5ate the epected rate of return on each a5ternative and

fi55 in the b5anks on the ro. for r< in the previous tab5e0

Answer: P3how 3/- and 3/-*? here.S 7he expected rate of return, r< , is

expressed as follows:

!

1i

iir#r< 0

Lere #iis the probabilit! of occurrence of the ith state, riis the

estiated rate of return for that state, and !is the nuber of states.

Lere is the calculation for Ligh 7ech:

r< "igh $ech *01)+70*-E*0)+70*-E*02)1(0*-E*0)&*0*-

E*01)2(0*-

1020

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1(00

C0 )- >hat tpe of risk is ,easured b the standard deviation@

Answer: P3how 3/-* through 3/-*5 here.S 7he standard deviation is aeasure of a securit!s )or a portfolios+ stand-alone risk. 7he larger

the standard deviation, the higher the probabilit! that actual realiCed

returns will fall far below the expected return, and that losses rather

than profits will be incurred.

C0 )&- Dra. a graph that sho.s rough5 the shape of the probabi5it

distributions for "igh $echB ;0S0 RubberB and $+bi55s0

Ans.er: P r o b a b i l i t y o f

O c c u r r e n c e

R a t e o f R e t u r n ( % )

T # 4 i ll s

U . S . R u ! ! e r

5 i + 6 T e / 6

- 6 0 - 4 5 - 3 0 - 1 5 0 1 5 3 0 4 5 6 0

(n the basis of these data, Ligh 7ech is the ost risk! investent,

7-bills the least risk!.

D0 Suppose ou sudden5 re,e,bered that the coefficient of

variation )C=- is genera55 regarded as being a better

,easure of stand+a5one risk than the standard deviation

.hen the a5ternatives being considered have .ide5

differing epected returns0 Ca5cu5ate the ,issing C=sB and

fi55 in the b5anks on the ro. for C= in the tab5e0 Does the C=

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produce the sa,e risk rankings as the standard deviation@

%p5ain0

Answer: P3how 3/-*9 through 3/-* here.S 7he coefficient of variation )"H+ is

a standardiCed easure of dispersion about the expected valueK it

shows the aount of risk per unit of return.

C= r< 0

C=$+bi55s *0*(0( *0*0

C="igh $ech *0*102 1040

C=Co55ections 1&010* 1&00

C=;0S0 Rubber 1808308 1030

C= 1(01*0( 1020

&hen we easure risk per unit of return, "ollections, with its low

expected return, becoes the ost risk! stock. 7he "H is a better

easure of an assets stand-alone risk than because "H considers

both the expected value and the dispersion of a distribution8a securit!

with a low expected return and a low standard deviation could have a

higher chance of a loss than one with a high but a high r< .

%0 Suppose ou created a +stock portfo5io b investing K(*B***

in "igh $ech and K(*B*** in Co55ections0

)1- Ca5cu5ate the epected return ) pr< -B the standard deviation

)p-B and the coefficient of variation )C=p- for this portfo5ioand fi55 in the appropriate b5anks in the tab5e0

Answer: P3how 3/-? through 3/- here.S 7o find the expected rate of return

on the two-stock portfolio, we first calculate the rate of return on the

portfolio in each state of the econo!. 3ince we have half of our

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one! in each stock, the portfolios return will be a weighted average

in each t!pe of econo!. 0or a recession, we have: rp ?.5)-/@+ =

?.5)/@+ ?@. &e would do siilar calculations for the other states

of the econo!, and get these results:

State #ortfo5io

%ecession ?.?@Gelow average .?Average /.5Above average .5Goo *.?

6ow we can ultipl! the probabilit! ties the outcoe in each

state to get the expected return on this two-stock portfolio, 9./@.

Alternativel!, we could appl! this forula,

r .iri *0()102- E *0()10*- 407Bwhich finds ras the weighted average of the expected returns of the

individual securities in the portfolio.

t is tepting to find the standard deviation of the portfolio as the

weighted average of the standard deviations of the individual

securities, as follows:

p.i)i- E .j)j- *0()*- E *0()1&0- 14040Lowever, this is not correct8it is necessar! to use a different forula, the

one for that we used earlier, applied to the two-stock portfolios returns.

7he portfolios depends 'ointl! on )*+ each securit!s and )+

the correlation between the securities returns. 7he best wa! to

approach the proble is to estiate the portfolios risk and return in

each state of the econo!, and then to estiate pwith the

forula. iven the distribution of returns for the portfolio, we can

calculate the portfolios and "H as shown below:

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p G)*0* H 407-)*01- E )&0* H 407-)*0- E )70( H 407-)*02- E )30( + 407-)*0- E )10* H 407-)*01-IJ &020

C=p &02407 *0(10

%0 )- "o. does the riskiness of this t.o+stock portfo5io co,pare

.ith the riskiness of the individua5 stocks if the .ere he5d in

iso5ation@

Answer: P3how 3/-M through 3/-B here.S ;sing either or "H as our stand-

alone risk easure, the stand-alone risk of the portfolio is significantl!

less than the stand-alone risk of the individual stocks. 7his is because

the two stocks are negativel! correlated8when Ligh 7ech is doingpoorl!, "ollections is doing well, and vice versa. "obining the two

stocks diversifies awa! soe of the risk inherent in each stock if it

were held in isolation, i.e., in a *-stock portfolio.

Optiona5 /uestion

Does the epected rate of return on the portfo5io depend on the

percentage of the portfo5io invested in each stock@ >hat about the

riskiness of the portfo5io@

Answer: ;sing a spreadsheet odel, its eas! to var! the coposition of the

portfolio to show the effect on the portfolios expected rate of return

and standard deviation:

"igh $ech #5us Co55ections

in "igh $ech

?@ *.?@ *.@

*? .* .? . 9.9? M.M .M? 5.9 ?.M5? 9./ .M9? /.B 9.B

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pr< p

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/? .? *?.*B? *?.* *.M? **. *9./

*?? *.M ?.?

7he expected rate of return on the portfolio is erel! a linearcobination of the two stocks expected rates of return. Lowever,

portfolio risk is another atter. pbegins to fall as Ligh 7ech and

"ollections are cobinedK it reaches near Cero at M?@ Ligh 7echK and

then it begins to rise. Ligh 7ech and "ollections can be cobined to

for a near Cero risk portfolio because the! are ver! close to being

perfectl! negativel! correlatedK their correlation coefficient is -?.5.

)6ote: ;nfortunatel!, we cannot find an! actual stocks withr -*.?.+

'0 Suppose an investor starts .ith a portfo5io consisting of one

rando,5 se5ected stock0 >hat .ou5d happen: )1- to the

riskiness and to the epected return of the portfo5io as ,ore

rando,5 se5ected stocks .ere added to the portfo5io@ )-

>hat is the i,p5ication for investors@ Dra. a graph of the

t.o portfo5ios to i55ustrate our ans.er0

Answer: P3how 3/- here.S

Density

0

Portfolio of Stocksit! r" # 10$5%

One Stock

10$5%

Density

0

Portfolio of Stocksit! r" # 10$5%

One Stock

10$5%

7he standard deviation gets saller as ore stocks are cobined in

the portfolio, while rp)the portfolios return+ reains constant. 7hus,

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b! adding stocks to !our portfolio, which initiall! started as a *-stock

portfolio, risk has been reduced.

n the real world, stocks are positivel! correlated with one

another8if the econo! does well, so do stocks in general, and vice

versa. "orrelation coefficients between stocks generall! range in the

vicinit! of =?.5. A single stock selected at rando would on

average have a standard deviation of about 5@. As additional

stocks are added to the portfolio, the portfolios standard deviation

decreases because the added stocks are not perfectl! positivel!

correlated. Lowever, as ore and ore stocks are added, each new

stock has less of a risk-reducing ipact, and eventuall! adding

additional stocks has virtuall! no effect on the portfolios risk as

easured b! . n fact, stabiliCes at about ?@ when M? or ore

randol! selected stocks are added. 7hus, b! cobining stocks into

well-diversified portfolios, investors can eliinate alost one-half the

riskiness of holding individual stocks. )6ote: t is not copletel!

costless to diversif!, so even the largest institutional investors hold

less than all stocks. Even index funds generall! hold a saller

portfolio that is highl! correlated with an index such as the 3W# 5??

rather than holding all the stocks in the index.+

7he iplication is clear: nvestors should hold well-diversified

portfolios of stocks rather than individual stocks. )n fact, individuals

can hold diversified portfolios through utual fund investents.+ G!

doing so, the! can eliinate about half of the riskiness inherent in

individual stocks.

0 )1- Shou5d the effects of a portfo5io i,pact the .a investors

think about the riskiness of individua5 stocks@

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"igh $ech 102 10&

arket 1*0( 10**

;0S0 Rubber 308 *088

$+6i55s (0( *0**

Co55ections 10* )*087-

)1- >hat is a beta coefficientB and ho. are betas used in risk

ana5sis@

Answer: P3how 3/- through 3/- here.S

i&! 'ec!

(slo"e # beta # 1$3)

arket(slo"e # beta # 1$0)

Return on Stock i%

Return on t!earket

40

0

-0 0 40

-0

'-*ills(slo"e # beta # 0)

)Draw the fraework of the graph, put up the data, then plot the

points for the arket )M5line+ and connect the, and then get the

slope as *.?.+ 3tate that an average stock, b! definition,oves with the arket. 7hen do the sae with Ligh 7ech and 7-bills.

Geta coefficients easure the relative volatilit! of a given stock vis-X-

vis an average stock. 7he average stocks beta is *.?. $ost stocks

have betas in the range of ?.5 to *.5. 7heoreticall!, betas can be

negative, but in the real world the! are generall! positive.

Getas are calculated as the slope of the 1characteristic4 line,which is the regression line showing the relationship between a given

stock and the general stock arket.

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90 $he ie5d curve is current5 f5atN that isB 5ong+ter, $reasur

bonds a5so have a (0( ie5d0 ConseMuent5B CD96 assu,es

that the risk+free rate is (0(0

)1- >rite out the Securit arket Line )SL- eMuationB use it toca5cu5ate the reMuired rate of return on each a5ternativeB and

graph the re5ationship bet.een the epected and reMuired

rates of return0

Answer: P3how 3/-M* through 3/-M here.S Lere is the 3$2 equation:

ri rR'E )rH rR'-bi0

"DG has estiated the risk-free rate to be rR' 5.5@. 0urther,

our estiate of r r< is *?.5@. 7hus, the required rates of return

for the alternatives are as follows:

"igh $ech: (0( E )1*0( H (0(-10& 101*0

arket: (0( E )1*0( H (0(-10** 1*0(*0

;0S0 Rubber: (0( E )1*0( H (0(-*088 303*0

$+bi55s: (0( E )1*0( H (0(-* (0(*0

Co55ections: (0( E )1*0( H (0(-+*087 101(0

90 )- "o. do the epected rates of return co,pare .ith the

reMuired rates of return@

Answer: P3how 3/-MM and 3/-M5 here.S &e have the following relationships:

%pected ReMuired

Return Return

Securit )r< - )r- Condition

Ligh 7ech *.M@ *.*@ ;ndervalued: r< F r$arket *?.5 *?.5 0airl!valued)arketequilibriu+;.3. %ubber .B . (vervalued: rF r

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