Topic Risk And Return Questions.pdf

Calculating the rate of return Amount Amountinvestedinvested ¿ Rate Rateof of retreturnurn= Amount r Amount receivedeceived− ¿ Amount Amountinvesteinvestedd Eam!le" #Tool Kit $%&' "up

RISK AND RETURN INTRODUCTION

What is an investment?

Investment is current commitment of dollars for a period of time in order to derivefuture payment that will compensate the investor for:

1 the time funds are committed

2 the expected rate of ination

3 the uncertainty of the future paymentsWhat are

What are investmeninvestment returns?t returns?

Investment returns measure the nancial results of an investment

Returns may be historical or prospective anticipated!

Calculating the rate of return

 Amount Amountinvestedinvested

¿

 Rate Rateof of retreturnurn= Amount r Amount receivedeceived− ¿ Amount Amountinvesteinvestedd

Eam!le" #Tool Kit $%&'

"uppose you pay #$%% for

"uppose you pay #$%% for an investment that returns #&%% in an investment that returns #&%% in one yearone year 'hat is the'hat is theannual rate of return(

Ra atte e o of f rreettu urrn n 2 20 0% %

If you have a )nown amount of the initial mar)et price 

If you have a )nown amount of the initial mar)et price  PP%!* reali+ed or planneddividend 

dividend DD1! and reali+ed or planned sellin, price ! and reali+ed or planned sellin, price  PP1!* and a holdin, period of security is not exceedin, one year* then the rate of return on investment in thissecurities can be calculated as follows:

 Rate Rateof of retureturnrn= D1+( P1− P0))

 P1

"o* this form

"o* this formula can ula can be be usused ed to to detdetererminmine e the histthe histororicaical l rarate te of of rereturturnn for thefor theholding period of one year 

holding period of one year  based on the historical data! as the expected rate of  based on the historical data! as the expected rate of 

returnreturn for the holding period of one year for the holding period of one year  based on expected future dividends and based on expected future dividends andprices!.

For holding periods longer than one year For holding periods longer than one year   we have to calculated internal rate of   we have to calculated internal rate of return* which is consistent with the principles of time value of money

 -he term in  -he term in brac)etbrac)ets is called as is called a capital gain or losscapital gain or loss durin, the period. durin, the period

Eam!les"

1 "uppose you buy 1% shares of a stoc) for # 1*%%% -he stoc) pays no dividends*but at the end of 1 year* you sell the stoc) for # 1*1%% 'hat is the return on your #1*%%% investment(

P%1*%%% #

P11*1%% #

D1% #n

n11 Rate Rateof of retreturnurn=1,100−1,000

1,000 =0.1=10

2 /t the be,in of the year* the investor has bou,ht a share of 0 -elecom on thedividend in the amount of %.6% 45 -he investor sold the share at the end of theyear for 36 45

year for 36 45 'hat is the rate of 'hat is the rate of returreturn on this investment(n on this investment(

P%2$ 45

D1%*6% 45

P136 45R

1+k 

1818k k 36.692$36.692$

1818k k 1.$6&1.$6&

k 

k %.$6& $6.&;%.$6& $6.&;

3 /nother invester has also bou,ht a share of 0 -elecom at a price of 2$ 45 e)ept the share for two years and sold it after two years for $$ 45 0 -elecom paiddurin, the rst year a dividend in the amount of %.6% 45 and durin, the secondyear a dividend in the amount of 1.&% 45 'hat is the rate of return on thisinvestment(

1818k k t t   ⇒  k k 1.$2=1%.$2 $2;1.$2=1%.$2 $2;

What is What is investment ris(?investment ris(?

7or all securities* except for the ris) free securities* the return expected may di>erfr

from om ththe e rretetururn n rreaealili+e+ed d "o"o* * we we dedenned ed ththee  ris(   ris(    as as ththe e chchanance ce ththat at sosomemeunfavorable event will occur /n event)sevent)s  !ro*a*ilit+!ro*a*ilit+ is dened as the chance that is dened as the chance thatth

the e evevenent t wiwill ll ococcucurr If If alall l popossssibible le evevenentsts* * or or ououtctcomomeses* * arare e liliststeded* * anand d if if aap

prroobababibilility ty is is asasssii,n,ned ed to to eaeacch h evevenentt* * tthe he llisistitinn, , is is ccaalllleded a a !r!ro*ao*a*il*ilit+it+,istri*ution

,istri*ution -he -he proprobabilbabilities musities must sum t sum to 1*% or to 1*% or 1%%;1%%; For risky securities theFor risky securities theactual rate of return can be viewed as a random variable that has a probability distribution

 -he ris) of a -he ris) of an assets casn assets cash ow can be ch ow can be consideronsidered:ed:

a! on a stan,-alone *asisstan,-alone *asis each asset by  each asset by itself!itself!   Stand  Stand−alone Risk b! in a !ortfolio contet!ortfolio contet* where the investment is * where the investment is combined with other assets andcombined with other assets andits ris) is reduced throu,h ,iversi.cation  Risk ∈a portfoa portfoliolio context context 

IN/EST0ENT IN ONE ASSET

E!ecte, rate of return # rr^^¿   

rrii= =¿ ¿  the the iith possible outcometh possible outcome

 Pii= =¿ ¿ the probability of thethe probability of the iith outcometh outcome

n= =¿ ¿  the number of  the number of possible outcomespossible outcomes

The tighter the probability distribution, the more likely is that the actual outcomewill be close to the expected value and conseuently, the less likely it is that theact

actual returual return n wilwill l end end up up far below the far below the expexpectected ed retretururn n ThThus, us, the the tigtightehter r thethe probability

 probability distributdistribution ion of of expected expected return, return, the the smaller smaller the the risk risk of of a a givengiveninvestment

The measure of tightness of the probability distribution is the standard deviation,which we calculate as follows!

Variance (    σ σ 2¿

1 12 2 & &3 3 2 & &3 3 1 12 2 4 43 3 5 52 2

S

St to oc c( ( A S St to oc c( ( 6

Returns #7'

producin, a 1%; returnin, a 1%; return* and a 3%; chance of produci* and a 3%; chance of producin, a =1$; returnn, a =1$; return 'hat is its'hat is itsexpected return(

expected return( 'hat is 'hat is its standard its standard deviation(deviation(

Probabilit

3

30 0% % 2 25 5% % 4

40 0% % 1 10 0% % 3

conse<uently "o if "o if a choice has a choice has to be made to be made between two investments that have thebetween two investments that have thesame expected returns but di>erent standard deviations* most people would choosethe one with lower standard deviation and therefore the lower ris)

1 Rate of return is continuous variable* we have lar,e number of iterations andbecause of that we will ta)e approximation with normal distribution

In order to perform comparisons of alternative investments with di>erent expectedrates of return* you cannot use the standard deviation In that case we use anothermeasure of ris) called the coeAcient of variation BC! which shows the ris) per unit

of return

Coe8cienCoe8cient of t of variation #variation #    !" !" ¿

!" =σ 

^^

rrEam!le" #Tool Kit $%$'/n investment has an expected return of 1$; and a standard deviation of 3%;.'hat is its coeAcient of variation(

!" =30

15 =2

-sing istorical data

'e described the procedure for ndin, mean and standard deviation when data are

in the form of a )nown probability distribution

0ut in case that 0ut in case that we have available only data we have available only data on returns realion returns reali+ed in some +ed in some past periodpast period

    ´´rrt ¿ available we use followin, formula:

 / stoc)Ds return has the followin, distributions:

Femand for theBompanyDs Groducts

Grobability of -his Femand

Hccurrin,

Rate of Return if -hisFemand Hccurs ;!'

11.40%

26.69%

  2.3?

 -he  -he avera,e avera,e investor investor is is ris) ris) averse* averse* which which means means that that he he or or she she must must bebecompensated for holdin, ris)y

compensated for holdin, ris)y assets -hereforassets -herefore* ris)ier e* ris)ier assets have assets have hi,her re<uirhi,her re<uirededreturns than less ris)y assets

IN/EST0ENT IN T9E :ORT;O<IO

:ortfolio:ortfolio is a combination of two or more securities* currencies* real estate or other is a combination of two or more securities* currencies* real estate or otherass

assets ets helheld d by by indindiviividuduals als or or comcompanpaniesies -h-he e ,oa,oal l of of crcreateatin, a in, a porportfotfolio is lio is totominimi+e ris) throu,h diversication

"o* portfolio of securities consists of two and 9 or more securities in which theinvestor invests money in specic ratios to reduce the ris) Diversi.cDiversi.cation of ation of ris( ris( 

is achieved by

is achieved by the successful combination of securities -he successful combinationthe successful combination of securities -he successful combination

of securities is done by selectin, the securities that are each wea)ly

of securities is done by selectin, the securities that are each wea)ly corrcorrelated withelated witheach* and whose yields move

each* and whose yields move inverselyinversely "o we "o we can use the benets of can use the benets of diversicatiodiversicationn

in terms of ris) reduction until the securities are not perfectly positively correlated

(( $ $ %%1))

E!ecte, return on a !ortfolio #

   rr^  p

¿

 -he  -he expected expected returreturn n on on a a portfolio portfolio is is wei,hted wei,hted mean mean of of the the expected expected returreturns ns of of individual investments that ma)e up the portfolio* where wei,hts are the shares of the money invested in each security

^

rrii= =¿ ¿

 the expected returns on the individual stoc)s

&ii= =¿ ¿ the wei,hts

n= =¿ ¿  the number of  the number of stoc)s in the portfoliostoc)s in the portfolio

Stan,ar, ,eviation of a !ortfolio #    σ σ  p¿

 -he  -he ris) ris) of of the the portfolio portfolio is is as as with with the the individual individual securities securities expresexpressed sed by by thethevariance and standard deviation of returns owever* the ris) of the portfolio is notsimply the wei,hted mean of individual securities standard deviations* because theris) of a portfolio depends not only on the ris)iness of the securities that ma)e upthe portfolio* but also the

the portfolio* but also the relationsrelationships that exist between these hips that exist between these securities.securities

&ii∙∙ && '∙!(" i'

&ii= =¿ ¿  share of total funds invested in security 

& '= =¿ ¿  share of total funds invested in security i

!(" i'= =¿ ¿  covariance between the yield of the  covariance between the yield of the i=th and =th securityi=th and =th security

Covariance *et=een stoc( i an, the stoc( > #    !(" !(" i'¿

 -his is a statistical measur -his is a statistical measure that indicates the de,ree to which two variables* in thise that indicates the de,ree to which two variables* in thiscase securitiesD rates of return* are movin, to,ether Gositive value means that* onavera,e* they are movin, in the same direction.

 can ran,e from 81*%* denotin, that twova

variriabableles s momove ve up up anand d dodown wn in in ppererfefect ct sysyncnchrhrononi+i+atatioion n ppererfefect ct poposisititivevecorrelation!* to =1*%* denotin, that the variables always move in exactly oppositedirections perfect ne,ative correlation! / correlation coeAcient of +ero indicatesthat two variables are not related to each other

 $i'=!(" i'

σ ii∙∙ σ σ  '

 -hus*

 -hus* the the ris) ris) of of the the portfolio portfolio or or portfolios portfolios or or standard standard deviation deviation of of a a portfolio portfolio 

σ  p¿   depends on:depends on:

= variances of individual securities and the

=

= covcovariariancance e betbetweeween n di>di>ererent ent paipairs rs of of secsecuriuritieties s whwhich ich arare e crcruciucial al for for larlar,e,eportfolios!

 -he  -he ,oal ,oal of of a a successsuccessful ful diversication diversication is is the the combination combination of of securities securities which which areareeach sli,htly with each dependent as measured by covariance and correlations!and whose rates of returns move inversely

 -he  -he e>ect e>ect of of diversication diversication is is present present when when thethe  $i' M1M1* * aand nd wwe e hahave ve tthheemaximum e>ect of

maximum e>ect of diversicatiodiversication whenn when  $i' =1

TWO-STOCK :ORT;O<IOSE!ecte, return on a !ortfolio #    rr^ p¿

^

rr p=&1∙∙rr^ 1+&2∙∙rr^ 2

Stan,ar, ,eviation of a !ortfolio #    σ σ  p¿

σ  P= √ √ &12σ 12+2&1((1−&1))!ov1,2+(1−&1))2

2

22%%%%66 22EE %%%% 22&& 33%%a! Balculate the avera,e rate of

a! Balculate the avera,e rate of returreturn for each stoc) durin, the $=year periodn for each stoc) durin, the $=year periodb! /ssume that someone held a portfolio consistin, of $%; of "toc) / and $%; of 

"toc) 0 'hat would have been the reali+ed rate of return on the portfolio in eachyear( 'hat would have been the avera,e return on the

year( 'hat would have been the avera,e return on the portfolio durin, this periodportfolio durin, this period((c! Balculate the standard deviation of returns for each stoc) and for the portfoliod! Balculate the coeAcient of variation for each stoc) and for

d! Balculate the coeAcient of variation for each stoc) and for the portfolio.the portfolio

e! If you are a ris)=averse investor* would you prefer to hold "toc) /* "toc) 0* or theportfolio( 'hy(

 -he answers  -he answers to a* b* c* and to a* b* c* and d are ,d are ,iven belowiven below::

 %.LL!

Eercise"

 -he mar) -he mar)et and "toc) P het and "toc) P have the followiave the followin, probn, probability distrability distributions:ibutions:

/n asset/n assetDs ris) consiDs ris) consists ofsts of ,iversi.a*le ris( #uns+stematic ris( or com!an+-s!e

s!eci.c ci.c ris(ris(''* * whwhicich h cacan n be be elelimimininatated ed by by didiveversrsiicacatitionon* * plplusus mar(mar(et et ris( ris( 

e>ects on a portfolio can be eliminated by diversicatiodiversication=bad events in one rm willn=bad events in one rm will

be o>set by ,ood events in another

coeAcient determincoeAcient determines how stoc) a>ects the es how stoc) a>ects the ris) of a diversied portfolio* beta is theris) of a diversied portfolio* beta is themost relevant measure of any stoc)Ds ris) If b e<ual 1.% then the stoc) is about asris)y as the mar)et* if held in a

ris)y as the mar)et* if held in a diversied portfoliodiversied portfolio If b If b is less than 1.%* the is less than 1.%* the stoc) isstoc) isless ris)y than the mar)et If beta

less ris)y than the mar)et If beta is ,reater than 1.%* the stoc) is ,reater than 1.%* the stoc) is more ris)y than ais more ris)y than amar)et -he beta of a portfolio is a wei,hted avera,e of the betas of the individualsecurities in the portfolio

ris(  -hat lo,ic is explained by the -hat lo,ic is explained by the Ca!ital Asset :ricing 0o,el #CA:0'%

B/G5 ,ives us the answer to the <uestion of the si+e of the re<uired rate of return

on ris)y asset Hn the other hand* if we have assessed the expected rate of return*then comparin, the expected and the re<uired rate of return implied by B/G5 wecan determine whether a property is undervalued* overvalued or

can determine whether a property is undervalued* overvalued or fair valued.fair valued

/ssu/ssumin, that min, that the unsystemthe unsystematic ris) atic ris) is is compcompletely removletely remove e by by diverdiversicsicationation** there@uire, rate of return

re@uire, rate of return on a stoc) on a stoc) ii is e<ual to is e<ual to ris(-free rateris(-free rate plus the plus the stoc()s

rrii= =¿ ¿ the rthe re<uired e<uired return return on on a a stoc)stoc) ii

rr , =t-e re<uired return on stoc) mar)et or on a portfolio consistin, of all stoc)s*which is called the mar)et portfolio!

bii  the beta of   the beta of stoc)stoc) ii

rr R+   the ris)=free rate

/ssume that the ris)=free rate is &; and the expected return on the mar)et is 13;.'hat is the re<uired rate of return on a stoc) that has a beta of %.E(

/ssume that the ris)=free rate is &; and the mar)et ris) premium is &;

a! 'hat is the re<uired rate of return for the overall stoc) mar)et(

b! 'hat is the re<uired rate of return on a stoc) that has a beta of 1.2(

"uppose r r RF $;*$;* r r 21%; and1%; and r r  ;12;

a! Balculate "toc) /Ds betab! If "toc) /Ds beta were 2.%* what would be /Ds new re<uired rate of return(

b!

b! r r  ;  $; 8 $;  $; 8 $;bb ;

b.2 rR7 decreases to L;

 -he slope of  -he slope of the "5S remaithe "5S remains constant.ns constant

ow would this a>ect

ow would this a>ect r r 2 and and r r ii(c! @ow assume rR7 remains 6; butc.1

c.1 r  r 2 increases to 1&; orc.2

c.2 r r 2 falls to 13;

 -he slope of  -he slope of the "5S doe not rthe "5S doe not remains constaemains constant.nt

ow would these chan,es a>ect

ow would these chan,es a>ect r r ii(

"toc) R has a beta of 1.$*

"toc) R has a beta of 1.$* "toc) " has a beta of "toc) " has a beta of %.E$* the expected rate of return on%.E$* the expected rate of return on

an avera,e stoc) is 1.3; and the ris)=free rate of return is E; 0y how much doesthe re<uired retur

the re<uired return on n on the ris)ier stoc) exceeds the the ris)ier stoc) exceeds the re<uirre<uired return on the ed return on the less ris)yless ris)ystoc)(

'e )now that'e )now that bbR  1.$%*  1.$%* bb8  %.E$*  %.E$* r r 2  13;*  13;* r r RF   E;

r ii =r  RF  + (r  M  - r  RF )∙bii   E; 8 13; = E;! E; 8 13; = E;!bbii

r R  E; 8   E; 8 &;1.$%!  1&.%;&;1.$%!  1&.%;