ESTI MATI NG RI SK AND RETURN ON AS SETS

SUGGESTED ANSWERS TO THE R EVIEW EVIEW QUESTIONS AND PROBLEMS I.

Questions

1. Risk is is the variability of an asset’s future future returns. When only one return is possible, there is no risk. When more than one return is possible, the asset is risky. 2. An objective probability distribution is generally based on past outcomes of similar events while a subjective probability probability distribution distribution is based on opinions or “educated guesses about the likelihood that an event will have a particular future outcome. !. A discrete probability probability distribution is an arrangement of the probabilities associated with the values of a variable that can assume a limited or probability fini finite te numb number er of valu values es "out "outco come mes# s# whil whilee a continuous probability distribution is an arrangement of probabilities associated with the values of a variable that can assume an infinite number of possible values "outcomes#. $. %he accura accuracy cy of foreca forecaste stedd returns returns general generally ly decreas decreases es as the length length of the pro&ect being forecast increases. %his increases the variability of an asset’s returns and therefore risk. '. %he expected value of return of a single asset is the weighted average of the returns, with the weights being the probabilities of each return. (. %he %he risk risk of a sing single le asset asset is meas measur ured ed by its its stan standa dard rd devia deviati tion on or coefficient of variation. %he standard deviation measures the variability of outcomes around the e)pected value and is an absolute measure of risk. risk. %he coefficient of variation is the ratio of the standard deviation to the e)pected value and is a relative measure of risk. *. %he %he char charac acte teri rist stic ic of of a normal curve is a bell+shaped distribution that is dependent upon the mean and the standard deviation of the population under investigation. ince the normal distribution is a continuous rather than a discrete distribution, it is not possible to speak of the probability 22-1

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Estimating Risk and Return on Assets

of a point but only of the probability of falling within some specified range of values. %hus, the area under the curve between any two points must must then then also also depe depend nd upon upon the the valu values es of the the mean mean and and stan standa dard rd deviation. -owever, it is possible to standardie any normal distribution so that it has a mean of ero and a standard deviation of one. /. 0ecisi 0ecision on makers makers may be classi classifie fiedd into three three catego categorie riess accordin accordingg to their risk preferences risk-averse, risk-neutral and risk-taker . inancial theory assumes that decision makers are risk+averse. 3. Portfolio risk is measured by the portfolio standard deviation. 4ortfolio risk is influenced by diversification. 5isk reduction is achieved through diversification whenever the returns of the assets combined in a portfolio are not perfectly positively correlated. 6orrelation measures the tendency of two variables to move together. 17. 8o to both 9uestions. 9uestions. %he portfolio portfolio e)pected return is a weighted weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. 11. alse. %he variance of of the individual assets assets is a measure of the the total risk. %he variance on a well+diversified portfolio is a function of systematic risk only. 12. :es, es, the standard standard deviation deviation can be less than that of every asset asset in the portfolio. -owever, β p cannot be less than the smallest beta because β p is a weighted average of the individual asset betas. II. Multiple Choice Questions Questions

1. 2.

; 6

!. $.

0 A

'. (.

; ;

*.

A

III. Problems III. Problems P"o$l!% &

"a# %he bar charts for tock A and tock ; are are shown in the the ne)t page. "b# tock A’ A’s probability probability distribu distribution tion is skewed to the left and tock ;’s probability distribution distribution is symmetrical. symmetrical. "c# tock A’ A’s range of returns returns is 2$ percentage percentage points "2' < 1# and tock ;’s ;’s range of returns is 27 points "!7 < 17#. 22-2

Estimating Risk and Return on Assets

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"d# tock tock A is riskier riskier than tock tock ; becaus becausee tock tock A has a wider wider range of returns and a flatter probability distribution. distribution. Stock A

0.60

0.50

i o a l i y r t b P b

0.40

0.30

0.20

0.10

10

40 -5

0

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35

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25

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40 -5

Return (%)

Stock B

0.60

0.50

0.40

b y o t r a i l P i b

0.30

0.20

0.10

10

-5

0

5

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15 Return 22-3 (%)

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Estimating Risk and Return on Assets

P"o$l!% #

%he e)pected value of the returns for each stock is Stock A ȓA

= "7.7'# "7.71# > "7.27# "7.7'# > "7.2'# "7.2'# "7.17# > "7.!'# "7.1'# > "7.1'# "7.2'# = 7.12'' or 12.''? 12.''?

Stock B ȓ;

= "7.17# "7.17# > "7.27# "7.1'# > "7.$7# "7.$7# "7.27# > "7.27# "7.27# "7.2'# > "7.17# "7.17# "7.!7# = 7.27 or 27? 27?

P"o$l!% '

"a# %he calculatio calculationn of the e)pected e)pected value can be set set up in tabular tabular form. i

i

"i () ()*

i "i ()*

1 2 ! $ '

7.1 7.2 7.$ 7.2 7.1

7 17 27 !7 $7

7 2.7 /.7 (.7 $.7 = 27.7?

ȓ

"b# %he calculatio calculationn of the standard standard deviation deviation can also also be set up in tabular form. %he s9uare root of the variance, @ 2, of 127 percent is 17.3' percent "rounded#. i

"i () ()*

1 2 ! $ '

7 17 2277 !7 $7

ȓ

()*

27 27 27 27 27

"i ()* + ȓ ()*

+ 27 + 17 7 17 27

()* ""i + ȓ* # ()

$77 177 7 177 $77

22-4

i

i ""i + ȓ* # ()*

7.1 7.2 7.$ 7.2 7.1

$7.7 27.7 7 27.7 $7.7 @ 2 = 127.7

Estimating Risk and Return on Assets

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127.7 = 17.3'? "c# %he coefficien coefficientt variati variation on is is 6F

=

17.3' 27.77 =

7.''

A coefficient coefficient of variation of 7.'' means that there is 7.'' percent risk for every 1 percent of return. P"o$l!% ,

"a# "a# 4ro& 4ro&ec ectt : is risk riskie ierr than than 4ro& 4ro&ec ectt B when when rank ranked ed by thei theirr stan standa dard rd deviations. -owever, the two pro&ects are e9ually risky when ranked by their coefficients of variation. "b# Cn this situation, situation, the coefficient coefficient variation variation is the more appropriate appropriate risk measure because the pro&ects have different net investments and e)pected values. %hus, a relative measure of risk "coefficient of variation# is needed rather than an absolute measure "standard deviation#. P"o$l!% -

"a# %he ranges ranges for 4ro&ec 4ro&ectt B are shown shown below below Expected Value

Standard Deiation !"#$###%

Range

4177,777 4177,777

D1 D2

4/7,777 E 4127,777 4(7,777 E 41$7,777

"b# Appro)im Appro)imately ately (/ percent percent of the returns should should lie between between D 1 standard standard deviation of the e)pected value and about 3' percent within D 2 standard deviations. P"o$l!%

"a# 6alculatin 6alculatingg the probabil probability ity that the return will e)ceed 41!7,777 41!7,777 re9uires re9uires three steps irst, compute the z value value as follows 22-5

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Estimating Risk and Return on Assets

=

41!7,777 < 4177,777 427,777

=

1.'7

8e)t, find 4r "7 G z G 1.'# which is 7.$!!2 or $!.!2 percent. %his probability is the chance of getting a return between the e)pected return of 4177,777 and a return of 41!7,777. inally, the probability of getting a return greater than 41!7,777 must be calculated. 5emember that in a normal distribution, '7 percent of the outcomes lies on each side of the e)pected value. %he probability of receiving a return of more than 41!7,777 is 7.7((/ "7.'777 E 7.$!!2# or (.(/ percent. "b# 0eterm 0etermini ining ng the probabil probability ity of a return return falling falling betwee betweenn 4'7,77 4'7,7777 and 41!7,777 re9uires several additional steps. irst, determine the z value value for a return between 4'7,777 and 4177,777.

=

4'7,777 < 4177,777 427,777

=

E 2.'7

8e)t, find 4r "+ 2.'7 G z G G 7# which is 7.$3!/ or $3.!/ percent. inally, the probability of a return between 4'7,777 and 41!7,777 is obtained by adding the probability of a return between 4'7,777 and 4177,777, or 7.$3!/, and the probability of a return between 4177,777 and 41!7,777, or 7.$!!2. %his produces a probability of 7.32*7 "7.$3!/ > 7.$!!2# or 32.*7 percent chance of obtaining a return between 4'7,777 and 41!7,777. P"o$l!% /

"a# %he e)pected e)pected portfo portfolio lio return return for each each plan is is Plan A ȓ p

= =

Plan B

"7.(# "7.2$# > "7.$# "7.7/# 7.1*( or 1*.(?

ȓ p

= =

"7.2 "7.2## "7. "7.2$ 2$## > "7./ "7./## "7. "7.7/ 7/## 7.112 or 11.2?

%he portfolio standard deviation for each plan is Plan A

@ p = =

"7.(# 2 "7.1(# 2 > "7.$# 2 "7.72# 2 > "2# "7.(# "7.$# "7.'# "7.1(# "7.72# 7.77322 > 7.7777( > 7.777** 22-6

Estimating Risk and Return on Assets

=

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7.7177'

= 7.1772' or 17.72'? Plan B

@ p =

"7.2# 2 "7.1(# 2 > "7./# 2 "7.72# 2 > "2# "7.2# "7./# "7.'# "7.1(# "7.72#

=

7.77172 > 7.7772( > 7.777'1

=

7.771*3

= 7.7$2!1 or $.2!1?

"b# As shown in 4lan 4lan ;, both both the e)pecte e)pectedd portf portfoli olioo return return and portfo portfolio lio standard deviation decrease as a greater proportion of the portfolio is invested in Hega Falue ood tores. %hus, the influence of Iigabyte 6omputer’s higher e)pected risk and return are replaced in the portfolio by Hega Falu Faluee ood tores’ lower e)pected risk and return. P"o$l!% 0

%he e)pected returns are &ust the possible returns multiplied by the associated probabilities J "5 A# = ".27 ) +.1'# > ".'7 ) .27# > ".!7 ) .(7# = 2'? J "5 ;# = ".27 ).27# > ".'7 ".'7 ) .!7# > ".!7 ".!7 ) .$7# = !1?

%he variances are given by the sums of the s9uared deviations from the e)pected returns multiplied by their probabilities @2A = .27 ) "+.1' E .2'# 2 > .'7 ) ".27 E .2'# 2 > .!7 ) ".(7 E .2'# 2 = ".27 ) +.$72# > ".'7 ) +.7' 2# > ".!7 ) .!' 2# = ".27 ) .1(# > ".'7 ) .772'# > ".!7 ) .122'# = .7*77

@2; = .27 ) ".27 E .!1# 2 > .'7 ) ".!7 E .!1# 2 > .!7 ) ".$7 E .!1# 2 = ".27 ) +.112# > ".'7 ) +.71 2# > ".!7 ) .73 2# = ".27 ) .7121# > ".'7 ) .7771# > ".!7 ) .77/1# 22-7

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Estimating Risk and Return on Assets

= .7*$3

%he standard deviations are thus @A =

.7*77

@; =

= 2(.$(?

.77$3 = *?

P"o$l!% 1

%he portfolio weights are 41',777K27,777 = .*' and 4',777K27,777 = .2'. %he e)pected return is thus J "5 4# = .*' ) J "5 A# > .2' ) J "5 ;# = ".*' ) 2'?# > ".2' ) !1?# = 2(.'?

Alternatively, Alternatively, we could calculate the portfolio’s return in each of the states State o& Econom'

5ecession 8ormal ;oom

Probabilit' o& State o& Econom'

.27 .'7 .!7

Port&olio Return i& State (ccurs

".*' ) E.1'# > ".2' ) .27# = E.7(2' ".*' ) .27# > ".2' ) .!7# = .22'7 ".*' ) .(7# > ".2' ) .$7# = .''77

)eighted Returns !*%

E 1.2'? 11.2'? 1(.'? 2(.'?

P"o$l!% &2

%he portfolio weight of an asset is total investment in that asset divided by the total portfolio value. irst, we will find the portfolio value, which is %otal %otal value = 1/7 "4$'# > 1$7 "42*# = 411,//7 %he portfolio weight for each stock is WeightA = 1/7 "4$'# K 411,//7 = .(/1/ 22-8

Estimating Risk and Return on Assets

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Weight; = 1$7 "42*# K 411,//7 = .!1/2 P"o$l!% &&

%he e)pected return of a portfolio is the sum of the weight of each asset times the e)pected return of each asset. %he total value of the portfolio is %otal %otal value = 42,3'7 > !,*77 = 4(,('7 o, the e)pected return of this portfolio is J "5 p# = "42,3'7K4(,('7#"7.11# > "4!,*77K4(,('7#"7.1'# "4!,*77K4(,('7#"7.1'# = .1!2! . 1!2! or 1!.2!? P"o$l!% &#

%he e)pected return of a portfolio is the sum of the weight of each asset times the e)pected return of each asset. o, the e)pected return of the portfolio is J "5 p# = .(7 ".73# > .2' ".1*# > .1' . 1' ".1!# = .11(7 or 11.(7? P"o$l!% &'

-ere, we are given the e)pected return of the portfolio and the e)pected return of each asset in the portfolio, and are asked to find the weight of each asset. We can use the e9uation for the e)pected return of a portfolio to solve this problem. ince the total weight of a portfolio must e9ual 1 "177?#, the weight of tock : must be one minus the weight of tock B. Hathematically speaking, this means J "5 p# = .12$ = .1$w B > .17'"1 < w B# We can now solve this e9uation for the weight of tock B as .12$ = .1$wB > .17' < .17'wB .713 = .7!'wB wB = 7.'$2/'* o, the amount invested in tock B is the weight of tock B times the total portfolio value, or or Cnvestment in B = 7.'$2/'* "417,777# = 4',$2/.'* 22-9

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Estimating Risk and Return on Assets

And the amount invested in tock : is Cnvestment in : = "1 < 7.'$2/'*# "417,777# = 4$,'*$.$! P"o$l!% &,

%he e)pected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. o, the e)pected return of the asset is J "5# = .2' "<.7/# > .*' ".21# = .1!*' or 1!.*'? P"o$l!% &-

%he e)pected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. o, the e)pected return of the asset is J "5# = .27 "<.7'# > .'7 ".12# > .!7 ".2'# ".2'# = .12'7 or 12.'7? P"o$l!% &

%he e)pected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. o, the e)pected return of each stock asset is J "5 A# = .1' ".7'# ".7'# > .(' ".7/# > .27 ".1!# = .7/'' or /.''? /.''? J "5 ;# = .1' "<.1*# > .(' ".12# > .27 .27 ".23# ".23# = .117' or 11.7'? 11.7'? %o calculate the standard deviation, we first need to calculate the variance. %o find the variance, we find the s9uared deviations from the e)pected return. We then multiply each possible s9uared deviation by its probability, then add all of these up. %he result is the variance. o, the variance and standard deviation of each stock is σA2 = .1'".7' < .7/''# 2 > .('".7/ < .7/''# 2 > .27".1! < .7/''# 2 = .777(7 σA = ".777(7# 1K2 = .72$( or 2.$(? σ;2 = .1'"<.1* < .117'# 2 > .('".12 < .117'#2 > .27".23 < .117'# 2 = .71/!7 22-10

Estimating Risk and Return on Assets

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σ; = ".71/!7# 1K2 = .1!'! or 1!.'!?

P"o$l!% &/

%he e)pected return of a portfolio is the sum of the weight of each asset times the e)pected return of each asset. o, the e)pected return of the portfolio is J "5 p# = .2'".7/# > .''".1'# > .27".2$# = .1'7' or 1'.7'? Cf we own this portfolio, we would e)pect to get a return of 1'.7' percent. P"o$l!% &0

"a# %o find find the e)pected e)pected return return of the portfolio, portfolio, we need to find the return of the portfolio in each state of the economy. %his portfolio is a special case since all three assets have the same weight. %o find the e)pected return in an e9ually weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the e)pected return of the portfolio in each state of the economy is ;oom J "5 p# = ".7* > .1' > .!!#K! = .1/!! or 1/.!!? 5ecession J "5 p# = ".1! > .7! −.7(#K! = .7!!! or !.!!? %o find the e)pected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. 0oing this, we find J "5 p# = .!'".1/!!# > .('".7!!!# = .7/'/ or /.'/? "b# %his portfoli portfolioo does not have an e9ual weight weight in each asset. asset. We still still need to find the return of the portfolio in each state of the economy. %o do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. economy. 0oing so, we get ;oom J "5 p# = .27".7*# >.27".1'# > .(7".!!# =.2$27 or 2$.27? 5ecession J "5 p# = .27".1!# >.27".7!# > .(7" −.7(# = <.77$7 or <7.$7? 22-11

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Estimating Risk and Return on Assets

And the e)pected return of the portfolio is J "5 p# = .!'".2$27# > .('" −.77$# = .7/21 or /.21? %o find the variance, we find the s9uared deviations from the e)pected return. We then multiply each possible s9uared deviation by its probability, probability, than add all of these up. %he result is the variance. o, the variance and standard deviation of the portfolio is σ p2 = .!'".2$27 < .7/21# 2 > .('"−.77$7 < .7/21# 2

=

.71!*(*

= 11.*!? P"o$l!% &1

"a# %his portfolio portfolio does does not have an e9ual weight weight in each asset. We We first need to find the return of the portfolio in each state of the economy. %o do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. economy. 0oing so, we get ;oom Iood 4oor 5ece 5ecess ssio ion n

J "5 p # = .!7".!# > .$7".$'# > .!7".!!# = .!(37 or !(.37? J "5 p# = .!7".12# > .$7".17# > .!7".1'# = .1217 or 12.17? J "5 p # = .!7".71# > .$7"<.1'# > .!7"<.7'# = <.7*27 or <*.27? J "5 p# = .!7"<.7(# > .$7"<.!7# > .!7"<.73# = <.1('7 or <1(.'7?

And the e)pected return of the portfolio is J "5 p# = .1'".!(37# > .$'".1217# .$'".1217# > .!'"<.7*27# .!'"<.7*27# > .7'"<.1('7# = .7*($ or *.($? "b# %o calculate the standard deviation, deviation, we first need to calculate the variance. %o %o find the variance, we find the s9uared deviations from the e)pected return. We then multiply each possible s9uared deviation by its probability, probability, than add all of these up. %he result is the variance. o, the variance and standard deviation of the portfolio is σ p2 = .1'".!(37 < .7*($# 2 > .$'".1217 < .7*($# 2 > .!'"<.7*27 < .7*($# 2 >

.7'"<.1('7 < .7*($# 2 22-12

Estimating Risk and Return on Assets

σp2 = .02436 σ p = .72$!(

= .1'(1 or 1'.(1?

22-13

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