SOLUTIONS MANUAL
w
, CHAPTERw2
PresentwValuewandwthewOpportunitywCostwofwCapital
AnswerswtowPracticewQuestions
1. LetwINVw=winvestmentwrequiredwatwtimewtw=w0w(i.e.,wINVw=w-
C0)wandwletwxw=wratewofwreturn.w Thenwxwiswdefinedwas:
xw=w(C1w–wINV)/INV
Therefore:
C1w =wINV(1w+wx)
Itwfollowswthat:
NPVw=wC0w+w{C1/(1w+wr)}
NPVw=w-INVw+w{[INV(1w+wx)]/(1w+wr)}
NPVw=wINVw{[(1w+wx)/(1w+wr)]w–w1}
a. Whenwxwequalswr,wthen
:w[(1w+wx)/(1w+r)]w–
w1w=w0wandwNPVwiswzer
o.
b. Whenwxwexceedswr,wthen
:w[(1w+wx)/(1w+wr)]w–
w1w>w0wandwNPVwiswpositi
ve.
2. Thewfacewvaluewofwthewtreasurywsecuritywisw$1,000.wIfwthiswsecuritywearnsw5%,wthe
nwinwonewyearwwewwillwreceivew$1,050.wThus:
NPVw=wC0w+w[C1/(1w+wr)]w=w-1000w+w(1050/1.05)w=w0
Thiswiswnotwawsurprisingwresult,wbecausew5wpercentwiswthewopportunitywcostwofwcapit
al,wi.e.,w5wpercentwiswthewreturnwavailablewinwthewcapitalwmarket.wIfwanywinvestmentwe
arnswawratewofwreturnwequalwtowthewopportunitywcostwofwcapital,wthewNPVwofwthatwinve
stmentwiswzero.
1
,3. NPVw=w-$1,300,000w+w($1,500,000/1.10)w=w+$63,636
SincewthewNPVwiswpositive,wyouwwouldwconstructwthewmotel.wAlte
rnatively,wwewcanwcomputewrwaswfollows:
rw=w($1,500,000/$1,300,000)w–w1w=w0.1538w=w15.38%
Sincewthewratewofwreturnwiswgreaterwthanwthewcostwofwcapital,wyouwwouldwconstructwth
ewmotel.
4.
Investment NPV Return
1) 18,000w 18,000w −w10,000w
−w10,000w +w =w $5,000 =w 0.80w =w 80.0%
1.20 10,000
2)
−w5,000w+w
9,000w
=w $2,500 9,000w−w5,000w
=w 0.80w =w 80.0%
1.20 5,000
3)
−w5,000w+w
5,700w
=w−$250 5,700w−w5,000w
=w 0.14w =w14.0%
1.20 5,000
4) 4,000w−w2,000w
=w1.00w =w100.0%
4,000w 2,000
−w2,000w+w =w$1,333.3
3
1.20
a. Investmentw1,wbecausewitwhaswthewhighestwNPV.
b. Investmentw1,wbecausewitwmaximizeswshareholders’wwealth.
5. a. NPVw=w(-
50,000w+w30,000)w+w(30,000/1.07)w=w$8,037.38wb. NPVw=w(-
50,000w+w30,000)w+w(30,000/1.10)w=w$7,272.73
Since,winweachwcase,wthewNPVwiswhigherwthanwthewNPVwofwthewofficewbuildingw($7,14
3),wacceptwE.wColi’swoffer.wYouwcanwalsowthinkwofwitwanotherwway.wThewtruewopportunit
ywcostwofwthewlandwiswwhatwyouwcouldwsellwitwfor,wi.e.,w$58,037w(or
$57,273).w Atwthatwprice,wthewofficewbuildingwhaswawnegativewNPV.
6.
Thewopportunitywcostwofwcapitalwiswthewreturnwearnedwbywinvestingwinwthewbestw
alternativewinvestment.wThiswreturnwwillwnotwbewrealizedwifwthewinvestmentwunderw
2
, considerationwiswundertaken.wThus,wthewtwowinvestmentswmustwearnwatwleastwthew
samewreturn.wThiswreturnwratewiswthewdiscountwratewusedwinwthewnetwpresentwvalu
ewcalculation.
3
w
, CHAPTERw2
PresentwValuewandwthewOpportunitywCostwofwCapital
AnswerswtowPracticewQuestions
1. LetwINVw=winvestmentwrequiredwatwtimewtw=w0w(i.e.,wINVw=w-
C0)wandwletwxw=wratewofwreturn.w Thenwxwiswdefinedwas:
xw=w(C1w–wINV)/INV
Therefore:
C1w =wINV(1w+wx)
Itwfollowswthat:
NPVw=wC0w+w{C1/(1w+wr)}
NPVw=w-INVw+w{[INV(1w+wx)]/(1w+wr)}
NPVw=wINVw{[(1w+wx)/(1w+wr)]w–w1}
a. Whenwxwequalswr,wthen
:w[(1w+wx)/(1w+r)]w–
w1w=w0wandwNPVwiswzer
o.
b. Whenwxwexceedswr,wthen
:w[(1w+wx)/(1w+wr)]w–
w1w>w0wandwNPVwiswpositi
ve.
2. Thewfacewvaluewofwthewtreasurywsecuritywisw$1,000.wIfwthiswsecuritywearnsw5%,wthe
nwinwonewyearwwewwillwreceivew$1,050.wThus:
NPVw=wC0w+w[C1/(1w+wr)]w=w-1000w+w(1050/1.05)w=w0
Thiswiswnotwawsurprisingwresult,wbecausew5wpercentwiswthewopportunitywcostwofwcapit
al,wi.e.,w5wpercentwiswthewreturnwavailablewinwthewcapitalwmarket.wIfwanywinvestmentwe
arnswawratewofwreturnwequalwtowthewopportunitywcostwofwcapital,wthewNPVwofwthatwinve
stmentwiswzero.
1
,3. NPVw=w-$1,300,000w+w($1,500,000/1.10)w=w+$63,636
SincewthewNPVwiswpositive,wyouwwouldwconstructwthewmotel.wAlte
rnatively,wwewcanwcomputewrwaswfollows:
rw=w($1,500,000/$1,300,000)w–w1w=w0.1538w=w15.38%
Sincewthewratewofwreturnwiswgreaterwthanwthewcostwofwcapital,wyouwwouldwconstructwth
ewmotel.
4.
Investment NPV Return
1) 18,000w 18,000w −w10,000w
−w10,000w +w =w $5,000 =w 0.80w =w 80.0%
1.20 10,000
2)
−w5,000w+w
9,000w
=w $2,500 9,000w−w5,000w
=w 0.80w =w 80.0%
1.20 5,000
3)
−w5,000w+w
5,700w
=w−$250 5,700w−w5,000w
=w 0.14w =w14.0%
1.20 5,000
4) 4,000w−w2,000w
=w1.00w =w100.0%
4,000w 2,000
−w2,000w+w =w$1,333.3
3
1.20
a. Investmentw1,wbecausewitwhaswthewhighestwNPV.
b. Investmentw1,wbecausewitwmaximizeswshareholders’wwealth.
5. a. NPVw=w(-
50,000w+w30,000)w+w(30,000/1.07)w=w$8,037.38wb. NPVw=w(-
50,000w+w30,000)w+w(30,000/1.10)w=w$7,272.73
Since,winweachwcase,wthewNPVwiswhigherwthanwthewNPVwofwthewofficewbuildingw($7,14
3),wacceptwE.wColi’swoffer.wYouwcanwalsowthinkwofwitwanotherwway.wThewtruewopportunit
ywcostwofwthewlandwiswwhatwyouwcouldwsellwitwfor,wi.e.,w$58,037w(or
$57,273).w Atwthatwprice,wthewofficewbuildingwhaswawnegativewNPV.
6.
Thewopportunitywcostwofwcapitalwiswthewreturnwearnedwbywinvestingwinwthewbestw
alternativewinvestment.wThiswreturnwwillwnotwbewrealizedwifwthewinvestmentwunderw
2
, considerationwiswundertaken.wThus,wthewtwowinvestmentswmustwearnwatwleastwthew
samewreturn.wThiswreturnwratewiswthewdiscountwratewusedwinwthewnetwpresentwvalu
ewcalculation.
3
