精算师FM考试Homework 1.docx
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精算师FM考试Homework 1.docx
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精算师FM考试Homework1
UniversityofConnecticut
Math5620:
FinancialMathematicsI
Fall2015
HomeworkAssignment1
DueThursday9/10/15Name_______________________________
1.BrandonIronsdeposits1,000intoasavingsaccountatt=0.
Theaccountearnssimpleinterestatarateof6%peryear.
Brandondepositsanadditional2,000intohissavingsaccountatt=3.
(a)Assumingthattherearenootherdepositsorwithdrawals,whatisthebalanceinBrandon’ssavingsaccountatt=10?
(b)NowassumethatBrandonwithdraws1,500fromthisaccountatt=7.Drawatimelineshowingthetwodeposits(att=0andt=3),thewithdrawalatt=7,andalsoafinalwithdrawalofamountXatt=10.
(c)Basedonthecashflowsshownonthetimelinein(b),andassumingtheaccountearnssimpleinterestat6%peryear,writetheequationofvalueforthisseriesoftransactions.(UseXastheamountofthewithdrawalatt=10thatwillreducetheaccountbalanceto0.)
(d)Solvetheequationofvaluein(c)tofindX,theamountofthewithdrawalatt=10thatwillreducetheaccountbalanceto0.
2.Supposethatanaccountearnssimpleinterestata5%annualrate,andthatthefollowingcashflowsoccur(whereapositivenumberisadepositintotheaccount,andanegativenumberisawithdrawalfromtheaccount).
Thecashflowsare:
Att=0:
+1,000.Att=1:
+1,800.Att=2:
-500.Att=3:
-300.
Att=4:
+600.Att=5:
-1,000.
(a)Giventhesecashflows,entervaluesintothefollowingtabletocalculatethebalanceintheaccountattheendofeachyearfor6years.
t
Prtbef.CF
CFt
Pr’tafterCF
CurrentyearIt+1
Cum.It+1
AVt+1
(b)Nowcalculatetheaccountbalanceatt=6byapplyingthesimpleinterestaccumulationfactor(
)toeachoftheabovecashflows.
Ifyouranswerisdifferentfromthevaluedevelopedinpart(a),explainwhy.
(c)Whateffectiveinterestratedoesthisaccountearnduringthe4thyear?
Iftheeffectiverateisgreaterorlessthanthe5%rateofsimpleinterest,
explainwhythatisthecase.
3.Supposethatanaccountearnscompoundinterestata5%annualeffectiverate,andthatthefollowingcashflowsoccur.(Thesecashflowsarethesameasinthepreviousproblem.)
Thecashflowsare:
Att=0:
+1,000.Att=1:
+1,800.Att=2:
-500.Att=3:
-300.
Att=4:
+600.Att=5:
-1,000.
(a)Giventhesecashflows,entervaluesintothefollowingtabletocalculatethebalanceintheaccountattheendofeachyearfor6years.
t
AVt
CFt
AV’t
It+1
AVt+1
(b)Nowcalculatetheaccountbalanceatt=6byapplyingthecompoundinterestaccumulationfactor
toeachoftheabovecashflows.
Ifyouranswerisdifferentfromthevaluedevelopedinpart(a),explainwhy.
(c)Whatsingledepositintothisaccountatt=0wouldhaveproducedthesameaccountbalanceatt=6thatwascalculatedinpart(b)?
4.DougGravesdeposits1,000intoaninterest-bearingaccountatt=0,anddepositsanadditional600intotheaccountatt=5.
(a)IfDoug’saccountearnssimpleinterestatalevelrateof8%peryear,whatisthebalanceintheaccountattheendof9years?
(b)IfDoug’saccountearnscompoundinterestatalevelannualeffectiverateof
8%peryear,whatisthebalanceintheaccountattheendof9years?
(c)IfDoug’saccountearnscompoundinterestatalevelforceofinterestof
0.08peryear,whatisthebalanceintheaccountattheendof9years?
(d)Whataccountsforthedifferencebetweenyouranswersfor(a)and(b)?
(Don’tdescribethecalculations.Describewhythedifferentinterestcreditingmethodologiescausethevaluestobedifferentfromeachother.)
Similarly,explainwhatcausesthedifferencebetweentheanswersfor(b)and(c).
5.MarvEllistakesoutaloanforanamountXatt=0.Theloanbearsinterestatanannualeffectiverateof8%.
Marvthenmakesaloanpaymentof600attheendofeachyearfor5years.Attheendof5years,theloanhasbeenfullyrepaid.
(a)Writetheequationofvalueforthisloanintwoways:
a.Attime0
b.Attime5
(b)SolveeitherequationofvaluetofindthevalueofX(theamountborrowed).
(c)Whatwastheoutstandingbalanceoftheloan(thatis,howmuchdidMarvstillowe)att=3,immediatelyafterhemadethethirdpaymentof600?
6.AccountAearnscompoundinterestatannualeffectiverateequaltox.
AccountBearnssimpleinterestatanannualrateequaltox.
Att=0,100isdepositedintoaccountA,and200isdepositedintoaccountB.
Inthe9thyear(fromt=8tot=9),AccountBearnsthesameamountofinterestas
AccountA.
(a)Whatisthevalueofx?
(b)Basedonthevalueofx(theinterestrate)calculatedin(a),whichaccount
(AorB)willhavealargerbalanceattheendof20years,andbyhowmuch?
(Thatis,whatisthedifferencebetweenthebalancesinthetwoaccountsattheendof20years?
)
7.Theaccumulationfunction
representstheaccumulatedvalueattime tofanamountof1investedatt=0ataforceofinterest,
of0.08.
(a)Sketchagraphofthisfunctionforvaluesoftfrom0toapproximately3.
(Showtonthex axisand
ontheyaxis.)
(b)Calculatetheslopeofthisgraphatt=0andatt=1,anditsaverageslopeovertheintervalt=0tot=1.
(c)Whatistheterm(thatis,whatarethewords)weusefortheaverageslopeofthiscurvebetween0and1?
Forwhatvalueoftistheslopeofthecurveexactlyequaltothataveragevalue?
8.Fillinthemissingvaluesofi,d,v,and
inthefollowingtable.Oneachrow,thevaluesofthesevariablesrepresentthesamerateofcompoundinterestgrowth.
Forexample,onthefirstrowofthetable,theannualeffectiveinterestrate,i,is6%.Asshown,thisisequivalenttoarateofdiscount,d,of0.05660,a(oneyear)presentvaluefactor,v,of0.94340,andaforceofinterest,
of0.05827.
Pleaseprovide4significantdigits,unlessthevalueisgivenexactly
(i.e.,thereisnoneedtoaddtrailingzeroesifallremainingdigitsarezeroes).
Ann.Eff.Int.Rate
i
Ann.Eff.RateofDisc.
d
PresentValueFactor
v
ForceofInterest
0.06
0.05660
0.94340
0.05827
0.07
0.06766
0.055
0.075
0.925
0.095
9.Consideraninvestmentof100att=0.Supposethatitcanbeinvestedinanyofthefollowing3ways:
(1)atsimpleinterestbasedona(constant)interestrateof0.07peryear
(2)atcompoundinterestbasedona(constant)forceofinterestof0.07peryear
(3)atcompoundinterestbasedona(constant)annualeffectiverateofinterestof
0.07peryear
Ineachcase,howmanyyears(tothenearest0.01year)wouldittakefortheinvestmenttogrowto200(thatis,fortheamountoftheinvestmenttodouble)?
Ineachofthe3cases,wouldtheanswerbethesameifwechoseadifferentamounttoinvestanddeterminedtheperiodoftimerequiredforthatamounttodoubleinvalue?
10.Att=0,adepositof1,000ismadeintoanaccountthatearnscompoundinterest.
Att=4,awithdrawalof600ismade.Therearenootherdepositsorwithdrawals.
Findthebalanceintheaccountatt=10undereachofthefollowingscenarios.
(a)Theaccountearnsinterestatanannualeffectiverate(i)of6%duringthefirst5years,and4.5%duringthelast5years.
(b)Theaccountearnsinterestbasedonaforceofinterest(
)of0.08duringthefirst
5years,andanannualeffectiverateofdiscount(d)of0.05duringthelast5years.
(c)Theaccountearnsinterestbasedona“one-yearpresentvaluefactor”(v)of0.95duringthefirst3.2years,andaforceofinterest(
)of0.075duringthelast6.8years.
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