CompoundInterest:Intro&PresentValues

Department of Mathematical SciencesFaculty of Science

SSCM4863Room: C10 336/C22 441

Tel: 34321/34274/019-7747457

http://science.utm.my/norhaiza/

TheoryofInterests

SIMPLE COMPOUND

• Interestcanbeeithersimpleorcompound

RecallSimpleinterest

Present Endof1st year Endof2nd year

P=1000

0

• Simpleinterestisaddedtoinvestmentatthematuritydate• Interestcomputedontheoriginalinvestment(principalonly)

P=1000 P=1000

Rateofinterest=10%

I=100

Rateofinterest=10%

I=100

CompoundInterest

Present Endof1st year Endof2nd year

P=1000+100=1100

0

P=1000 P=1100+110=1210

Rateofinterest=10%

I=100

Rateofinterest=10%

I=110

Rateofinterest=10%

I=121

Endof3rd year

P=1210+121=1331

• RecallSimpleinterestiscalculatedbasedontheoriginalprincipal.• Incontrast,ifinterestisaddedtotheinvestmentattheendofeach

period,andthereafteralsoearnsinterest.è Here,theinvestmentearnscompoundinterest.

• Thefuturevalueisthesumoforiginalinvestment(principal)andthecompoundinterest.

• Thetimeperiodbetweentwosuccessiveinterestratecalculationsistheinterestperiod.

Example1Calculatethecompoundinterestearnedandthefuturevalue,ifinvestedfor3yearsat8%pacompoundinterest.

At endof Compound Interest Future Value

1st year 1000(0.08)=RM80 RM1080

2nd year 1080(0.08)=RM86.40 RM1080+RM86.40=RM1166.40

3rd year 1166.40(0.08)= RM93.31 RM1166.40+RM93.31=RM1259.71

è Thecompoundinterestearnedinthe3yearsisRM259.71.è ThefuturevalueisRM1259.71

Inthisexample,consider:P representtheprincipal atthebeginningofthefirstyear.i representthecompoundinterestrateperannum

At endof Compound Interest Future Value

1st year 1000(0.08)=RM80 RM1080

2nd year 1080(0.08)=RM86.40 RM1080+RM86.40=RM1166.40

3rd year 1166.40(0.08)= RM93.31 RM1166.40+RM93.31=RM1259.71

Attheendofthe1styear:Theinterestdue. =Piè Thefuturevalue =P+Pi

=P(1+i)

Attheendofthe2ndyear:Theinterestdue. =[P(1+i)]iè Thefuturevalue =P(1+i) + [P(1+i)]i

=P(1+i)(1+i)=P(1+i)2

Attheendofthe3rdyear:Theinterestdue. =[P(1+i)2]iè Thefuturevalue=P(1+i)2 +[P(1+i)2]i

=P(1+i)2 (1+i)=P(1+i)3

à Thefuturevaluesattheendofsuccessiveyearsfornyearsis:

𝑆 = 𝑃 1 + 𝑖 𝑛 Eq.7

• (1+i)n istheCompoundInterestFactor

• (1+i)nisalsoknownasFutureValueInterestFactor

Exponentialgrowth

Lineargrowth

Example2CalculatethefuturevalueofRM100at12%pacompoundinterestin(a)5years,(b)25years

(a) 5yearsGiven:P=100n=5r=0.12(peryear)S=?

S=P(1+i)n=100(1+0.12)5=RM176.23

• ThefuturevalueofRM100at12%pacompoundinterestin5yearsisRM176.23

• ThecompoundinterestonRM100at12%pafor5yearsisRM76.23

(b)25yearsGiven:P=100n=25r=0.12(peryear)S=?

S=P(1+i)n=100(1.12)25=RM1700.01

• ThefuturevalueofRM100at12%pacompoundinterestin25yearsisRM1700.01

• ThecompoundinterestonRM100at12%pafor25yearsis_________

Whatistheinterestearnediftheinvestmenthadbeenat12%pasimpleinterestin25years?

EffectoftimeandrateofcompoundinterestFuturevalueofRM100atvariousratesofcompound interest

PerannuminterestratesYears 0.06 0.08 0.1 0.12

5 133.82 ? ? 176.2310 179.08 215.89 259.37 310.5815 239.66 317.22 417.72 547.3620 ? 466.10 672.75 964.6325 429.19 684.85 1083.47 ?30 574.35 1006.27 1744.94 2995.9935 768.61 1478.53 2810.24 5279.9640 1028.57 2172.45 ? 9305.1045 1376.46 3192.04 7289.05 16398.7650 1842.02 4690.16 11739.09 28900.22

Theexampleaboveistheinterestperiodof1year.Sometimes interestperiod iscompoundedhalf-yearly orpayablequarterlyTheserateswhenexpressedperannumareknownas‘nominalrates’

Example3FindthecompoundinterestearnedonRM1000investedfor2yearsat10%pacompoundedhalf-yearly.

Compounded halfyearly.è InvestPeriod=2yearsè Interestperiod=6months(i.e halfyearly)è Frequencyofearnedinterestinthe investperiod=Thereare4interestperiods in2yearsè %Interestearnedperperiod=5%perperiodin1year(10%peryear)

è ThecompoundinterestearnedonRM1000at10%pafor2yearscompounded halfyearly isRM215.51

At endof CompoundInterest

Future Value

1st interestperiod

1000(0.05)=RM50

RM1050

2nd interestperiod

1050(0.05)=RM52.50

RM1050+RM52.50=RM1102.5

3rd interestperiod

1102.5(0.05)=RM55.13

RM1102.5+RM55.13=RM1157.63

4th interestperiod

1157.63(0.05)=RM57.88

RM1157.63+RM57.88=RM1215.51

S=P(1+i)n=1000(1+0.05)4=RM1215.51

orGiven:P=1000n=4i=0.05(peryear)S=?

Notations

Notation Details

P Principal ie. Present value of S

I Total simple interest

S Future value ie. Maturity value of P

r Rate of interest per period (typically per annum)

t Time in periods (eg.Years)

Recall:SimpleInterests

Notation Details

P Principal ie. Present value of S

S Future value of P

n The number of interest periods involved

m The number of interest periods per year

i.e. the frequency of compounding

jm The nominal interest rate pa compounded (payable, convertible)

m times per year

i The effective interest rate per period=jm/m

Compound Interests

Eg.j12=9%è m,thenumberofinterestperiods peryear(i.e.thefreq ofcompounding) is12.

i.e everymonthoftheyearè ayearlyrateof9%iscompounded (ie.Converted, payable)12timesperyearè theeffective(i.e real)interestrateis9/12%i.e 0.75%permonthi.e i=0.0075

Example4ApersondepositsRM1000intoasavingsaccountthatearnsinterestat12.25%papayablemonthly.Howmuchinterestwillbeearned(a)duringthe1styear(b)duringthe2nd year

(a) Interestearnedduring1styear

S=P(1+i)n=1000(1+0.0102)12=RM1129.51

Compounded monthly.jm=12.25%è InvestPeriod=1yearè Interestperiods involvedwithininvestperiod(1year),12months(i.emonthly);n=12è FrequencyofearnedinterestwithintheinvestperiodPERyear=Thereare12interestperiodsin1year;m=12è %Interestearnedperperiodin1year= jm /12=12.25%/12.=0.1225/12Given:

P=1000n=12m=12jm=j12=12.25%i=j12/m=0.1225/12=0.0102

Thecompound interestearnedduring the1st yearisRM129.51

Example4(exercise)ApersondepositsRM1000intoasavingsaccountthatearnsinterestat12.25%papayablemonthly.Howmuchinterestwillbeearned(a)duringthe1styear(b)duringthe2nd year

(b)Interestearnedduring2ndyear

S=RM1275.78

Exercise1. Calculatethefuturevalueandtheamountofcompound

interestearned,foraprincipalofRM500at111/4%pacompoundedmonthly,investedfor2years.(625.51,125.51)

2. Howmuchmoneywillberequiredon31Dec2003torepayaloanofRM2000madeon31Dec2000ifj4=12% (2851.52)

3. FindthecompoundinterestearnedonaninvestmentofRM10,000for10yearsatanominalrateof12%pacompoundedwithfrequenciesm=1,2,4,12,52,365

4. SetupatableandplotthegraphshowingthegrowthofRM100atcompoundinterestratesj365 of4%,8%,12%,16%andover5,10,15,20and25years