Fin118 Unit 9

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Al-Imam Muhammad Ibn Saud Islamic University

College of Economics and Administration Sciences

Department of Finance and Investment

Financial MathematicsCourse

FIN 118Unit course

9Number Unit

Compound Interest

Non annual Compound Interest

Continuous Compound Interest

Unit Subject

Dr. Lotfi Ben Jedidia

Dr. Imed Medhioub

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!!!remember what we saw last time

The relationship between time and money.

The simple interest rate and the interest

amount

The present value of one future cash flow

The future value of an amount borrowed or

invested.

The relationship between Real Interest Rate,

Nominal Interest Rate and Inflation.

2

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we will see in this unit

The compound interest rate and the interest

amount

How to Calculate the future value of a singlesum of money invested today for several

periods.

How to Calculate the interest rate or the

number of periods or the principal that achievea fixed future value.

3

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Learning Outcomes

4

At the end of this unit, you should be able to:

1. Understand compound interest, includingaccumulating, discounting and making comparisons

using the effective interest rate.

2. Distinguish between compound interest.

3. Identify variables fundamental to solvinginterest problems.

4. Solve problems including future and presentvalue.

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Definition1: In each subsequent period, the interest

amount computed is used to form a new principal sum,which is used to compute the next interest due.

As we said, Compound Interest uses the Sum ofPrincipal & Interest as a base on which to calculate new

Interest and new Principal !

n periodsiiiiPVFV

odsthree periiiiPViFFV

stwo periodiiPViFFV

one periodiPVFV

nn )1()1)(1)(1(

)1)(1)(1()1(V

)1)(1()1(V

)1(

321

321323

21212

11

The Compound Interest

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6

Definition2:If the interest rate is constant overdifferent periods we have:

and

!!! Remember

nn iPVFV 1

nn

i

FVPV

11

1

nn

PV

FVi

i

PVFVn n

1ln

ln

IPVFVn

FVn= Principal(1+ Interest Rate)number of Periods


iiiiin ....

321

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7

The compound interest rate is a geometric sequencesbut the simple interest is an arithmetic sequences.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14 16 18 20

Value

YearsSimple Interest

Compound Interest

Simple interest: Linear growth i = 0.15Compound interest: Geometric growth i = 0.15

Property1:


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Example1:Future Value and InterestHow much money would you pay in interest if youborrowed $1600 for 3 years at 16% compoundinterest per annum?

Solution:

Convert the percent to a decimal: 16% = 0.16

434.249716.0116001 333 iPVFV

PVFVIIPVFV nn

More ExamplesThe Compound Interest

4336.8971600434.2497 I

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More ExamplesExample3: Interest rateAssume that the initial amount to invest isPV = $100 and the interest rate is constant overtime. What is the compound interest rate in

order to have $150 after 5 years?Solution:

PV = $100 and FV5= $150

%4.8

084.01084.11100

150

11)1(

5

1

5

1

55

1

555

i

i

PVFVi

PVFViiPVFV


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11

More ExamplesExample4: The number of periods (n)Find the number of periods to double yourinvestment at 6% compound interest per annum .

Solution:

PV = x and FVn= 2x

Convert the result:11 years + 0.895 12 months= 11 years + 10.74 months

11 years + 10 months + 0.74 30 days

n = 11 years + 10 months + 22 days

years

xx

i

PVFVn

PVFVinPV

FViiPVFV

n

nnnn

n

895.1106.1ln

2ln

06.01ln

2ln

1ln

ln

ln1ln1)1(


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Question 1 ? How to calculate the FV if we have more thanone compounding periods per year ?

Response:

The table shows some common compounding periodsand how many times per year interest is paid for them.

And

If t=1 we retrieve the old formula

Compounding Periods Times per year (t)

Annually 1

Semi-annually 2

Quarterly 4

Monthly 12

tn

tnt

iPVFV

1,

!!!

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Example1:Future Value semi-annuallyYou invested $1800 in a savings account that pays4.5% interest compounded semi-annually. Find thevalue of the investment in 12 years.

Solution:Convert the percent to a decimal: 4.5% = 0.045

212

2,12

,

2

045.011800

1

FV

t

iPVFV

tn

tn

= 1800(1 + 0.0225)24

= 1800(1.0225)24 =$3070.38


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Example2:Future Value Quarterly

You invested $3700 in a savings account that pays2.5% interest compounded quarterly. Find the value ofthe investment in 10 years.

Solution:

Convert the percent to a decimal: 2.5% = 0.025

410

4,10

,

4

025.013700

1

FV

t

iPVFV

tn

tn

= 3700(1 + 0.00625)40

= 3700(1.00625)40 =$4747.2


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Example3:Future Value monthly

You invested $1700 in a savings account that pays 1.5%interest compounded monthly. Find the value of theinvestment in 15 years.

Solution:

Convert the percent to a decimal: 1.5% = 0.015

1215

12,15

,

12

015.011700

1

FV

t

iPVFV

tn

tn

= 1700(1 + 0.00125)180

= 1700(1.00125)180 =$2128.65


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Example4:Present Value

You expect to need $1500 in 3 years. Your bankoffers 4% interest compounded semiannually. Howmuch money must you put in the bank today (PV) toreach your goal in 3 years?

Solution:

Convert the percent to a decimal: 4% = 0.04

tn

tntn

tn

ti

FVPV

t

iPVFV

1

1,

,

957.1331$

02.1

1500

2

04.0

1

1500623

PV


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Example5:Making a choiceSuppose a bank quotes nominal annual interest

rates on five-year of: 6.6% compounded annually, 6.5% compounded semi-annually, and

6.4% compounded monthly.Which rate should an investor choose for aninvestment of $10000?Solution:

Convert the percent to a decimal:

6.6% = 0.066; 6.5% = 0.065 and 6.4% = 0.064

The times per year is respectively t= 1; t = 2 and

t = 12.


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Solution : continued

First proposition:

Second proposition:

Third proposition:

31.137651

066.0110000

15

1,5

FV

943.137682

065.0110000

25

2,5

FV

572.1375912

064.0

110000

125

12,5

FV

We choose the second proposition.6.5% compounded semi-annually provides the best

return on investment.

tn

tntiPVFV

1,


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Question 2 ?What would happen to our money ifwe compounded a really large number of times?

Response:

We would have to compound not just every hour, or

every minute or every second but for everymillisecond. We have:

Then with Continuous compounding interest wehave:

tn

tn

t

iPVFV

1,

inePV t

intn ePVFV

,19

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Example1:Future Value

If you invest $1000 at an annual interest rate of5% compounded continuously, calculate the finalamount you will have in the account after fiveyears.

Solution:Convert the percent to a decimal 5% =0.05

With continuous compounding formula we obtain

02.1284$1000 05.055 eePVFV in


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Example2:Finding the time

How long will it take an investment of $10000 togrow to $15000 if it is invested at 9% compoundedcontinuously?

Solution:

Convert the percent to a decimal 9% =0.09With continuous compounding formula we obtain

= 4 ears + 6 months + 2 da s

years505.4

09.0

)5.1ln(

lnln

n

i

PVFVnPVFVin

PV

FVeePVFV

nn

nininn


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Example3:Finding the interest rate

What is the interest rate compounded continuouslyof an investment of $10000 to grow to $20000 ifit is invested for 7 years?

Solution:

%9.9099.0

7

)2ln(

lnln

ii

nPVFViPVFVin

PV

FVeePVFV

nn

nininn


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Example4:Making a choice

What amount will an account have after 5 years if$100 is invested at an annual nominal rate of 8%compounded annually? Semiannually? continuously?

Solution:

Compounded annually:

Compounded semi-annually:

Compounded continuously:

93.14608.01100 55 FV

02.148

2

08.01100

25

2,5

FV

18.149100 508.05 eFV

We choose the third proposition.8% compounded continuously provides the best return

on investment.

Compound Interest

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Its time to review

Compound interestSimple Interest

Continuous Compound

InterestMore than one compounding

periods per year

IPVFVn

nn iPVFV 1

tn

tnt

iPVFV

1,

Real Interest Rate = Nominal Interest Rate - Inflation

inn ePVFV

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we will see in the next unit

Meant of simple Annuity

Simple Annuity: Ordinary Annuity,Annuity Due (unit10)

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Fin118 Unit 9

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