mathematics of finance - inter · barnett/ziegler/byleen college mathematics 12e 2 learning...

Chapter 3

Mathematics of

Finance

Section 2

Compound and

Continuous Interest

2 Barnett/Ziegler/Byleen College Mathematics 12e

Learning Objectives for Section 3.2

The student will be able to compute compound and

continuous compound interest.

The student will be able to compute the growth rate of a

compound interest investment.

The student will be able to compute the annual

percentage yield of a compound interest investment.

Compound and

Continuous Compound Interest


Table of Content

Compound Interest


Growth and Time

Annual Percentage Yield


Terms

compound interest

rate per compounding period

continuous compound interest

APY (Annual Percentages Yield)


Compound Interest

Unlike simple interest, compound interest on an amount

accumulates at a faster rate than simple interest. The basic

idea is that after the first interest period, the amount of

interest is added to the principal amount and then the

interest is computed on this higher principal. The latest

computed interest is then added to the increased principal

and then interest is calculated again. This process is

completed over a certain number of compounding periods.

The result is a much faster growth of money than simple

interest would yield.


Example

Suppose a principal of $1,000 was invested in an

account paying 8% annual interest compounded

quarterly. How much would be in the account after one

year?

Compounding quarterly means that the interest is paid

at the end of each 3-month period and the interest as

well as the principal earn interest for the next quarter.


Solution

Solution: Using the simple interest formula A = P (1 + rt)

we obtain:

At the end of the first quarter will have:

A = P (1 + rt)

A = 1000[1 + 0.08(1/4)]

A = 1000(1.02) = $1.020

Now the principal is $1,020. At the end of the second

quarter will have :

A = 1020[1 + 0.08(1/4)]

A = 1020(1.02) = $1.040.40


Solution (cont.)

At the end of the third quarter will have :

A = 1.040.40[1 + 0.08(1/4)]

A = 1.040.40(1.02) = $1.061.21

At the end of the fourth quarter will have :

A = 1.061.21[1 + 0.08(1/4)]

A = 1.061.21(1.02) = $1.082.43

• Compare with simple interest:

A = P(1 + rt) = $1,000[1 + 0.08(1)]

A = $1,000[10.08] = $1,080 $2,43 more


Solution (cont.)

Looking over the calculations …

A = 1.000(1.02) End 1st quarter

A = [1.000(1.02)](1.02) = 1,000(1.02)2 End 2nd quarter

A = [1.000(1.02)2](1.02) = 1,000(1.02)3 End 3rd quarter

A = [1.000(1.02)3](1.02) = 1,000(1.02)4 End 4th quarter

• At the end of n quarters:

A = 1.000(1.02)n End nth quarter

or A = 1.000[1 + 0.08(1/4)]n

A = 1.000[1 + 0.08/4]n

Rate per compounding period = annual nominal rate / number of

compounding periods per year


Solution (cont.)

In general

A = P(1 + i) End 1st period

A = [P(1 + i)](1 + i) = P(1 + i)2 End 2nd period

A = [P(1 + i)2](1 + i) = P(1 + i)3 End 3rd period

….

A = [P(1 + i)(n-1)](1 + i) = P(1 + i)n End nth period

Then it can be summarized …


This formula can be generalized to

where A is the future amount, P is the principal, r is the interest

rate as a decimal, m is the number of compounding periods in

one year, and t is the total number of years. To simplify the

formula,

General Formula

1

mtr

A Pm

1n

A P i r

im

n mt


Compound Interest General Formula

where i = r/m

A = amount (future amount) at the end of n periods

P = principal (present value)

r = annual nominal rate

m = number of compounding periods per year

i = rate per compounding period

t = time (years)

n = total number of compounding periods = mt

1n

A P i


Graphical Illustration of

Compound Interest

Growth of 1.00 compounded monthly at 6% annual interest

over a 15 year period (Arrow indicates an increase in value of

almost 2.5 times the original amount.)

0

0.5

1

1.5

2

2.5

3

Growth of 1.00 compounded monthly at 6% annual interest

over a 15 year period


Example

Find the amount to which $1500 will grow if compounded

quarterly at 6.75% interest for 10 years.


Example

Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years.

Solution: Use

Helpful hint: Be sure to do the arithmetic using the rules for order of operations.

1n

A P i 10(4)

0.06751500 1

4

2929.50

A

A


Same Problem Using

Simple Interest

Using the simple interest formula, the amount to which $1500

will grow at an interest of 6.75% for 10 years is given by

A = P (1 + rt)

= 1500(1+0.0675(10)) = 2512.50

which is more than $400 less than the amount earned using

the compound interest formula.


Changing the number of


To what amount will $1500 grow if compounded daily at

6.75% interest for 10 years?






Solution: = 2945.87

This is about $15.00 more than compounding $1500 quarterly

at 6.75% interest.

Since there are 365 days in year (leap years excluded), the

number of compounding periods is now 365. We divide the

annual rate of interest by 365. Notice, too, that the number of

compounding periods in 10 years is 10(365)= 3650.

10(365)0.0675

1500 1365

A




Example 1

If $1,000 is invested at 8% compound:

A. Annually

B. Semiannual

C. Quarterly

D. Monthly

What is the amount after 5 years?




Example 1 - Solution

If $1,000 is invested at 8% compound, what is the amount after

5 years?

n = 5 ; I = r = 0.08

A = P(1 + i)n

= 1,000(1 + 0.08)5

= 1,000(1.469328)

= $1,469.33

A. Annually

n = (2)5 = 10

I = r/n = 0.08/2 = 0.04

A = P(1 + i)n

= 1,000(1 + 0.04)10

= 1,000(1.480244)

= $1,480.24

B. Semiannual




Example 1 - Solution

If $1,000 is invested at 8% compound, what is the amount after

5 years?

n = (12)5 = 60

I = r/n = 0.08/12 = 0.006666

A = P(1 + i)n

= 1,000(1 + 0.00666)60

= 1,000(1.489846)

= $1,489.85

D. Monthly

n = (4)5 = 20

I = r/n = 0.08/4 = 0.02

A = P(1 + i)n

= 1,000(1 + 0.02)20

= 1,000(1.485947)

= $1,485.95

C. Quarterly


Effect of Increasing the

Number of Compounding Periods

If the number of compounding periods per year is

increased while the principal, annual rate of interest and

total number of years remain the same, the future amount

of money will increase slightly.



Previously we indicated that increasing the number of

compounding periods while keeping the interest rate,

principal, and time constant resulted in a somewhat higher

compounded amount. What would happen to the amount if

interest were compounded daily, or every minute, or every

second?


Answer to Continuous Compound

Interest Question

As the number m of compounding periods per year increases

without bound, the compounded amount approaches a limiting

value. This value is given by the following formula:

limm P(1

r

m)mt Pert

A Pert

Here A is the compounded amount.


Example of

Continuously Compounded Interest

What amount will an account have after 10 years if $1,500

is invested at an annual rate of 6.75% compounded

continuously?


Example of


What amount will an account have after 10 years if $1,500

is invested at an annual rate of 6.75% compounded

continuously?

Solution: Use the continuous compound interest formula

A = Pert with P = 1500, r = 0.0675, and t = 10:

A = 1500e(0.0675)(10) = $2,946.05

That is only 18 cents more than the amount you receive by

daily compounding.






Solution: = 2945.87

That is only 18 cents more than the amount you receive by

daily compounding.

10(365)0.0675

1500 1365

A



Example 2

What amount will an account have after 2 years if $5,000 is

invested at an annual rate of 8% compoundly

A. daily?

A = P(1 + r/m)mt

P = 5,000; r = 0.08; m = 365

t = 2

A = 5,000(1+ 0.08/365)(365)(2)

= $5,867.55

A. continuously?

A = Pert

P = 5,000; r = 0.08; t = 2

A = 5,000e(0.08)(2)

= $5,867.55


Growth & Time

How much at a future date?

What annual rate has been earned?

How long take to double the investment?

The formulas for the compound interest and the

continuous interest can be used to answer such

questions.


Growth & Time

How long will it take for $5,000

to grow to $15,000 if the money

is invested at 8.5%

compounded quarterly?


Growth & Time

Solution

How long will it take for $5,000

to grow to $15,000 if the money

is invested at 8.5%

compounded quarterly?

1. Substitute values in the

compound interest formula.

2. divide both sides by 5,000

3. Take the natural logarithm of

both sides.

4. Use the exponent property of

logarithms

5. Solve for t.

Solution:

40.08515000 5000(1 )

4

t

3 = 1.021254t

ln(3) = ln1.021254t

ln(3) = 4t*ln1.02125

ln(3)13.0617

4ln(1.02125)t


Growth & Time

Example 3

Finding Present Value

How much will you invest now at 10% to have $8,000 toward

the purchase of a car in 5 years if interest is

A. compounded quarterly?

P = ? A = 8,000 i = 0.10/4

n = 4(5) = 20

A = P(1 + i)n

8,000 = P(1 + 0.025)20

P = 8,000 / (1 + 0.025)20

P = 8,000 / (1.638616)

= $4,882.17

B. compounded continuously?

P = ? A = 8,000 r = 0.10

t = 5

A = Pert

8,000 = Pe(0.10)(5)

P = 8,000 /e(0.10)(5)

P = $4,852.25


Growth & Time

Example 4

Computing Growth Rate

An investment of $10,000 in a growth-oriented fund over a 10-

years period would have grown in $126,000. What annual

nominal rate would produce the same growth if interest was:

C. compounded quarterly?

A = P(1 + i)n

126,000 = 10,000 (1+ r)10

12.6 = (1+ r)10

Raiz10(12.6) = 1+ r

r = Raiz10(12.6) – 1

r = 0.28836 ó 28.836%

D. compounded continuously?

A = Pert

126,000 = 10,000e10r

12.6 = e10r

ln 12.6 = 10r

r = ln 12.6/10

r = 0.25337 ó 25.337%


Growth & Time

Example 5

Computing Growth Time

How long will it take $10,000 to grow to $12,000 if it is invested

at 9% compounded monthly?

Using logarithms & calculator:

A = P(1 + i)n

12,000 = 10,000 (1+ 0.09/12)n

1.2 = 1.0075n

Solve for n by taking logarithms

of both sides:

ln 1.2 = ln 1.0075n

ln 1.2 = nln 1.0075

n = ln 1.2 / ln 1.0075

n = 24.40 ó 25 months

or 2yr 1mth



The simple interest rate that will produce the same amount as a

given compound interest rate in 1 year is called the annual

percentage yield (APY). To find the APY, proceed as follows:

(1 ) 1

1 1

1 1

m

m

m

rP APY P

m

rAPY

m

rAPY

m

This is also called

the effective rate, as

well as APR.

Amount at simple interest APY after one year

= Amount at compound interest after one year



Example

What is the annual percentage yield for money that is

invested at 6% compounded monthly?



Example

What is the annual percentage yield for money that is

invested at 6% compounded monthly?

General formula:

Substitute values:

Effective rate is 0.06168 = 6.168

1 1

mr

APYm

120.06

1 1 0.0616812

APY


Computing the Annual Nominal

Rate Given the APY

What is the annual nominal rate

compounded monthly for a CD

that has an annual percentage

yield of 5.9%?


Computing the Annual Nominal

Rate Given the APY

What is the annual nominal rate compounded monthly for a CD that has an annual percentage yield of 5.9%?

1. Use the general formula for APY.

2. Substitute value of APY and 12 for m (number of compounding periods per year).

3. Add one to both sides

4. Take the twelfth root of both sides of equation.

5. Isolate r (subtract 1 and then multiply both sides of equation by 12.

1 1

mr

APYm

12

12

12

12

12

0.059 1 112

1.059 112

1.059 112

1.059 112

12 1.059 1

0.057

r

r

r

r

r

r



EXAMPLE 6 - Using APY to Compare Investments.

Find the APY for each of the bank in the table and compare

these CD’s.

Certificates of Deposits (CDs)

Bank Rate Compounded

Advanta 4.93% monthly

DeepGreen 4.95% daily

Charter One 4.97% quarterly

Liberty 4.94% continuously



EXAMPLE 6 - Using APY to Compare Investments (contd.)

Find the APY for each of the bank in the table and compare these

CD’s.

Bank Calculations [APY = (1 + r/m)m – 1] APY

Advanta

= (1 + 0.0493/12)^12 – 1 = 0.05043 5.043%

DeepGreen

= (1 + 0.0495/365)^365 – 1 = 0.05074 5.074%

Charter One

= (1 + 0.0497/4)^4 – 1 = 0.05063 5.063%

Liberty

= e^(0.0494) – 1 = 0.05064 5.064%


Example of


EXAMPLE 7 - Computing the Annual Nominal Rate given

the APY

A savings & loans wants to offer a CD with a monthly

compounding rate that has an APY of 7.5%. What annual

nominal rate compounded monthly should they use?.


Example of


EXAMPLE 7 - Computing the Annual Nominal Rate given the

APY (continued)

APY = (1 + r/m)m - 1

0.075 = (1+ r/12)12 - 1

1.075 = (1+ r/12)12

Raiz12(1.075) = 1+ r/12

Raiz12(1.075) – 1 = r/12

r = 12(Raiz12(1.075) – 1)

Using the calculator:

r = 0.072539

r = 7.254%

Chapter 3

Mathematics of

Finance

Section 2

Compound and

Continuous Interest

END Last Update: Marzo 19/2013

mathematics of finance - inter · barnett/ziegler/byleen college mathematics 12e 2 learning...

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