Changing compounding frequency Using annual, semiannual, and quarterly compounding periods for each of the following, (1) calculate the future value if $5,000 is deposited initially, and (2) determine the effective annual rate (EAR).

a. At 12% annual interest for 5 years.

b. At 16% annual interest for 6 years.

c. At 20% annual interest for 10 years.

a). Future value=$8,811.71 annually, and the effective annual rate is=12%

Future value = $8,954.23 semiannually, and the effective annual rate=12.36%

Future value quarterly=$9,030.56, and the effective annual rate=12.55%

b). Future value annually=$12,181.98, and the effective annual rate=16%

Future value semiannually=$12,590.85, and the effective annual rate=16.64%

Future value quarterly=$12,816.52, and effective annual rate=16.99%

c). Future value annually=$30,958.68, and the effective annual rate=20%

Future value semi-annually=$33,637.49, and the effective annual rate=21%

Future value quarterly=$35,199.94, and the effective annual rate=21.55%

Explanation:

a). At 12% annual interest for 5 years

Compounded annually

A=P (1+r/n) ^nt

where;

A=future value

P=initial value=5,000

n=1

r=annual interest rate=12%=12/100=0.12

t=number of years=5

A=5,000 (1+0.12/1) ^5=8,811.71

Future value when interest is compounded annually=$8,811.71

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=12/100=0.12

n=number of compounding periods in a year=1

replacing;

EAR={ (1+0.12/1) ^1}-1

EAR=0.12*100

Effective annual rate when compounding is done annually=12%

Compounded semiannually

P=initial value=5,000

n=2

r=annual interest rate=12%=12/100=0.12

t=number of years=5

A=5,000 (1+0.12/2) ^ (5*2) = 8,954.23

Future value when interest is compounded semiannually=$8,954.23

i=stated interest rate=12/100=0.12

n=number of compounding periods in a year=2

replacing;

EAR={ (1+0.12/2) ^2}-1

EAR=0.1236*100

Effective annual rate when compounding is done semi-annually=12.36%

Compounded quarterly

P=initial value=5,000

n=4

r=annual interest rate=12%=12/100=0.12

t=number of years=5

A=5,000 (1+0.12/4) ^ (5*4) = 9,030.56

Future value when interest is compounded quarterly=$9,030.56

i=stated interest rate=12/100=0.12

n=number of compounding periods in a year=4

replacing;

EAR={ (1+0.12/4) ^4}-1

EAR=0.1255*100

Effective annual rate when compounding is done quarterly=12.55%

b). At 16% annual interest for 6 years

Compounded annually

P=initial value=5,000

n=1

r=annual interest rate=16%=16/100=0.16

t=number of years=6

A=5,000 (1+0.16/1) ^6=12,181.98

Future value when interest is compounded annually=$12,181.98

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=16/100=0.16

n=number of compounding periods in a year=1

replacing;

EAR={ (1+0.16/1) ^1}-1

EAR=0.16*100

Effective annual rate when compounding is done annually=16%

Compounded semi-annually

P=initial value=5,000

n=2

r=annual interest rate=16%=16/100=0.16

t=number of years=6

A=5,000 (1+0.16/2) ^6*2=12,590.85

Future value when interest is compounded semiannually=$12,590.85

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=16/100=0.16

n=number of compounding periods in a year=2

replacing;

EAR={ (1+0.16/2) ^2}-1

EAR=0.1664*100

Effective annual rate when compounding is done semi-annually=16.64%

Compounded quarterly

P=initial value=5,000

n=4

r=annual interest rate=16%=16/100=0.16

t=number of years=6

A=5,000 (1+0.16/4) ^6*4=12,816.52

Future value when interest is compounded quarterly=$12,816.52

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=16/100=0.16

n=number of compounding periods in a year=4

replacing;

EAR={ (1+0.16/4) ^4}-1

EAR=0.1699*100

Effective annual rate when compounding is done quarterly=16.99%

c). At 20% annual interest for 10 years

Compounded annually

P=initial value=5,000

n=1

r=annual interest rate=20%=20/100=0.2

t=number of years=10

A=5,000 (1+0.2/1) ^10=30,958.68

Future value when interest is compounded annually=$30,958.68

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=20/100=0.2

n=number of compounding periods in a year=1

replacing;

EAR={ (1+0.2/1) ^1}-1

EAR=0.2*100

Effective annual rate when compounding is done annually=20%

Compounded semi-annually

P=initial value=5,000

n=1

r=annual interest rate=20%=20/100=0.2

t=number of years=10

A=5,000 (1+0.2/2) ^10*2=33,637.49

Future value when interest is compounded semi-annually=$33,637.49

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=20/100=0.2

n=number of compounding periods in a year=2

replacing;

EAR={ (1+0.2/2) ^2}-1

EAR=0.21*100

Effective annual rate when compounding is done semiannually=21%

Compounded quarterly

P=initial value=5,000

n=1

r=annual interest rate=20%=20/100=0.2

t=number of years=10

A=5,000 (1+0.2/4) ^10*4=35,199.94

Future value when interest is compounded quarterly=$35,199.94

The effective annual rate formula is expressed as;

Effective annual rate = ((1+i/n) ^n}-1

where;

i=stated interest rate=20/100=0.2

n=number of compounding periods in a year=4

replacing;

EAR={ (1+0.2/4) ^4}-1

EAR=0.2155*100

Effective annual rate when compounding is done quarterly=21.55%