Answer:
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%
Step-by-step 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%