Some Interest problems?

Question

1. You have set up an ordinary annuity that will pay you $650.00 a month for the next 25 years. You will earn interest at a rate of 5.5% compounded monthly. What amount did you invest to accomplish this goal?

2. If an annuity was set up for semi-annual payments at the end of each period in the amt of $1350, what would be the value of this annuity after 15 1/2 yrs with interest compounded semiannually at a rate of 4%?

3. A company requires the amount of $850,000 in twenty(20) years to retire a bond issue. Assume they earn 5% interest compounded quarterly. What amount would they have to pay quarterly to be able to retire this debt in 20 years?

Answer 1

1. In this question, you are looking for the present value of the money you invested.

Present Value = ?
Future Value = 0
Payments = 650.00
Interest = 5.5%
Compounding Periods = monthly = 12
Payment Periods = monthly = 12
Number of Payments = 12 x 25 = 300

The formula is:

PV = PMT( [1 - (1+i)^-n] / i )

So plugging our information into the formula we have:

PV = 650( [ 1 - (1 + (0.055/12)^-300 ] / (0.055/12) )
PV = 650( [ 1 - (1.004583333)^-300 ] / 0.004583333 )
PV = 650(0.746365 / 0.004583333)
PV = 105 848.11

Therefore, you invested $105 848.11

2. This time, you are looking for the future value of the money.

Present Value = 0
Future Value = ?
Payments = 1350
Interest = 4%
Compounding Periods = semi-annually = 2
Payment Periods = semi-annually = 2
Number of Payments = 2 x 15.5 = 31

The formula in this case is:

FV = PMT( [(1+i)^n - 1] / i )

So plugging in our values we get:

FV = 1350( [(1+(0.04/2))^31 - 1] / (0.04/2) )
FV = 1350( [(1+(0.02))^31 - 1] / (0.02) )
FV = 1350( [(1+(0.02))^31 - 1] / (0.02) )
FV = 1350( 0.847588816 / (0.02) )
FV = 57 212. 25

Therefore, after 15.5 years of 1350 semi annually, you would have accumulated $57,212.25

3. Now you are looking for the value of the payments.

Present Value = 850 000
Future Value = 0
Payments = ?
Interest = 5%
Compounding Periods = quarterly = 4
Payment Periods = quarterly = 4
Number of Payments = 20 x 4 = 80

We have to rearrange the formula for the present value.

PV = PMT( [1 - (1+i)^-n] / i )
PMT = PV / ( [1 - (1+i)^-n] / i )

Then, subbing in the values we get:

PMT = 850000 / ( [1 - (1+(0.05/4))^-80] / (0.05/4) )
PMT = 850000 / ( [1 - (1+(0.0125))^-80] / (0.0125) )
PMT = 850000 / ( [0.629833213] / (0.0125) )
PMT = 850000 / 50.38665706
PMT = 16869.55

Therefore, they would have to make payments of $16 869.55 in order to retire the debt in 20 years.