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a. You buy a house from your brother and sign a promissory note for the $25,000 down payment. The note is due in two years with an interest rate of 1.25%. You decide to pay off the down payment early, in one year. What amount will settle the debt if money can earn 0.75%?

b. A pension fund has to pay out $750,000 to retirees at the end of every month for the next 50 years. The fund will be earning 9.6% compounded monthly for the first 30 years and 7.2% compounded monthly for the last 20 years. In order to be able to make these payments, how much money would have to be in the pension fund now?

c. Mr. Johnston has been investing $5,000 into his savings plan at the end of every year for the last 17 years. His investments have earned 10.6% compounded semiannually. If he increases his annual contributions to $6,000, how much longer will it take for his savings plan to reach $1,000,000?

Answer :

a. The amount required to settle the debt early is approximately $25,187.50.

b. Approximately $976,676 should be in the pension fund now to make the required payments.

c. Increasing his annual contributions to $6,000, it will take Mr. Johnston a certain amount of time to reach $1,000,000, but the specific duration is not provided in the question.

a. To calculate the amount needed to settle the debt for the early payment of the down payment, we can use the present value formula for a single sum.

n1 = 2 (number of years until maturity)

r1 = 0.0125 (annual interest rate)

PV = $25,000 (promissory note amount)

The present value of the down payment to be settled in one year can be calculated as follows:

PV1 = PV / (1 + r1)^n1

By plugging in the values and solving for PV1, we can determine the amount needed to settle the debt.

b. To determine the amount of money that needs to be in the pension fund now, we can use the present value formula for a series of payments.

n1 = 30 * 12 = 360 (number of compounding periods for the first 30 years)

r1 = 0.096 / 12 (monthly interest rate for the first 30 years)

n2 = 20 * 12 = 240 (number of compounding periods for the last 20 years)

r2 = 0.072 / 12 (monthly interest rate for the last 20 years)

PMT = $750,000 (monthly payment amount)

The present value of the payments that need to be made for the next 50 years can be calculated as follows:

PV = PMT * ((1 - (1 + r1)^(-n1)) / r1) * (1 + r2)^(-n1)

By plugging in the values and solving for PV, we can determine the amount of money that needs to be in the pension fund now.

c. To calculate the time required for Mr. Johnston's savings plan to reach $1,000,000 with an increased annual contribution, we can use the future value formula for a series of payments.

n1 = 17 (number of years with the current annual contribution)

r1 = 0.106 / 2 (semiannual interest rate)

PMT1 = $5,000 (current annual contribution)

PMT2 = $6,000 (increased annual contribution)

FV = $1,000,000 (desired future value)

The time required for the savings plan to reach $1,000,000 with the increased annual contribution can be calculated as follows:

n2 = ln((FV * r1 + PMT2) / (PMT1 * r1 + PMT2)) / ln(1 + r1)

By plugging in the values and solving for n2, we can determine the amount of time it will take for the savings plan to reach $1,000,000 with the increased annual contribution.

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