Homework answers / question archive /   SCHOOL OF MATHEMATICAL SCIENCES                        Question 1 (Example 2) A loan is being repaid with 24 quarterly payments, where the first 10 payments are each 300 and the last 14 payments are each 500

  SCHOOL OF MATHEMATICAL SCIENCES                        Question 1 (Example 2) A loan is being repaid with 24 quarterly payments, where the first 10 payments are each 300 and the last 14 payments are each 500

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SCHOOL OF MATHEMATICAL SCIENCES                     

 

Question 1 (Example 2)

A loan is being repaid with 24 quarterly payments, where the first 10 payments are each 300 and the last 14 payments are each 500. If the nominal rate of interest is 8% convertible quarterly, use both the prospective method and the retrospective method to find the outstanding loan balance immediately after the first eight payments are made.

Question 2 (Example 3)

Molly is buying a car for 50,000 using a 60-month car loan with an interest rate of 12% compounded monthly. For the first three years, Molly makes the required payment. Beginning with the first payment  in the fourth year, Molly begins paying twice the required payment. Molly will completely pay off her loan by making a smaller final payment. Determine the total number of payments that Molly will make. The original required payment is P, where 50,000 ?Pa ?P?1,112.22.

             

Question 3 (Example 5)

Tom and Jerry have a 35-year RM300,000 mortgage with a rate of interest of 6% convertible monthly. Immediately after the 180^th payment, they refinance the mortgage. The interest rate is reduced to 4% convertible monthly, and the term is reduced to 25 years (so there are 10 years of payments remaining). They also make an additional payment of RM50,000 at the time of refinancing. Calculate their new monthly payment.

Question 4 (Example 6)

Thor has a 25-year RM100,000 mortgage with a nominal rate of interest of 12% convertible monthly. The first payment is due one month after the mortgage is taken out. Twelve years after taking out the mortgage (after making his 144^th payment), he refinances with a new nominal rate of interest of 8% convertible monthly. The new mortgage will be paid off on the same date as the original one. Calculate the difference in the monthly mortgage payment after refinancing. 

Question 5

A loan of 1 was originally scheduled to be repaid by 25 equal annual payments at the end of each year.

An extra payment K with each of the 6^th through the 10^th scheduled payments will be sufficient to

a ?a

repay the loan 5 years earlier than under the original schedule. Show that K? ^{20   15} . a a

25 5

Question 6

Rafael buys a house and takes out a 30-year RM150,000 mortgage with a nominal rate of interest of i convertible monthly. He makes monthly payments of RM1,400 for the first 3 years. He pays RM X

 

36

360

36

a

a

?

150,000 ?1400a

 

per month for the remaining 27 years. Show that X ?.

Question 7 (Example 7, 8)

Interest rate	6%
Years of repayment	5

Create an amortization schedule for a loan of RM1,000 repaid over five years if the annual effective rate of interest is 6%.

Question 8 (Example 9)

Mary takes out a 30-year loan on 1 January 2005 for RM30,000 at an annual effective rate of interest of 6%. Payments are made at the end of each year. On 1 January 2015, Mary takes out a 20-year loan for RM50,000 at an annual effective rate of interest of 7%. Payments are also made at the end of each year. Calculate the total amount of principal repayment during the year 2016 on both loans.  

Question 9 (Example 10)

Sandra borrows L for five years at an annual effective rate of interest of 6%, to be repaid with equal payments at the end of each year. The outstanding balance at the end of the third year is 1,671.32 and at the end of the fourth year is 860. Calculate the principal repayment in the third and fourth payment.

Question 10 (Example 13)

A loan of RM2,000 at a nominal rate of 6% convertible monthly is to be repaid by eight monthly payments with the first payment due at the end of 1 month. The first three payments are X each and the final five payments are 5X each. Determine

1. The principal repayment in the fourth payment
2. The interest paid in the sixth payment

   Question 11 (Example 16)

   Smith borrows RM5,000 and agrees to establish a sinking fund to repay the loan at the end of 12 years. Interest at 8% on the debt is paid annually as it falls due. Level payment deposits to the sinking fund are made at the end of each year, with interest accumulating at an annual effective rate of 6% for the first 5 years and 5% thereafter. Determine the size of the sinking fund payment.

   Question 12

   A 10-year loan of RM10,000 is to be repaid with payments at the end of each year consisting of interest on the loan and a sinking fund deposit. Interest on the loan is charged at an annual effective rate of 12%. The sinking fund’s annual effective rate of interest is 8%. However, beginning in the 6^th year, the annual effective rate of interest drops to 6%. As a result, the annual payment to the sinking fund is then increased by X. Calculate X.

   Question 13 (Example 17)

   Joe repays a loan of RM10,000 by establishing a sinking fund and making 25 equal payments at the end of each year. The sinking fund earns an annual effective rate of interest of 8%. Immediately after the 10^th payment, the sinking fund interest increases to an annual effective rate of interest of 9%. At that time, Joe adjusts his sinking fund payment to X so that the sinking fund will still accumulate to RM10,000, 25 years after the original loan date. Determine X.

   Question 14 (Example 20)

   A borrower is repaying a loan with payments at the end of each year for 10 years, such that the payment the first year is 300, the second year is 290, and so forth, until the 10^th year it is 210. Find an expression of the amount of the loan.

   Question 15 (Example 18)

   Two options of repayment for a 10-year RM5,000 loan at an annual effective rate of interest of 6%. 

   (Option I)        Make level payments at the end of the year for ten years

   (Option II)  Accumulate the principal by making equal payments at the end of each year into a sinking fund. An annual interest payment is paid to the creditor.

   The sinking fund earns an annual effective rate of interest of 5%. Determine how much more the repayment is made in total for Option II than for Option I.

   Question 16 (Example 21b)

   Warren has a loan with an annual effective rate of interest of 4%. He makes payments at the end of each year for 10 years. The first payment is 300, and each subsequent payment increases by 20 per year. Calculate the interest portion in the 6^th payment.

   Question 17

   A loan is being repaid in 20 increasing annual instalments of 1, 2, 3, …, 20. The payments begin one year after the loan is made. Find the principal repayment in the 10^th payment, if the annual effective rate of interest is 4%.

   Question 18 (Example 21c)

   A loan is repaid with six annual payments. The first three payments are RM300. The 4^th and 5^th payments are RM500. The final payment is RM700. The annual effective rate of interest is 6%.

   Determine the interest paid in the 3^rd payment.

   Question 19 (Example 21d)

   Joe negotiates a RM80,000 mortgage on a house with monthly payments of RM700 for the first year, RM800 for the second year, and RM900 for the third year, and RM1,000 until the final payment. The first payment is due one month after the loan. A nominal rate of interest of 12% convertible monthly is charged. Find the outstanding loan balance on Joe’s mortgage immediately after the 48^th payment.

   Question 20 (Example 22)

   A 30-year loan is repaid by a decreasing annuity of 30, 29, 28, etc. Payments are made at the end of the year. The annual effective rate of interest is 8%. Determine the row of an amortization schedule associated with the 15^th payment.

   Question 21

   Joe negotiates an 8-year loan which requires him to pay RM1,200 per month for the first 4 years and RM1,500 per month for the remaining years. The nominal rate of interest is 13% convertible monthly, and the first payment is due in one month. Determine the principal repayment in the 17^th payment. The outstanding loan balance at time 16 is, prospectively