Financial Maths - Series - Loans - changing interest rates and payments.
Test Yourself 1.
The questions on this page focus on: |
1. Changing the interest rate. |
2. Increasing the regular repayment amount. |
3. Making a special contribution. |
Strategy: When approaching questions involving changes to rates, deposits or making special deposits, it is usually a good strategy to sove the expression you develop for the last period before the change. Then carry the amount you calculate into the next phase rather than try to extend the series pattern. Less writing and less chance of errors
Changing the interest rate. | 1. It is a common practice to provide loans at a special "introductory rate" for some time and then revert to a "normal" rate.
Phoebe was offered such an incentive when she borrowed $150,000 on a 15 year monthly reducible loan to buy her first apartment. The introductory rate was 4% for two years rising to the usual 6% after 24 months. She felt she could afford $1,500 per month for repayments during the initial period. Answer.(i) $124,942.11 owing at end Y2. (ii) $1,361.47 p.m. (iii) Loan ends 24 months earlier. |
2. One year ago, Harry and Sally borrowed $350,000 to purchase a house. The terms of the loan were a 25 year period with 9% monthly reducible interest rate. The monthly installments were set at $2,937.
Answer.(i) $346,095 owing at end Y1. (ii) $2,270.31 p.m. (iii) Loan ends after 15 years - so 10 years early!! (iv) Saving of $128,126. |
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3. A loan for $L was taken out by a small business just starting up its operations. The conditions of the loan were:
If $An is the amount of the loan still owing after the nth repayment:
At the end of the three months initial period, the interest rate was increased to 9% p.a. and that rate was to be maintained for the remainder of the loan. The loan was then to be repaid in equal monthly repayments of $4,800 for the remainder of the three year period. Answer.(iii) Loan of $149,700. |
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Increasing the regular repayment amount. | 6. Pete borrows $150,000 from his bank and agrees to repay the loan in equal monthly installments of $1,075 at the end of each month over 20 years.
The interest rate is 6% p.a. Answer.(i) $149,348.38. (iii) 9 years 3 months. (iv) |
7. Ben borrows $200,000 from a bank. He wants to make payments of $P at the end of each month over 25 years. The bank offers him interest at 6% p.a. monthly reducible.
If $Bn is the balance of the loan after n months: Answer.(ii) $1,288.60 monthly. (iii) 7 years 1 months + $175. |
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8. Paulina charged $5000 on her credit card at the beginning of January 2019. Her debt was to be repaid by equal monthly installments of $100. The first installment was due on 1 February.
Interest rate on the credit card is 18% p.a. calculated at the beginning of each month on the balance at the end of the previous month. The interest is added to the outstanding amount before the monthly repayment installment is paid. Answer.(i) 93.11 months = 7 years 10 months. (iii) $392.77 p.m. (iv) Saving = $2,197.76!!! |
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9. Ellen borrowed $600,000 for the purchase of her home. The interest rate charged was 3.6% fixed over the 30 year term of the loan.
Let $A be the amount owing at the end of n months and $M be the monthly payment amount. (i) Show that . (ii) Show that the monthly repayments will be $2,728 (to the nearest $) if Ellen is to repay the loan at the end of 30 years. (iii) Show that Ellen still owes $274,513 after 20 years. (iv) At the end of 20 years, Ellen decides to increase her repayment amount by 0.1% per month so that she can reduce the term of the loan. Show that M243 = 27451×1.0033 - 2728 × 1.001(1.0032 + 1.003 × 1.001+ 1.0012) (v) By factorising out a "common factor" of 1.0033 from the bracketed term, write the second part of this expression as a geometric series. (vi) Hence calculate the balance of Ellen's loan |
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10. Ted wishes to borrow $10,000 from his Bank. The bank sets the conditions that his rate is 6% monthly reducible interest and the repayment period is 24 months.
Answer.(ii) $443.21 monthly so a total of $10,637.04. (iii) First 3 months are $395.29, $399.24, $403.24. (iv) Total = $10,662.34. |
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Making a special contribution. | 11. Jonah borrowed $26,000 at 9% monthly reducible interest. He intends to repay the loan in quarterly repayments of $R over two years.
If An is the amount outstanding on the loan immediately after the nth payment: Answer.(iii) Final payment is reduced by $2,185.66. (iii) First 3 months are $395.29, $399.24, $403.24. (iv) Total = $10,662.34. |
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