Finance - Periods TYS 1

Although the context here is on loans, some of these questions can focus on analogous situations where an investment fund has been established and regular payments are made from that fund. This would be the case with many superannuation accounts for example. Interest is paid on the balance in the fund but there is no further deposit of capital into the fund. Prize funds are similar.

The questions on this page focus on:

1. finding the number of periods for repayment.

2. Implications of a repayment free period.

3. implications of an interest free period.

4. missing payments.

5. Payment in advance.

Finding the number of periods for repayment.	1. A sum of $26,000 is borrowed at 6% p.a. reducible yearly. Annual repayments are set at $4,200 beginning at the end of the first year. (i) Show that the amount owing at the end of the second year is $A₂ = 26000 × 1.06² - 4200 (1 + 1.06). (ii) Find an equation for the amount owing after n years. (iii) Find the number of years required to repay the loan fully. Answer.(iii) 8 years.
	2. A boat building business borrowed $2,000,000 to finance expansion of the business at the beginning of 2020. The annual interest rate was 8% reducible annually. The risks associated with the business attracted loan conditions a little different to the usual. At the beginning of each year, interest was calculated on the balance of the loan at that time and added to the balance. Repayments were scheduled at the end of each year as a fixed amount of $300,000. The first repayment was scheduled at the end of 2020. (i) Show that the balance owing at the end of 2021 is B₂₀₂₁ = 2,000,000 × 1.09² = 300,000(1 + 1.09) (ii) At the end of what year will the boat builder repay the loan? Answer.(ii) At the end of 2029.
	3.
Repayment free period.	4. Milka wants to borrow $5,000 from her bank. She accept the special offer of a four year loan at 3% p.a. monthly reducible interest with no repayment for 4 months but with interest accruing during that time. (i) How much does Milka owe at the end of the 4 months repayment free period? (ii) What is Milka's monthly repayment? (iii) In the statement for the amount owing at the end of 6 months, how can the term $5,000 × 1.0025⁶ be interpreted? (iv) How much interest did Milka pay in total? Answer.(i) Balance now $4040.15. (ii) R = $121.35. (iii) Interest = $339.40.
Interest free period.	5. A local car dealership is offering a special deal on a run-out model new car for $20,000. It is to purchased through a loan over 3 years but with the first six months interest free. After that period, interest will be 12% p.a. monthly reducible. The monthly payments across all 3 years are $R. If the balance owing on the loan after n months is $B_n: (i) Develop an expression for the amount owing at the end of the 6 months interest free period. (ii) Show that at the end of the 8th month of the loan, the amount owing is $B₈ = (20000 - 6M)×1.01² - R(1 + 1.01) (iii) Find the monthly amount $R to be paid to service the loan. Answer.(iii) Repayment = $659.80.
	6. At a large well-known store near you, the retailer offers 3 months a "interest free period". During that time, you pay no interest but you do have to meet the schedule of monthly repayments. You take up this offer to buy something precious and it costs $1,500. The interest rate being charged is 1% per month. The amount due (cost + interest) must be repaid in 3 years with the monthly repayment schedule. (i) How much will the monthly repayments be - given the interest free period? (ii) How much interest will be paid? Answer.(i) $48.40 per week. (ii) $242.40 in interest (don't use the calculator value).
	7. Alexandra borrows $30,000 over four years to purchase a sporty car to reward herself for her efforts to help her teenagers finish secondary school. The local car dealer offers her an interest free period covering the first 8 fortnights. After that grace period, the interest rate of 12% p.a. applies to her balance reducing loan every 2 weeks. The fortnightly repayment will be $F. If $A_n is the amount still owing in the loan after n payments: (i) Develop an expression for the amount owing at the end of the 8 interest-free fortnights. (ii) Show that at the end of the 10th fortnight of the loan, the amount owing is $A₁₀ = (30000 - 8F)×1.004615² - F(1 + 1.004615) (iii) Find the fortnightly amount $F paid by Alexandra. Answer.(iii) F = $351.23.
	8. Sam needs to take out a loan for $80,000. She is to be charged interest at the rate of 4.8% compounded monthly and she will repay the loan in monthly installments of $R over an 8 year period. The lending institution also offers Sam a 6 month interest free period as an incentive. (i) Write down what Sam owes on the loan $M₆ at the end of six months. (ii) Develop an expression for what Sam owes at the end of eight months $M₈. (iii) Calculate the value of $R Sam will need to pay for her monthly repayments. Answer.(iii) R = $992.00.
	9. Tania borrows $150,000 from her Building Society at 9% with monthly reducible interest and monthly repayments for the 25 years term of the loan. She has however secured a "special customer" deal and no repayments are required for the first 9 months although interest is charged. After that both interest and repayments are due each month. (i) Write down an expression for the amount owing at the end of the 10th month. (ii) Extend this statement, with support, to show the amount owing at the end of n months. (iii) Calculate, to the nearest dollar, the amount required for each monthly repayment. Answer.(iii) R = $1,179.47.
Missing payments.	10. Ruby takes out a loan for $5,500 over 5 years. The compound interest rate she is charged is 8% and payments are to be made at the end of each month. (i) Calculate the monthly repayment required. (ii) Ruby unfortunately becomes sick and is unable to make any payments for months 4 and 5. She is however reluctant to extend the life of her loan beyond 5 years. If she is to complete the loan in 5 years, what will be her new monthly repayment amount after month 5? Answer.(i) R = $111.63. (ii) Revised repayment = $116.51.
	11. Matilda borrows $4,000 to buy a new TV and some hi-fi equipment. The borrowing rate is 9% monthly reducible and she is to make monthly repayments of $M over four years. She makes the first two monthly repayments on time but, due to her Uni studies, she misses three payments. She therefore has to refinance her loan. (i) What is the original monthly repayment Matilda had agreed to make in her contract? (ii) What is the new amount for monthly repayments covering the remaining time of the contract to enable Matilda to pay off her loan in the four years (and so avoid any penalty)? Answer.(i) R = $99.04. (ii) Revised repayment = $107.75.
Payment in advance.	12. An Art Gallery in Tasmania requires a loan of $P for extensions. The terms are that there is to be a special interest rate for the first three months of 6% p.a. and then that rate will rise from the beginning of the fourth month to 9% p.a.. The rate is guaranteed to be held at 9% p.a. for the remaining 2.75 years of the 3 year loan period. For the first three months, the repayment is $4,000 per month. After that, the repayment increases to $4,800 per month. If $A_n represents the balance of the loan after n months: (i) find $A₂. (ii) Find an expression for $A₄. (iii) Find the amount of the loan ($P). Answer.(iii) Loan amount is (say) $150,000.