# Reducing balance loans

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1 Reducing balance loans 5 VCEcoverage Area of study Units 3 & 4 Business related mathematics In this chapter 5A Loan schedules 5B The annuities formula 5C Number of repayments 5D Effects of changing the repayment 5E Frequency of repayments 5F Changing the rate 5G Reducing balance and flat rate loans

2 7 Further Mathematics Introduction When we invest money with a financial institution the institution pays us interest because it is using our money to lend to others. Conversely, when we borrow money from an institution we are using the institution s money and so it charges us interest. In reducing balance loans, interest is usually charged every month by the financial institution and repayments are made by the borrower also on a regular basis. These repayments nearly always amount to more than the interest for the same period of time and so the amount still owing is reduced. Since the amount still owing is continually decreasing and interest is calculated on a daily balance but debited monthly, the amount of interest charged decreases as well throughout the life of the loan. This means that less of the amount borrowed is paid off p in the early stages of the loan compared to the end. If we graphed the amount owing against time for a loan it would look like the graph at right. That is, the rate at which the loan is paid off increases as the loan progresses. Time The terms below are often used when talking about reducing balance loans: Principal, P amount borrowed (\$) Balance, A amount still owing (\$) Term life of the loan (years) To discharge a loan to pay off a loan (that is A \$0) It is possible to have an interest only loan account whereby the repayments equal the interest added and so the balance doesn t reduce. This option is available to a borrower who wants to make the smallest repayment possible. Though the focus of this chapter is reducing balance loans, note that the theory behind reducing balance loans can also be applied to other situations such as superannuation payouts, for people during retirement, and bursaries. In each of these situations a lump sum is realised at the start of a period of time and regular payments are made during that time. Regular payments are called annuities. So these situations are often called annuities in arrears because the annuity follows the realisation of the lump sum. Amount owing

3 Chapter 5 Reducing balance loans 73 Loan schedules The first amount of interest is added to the balance of a loan account one month after the funds are provided to the customer. The first repayment is usually made on the same day. Consider a loan of \$800 that is repaid in 5 monthly instalments of \$65.8 at an interest rate of.% per month, interest debited each month. A loan schedule can be drawn for this information, showing all interest debits and repayments. From the schedule the amount owing after each month is shown and the total interest charged can be calculated. For any period of the loan: Total repayments Interest paid + Principal repaid Month Balance at start of month (\$) Interest (.% of monthly starting balance) (\$) Total owing at end of month (\$) Repayment (\$) Balance after repayment (\$) Each month interest of.% of the monthly starting balance is added to that balance and the repayment value is subtracted, leaving the starting balance for the next month. This process continues until the loan is paid off after the 5 months. Note that the amount of interest charged falls each month and so the amount of principal paid each month increases as outlined earlier. Another method can be used to analyse this account, but it doesn t display interest amounts. Since the interest rate is.% per month the balance increases by this rate each month. Recalling the work covered in the previous chapter about the growth factor, we can write: r Growth factor, R where represents the original amount and r represents the increase per period.0 So: Balance at start of second month balance at start of first month R repayment A Α R Q where Q is the regular repayment.

4 74 Further Mathematics WORKED Example An \$800 loan is repaid in 5 monthly instalments of \$65.8 at an interest rate of.% per month, interest debited each month. Calculate: a the amount still owing after the 4th month b the total interest charged during the 5 months. THINK a 3 4 Calculate the growth factor. R + r Find the balance, A, at the start of the nd month. A 0 starting principal A 0 \$800 Find the balance, A, at start of the 3rd month. Continue this process to find A 3, A 4 and A 5. 5 The amount still owing at the end of the 4th month is A 4. b Total interest Total repayments Principal repaid WRITE r a R A A 0 R Q 800(.0) 65.8 A \$ A A R Q (.0) 65.8 A \$485.7 A 3 A R Q 485.7(.0) 65.8 A 3 \$35.73 A 4 A 3 R Q 35.73(.0) 65.8 A 4 \$63.83 The amount still owing at the end of the 4th month is \$63.83 b Total interest \$9.05 As mentioned earlier, institutions usually debit a loan account with interest each month. In this chapter we also consider situations in which interest is debited fortnightly and quarterly. The frequency with which a customer can make repayments may be weekly, fortnightly or monthly, and we also consider quarterly repayments. In all cases in this chapter the frequency of debiting interest will be the same as the frequency of making repayments, although this is not necessary in practice. It simply makes calculations easier. The calculations outlined for monthly repayments would follow exactly the same pattern for other repayment frequencies. In worked example, the loan was paid off with only a few repayments. In practice, the repayment of most loans takes considerably longer than this. The process outlined in the example continues throughout any part of the term of the loan.

5 Chapter 5 Reducing balance loans 75 WORKED Example A loan of \$6 000 is repaid by monthly instalments of \$ over 4 years at an interest rate of.% per month, interest debited monthly. Calculate: a the amount still owing after the 5th repayment b the decrease in the principal during the first 5 repayments c the interest charged during this time. THINK WRITE a Calculate the growth factor, R. a R + r Find the balance, A, at the end of the st month (or the start of the nd month). A , Q A A 0 R Q 6 000(.0) A \$ (a) Find A from A. (b) Repeat until A 5 is found. (A 5 is the balance at the end of the 5th month.) A A R Q (.0) A \$ A (.0) \$ A (.0) \$ A (.0) \$ Write a statement. The amount owing after 5 months is \$ b The decrease in the principal is the difference between the amount owing initially, A 0, and after the 5th month, A 5. b Decrease in principal A 0 A \$30.49 Write a statement. The principal has decreased by \$30.49 in the first 5 months of the loan. c Interest charged Total repayments Principal repaid c Interest charged \$85.66 Write a statement. The interest charged during the first 5 months is \$85.66.

6 76 Further Mathematics a b More often than not a financial institution provides the nominal interest rate per year rather than the interest rate per period. As outlined in the previous chapter in the compound interest formula section, the rate per period can be obtained from the nominal annual rate as follows: Nominal interest rate per annum Interest rate per period, r Number of interest periods per year It is important to note that while a loan can be drawn at a certain interest rate, that rate will generally not remain the same for the life of the loan. This means that when we consider borrowing we should be aware that the amount of the repayments may increase (due to an increase in the interest rate) during the term of the loan and we should be confident that repayments can be met even if the rate rises. It has been said that if a potential borrower can maintain repayments for a rate of % p.a. over the term of the loan then the borrower can withstand rate changes that may range from perhaps 5% p.a. to 7% p.a. Let us now look at how quickly the principal decreases at the end of a loan compared with the earlier stages. WORKED Example 3 A family take out a loan of \$ to extend their home. The loan is made at a rate of interest of 0% p.a. (debited monthly) and is repaid over 0 years by monthly instalments of \$ For the 3rd repayment find: i the amount of principal repaid ii the amount of interest paid. After 8 years the amount still owing is \$ Assuming the same conditions apply as in part a, for the 97th repayment find: i the principal repaid ii the interest paid. THINK WRITE a i Calculate the monthly interest rate, r. air (a) Calculate the monthly growth factor, R. (b) Store in your calculator memory if it is recurring. Calculate the amount owing after each of the first 3 months A, A and A r % per month R A A 0 R Q ( ) A \$ A ( ) \$ A ( ) \$

7 Chapter 5 Reducing balance loans 77 THINK WRITE 4 Principal repaid A A 3 (3rd repayment) ii Interest paid Total repayments Principal repaid b i Monthly repayment 8 years payments/year 96. So, A 96 \$ Find A 97. Principal repaid A 96 A 97 (97th repayment) ii Interest paid Repayments Principal repaid b Principal repaid \$98.53 ii Interest \$ i A 97 A 96 R Q 455.7( ) \$ 0.57 Principal repaid \$433.4 ii Interest \$95.46 As mentioned in the introduction, a greater percentage of each repayment made in the early part of a loan is interest, compared with the repayments toward the end. This is confirmed by the calculations made in the last example. In summary, with each of 0 repayments being \$58.60; for the 3rd repayment: interest \$330.07, principal repaid \$98.53 for the 97th repayment: interest \$95.46, principal repaid \$ That is, the principal decreases faster towards the end of the loan. remember remember. In a loan schedule: (a) the interest charged each period increases the amount owed (b) the repayment each period decreases the amount owed. r. Growth factor, R where represents the original amount and r represents the increase per period in %. 3. Balance at the end of the month balance at start of the month R Q A n + A n R Q where Q repayment 4. Total repayments Interest paid + Principal repaid 5. Interest rate per period, r Nominal interest rate per annum Number of interst periods per year

8 78 Further Mathematics 5A Loan schedules EXCEL Spreadsheet WORKED Example Mathcad Reducing balance loans A loan of \$0 is repaid in five monthly instalments of \$06.04 at a rate of % per month, interest debited monthly. Calculate: a the amount still owing after the 4th repayment b the total interest charged during the 5 months. Dimitri takes out a loan of \$500 and repays it in five monthly instalments of \$ at a rate of.% per month, interest debited monthly. Calculate: a the amount still owing after the 4th repayment b the total interest charged during the 5 months. 3 A loan of \$000 is repaid in four quarterly instalments of \$55.5 at a rate of % per quarter, interest debited quarterly. Calculate: a the amount still owing after the 3rd repayment b the total interest charged during the 4 quarters. 4 Gaetana borrows \$900 which she repays in five quarterly instalments of \$93.7 at a rate of.5% per quarter, interest debited quarterly. Calculate: a the amount still owing after the 4th repayment b the total interest charged during the 5 quarters. 5 Josh s loan of \$3000 is repaid in four halfyearly instalments of \$ at a rate of 3% per half-year, interest debited half-yearly. Calculate: a the amount still owing after the 3rd repayment b total interest charged during the 4 repayments. 6 Rebecca takes out a loan of \$500 to purchase a new computer. The loan is repaid in four 6-monthly instalments of \$ at a rate of 4.5% per 6-months, interest debited 6-monthly. Calculate: a the amount still owing after the 3rd repayment b the total interest charged during the 4 repayments.

9 Chapter 5 Reducing balance loans 79 WORKED Example 7 a A loan of \$0 000 is repaid by monthly instalments of \$ over 5 years at an interest rate of % per month, interest debited monthly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. b A loan of \$0 000 is repaid by quarterly instalments of \$344.3 over 5 years at an interest rate of 3% per quarter, interest debited quarterly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. 8 a Jose borrows \$ which he repays in fortnightly instalments of \$06.45 over 0 years at an interest rate of 0.5% per fortnight, interest debited fortnightly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. b A loan of \$ is repaid by quarterly instalments of \$ over 0 years at an interest rate of 3.5% per quarter, interest debited quarterly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. 9 a Angela takes out a loan of \$0 000 to set up a catering business. The loan is repaid by monthly instalments of \$664.9 over 3 years at an interest rate of % per month, interest debited monthly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. b Emad borrows \$0 000 to establish a pet-minding business. The loan is repaid by monthly instalments of \$35.06 over 8 years at an interest rate of % per month, interest debited monthly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. c Hank takes out a loan of \$0 000 which he repays in monthly instalments of \$86.94 over 0 years at an interest rate of % per month, interest debited monthly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. d In parts a c above the three loan accounts are the same except for the term. As the term of the loan increases how does this affect: i the repayment? ii the amount still owing after the 5th repayment? iii the amount of interest paid during the 5 repayments?

10 70 Further Mathematics 0 a Jaques borrows \$ which he repays in quarterly instalments of \$ over 8 years at an interest rate of.5% per quarter, interest debited quarterly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. b Isabel borrows \$ and repays it by quarterly instalments of \$95.09 over 0 years at an interest rate of.5% per quarter, interest debited quarterly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments iii the interest charged during this time. c George takes out a loan of \$ which he repays in quarterly instalments of \$080.8 over years at an interest rate of.5% per quarter, interest debited quarterly. Calculate: i the amount still owing after the 5th repayment ii the decrease in the principal during the first 5 repayments d iii the interest charged during this time. In parts a c above the 3 loan accounts are the same except for the term. As the term of the loan increases how does this affect: i the repayment? ii the amount still owing after the 5th repayment? iii the amount of interest paid during the 5 repayments? In questions 3 find: i the amount still owing after the 4th repayment ii the decrease in the principal during the first 4 repayments iii the total interest paid during this time. A loan of \$ is to be paid by monthly instalments of: a \$55.3 over 5 years at 0.8% per month (interest debited monthly) b \$487.3 over 8 years at 0.8% per month (interest debited monthly) c \$ over 5 years at 0.8% per month (debited monthly) d \$639. over 5 years at.% per month (debited monthly) e \$607.5 over 8 years at.% per month (debited monthly) f \$57.46 over 5 years at.% per month (debited monthly). A loan of \$ is to be repaid by monthly instalments of: a \$49.86 over 0 years at 0.5% per month (interest debited monthly) b \$47.4 over 0 years at 0.6% per month (interest debited monthly) c \$56.90 over 0 years at 0.7% per month (interest debited monthly) d \$563.0 over 0 years at 0.8% per month (interest debited monthly) e \$ over 0 years at 0.95% per month (interest debited monthly) f \$685.9 over 0 years at.05% per month (interest debited monthly). 3 A loan of \$ is to be repaid by quarterly instalments of: a \$9.90 over 0 years at.5% per quarter (debited quarterly) b \$4.0 over 0 years at.8% per quarter (debited quarterly) c \$ over 0 years at.% per quarter (debited quarterly) d \$ over 0 years at.4% per quarter (debited quarterly) e \$9.89 over 0 years at.85% per quarter (debited quarterly) f \$06.53 over 0 years at 3.5% per quarter (debited quarterly).

11 Chapter 5 Reducing balance loans 7 WORKED Example 3 4 The loan accounts outlined in question are the same except for the interest rate. The same applies to question 3. In these cases, as the interest rate increases, what happens to: a the repayment? b the amount still owing after the 4th repayment? c the amount of interest paid during the 4 repayments? 5 a Madako s loan of \$ has interest charged at a rate of 9% p.a. (debited monthly) and it is repaid over 0 years by monthly instalments of \$ For the 3rd repayment find: i the principal repaid ii the interest paid. b After 8 years the amount still owing is \$ Assuming the same conditions apply as in part a, for the 97th repayment find: i the principal repaid ii the interest paid. 6 a Pina s loan of \$ has interest charged at a rate of 8% p.a. (debited monthly) and it is repaid over 0 years by monthly instalments of \$ For the 3rd repayment find: i the principal repaid ii the interest paid. b After 8 years the amount still owing is \$ Assuming the same conditions apply as in part a, for the 7th repayment find: i the principal repaid ii the interest paid. 7 a Katharine s loan of \$ has interest charged at a rate of % p.a. (debited quarterly) and it is repaid over 0 years by quarterly instalments of \$ For the 3rd repayment find: i the principal repaid ii the interest paid. b After 8 years the amount still owing is \$ Assuming the same conditions apply as in part a, for the 73rd repayment find: i the principal repaid ii the interest paid. 8 a Tony and Marietta take out a loan of \$ as part payment on their new house. The loan is to be repaid over 5 years at 3% p.a. (debited fortnightly) and with fortnightly instalments of \$ For the 3rd repayment find: i the principal repaid ii the interest paid. b If the principal is reduced to \$ after 0 years (use the same conditions as in part a), for the 6st repayment find: i the principal repaid ii the interest paid. c If the principal is reduced to \$ after 0 years (use the same conditions as in part a), for the 5st repayment find: i the principal repaid ii the interest paid.

12 7 Further Mathematics 9 multiple choice If the quarterly instalments for a \$5 000 loan, which is to be repaid over 4 years, are \$48.98 and interest is debited quarterly at.5% per quarter, the decrease in the principal in the first year would be (to the nearest dollar): A \$ 786 B \$34 C \$38 D \$774 E \$375 0 multiple choice John s \$3 000 loan has interest charged at 9% p.a., debited fortnightly, and is repaid over 8 years by fortnightly instalments of \$ For the 3rd repayment the amount of interest paid is: A \$3.98 B \$75.95 C \$76. D \$79.09 E \$55.30 multiple choice The term of a loan is 0 monthly instalments. Which of the following repayments will reduce the principal by the greatest amount? A 0th B 0th C 30th D th E 0th multiple choice Which of the following loan terms would have the greatest amount of interest debited? (Assume other conditions are the same.) A 0 years B years C 4 years D years E 0 years 3 Voula s loan of \$ starts with quarterly repayments of \$ and is due to run for 5 years at 6% p.a., interest debited quarterly. However, after year the interest rate rises to 7% p.a. and consequently the quarterly repayments rise to \$48.84 to maintain the 5 year term. a What amount is still owing after years? b What amount would have still been owing after years if the rate had remained at 6% p.a.? c What would be the difference in interest charged between the two scenarios? 4 Cynthia takes out a loan of \$ to set up an outdoors adventure business. She starts with quarterly repayments of \$300.4 and the loan is due to run for 0 years at 9% p.a., interest debited quarterly. However, after year the interest rate falls to 8% p.a. and consequently the quarterly repayments fall to \$43.88 to maintain the 0 year term. a What amount is still owing after years? b What amount would have still been owing after years if the rate had remained at 9% p.a.? c What would be the difference in interest charged between the two scenarios?

13 Chapter 5 Reducing balance loans 73 The annuities formula In the previous section step-by-step calculations were made to determine the amount still owing. The process was restrictive in that the previous balance was needed to calculate subsequent balances. A method is needed to enable calculation of the amount still owing at any point in time during the term of the loan. An annuities formula can be used to enable such calculations to be made. An annuity is a regular payment. When a consumer borrows money from a financial institution that person contracts to make regular payments or annuities in order to repay the sum borrowed over time. Let us now use, in general terms, the process adopted in the previous section to develop this annuities formula. Let P amount borrowed (principal) R growth factor for amount borrowed r (r interest rate period) n number of repayments Q amount of regular repayments made per period A n amount owing after n repayments Assuming interest is debited to the account before a repayment is credited, then: A 0 P A A 0 R Q A A R Q (PR Q)R Q PR Q A 3 A R Q A 4 A 3 R Q PR QR Q PR Q(R + ) (PR QR Q)R Q PR 3 QR QR Q PR 3 Q(R + R + ) (PR 3 QR QR Q)R Q PR 4 QR 3 QR QR Q PR 4 Q(R 3 + R + R + ) In general, A n PR n Q(R n R + R + ) The term in the bracket (R n R + R + ) is the sum of n terms of a geometric progression (GP) (refer to chapter 6: Arithmetic and geometric sequences). First term, a Common ratio, r R Now, the sum of n terms of a geometric progression is: ar ( S n n ) r Hence, in this case, + R + R R n ( R n ) R A n PR n QR ( n ) R

14 74 Further Mathematics So, in general, the amount owing in a loan account for n repayments is given by the annuities formula: Number of repayments made Amount still owing A PR n QR ( n ) R Repayment value Interest rate per period where R + r Amount borrowed WORKED Example 4 Growth factor A loan of \$ is taken out over 0 years at a rate of 6% p.a. (interest debited monthly) and is to be repaid with monthly instalments of \$358.. Find the amount still owing after 0 years. THINK WRITE State the loan amount, P, and regular repayment, Q. P Q 358. Find the number of payments, n, interest rate per month, r, and growth factor, R. n r r R Substitute into the annuities formula. A PR n QR ( n ) R (.005) ( ) Evaluate A. A \$ Write a statement. The amount still owing after 0 years will be \$ Note: If R is a recurring decimal, place the value in the calculator memory and bracket R if needed when evaluating A.

15 Chapter 5 Reducing balance loans 75 Note that, even though 0 years is the halfway point of the term of the loan, more than half of the original \$ is still owing. When we consider borrowing money we usually know how much is needed and we choose a term which requires a repayment that we can afford. To find the repayment value, Q, the annuities formula is used where A is zero, that is, the loan is fully repaid. Q is then isolated. A PR n QR ( n ) R When A 0, 0 PR n QR ( n ) R QR ( n ) PR n R PR Q n ( R ) R n WORKED Example 5 Rob wants to borrow \$800 for a new hi-fi system from a building society at 7.5% p.a., interest adjusted monthly. a What would be Rob s monthly repayment if the loan is fully repaid in -- years? b What would be the total interest charged? THINK WRITE a (a) Find P, n, r and R. (b) Store in your calculator memory the growth factor, R. b 3 4 a P 800 n r R PR Substitute into the annuities formula to Q n ( R ) find the regular monthly repayment, Q. R n 800(.0065) 8 (.0065 ) Evaluate Q. Q \$64.95 Write a statement. The monthly regular payment is \$64.93 over 8 months. Total interest Total repayments Amount borrowed Write a statement. b Total interest \$69.0 The total interest on a \$800 loan over 8 months is \$69.0.

16 76 Further Mathematics Alternative method using a graphics calculator The Texas Instrument graphics calculators TI 83 and TI 86 have a FINANCE function: TVM Solver. This allows quick analysis of reducing balance loans using the annuities formula. To use the TVM Solver, press nd [FINANCE] and select :TVM Solver. From this screen we define the following: where N the number of repayments I% the nominal interest rate (must enter as % per annum) PV the amount borrowed or the current amount owed (enter as a positive number as cash is flowing to you from the bank; a positive cashflow) PMT regular payment amount (enter as a negative number as the cash is FV WORKED Example flowing from you to the bank; a negative cashflow) the final amount owing (enter as 0 if the loan is fully repaid or enter the amount still owing as a negative number) P/Y number of payments per year, for example quarterly; P/Y 4. C/Y number of compounds per year, for example monthly adjusted C/Y (Note: In this chapter, P/Y and C/Y are to be of the same frequency.) PMT:END BEGIN Leave END highlighted as normally interest is charged at the end of the month. 6 Josh borrows \$ 000 for some home office equipment. He agrees to repay the loan over 4 years with monthly instalments at 7.8% p.a. (adjusted monthly). Find: a the instalment value b the principal repaid and interest paid during the: i 0th repayment ii 40th repayment. THINK WRITE a (a) Find P, n, r and R. a P 000 n r R (b) Store R in your calculator memory PR Substitute into the annuities formula to Q n ( R ) find the monthly repayment, Q. R n 000(.0065) 48 (.0065 ) Evaluate Q. Q \$9.83

17 Chapter 5 Reducing balance loans 77 THINK WRITE/DISPLAY 4 If using the TVM Solver on the TI 83, enter the appropriate values. Identify A, P, r and R. N 48 r ( I%) 7.8 P( PV) 00 Q( PMT) unknown A( FV) 0 P/Y C/Y Place cursor on PMT. Press ALPHA [SOLVE] to solve. Write a statement. The monthly repayment over a 4-year period is \$9.83. b i Find the amount owing after 9 months. b i (a) State P, n, R. P 000, n 9, R.0065 (b) Substitute into the annuities formula. A PR n QR ( n ) R 000(.0065) ( ) Evaluate A 9. A 9 \$ If using the TVM Solver on the TI 83, enter the appropriate values. Place cursor on FV. Press ALPHA [SOLVE] to solve. 3 Find the amount owing after 0 months. Substitute (change n 9 to n 0) and evaluate. A 0 000(.0065) (.0065 A 0 0 ) A 0 \$ Continued over page

18 78 Further Mathematics THINK WRITE/DISPLAY If using the TVM Solver on the TI 83, enter the appropriate values. Place cursor on FV. Press ALPHA [SOLVE] to solve Principal repaid A 9 A 0 Principal repaid \$6.67 Interest paid Total repayments Principal repaid Write a statement. Total interest \$ \$65.6 In the 0th repayment \$6.67 principal is repaid and \$65.6 interest is paid. bii Repeat steps 6 for A 39 and A 40. bii A (.0065) 39 A ( ) A 39 \$543.0 A 40 \$67.80 Principal repaid A 39 A \$75.30 Interest \$6.53 Write a statement. In the 40th repayment \$75.30 principal is repaid and \$6.53 interest is paid. remember remember. To calculate the amount in a loan account use the formula: A PR n QR ( n ) R. To calculate the repayment value use the formula: PR Q n ( R ) R n where P amount borrowed (principal) (\$) R growth factor for amount borrowed r (r interest rate per period) n number of repayments Q amount of regular repayments made per period (\$) A n amount owing after n repayments (\$)

19 Chapter 5 Reducing balance loans 79 5B The annuities formula WORKED Example 4 Use the annuities formula to find A, given: a P \$50 000, n, Q \$550, r b P \$50 000, n 00, Q \$550, r c P \$60 000, n, Q \$650, r d P \$60 000, n 00, Q \$650, r e P \$0 000, n 50, Q \$300, r 0.5 f P \$40 000, n, Q \$400, r 0.8 g P \$80 000, n 50, Q \$700, r 0.75 h P \$ 000, n 00, Q \$70, r A loan of \$ is taken out over 0 years at a rate of % p.a. (interest debited monthly) and is to be repaid with monthly instalments of \$75.7. Find the amount still owing after: a 5 years b 0 years c 5 years d 8 years. 3 Matthew takes out a reducing balance loan of \$ over 5 years at a rate of 0% p.a. (interest debited quarterly) and is to be repaid with quarterly instalments of \$ Find the amount still owing after: a 5 years b 0 years c 5 years d 0 years. 4 A loan of \$5 000 is taken out over 5 years at a rate of 3% p.a. (interest debited fortnightly) and is to be repaid with fortnightly instalments of \$ Find the amount still owing after: a 4 years b 8 years c years d 4 years. 5 Link borrows \$48 000, taken out over 0 years and to be repaid in monthly instalments. (Note: As the interest rate increases, the monthly repayment increases if the loan period is to remain the same.) Find the amount still owing after 5 years if interest is debited monthly at a rate of: a 6% p.a. and the repayment is \$53.90 b 9% p.a. and the repayment is \$ c % p.a. and the repayment is \$ d 5% p.a. and the repayment is \$ A loan of \$0 000 has interest charged monthly at a rate of 9% p.a. What will be the amount still owing after 3 years if the term of the loan is: a 4 years and monthly repayments of \$ are made? b 5 years and monthly repayments of \$45.7 are made? c 6 years and monthly repayments of \$360.5 are made? d 7 years and monthly repayments of \$3.78 are made? e 8 years and monthly repayments of \$93 are made? 7 Pablo s loan of has interest charged quarterly at a rate of 0% p.a. What will be the amount still owing after 5 years if the term of the loan is: a 6 years and quarterly repayments of \$ are made? b 7 years and quarterly repayments of \$50.64 are made? c 8 years and quarterly repayments of \$ are made? d 9 years and quarterly repayments of \$73.55 are made? e 0 years and quarterly repayments of \$95.09 are made? EXCEL Spreadsheet Mathcad Reducing balance loans

20 730 Further Mathematics 8 multiple choice Peter wants to borrow \$8000 for a second-hand car and his bank offers him a personal loan for that amount at an interest rate of 3% p.a., interest debited fortnightly, with fortnightly repayments of \$4. over 3 years. After years he wants to calculate how much he still owes by using the annuities formula. a Which of the following equations should he use? A A 8000(.005) 78 4.( ) B A 8000(.05) 5 4.(.05 5 ) C A 8000(.005) 5 4.( ) D A 8000(.05) 78 4.( ) b E A 8000( 0.005) 5 4.( ) The actual amount that Peter still owes after years is closest to: A \$500 B \$3000 C \$3500 D \$4000 E \$4500

21 Chapter 5 Reducing balance loans 73 WORKED Example 5 WORKED Example 5 9 multiple choice Gwendoline has borrowed \$4 000 for renovations to her house. The terms of this loan are monthly instalments of \$97.46 over 5 years with interest debited monthly at 0% p.a. of the outstanding balance. a The amount still owing after 3 years is given by: A B C D E A 4 000( ) 97.46( ) A 4 000( ) 97.46( ) A 4 000(.) 97.46(. 60 ) A 4 000( ) 97.46( ) A 4 000( ) 97.46( ) b The actual amount that Gwendoline still owes after 3 years is closest to: A \$5000 B \$5500 C \$6000 D \$6500 E \$ multiple choice Ben took out a loan for \$0 000 to buy a new car. The contract required that he repay the loan over 5 years with monthly instalments of \$4.0. After -- years Ben used the annuities formula to obtain the expression below to calculate the amount he still owed. A 0 000(.008) 4.0( ) The interest rate per annum charged by the bank for this reducing balance loan is: A.008% B 0.008% C 0.096% D 9.6% E.096% Use the annuities formula to find the repayment value, Q, given: a P \$5000, r, n b P \$3000, r, n 8 c P \$500, r 3, n 4 d P \$9000, r 0.5, n 30 e P \$4 000, r 0.8, n 4 f P \$0 000, r 0.6, n 40 g P \$95 000, r.5, n h P \$64 000, r 0.5, n 50. Sergio s reducing balance loan of \$ 000 has interest charged at 9% p.a., interest adjusted monthly. Find: i the monthly repayment ii the total interest charged if the loan is fully repaid in: a years b 3 years c 4 years d 4 -- years e 5 -- years. 3 Conchita s loan of \$ is charged interest at 7% p.a., interest adjusted monthly. Find: i the monthly repayment ii the total interest charged if the loan is fully repaid in: a 0 years b years c 5 years d 8 years e 0 years f 5 years.

22 73 Further Mathematics 4 In each of questions and 3 the only quantity which varied was the term of the loan. As the term of the loan increases, what happens to: a the repayment value? b the amount of interest paid? 5 Declan borrows \$3 000 and contracts to repay the loan over 0 years. Find: i the repayment value ii the total interest charged if the loan is repaid quarterly at: a 6% p.a., interest charged quarterly b 8% p.a., interest charged quarterly c 0% p.a., interest charged quarterly d 0.5% p.a., interest charged quarterly e % p.a., interest debited quarterly f.5% p.a., interest debited quarterly. 6 Felice borrows \$ and contracts to repay the loan over 5 years. Find: i the repayment value ii the total interest charged if the loan is repaid fortnightly, with interest adjusted fortnightly at: a 6% p.a. b 8% p.a. c 0% p.a. d 0.5% p.a. e % p.a. f.5% p.a. 7 A loan of \$ is to be repaid over 0 years. Find: i the repayment value ii the total interest charged if the loan is repaid: a weekly at 3% p.a., interest adjusted weekly b fortnightly at 3% p.a., interest adjusted fortnightly c monthly at 3% p.a., interest adjusted monthly d quarterly at 3% p.a., interest adjusted quarterly e weekly at 6.5% p.a., interest adjusted weekly f fortnightly at 6.5% p.a., interest adjusted fortnightly. 8 Based on your answers to question 7 a d when the frequency of repayments (and interest charged) decreases, how does this affect: a the repayment value? b the total interest paid? 9 multiple choice Which of the following would decrease the total amount of interest paid during the life of a loan? (There may be more than one answer.) A A fall in the interest rate B A decrease in the frequency of repayment (repay less often) C A greater amount borrowed D A decrease in the term of the loan E A rise in the interest rate 0 multiple choice Which of the equations below would enable the quarterly repayment value, Q, to be determined for a loan of \$6 000 to be repaid over 5 years at 7.8% p.a., interest debited quarterly? A ( 0.095) 0 Q( )

23 Chapter 5 Reducing balance loans 733 WORKED Example 6 B (.078) 5 Q( ) C (.095) 5 Q( ) D (.095) 0 Q( ) E (.078) 0 Q( ) Grace has borrowed \$8 000 to buy a car. She agrees to repay the reducing balance loan over 5 years with monthly instalments at 8.% p.a. (adjusted monthly). Find: a the instalment value b the principal repaid and the interest paid during: i the 0th repayment ii the 50th repayment. Tim has borrowed \$ to buy a house. He agrees to repay the reducing balance loan over 5 years with monthly instalments at 9.3% p.a. (adjusted monthly). Find: a the instalment value b the principal repaid and the interest paid during: i the 0th repayment ii the 50th repayment. 3 Gail has agreed to repay a \$ reducing balance loan with fortnightly instalments over 0 years at 9.75% p.a. (adjusted fortnightly). Find: a the instalment value b the principal repaid and the interest paid during: i the st repayment ii the 500th repayment. 4 Terry is repaying a \$5 000 loan over 5 years with quarterly instalments at 6.5% p.a. (adjusted quarterly). Currently, 5 -- years have passed since the loan was drawn down (money borrowed). How much does Terry still owe? 5 Stefanie borrowed \$8 000 exactly 3 -- years ago. The reducing balance loan was for a term of 5 years and was to be repaid in monthly instalments of 0.% p.a. (adjusted monthly). How much does Stefanie still owe? Questions 6 and 7 refer to the following information. The interest charged to a housing loan account during a financial year ( July 30 June) is a tax deduction against income if the house is rented to tenants. 6 Bruce borrowed \$ to finance the purchase of a rental property and he is repaying the loan over 0 years by quarterly instalments at 8.6% p.a. (adjusted quarterly). By July last year he had made 4 repayments. Find: a the amount Bruce owed after 4 repayments b c the amount he owes by 30 June this year the total interest that Bruce can claim as a tax deduction for this particular financial year. 7 Lyn is repaying a \$ housing loan over 5 years by monthly instalments at 9.9% p.a. (adjusted monthly). By July last year she had made 4 repayments. If Lyn rented the house to tenants, find: a the amount she owed after 4 repayments b c the amount she owes by 30 June this year the total interest that Lyn can claim as a tax deduction for this particular financial year. WorkSHEET5.

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