Chapter 15B and 15C - Annuities formula Finding the amount owing at any time during the term of the loan. A = PR n Q Rn 1 or TVM function on the Graphics Calculator Finding the repayment amount, Q Q = PRn R 1 R n 1 or TVM function on the Graphics Calculator Finding the number of repayments, n A = PR n Q Rn 1 and Trial and Error. or TVM function on the Graphics Calculator
Exercise 15B p.729 - Reducing balance loans (Find repayment value, Q) 13e Megan s loan of $85 000 is charged interest at 7% p.a., interest adjusted monthly. Find the monthly repayment if the loan is fully repaid in: e) 20 years 15c John borrows $32 000 and contracts to repay the loan over 10 years. Find the repayment value if the loan is repaid quarterly at: c) 10% p.a., interest charged quarterly Exercise 15C p.739 - Reducing balance loans (Find number of repayments, n) 4a Jim has a reducing balance loan of $3500 that he is using for a holiday and has agreed to repay it by monthly instalments of $206.35 at a rate of 7.6% p.a. (interest debited monthly). Find the number of repayments needed to repay in full and this time in years. 6a Ben has contracted to repay a reducing balance loan of $9200 by fortnightly instalments of $156.76. Interest is charged at 8.2% p.a. and adjusted fortnightly. Find how long it will take Ben to repay the loan in full. 9 Some time ago, Elizabeth took out a loan of $25 000. Interest has been charged at 10.5% p.a. (adjusted monthly) and monthly repayments of $537.35 have serviced the loan. If the amount still owing is $11 586.64: a how long ago was the loan taken out? b what was the term of the loan? 18 Grace has borrowed $48 000 to set up a travel agency and is repaying the reducing balance loan by monthly instalments of $401.17 at 5.85% p.a. (adjusted monthly). At present she still owes $36 381.40. a How much longer will it take to reduce the amount Grace owes to $15 203.19? b When will she have repaid the loan in full? c What was the term of the loan?
Annuities formula using a standard calculator (finding repayment amount, Q) Q = PRn R 1 R n 1 Rob wants to borrow $2800 for a new hi-fi system from a building society at 7.5% p.a., interest adjusted monthly. What would be Rob s monthly repayment if the loan is fully repaid in 1.5 years? P = 2800, Q =?, n = 1.5 12 = 18 R = (1 + r 7.5/12 ) = (1 + ) = 1.00625 100 100 Q = PRn R 1 R n 1 = 2800 x 1.0062518 1.00625 1 1.00625 18 1 = 164.95 Using a standard calculator option: 2800 x 1.00625^18 x (1.00625-1) / (1.00625^18-1)
Annuities formula using Casio fx-9860g AU Graphics Calculator (finding repayment amount, Q) Rob wants to borrow $2800 for a new hi-fi system from a building society at 7.5% p.a., interest adjusted monthly. What would be Rob s monthly repayment if the loan is fully repaid in 1.5 years? [$164.95] Calculator values: Variable Use n The Number of repayments I% Annual (yearly) interest rate (% pa) PV Principal [original amount borrowed], but entered as NEGATIVE number PMT The repayment amount (regular repayments) FV The amount owing on the loan after the n repayments P/Y The number of repayments each year C/Y Number of times interest is debited each year Set calculator to Compound Interest mode Enter the values: n: Enter 1.5x12 PMT: Enter 0 (unknown) Press F4 to calculate the Payment
1 - Annuities formula using Casio fx-9860g AU Graphics Calculator (finding number of repayments) A reducing balance loan of $60 000 is to be repaid with monthly instalments of $483.36 at an interest rate of 7.5% p.a. (debited monthly). Find the number of monthly repayments. Set calculator to Compound Interest mode Enter the values: n: Enter 0 (unknown) PV: Enter 60000 PMT: Enter -483.36 Press F1 to calculate the number of repayments n is 240 months It is 240/12 years = 20 years
1 - Annuities formula using a standard calculator (finding number of repayments) A = PR n Q Rn 1 R 1 A reducing balance loan of $60 000 is to be repaid with monthly instalments of $483.36 at an interest rate of 7.5% p.a. (debited monthly). Find the number of monthly repayments. P = 60000, Q = 483.36, n =?, R = (1 + r 7.5/12 ) = (1 + ) = 1.00625 100 100 1. Take a guess, say 200: A 200 = PR n Q Rn 1 = 60000 x 1.00625 200 483.36 1.00625200 1 1.00625 1 = 17058.38 2. Not less than 0, take another guess, say 220: A 220 = PR n Q Rn 1 = 9058.86 Using a standard calculator option: 60000 x (1.00625^200) (483.36 x (1.00625^200-1)) / (1.00625-1) 3. Not less than 0, take another guess, say 240: A 240 = PR n Q Rn 1 = -2.26 n is 240 months It is 240/12 years = 20 years
2 - Annuities formula using Casio fx-9860g AU Graphics Calculator (finding number of repayments) Some time ago Petra borrowed $14 000 to buy a car. Interest on this reducing balance loan has been charged at 9.2% p.a. (adjusted monthly) and she has been paying $446.50 each month to service the loan. Currently she still owes $9753.92. How long ago did Petra borrow the money? Set calculator to Compound Interest mode Enter the values: n: Enter 0 (unknown) PV: Enter 14000 PMT: Enter -446.50 FV: Enter -9753.92 Press F1 to calculate the number of repayments n is 12 months Borrow the money 1 year ago.
2 - Annuities formula using a standard calculator (finding number of repayments) A = PR n Q Rn 1 R 1 Some time ago Petra borrowed $14 000 to buy a car. Interest on this reducing balance loan has been charged at 9.2% p.a. (adjusted monthly) and she has been paying $446.50 each month to service the loan. Currently she still owes $9753.92. How long ago did Petra borrow the money? Do trail and error to get just over or equal 9753.92 P = 14000, Q = 446.50, n =?, R = (1 + r 9.2/12 ) = (1 + ) = 1.0076667 100 100 1. Take a guess, say 20: A 20 = PR n Q Rn 1 = 14000 x 1.0076667 20 446.50 1.007666720 1 1.0076667 1 = 6699.13 2. Not close to 9753.92, take another guess, say 11: A 11 = PR n Q Rn 1 = 10122.81 3. Close, take another guess, say 12: A 12 = PR n Q Rn 1 = 9753.92 n is 12 months, so she borrowed the money 1 year ago. Using a standard calculator option: 14000 x (1.0076667^20) (446.50 x (1.0076667^20-1)) / (1.0076667-1)
End of Lesson