Amortisation and annuities (using GDC): notes and practice questions
- Amortisation: Process of repaying a loan over a fixed period.
- Annuity: Fixed sum of money paid at specified intervals from an initial investment.
- Use GDC Finance / TVM Solver for problems.
- GDC Sign Convention: Money in is positive, money out is negative.
- TVM Solver Variables:
- N: Total number of payment periods ().
- I%: Nominal annual interest rate (entered as a percentage).
- PV: Present Value. Positive for amortisation (loan received), negative for annuity (initial investment).
- PMT: Payment amount per period. Negative for amortisation (paying back), positive for annuity (receiving).
- FV: Future Value. Typically for fully repaid loans or exhausted annuities.
- P/Y: Number of payments per year.
- C/Y: Number of compounding periods per year.
- PMT@: Payment timing. Assume END for amortisation, START for annuities, unless specified.
- To find an unknown variable, leave it blank in the GDC and solve.
- For amortisation, total amount paid: .
How it is examined
Pure technology. The question gives a loan or an annuity in words and asks for a payment, a term, a rate or a total interest figure, and the student is expected to drive the GDC's finance solver. Sign conventions are where marks go: a payment out and a balance owed have opposite signs on most calculators, and a student who gets that wrong produces a plausible but wrong number. Because payments are always at the end of the period, a question that hinges on beginning-of-period payments is out of syllabus.
Handle amortization and annuities using technology.
Knowledge of the annuity formula will enhance understanding but will not be examined. The route through this subtopic is the finance solver, not algebra.
Linking questions
- Other contexts: evaluating the real value of an investment when interest and inflation both act on it. Credit card debt, student loans, retirement planning.
- Links to other subjects: exchange rates (economics), loans (business management).
- Aim 8: ethical perceptions of borrowing and lending money. Short-term loans at high interest rates, and how knowing the mathematics protects people from extortion.
- International-mindedness: do all societies view investment and interest the same way?
Practice questions
7 questions · 4 medium · 3 hardQuestion 1
MediumPaper 1 · calculator6 marksA small business, "The Daily Grind", takes out a loan of $25,000 to purchase a new industrial espresso machine. The loan is to be repaid over 5 years, with an annual interest rate of 6.2% compounded monthly.
(a) Calculate the monthly payment required to repay the loan over this period.
(b) The business decides they can only afford to pay $450 per month.
Calculate how many full months it will take to repay the loan with this new monthly payment.
Use the financial application on your GDC. You know the total loan amount (the Present Value, PV), the time period, and the interest rate. You need to find the recurring payment (PMT).
The setup is similar to part (a), but this time the monthly payment (PMT) is known, and you need to find the total number of payments (N). Remember that the number of months must be a whole number, so consider how to round your answer.
Question 2
HardPaper 2 · calculator17 marksP2: In this question, give all answers to two decimal places.
Maria plans to expand her bakery business and needs a loan of Euros (€) to purchase new equipment. Her bank offers two financing options.
Option A: A four-year loan with an annual interest rate of compounded quarterly. No deposit is required.
If Maria chooses option A, find:
the repayment she makes each quarter.
the total amount she pays back.
the total interest she pays on the loan.
Option B: A four-year loan with an annual interest rate of compounded monthly. Terms of the loan require a deposit and monthly repayments of €.
If Maria chooses option B, find:
the total amount Maria pays to the bank.
the annual interest rate, .
State which option Maria should choose. Justify your reasoning.
Use a financial calculator (TVM solver) or the loan payment formula. Remember to adjust the interest rate and number of payments for quarterly compounding.
Multiply the quarterly repayment by the total number of quarters over the loan term.
Subtract the original loan amount from the total amount paid back.
Calculate the deposit first, then add it to the total of all monthly repayments.
First, determine the actual loan amount after the deposit. Then, use the TVM solver to find the monthly interest rate, and convert it to an annual rate.
Compare the total amount paid to the bank for each option.
Question 3
MediumPaper 1 · calculator9 marksIn this question, give all answers correct to 2 decimal places.
A small business owner, Mr. Chen, takes out a loan of 250000 Australian dollars (AUD) from a bank to expand his cafe. The loan is for 25 years and the annual interest rate for the loan is 4.2%, compounded monthly. He will pay the loan in fixed monthly instalments at the end of each month.
Find the amount Mr. Chen will pay the bank each month.
Find the amount Mr. Chen will still owe the bank at the end of the first 8 years.
Using your answers to parts (a) and (b), calculate how much interest Mr. Chen will have paid in total during the first 8 years.
Use a financial app on your GDC. Remember to correctly input the present value (PV), future value (FV), interest rate (I%), number of periods (N), and payment per year (P/Y) and compounding per year (C/Y).
To find the remaining balance, you need to calculate the future value of the loan after a certain number of payments have been made. Use the monthly payment found in part (a).
The total interest paid is the difference between the total amount paid in instalments and the amount by which the principal loan has been reduced.
Question 4
HardPaper 2 · calculator18 marks(a) Swift Deliveries, a small logistics company, plans to purchase a new fleet of electric delivery vans. The total cost of the fleet is USD. They secure a loan from a bank, but must make an initial down payment equal to of the total cost. The bank offers a -year loan for the remaining balance, with a nominal interest rate per annum, compounded monthly. Swift Deliveries will make fixed payments at the end of each month.
(a.i) Calculate the initial loan amount Swift Deliveries will need to take from the bank.
(a.ii) Calculate Swift Deliveries' monthly payment for this loan, to two decimal places.
(b) Using your answer from part (a)(ii), calculate the total amount Swift Deliveries will pay over the life of the loan, to the nearest dollar. Do not include the initial down payment.
(c) Swift Deliveries decides to repay the loan faster by increasing their monthly payments to USD. Find the total number of monthly payments they will need to make to pay off the loan.
(d) This strategy will result in Swift Deliveries' final payment being less than USD. Determine the amount of Swift Deliveries' final payment, to two decimal places.
(e) Hence, determine the total amount Swift Deliveries will save, to the nearest dollar, by making the higher monthly payments.
First, calculate the amount of the down payment. Then, subtract this from the total cost to find the loan amount.
Use the financial application (TVM solver) on your GDC. Identify the values for N, I%, PV, FV, P/Y, and C/Y.
Multiply the monthly payment by the total number of months in the loan term.
Adjust the PMT value in your GDC's financial app and solve for N. Remember to round up to the nearest whole number of payments.
Use the number of full payments from part (c) to calculate the future value (remaining balance) after those payments. Then, add one month's interest to this remaining balance to find the final payment.
Calculate the total amount paid with the increased payments (including the final payment). Then, find the difference between this and the total amount paid with the original payments (from part b).
Question 5
MediumPaper 1 · calculator4 marks(a) A university student takes out a loan of USD to cover tuition fees. The loan is to be repaid over years with an annual interest rate of , compounded monthly. Payments are made at the end of each month.
Calculate the monthly payment the student must make.
Use a financial solver on your GDC. Identify the correct values for N, I%, PV, FV, P/Y, and C/Y before solving for PMT. Remember that for a loan, the present value (PV) is positive and the future value (FV) is zero.
Question 6
HardPaper 2 · calculator18 marksIn this question, give all answers to two decimal places.
A small business owner, Ms. Chen, decides to purchase new equipment for her bakery, priced at $22000. She cannot afford the full amount upfront.
The equipment supplier offers two options to finance a loan.
Finance option A:
A 5-year loan at a nominal annual interest rate of 10% compounded quarterly.
No deposit required and repayments are made each quarter.
Find the repayment made each quarter.
Find the total amount paid for the equipment.
Find the interest paid on the loan.
Finance option B:
A 5-year loan at a nominal annual interest rate of % compounded monthly. Terms of the loan require a 10% deposit and monthly repayments of $350.
Find the amount to be borrowed for this option.
Find the annual interest rate, .
State which option Ms. Chen should choose. Justify your answer.
Ms. Chen chooses option B. The equipment supplier invests the money Ms. Chen pays as soon as they receive it.
If they invest it in an account paying 0.5% interest per month and inflation is 0.15% per month, calculate the real amount of money the equipment supplier has received by the end of the 5-year period.
Use the financial application (TVM solver) on your GDC. Remember to set the correct values for N, I%, PV, FV, P/Y, and C/Y. The Present Value (PV) should represent the initial loan amount.
Multiply the quarterly repayment by the total number of quarters over the loan term.
The interest paid is the difference between the total amount paid and the original price of the equipment.
First, calculate the deposit amount, then subtract it from the original price to find the loan amount.
Use the financial application (TVM solver) on your GDC. You know N, PV, PMT, FV, P/Y, and C/Y, and you need to solve for I%.
Compare the total cost of each option, including any deposits. Alternatively, consider the initial cash outlay.
First, calculate the real interest rate per month by subtracting the inflation rate from the investment interest rate. Then, calculate the future value of the deposit and the future value of the annuity (monthly payments) separately, using the real interest rate.
Question 7
MediumPaper 1 · calculator4 marks(a) Maya takes out a car loan of . The loan is for four years at an annual interest rate of , compounded monthly.
Calculate Maya's monthly payments.
Use the financial solver on your GDC. Ensure you correctly identify the present value (PV), future value (FV), number of payments (N), and the interest rate (I%) per annum. Remember to set payments per year (P/Y) and compounding periods per year (C/Y) to 12 for monthly payments.
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Every Amortisation and annuities (using GDC) question, marked for you
Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.
Where marks are lost
- Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three. Common wherever a GDC's full decimal display gets copied straight down.
- Rounding an intermediate value and then using it in a later part. Costs a mark every time, and AI's multi-part modelling questions give it more chances to happen than AA's shorter, more self-contained ones.
- Writing the answer and nothing else, where the mark scheme has an explicit M1 rather than an implied one. A bare answer cannot score full marks there.
