Annuities and Loan Repayments
Understand the present and future value of annuities, calculate loan repayments, and build amortisation tables for real-world financial problems.
What is an Annuity?
An annuity is a series of equal payments made at regular intervals. Examples include loan repayments, superannuation contributions, and rental payments. There are two key types:
Future Value Annuity
Regular deposits are made into an account. We calculate how much the total will grow to (e.g., saving for retirement).
FV = M x (1+r)n - 1r
Present Value Annuity
Regular payments are made from a lump sum. We calculate the initial amount needed, or the repayment amount (e.g., home loan).
PV = M x 1 - (1+r)-nr
Calculating Loan Repayments
To find the regular repayment M needed to pay off a loan of present value PV, rearrange the present value annuity formula:
M = PV x r1 - (1+r)-n
where PV = loan amount, r = interest rate per period, n = total number of payments
For example, a home loan of $400,000 at 6% p.a. compounded monthly over 30 years uses r = 0.005 and n = 360 monthly payments.
Amortisation Tables
An amortisation table tracks each payment, showing how much goes toward interest and how much reduces the principal. Early payments are mostly interest; later payments are mostly principal.
Example: $10,000 loan at 8% p.a. over 4 years (annual repayments)
| Year | Payment | Interest | Principal | Balance |
|---|---|---|---|---|
| 0 | - | - | - | $10,000.00 |
| 1 | $3,019.21 | $800.00 | $2,219.21 | $7,780.79 |
| 2 | $3,019.21 | $622.46 | $2,396.75 | $5,384.04 |
| 3 | $3,019.21 | $430.72 | $2,588.49 | $2,795.55 |
| 4 | $3,019.21 | $223.64 | $2,795.57 | $0.00 |
Notice how the interest portion decreases and the principal portion increases with each payment.
Key Vocabulary
Annuity
A series of equal payments made at regular intervals over a fixed period of time.
Present Value (PV)
The current worth of a future series of payments, discounted at the given interest rate.
Amortisation
The process of gradually paying off a debt through regular repayments of principal and interest.
Principal
The original amount borrowed or the outstanding balance on which interest is calculated.
Worked Examples
Find the monthly repayment on a $300,000 home loan at 6% p.a. compounded monthly over 25 years.
Step 1: Identify values: PV = $300,000, r = 0.06/12 = 0.005 per month, n = 25 x 12 = 300 months.
Step 2: M = PV x r / [1 - (1+r)-n] = 300000 x 0.005 / [1 - (1.005)-300].
Step 3: (1.005)-300 = 0.22396..., so denominator = 1 - 0.22396 = 0.77604.
Step 4: M = 1500 / 0.77604 = $1,932.90.
Answer: The monthly repayment is $1,932.90.
$500 is deposited at the end of each quarter into a superannuation fund earning 8% p.a. compounded quarterly for 20 years. Find the future value.
Step 1: M = $500, r = 0.08/4 = 0.02 per quarter, n = 20 x 4 = 80 quarters.
Step 2: FV = 500 x [(1.02)80 - 1] / 0.02.
Step 3: (1.02)80 = 4.87544..., so FV = 500 x [4.87544 - 1] / 0.02 = 500 x 193.772.
Answer: The future value is $96,886.00.
A $5,000 loan at 12% p.a. compounded monthly is repaid in 2 years. Find the interest paid in the first month.
Step 1: Monthly rate r = 0.12/12 = 0.01.
Step 2: Interest in month 1 = Balance x r = $5,000 x 0.01 = $50.00.
Step 3: The first month's repayment includes $50 interest. The rest reduces the principal.
Answer: The interest paid in the first month is $50.00.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
A loan of $20,000 is repaid in equal annual instalments over 5 years at 10% p.a. What is the annual repayment (to the nearest dollar)?
Question 2
In an amortisation table, what happens to the interest portion of each payment over time?
Question 3
The present value annuity formula is PV = M x [1 - (1+r)-n] / r. What does n represent?
Question 4
A car loan of $25,000 at 9% p.a. compounded monthly has monthly repayments of $518.96 over 5 years. What is the total amount paid?
Question 5
$1,000 is deposited at the end of each year for 8 years at 5% p.a. What is the future value (to the nearest dollar)?
Key Concepts Summary
- ●An annuity is a series of equal payments at regular intervals.
- ●The future value formula calculates how much regular deposits will grow to: FV = M x [(1+r)n - 1] / r.
- ●The present value formula calculates the current worth of future payments: PV = M x [1 - (1+r)-n] / r.
- ●Amortisation tables show the breakdown of each payment into interest and principal components.
- ●Always ensure r and n are in matching units (e.g., both monthly or both annual).