# Annuities and Loans

> Regular deposits and regular repayments. Future value, present value, interest factor tables and reducing balance loans.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/annuities-and-loans/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The first car loan
Sienna borrows 15 000 dollars to buy her first car. The bank charges 9% per year, and she pays back a fixed amount every month for five years. Each payment covers that month's interest first, and only the rest reduces what she owes.

At the start, most of her payment is eaten by interest. Near the end, almost all of it pays off the loan. Banks use one formula to set that monthly payment. It comes from the geometric series you have already met.

**In short:** An annuity is a series of equal regular payments that earn compound interest, like saving into an account every month. A loan works the same way backwards: the amount borrowed is the present value of all its repayments.

## Starting point: Years 9 and 10

### Building a balance table

An **annuity** is a series of equal payments made at regular times, with compound interest. Savings plans, superannuation and loans are all annuities.

Suppose you deposit 1000 dollars at the **end** of each year into an account paying 5% per year.

| Year | Start | Interest (5%) | Deposit | End |
|---|---|---|---|---|
| 1 | 0 | 0 | 1000 | 1000.00 |
| 2 | 1000.00 | 50.00 | 1000 | 2050.00 |
| 3 | 2050.00 | 102.50 | 1000 | 3152.50 |

After 3 years the account holds 3152.50 dollars. You paid in 3000 dollars, so 152.50 dollars is interest.

### Calculator tip: Build the table with Ans
Each row is "previous balance × growth factor + payment" (or minus a repayment for a loan). Type the starting balance and press `=`, then type `Ans × 1.005 + 200` and press `=` once for each period. Write down each line as you go.

### A loan table

A loan works the other way. Interest is **added** each period, then the repayment is **taken away**.

Borrow 2000 dollars at 1% per month and repay 700 dollars a month:

| Month | Start | Interest (1%) | Repayment | End |
|---|---|---|---|---|
| 1 | 2000.00 | 20.00 | 700 | 1320.00 |
| 2 | 1320.00 | 13.20 | 700 | 633.20 |
| 3 | 633.20 | 6.33 | 639.53 | 0.00 |

The last payment is smaller: just enough to clear the debt.

### Key idea: Two kinds of annuity
- **Future value (FV):** how much regular deposits will be worth at the end.
- **Present value (PV):** the single amount today that is worth the same as the regular payments. A loan amount is a present value.

### Watch out: Interest first
In a loan, interest is calculated on the balance **before** the repayment is taken off.

**Mind map**

```text
Annuities
  Regular payments
    Deposits: savings
    Repayments: loans
  Tables
    Interest then payment
  Future value
    FV = M((1+r)ⁿ - 1)/r
  Present value
    PV = FV / (1+r)ⁿ
  Loans
    Reducing balance
    Repayment formula
  Interest factor tables
    Value of 1 dollar
```

### Year 11: Recurrence rules
A balance table can be written as a rule. If $A_n$ is the balance after $n$ periods, $r$ is the rate per period and $M$ is the payment:

- Savings: $A_{n+1} = A_n(1 + r) + M$
- Loan: $A_{n+1} = A_n(1 + r) - M$

Spreadsheets use this rule to build tables quickly, and it shows how a balance changes step by step.

### Year 12: Future and present value formulas
**Future value.** Deposits of $M$ at the end of each period for $n$ periods. The first deposit earns interest for $n - 1$ periods, the last for none. The total is a geometric series:

$$FV = M + M(1 + r) + \dots + M(1 + r)^{n-1} = M\left[\frac{(1 + r)^n - 1}{r}\right]$$

Check the table: $1000 \times \dfrac{1.05^3 - 1}{0.05} = 3152.50$.

**Present value.** Bring the future value back to today:

$$PV = \frac{FV}{(1 + r)^n} = M\left[\frac{1 - (1 + r)^{-n}}{r}\right]$$

**Interest factor tables.** Exams often give a table of future or present values of an annuity of **1 dollar**. Multiply the table value by $M$. If the future value table shows 6.3081 for $r = 2\%$ and $n = 6$, then six deposits of 500 dollars grow to $500 \times 6.3081 = 3154.05$ dollars.

**Sienna's loan.** The loan amount is the present value of her repayments, with $r = \dfrac{0.09}{12} = 0.0075$ and $n = 60$.

$$15\,000 = M\left[\frac{1 - 1.0075^{-60}}{0.0075}\right] \;\Rightarrow\; M \approx 311.38 \text{ dollars per month}$$

She pays $60 \times 311.38 = 18\,682.80$ dollars in total, so the interest is about 3683 dollars.

The same result written as one formula is

$$M = \frac{Pr(1 + r)^n}{(1 + r)^n - 1}$$

**The balance after $n$ repayments.** Following the loan rule $A_{n+1} = A_n(1 + r) - M$ from $A_0 = P$ gives
$$A_n = P(1 + r)^n - M\left[\frac{(1 + r)^n - 1}{r}\right]$$
The loan is repaid when $A_n = 0$. To find how long that takes, rearrange to get $(1 + r)^n$ alone and use logs. For 20 000 dollars at 1% per month with 400 dollar repayments: $20\,000(1.01)^n = 40\,000(1.01^n - 1)$ gives $1.01^n = 2$, so $n = \dfrac{\ln 2}{\ln 1.01} \approx 69.7$, that is 70 months.

**Repayments that never finish.** In the first month the interest is $Pr$. If the repayment $M$ is no more than $Pr$, the balance never falls. A 400 000 dollar loan at 0.6% a month has 2400 dollars of interest in month one, so repayments of 2400 dollars or less never repay it.

**Total interest** = total repaid minus the amount borrowed: $nM - P$.

**Deposits at the start of each period.** Each deposit earns one extra period of interest, so multiply the usual future value by $(1 + r)$.

**Withdrawals.** Drawing $W$ each month from savings works like a loan: $A_n = P(1 + r)^n - W\left[\dfrac{(1 + r)^n - 1}{r}\right]$. The money lasts forever only if $W \leq Pr$, the interest earned each month.

## Quick summary

- An annuity is a set of equal regular payments with compound interest.
- Savings tables: add interest, then add the deposit. Loan tables: add interest, then subtract the repayment.
- $FV = M\left[\dfrac{(1 + r)^n - 1}{r}\right]$ and $PV = \dfrac{FV}{(1 + r)^n}$.
- A loan amount is the present value of its repayments.
- With a table of values for 1 dollar, multiply by the payment.

## Quiz

Try these questions from the chapter quiz.

1. An annuity is:
   - A. a single deposit
   - B. a set of equal regular payments with compound interest
   - C. a type of simple interest
   - D. a tax
   - Answer: B. a set of equal regular payments with compound interest. Savings plans and loans are annuities.

2. You deposit 500 dollars at the end of each year at 10%. What is the balance after 2 years?
   - A. 1000 dollars
   - B. 1050 dollars
   - C. 1100 dollars
   - D. 1155 dollars
   - Answer: B. 1050 dollars. Year 1: 500. Year 2: $500 \times 1.1 + 500 = 1050$.

3. A loan of 1000 dollars has 2% interest per month. After one month and a 300 dollar repayment, the balance is:
   - A. 700 dollars
   - B. 720 dollars
   - C. 714 dollars
   - D. 680 dollars
   - Answer: B. 720 dollars. $1000 \times 1.02 - 300 = 720$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/annuities-and-loans/#quiz