Annuity as a Recurrence Relation Revision Notes for HSC SSCE Mathematics Standard

Annuity as a Recurrence Relation

What is an annuity?

An annuity is a financial arrangement involving regular, repeated contributions of money over time. Common examples include making regular deposits into a superannuation fund or making monthly loan repayments.

When you invest through an annuity, the future value represents the total worth of your investment at a specified end date. This future value combines two components:

The total money you contributed
The compound interest earned on those contributions

Understanding future value helps you see how your regular savings grow over time through the power of compound interest.

Calculating the future value of an annuity

Let's explore how an annuity works through a concrete example.

lightbulbExample

Worked Example: Four-Year Investment Plan

Scenario: You invest AUD 1000 at the end of each year for $4$ years. The investment earns $10\%$ per annum compound interest.

Year-by-year breakdown

End of first year:

Interest earned: AUD 0 (payment made at year's end, so no time to earn interest)
Calculation: $FV = PV(1 + r)^n + 1000$
$= 0 \times (1 + 0.10)^1 + 1000$
$= 1000$

End of second year:

Interest earned: $(1000) \times 0.10 = 100$
Calculation: $FV = PV(1 + r)^n + 1000$
$= 1000 \times (1 + 0.10)^1 + 1000$
$= 2100$

End of third year:

Interest earned: $(2100) \times 0.10 = 210$
Calculation: $FV = PV(1 + r)^n + 1000$
$= 2100 \times (1 + 0.10)^1 + 1000$
$= 3310$

End of fourth year:

Interest earned: $(3310) \times 0.10 = 331$
Calculation: $FV = PV(1 + r)^n + 1000$
$= 3310 \times (1 + 0.10)^1 + 1000$
$= 4641$

Analysis of results

After $4$ years, the future value of this annuity is AUD 4641. Let's break this down:

Total money contributed: $4 \times 1000 = 4000$
Compound interest earned: $4641 - 4000 = 641$

chatImportant

Key Observation About Compound Interest

Notice how the interest increases each year (AUD 0, 100, 210, 331). This demonstrates compounding - you earn interest not just on your deposits, but also on previously earned interest. The interest doesn't simply increase by AUD 100 each year; it accelerates as the investment balance grows.

Understanding recurrence relations

The calculation method demonstrated above is called a recurrence relation. This mathematical approach builds on each previous calculation to determine the next value.

infoNote

In our annuity example, this means:

The future value at the end of year $1$ (1000) becomes the present value for year $2$
The future value at the end of year $2$ (2100) becomes the present value for year $3$
This pattern continues throughout the investment period

Using subscript notation

We can represent this recursive pattern using subscript notation, where $V$ represents the value of the investment and the subscript indicates the time period (in this case, years).

This can be expressed as:

End of first year: $V_1 = V_0(1 + 0.10) + 1000 = 1000$
End of second year: $V_2 = V_1(1 + 0.10) + 1000 = 2100$
End of third year: $V_3 = V_2(1 + 0.10) + 1000 = 3310$
End of fourth year: $V_4 = V_3(1 + 0.10) + 1000 = 4641$

General formula

This recurrence relation can be generalised using a variable $n$ to represent any time period:

V_{n+1} = V_n(1 + 0.10) + 1000

This formula states: "The value after $(n+1)$ periods equals the value after $n$ periods, multiplied by the growth factor, plus the regular payment."

Recurrence relation formulas

A recurrence relation uses each previous result to generate the next value in ongoing calculations.

chatImportant

Essential Formulas for Annuities

For investments (depositing money):

V_{n+1} = V_n(1 + r) + D

For loans (repaying borrowed money):

V_{n+1} = V_n(1 + r) - D

Variable definitions

$V_{n+1}$ = Value after $(n+1)$ payments
$V_n$ = Value after $n$ payments
$r$ = Interest rate per period (decimal)
$D$ = Payment per period

Memory aid: Investment formulas use +D, loan formulas use -D.

Converting interest rates

When the interest rate is given as a percentage per annum but the compounding period is monthly or quarterly, you must convert the rate:

Monthly: divide by $12$
Quarterly: divide by $4$

lightbulbExample

Converting Annual to Monthly Rate

For $12\%$ per annum compounded monthly:

r = \frac{0.12}{12} = 0.01

Worked example 1: Modelling a loan using a recurrence relation

lightbulbExample

Problem: Alyssa borrows AUD 1000 at $15\%$ p.a., compounding monthly. She makes $4$ monthly payments of AUD 257.85.

a) Recurrence relation

V_{n+1} = V_n(1 + r) - D

r = \frac{0.15}{12} = 0.0125

D = 257.85

V_{n+1} = V_n(1.0125) - 257.85

b) Recursive results

1st month: $1000(1.0125) - 257.85 = 754.65$
2nd month: $754.65(1.0125) - 257.85 \approx 506.23$
3rd month: $506.23(1.0125) - 257.85 \approx 254.71$
4th month: $254.71(1.0125) - 257.85 \approx 0.04$

c) Balance after two payments: AUD 506.23

d) Final adjustment:

\text{Last payment} = 257.85 + 0.04 = 257.89

Worked example 2: Modelling an investment using a recurrence relation

lightbulbExample

Problem: Spencer invests AUD 1000 at $7.2\%$ p.a., compounded quarterly, with AUD 100 added each quarter.

V_{n+1} = V_n(1 + r) + D

r = \frac{0.072}{4} = 0.018

V_1 = 1000(1.018) + 100

Results:

1st: $1118$
2nd: $\approx 1238.12$
3rd: $\approx 1360.41$
4th: $\approx 1484.90$

$n+1$	$V_n$	$D$	$V_{n+1}$
$1$	$V_0 = 1000$	100	$1118$
$2$	$1118$	100	$1238.12$
$3$	$1238.12$	100	$1360.41$
$4$	$1360.41$	100	$1484.90$

I = 1484.90 - 1000 - (4 \times 100)

= 84.90

Therefore, the interest earned is AUD 84.90.

Remember!

bookmarkSummary

An annuity = regular payments over time
Recurrence relations build values step-by-step
Always convert interest rates correctly
Compound interest accelerates growth
Use recursive calculator methods efficiently

Annuity as a Recurrence Relation (HSC SSCE Mathematics Standard): Revision Notes