Note: if you cannot remember the recurrence relation formula for a compound interest system, revise notes for 2.2 Analysis of Compound Interest.

### Modelling using a Recurrence Relation

- A
**compound interest system with regular withdrawals**describes a system which has a**positive interest rate**, and**withdrawals (negative)**made at the end of each compounding period. - We can use the
**simple recursion formula**to model this system:

A_{n+1}=d+(1+I) A_{n}

Where d<0 and I>0.

Example

**A bank account with an annual interest rate of 12% is compounded quarterly. It initially contains $1000 and at the end of each compounding period, $10 is withdrawn from the account. We wish to find the value of the account after 5 quarters.** To do this, we use the recursive formula:

\begin{aligned} A_{n+1}=-10+\left(1+\frac{0.12}{4}\right) A_{n} &=-10+1.03 A_{n}, A_{0}=\$ 1000 \\ A_{1} &=-10+1.03 * 1000=\$ 1020 \\ A_{2} &=-10+1.03 * 1020=\$ 1040.6 \\ A_{3} &=-10+1.03 * 1040.6=\$ 1061.82 \\ A_{4} &=-10+1.03 * 1061.82=\$ 1083.67 \\ A_{5} &=-10+1.03 * 1083.67=\$ 1106.18 \end{aligned}

### Modelling Graphically

- Non-recursive formulas for this type of system are beyond the scope of Further Maths. Thus, to graph a compound interest system with regular withdrawals, we use the recursive formula to generate data points which are then plotted.
- These graphs are a type of time series plot (i.e. time is plotted along the x-axis).
- If the length of the compounding period is not known, the number of compounding periods can be used as the independent variable.
- These systems will either increase, decrease, or remain the same over time. This behaviour is dependant on the interest rate, initial value and the value of withdrawals.
- If A_{0}-(1+I) A_{0}=-I A_{0}=d, the value of the system will not change in value over time.

Note: remember d is negative for a reducing balance system.

- The slope of compound interest graphs with regular withdrawals increases over time (unless their value doesn’t change over time).

Example

The following graphs all model the system A_{n+1}=-30+1.05 A_{n} (i.e. a system increasing by 5% each compounding period with regular withdrawals of $30) with differing initial values, causing the behaviour of the system to change:

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