Modelling Compound Interest Investments With Additions to the Principal Revision Notes for VCE SSCE General Mathematics

Modelling Compound Interest Investments With Additions to the Principal

Introduction

When you make an investment that earns compound interest and also add regular payments to it, you're dealing with a situation that combines two types of growth:

Geometric growth: Your investment grows by a percentage each period (compound interest)
Linear growth: Your investment grows by a fixed amount each period (regular additions)

This type of investment, where you regularly add money to a compound interest account, is known as an annuity investment.

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Recurrence relations are powerful mathematical tools that allow us to model complex financial situations. They show us exactly how the value changes from one period to the next, making it easier to predict future values and understand growth patterns.

We use recurrence relations to model these situations mathematically. A recurrence relation shows us how the value changes from one period to the next.

The general recurrence relation

For situations involving both geometric and linear growth or decay, we use a recurrence relation of this form:

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General Recurrence Relation Form

$V_0 = \text{starting value}, \quad V_{n+1} = R \times V_n \pm D$

Where:

$V_n$ is the value after $n$ periods
$R$ is the growth multiplier (for geometric growth)
$D$ is the fixed amount added or subtracted each period (for linear growth)
The $\pm$ symbol means we can either add or subtract $D$

This general form helps us represent financial situations where we have both types of growth happening together. The multiplication by R creates the geometric (percentage) growth, while the addition or subtraction of D creates the linear (fixed amount) growth.

Generating sequences from recurrence relations

To find the terms in a sequence, we start with the initial value and repeatedly apply the rule. Each new term depends on the previous term, following the pattern defined by the recurrence relation.

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Worked Example: Generating a sequence

Question: Write down the first five terms of the sequence generated by the recurrence relation $V_0 = 3, \, V_{n+1} = 4V_n - 1$

Solution:

Starting value: $3$

Apply the rule (multiply by 4, then subtract 1) to generate more terms:

$V_1 = 3 \times 4 - 1 = 11$
$V_2 = 11 \times 4 - 1 = 43$
$V_3 = 43 \times 4 - 1 = 171$
$V_4 = 171 \times 4 - 1 = 683$

The first five terms are: 3, 11, 43, 171, 683

Modelling annuity investments

For compound interest investments where you make regular additions, we use this specific form of the recurrence relation:

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Annuity Investment Recurrence Relation

$V_0 = \text{the principal}, \quad V_{n+1} = RV_n + D$

Where:

$V_n$ is the value of the investment after $n$ additional payments
$V_0$ is the principal (initial investment)
$D$ is the additional payment made each period
$R$ is the growth multiplier, calculated using: $R = 1 + \frac{r}{100 \times p}$
$r$ is the annual interest rate (as a percentage)
$p$ is the number of compounding periods per year

Understanding the growth multiplier

The growth multiplier $R$ tells us what we multiply the current balance by to account for interest.

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The growth multiplier always starts at 1 (representing 100% of the current value), then adds the interest fraction. This is why the formula is $R = 1 + \frac{r}{100 \times p}$ rather than just $\frac{r}{100 \times p}$ .

For example:

If the interest rate is 4% per year compounding annually, then $R = 1 + \frac{4}{100 \times 1} = 1.04$
This means each year the balance is multiplied by 1.04 (grows by 4%)

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Worked Example: Annual compounding

Question: Fred has saved $5000 and invests this in a compound interest account paying 4% per annum, compounding yearly. He also adds an extra $1000 each year. Model this investment using a recurrence relation where $V_n$ is the value of the investment after $n$ years.

Solution:

Step 1: Identify the principal and additional payment

$V_0 = 5000$ (initial investment)
$D = 1000$ (amount added each year)

Step 2: Calculate the growth multiplier

Using $R = 1 + \frac{r}{100 \times p}$ where $r = 4$ and $p = 1$ (annual compounding):

$R = 1 + \frac{4}{100 \times 1} = 1.04$

Step 3: Write the recurrence relation

$V_0 = 5000, \quad V_{n+1} = 1.04V_n + 1000$

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Worked Example: Monthly compounding

Question: Nor invests $1200 and plans to add an extra $50 each month. The account pays interest at a rate of 3% per annum, compounding monthly. Model this investment using a recurrence relation where $V_n$ is the value of the investment after $n$ months.

Solution:

Step 1: Identify the principal and additional payment

$V_0 = 1200$ (initial investment)
$D = 50$ (amount added each month)

Step 2: Calculate the growth multiplier

Using $R = 1 + \frac{r}{100 \times p}$ where $r = 3$ and $p = 12$ (monthly compounding):

$R = 1 + \frac{3}{100 \times 12} = 1.0025$

Step 3: Write the recurrence relation

$V_0 = 1200, \quad V_{n+1} = 1.0025V_n + 50$

Compounding periods

When interest compounds at intervals other than a year, we need to know how many compounding periods there are per year. Use these standard values:

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Standard Compounding Periods

Compounding frequency	Number of periods per year ( $p$ )
Monthly	12
Quarterly	4
Fortnightly	26
Weekly	52
Daily	365

These values are used in financial mathematics even though some are approximations (e.g., there are slightly more than 52 weeks in a year).

Using recurrence relations to analyse investments

Once we have a recurrence relation, we can use it to:

Calculate the value after a specific number of periods
Create graphs showing how the investment grows over time

These tools help us understand the long-term behavior of our investments and make informed financial decisions.

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Worked Example: Analysing an investment

Question: Albert has an investment that can be modelled by the recurrence relation:

$V_0 = 400, \quad V_{n+1} = 1.005V_n + 30$

where $V_n$ is the value of the investment after $n$ months.

a) State the value of the initial investment.

b) Determine the value of the investment after Albert has made three extra payments. Round your answer to the nearest cent.

c) What will be the value of his investment after 6 months? Round your answer to the nearest cent.

d) Plot the points for the value of the investment after 0, 1, 2 and 3 months on a graph.

Solution:

a) The initial investment was $400 (this is $V_0$ ).

b) To find the value after three payments, we calculate $V_1$ , $V_2$ , and $V_3$ :

$V_0 = `$ 400$`
$V_1 = 1.005 \times 400 + 30 = `$ 432$`
$V_2 = 1.005 \times 432 + 30 = `$ 464.16$`
$V_3 = 1.005 \times 464.16 + 30 = `$ 496.48$`

The value of Albert's investment after three extra payments is $496.48.

c) We can continue the calculations manually or use a calculator:

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Following the pattern through to $V_6$ :

$V_4 = 1.005 \times 496.48 + 30 = `$ 528.96$`
$V_5 = 1.005 \times 528.96 + 30 = `$ 561.61$`
$V_6 = 1.005 \times 561.61 + 30 = `$ 594.42$`

The value of Albert's investment after 6 months is $594.42.

d) Plotting the points $(n, V_n)$ for $n = 0, 1, 2, 3$ :

The graph shows how the investment value increases over time. Notice that the growth accelerates because you're earning interest on both the original amount and all previous additions.

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Calculator Tip: Efficient calculation of successive terms

You can use your CAS calculator to do these calculations efficiently:

Type 400 and press ENTER (or EXE)
Type $\times 1.005 + 30$ and press ENTER (or EXE)
Keep pressing ENTER (or EXE) to generate successive terms

This method is much faster than calculating each term manually!

Benefits of regular additions

Making additional payments on a regular basis causes your investment to grow faster. This is particularly beneficial when you make additions early, because compound interest is earned on those additional payments for a longer time. This makes annuity investments an excellent strategy for long-term goals like saving for retirement.

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The timing of your additional payments matters! Early contributions benefit from compound interest for longer periods, resulting in significantly more growth compared to later contributions of the same amount.

Finding the annual interest rate from a recurrence relation

Sometimes you're given a recurrence relation and need to work backwards to find the annual interest rate. We use the formula $R = 1 + \frac{r}{100 \times p}$ and solve for $r$ .

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Worked Example: Determining annual interest rates

Question: Determine the annual interest rates for each of the following investments.

a) An investment given by the recurrence relation: $A_0 = 400, \quad A_{n+1} = 1.005A_n + 30$ where $A_n$ is the value of the investment after $n$ months.

b) An investment given by the recurrence relation: $W_0 = 2000, \quad W_{n+1} = 1.012W_n + 500$ where $W_n$ is the value of the investment after $n$ quarters.

Solution:

a) We know that interest compounds monthly, so $p = 12$ .

From the recurrence relation, we can see that $R = 1.005$ .

We need to solve: $1.005 = 1 + \frac{r}{100 \times 12}$

Rearranging:

\begin{aligned} 1.005 - 1 &= \frac{r}{1200} \\ 0.005 &= \frac{r}{1200} \\ r &= 0.005 \times 1200 = 6 \end{aligned}

The annual interest rate is 6%.

b) We know that interest compounds quarterly, so $p = 4$ .

From the recurrence relation, we can see that $R = 1.012$ .

We need to solve: $1.012 = 1 + \frac{r}{100 \times 4}$

Rearranging:

\begin{aligned} 1.012 - 1 &= \frac{r}{400} \\ 0.012 &= \frac{r}{400} \\ r &= 0.012 \times 400 = 4.8 \end{aligned}

The annual interest rate is 4.8%.

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Key Points to Remember:

An annuity investment is a compound interest investment where you make regular additions to the principal.
The recurrence relation for annuity investments is: $V_0 = \text{principal}, \, V_{n+1} = RV_n + D$ where $D$ is the regular addition and $R$ is the growth multiplier.
The growth multiplier is calculated using: $R = 1 + \frac{r}{100 \times p}$ where $r$ is the annual interest rate and $p$ is the number of compounding periods per year.
Common compounding periods: monthly ( $p = 12$ ), quarterly ( $p = 4$ ), fortnightly ( $p = 26$ ), weekly ( $p = 52$ ), daily ( $p = 365$ ).
To find successive values, apply the recurrence rule repeatedly, or use your calculator's ENTER/EXE function for efficiency.
Making additional payments early is particularly beneficial because compound interest is earned on those additions for longer.

Modelling Compound Interest Investments With Additions to the Principal (VCE SSCE General Mathematics): Revision Notes