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6. Reducing Balance Loans and Annuities
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VCE 12 General 2023

6.05 Annuity investment

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Lesson

Introduction

An annuity investment is an investment or savings plan through which compound interest is earned and additional payments or deposits are regularly made. This is similar to an annuity, except that the payments are made into the account in each time period, rather than being withdrawn from the account.

Annuity investment with a recurrence relation

Interest on an annuity investment can be modelled using the following recurrence relation: V_{n+1}=RV_n+d,\,V_0=P where V_{n+1} is the value of the investment after n+1 time periods, R equals 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate, d is the amount added per time period, and P is the initial value of the investment (the principal value).

Examples

Example 1

A deposit of \$3000 is made on June 1, 2007 into an investment account and a deposit of \$200 is made each year on May 31. The balance at the end of each 12-month period for this investment, where interest is compounded annually, is given by A_{n+1}=1.06A_n+200, and A_0=3000.

a

State the annual interest rate.

Worked Solution
Create a strategy

Identify the compounding factor, R, and subtract 1 from it.

Apply the idea

In the recurrence relation A_{n+1}=1.06A_n+200, 1.06 represents the compounding factor and is equal to 1+r, where r is the interest rate per year. Therefore, r=R-1.

\displaystyle \text{Annual interest rate}\displaystyle =\displaystyle 1.06-1Substitute R=1.06
\displaystyle =\displaystyle 0.06Evaluate
b

Determine the balance on June 1, 2008.

Worked Solution
Create a strategy

Since 2008 is 1 year after 2007, we want to calculate the value of A_1.

Apply the idea

Using the recurrence relation, we know that A_1=1.06A_0+200, and we also know the initial value A_0 was given to be 3000.

\displaystyle A_1\displaystyle =\displaystyle 1.06 \times 3000 + 200Substitute A_0=3000
\displaystyle =\displaystyle \$3380Evaluate using your calculator
c

Determine the value of the investment on June 1, 2014. Round your answer to the nearest cent.

Worked Solution
Create a strategy

Since 2014 is 7 years after 2007, we want to calculate the value of A_7.

Apply the idea

Using the sequence facility on your calculator, enter the general rule for the sequence along with the initial term and value of n that you want to find.

By entering the rule, initial term and n=7 you should get:A_{7}=\$6189.66

Idea summary

Interest on a perpetuity can be modelled using the following recurrence relation:

\displaystyle V_{n+1}=R V_n + d,\,V_0=P
\bm{V_{n+1}}
is the annuity value after n+1 compounding periods
\bm{R}
is 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate
\bm{d}
is the amount added per time period
\bm{P}
is the initial value of the investment (the principal value

Annuity investment with an amortisation table

The amortisation table for an annuity investment looks similar to that of an annuity, but this time there is an increase in principal rather than a decrease, as the payments are being made into the account. As a result, the balance increases over time.

Examples

Example 2

Consider the following table.

PeriodPaymentInterest earnedIncrease in principalBalance
000010\,450.00
1250.00209.00459.0010\,909.00
2250.00218.18468.1811\,377.18
3250.00227.54477.5411\,854.72
4250.00237.09487.0912\,341.81
a

What is the quarterly interest rate for this investment?

Worked Solution
Create a strategy

To solve for the interest rate, we divide the interest earned by the previous balance.

Apply the idea

Using the initial balance and the interest earned in the first time period, we have that:

\displaystyle r\displaystyle =\displaystyle \dfrac{209}{10\,450}Substitute the values
\displaystyle =\displaystyle 0.02Evaluate
\displaystyle =\displaystyle 2\%Write as percentage

So the quarterly interest rate is 2\% (and thus the annual interest rate is 8\%).

b

Write a recurrence relation for B_{n+1} that gives the balance of the investment at the end of each quarter.

Worked Solution
Create a strategy

To use the recursive rule V_{n+1}=R \times V_{n}+ d, \, \, V_1=P, we need to find R, \, d, and P.

Apply the idea

The balance is increased by r=2\%=0.02 each quarter, and we substitute it in R=1+r=1+0.02. Each quarter the a payment of d=250 is added. The original investment was P=\$10\,450.

\displaystyle B_{n+1}\displaystyle =\displaystyle R\times B_n+d, \, B_0=PWrite the recursive ule
\displaystyle =\displaystyle (1+0.02) \times B_n+250, \, B_0=10\,450Substitute R, \,d, \,P
\displaystyle B_{n+1}\displaystyle =\displaystyle 1.02\times B_n+250, \, B_0=10\,450Simplify
Idea summary

The amortisation table for an annuity investment looks similar to that of an annuity, but this time there is an increase in principal rather than a decrease, as the payments are being made into the account. As a result, the balance increases over time.

Annuity investment with the finance application

A graphics or CAS calculator is a powerful tool for financial problems when used correctly. The questions above can also be answered using the financial application.

Examples

Example 3

Katrina has \$150\,000 to invest. She wishes to withdraw \$1400 each month after the interest is paid. Assume that the interest is compounded monthly.

We will use the financial solver on our CAS calculator to determine what annual interest rate Katrina needs if she wants her investment to last 30 years.

a

Fill in the value for each of the following. Type an X next to the variable we wish to solve for.

Value
N⬚
I\%⬚\%
PV⬚
PMT⬚
FV⬚
P/Y⬚
C/Y⬚
Worked Solution
Create a strategy

Use the given information to find each value, keeping in mind that the values of PV and PMT should be opposite in sign, and the FV value should be zero.

Apply the idea

N is the total number of payments. Since the interest is compounded monthly, we can multiply 30 by 12, which is 360.

I\% is our annual interest rate which is the value we are looking for.

PV is the present value of her investment which is initially 150\,000. Since she is investing her money, she no longer has it, so we can think of it as she is losing it. So we should make it negative: -150\,000.

PMT is the amount Katrina withdraws each time period which is \$1400. It represented by a positive value, as it functions as income for Katrina.

FV is the future value of our investment, and we're looking to solve for when the money runs out so it should be zero.

P/Y is the number of contributions per year which is 12 since she makes a monthly payment.

C/Y is the number of compounding periods each year which is also 12.

Value
N360
I\%X
PV-150\,000
PMT1400
FV0
P/Y12
C/Y12
b

Determine the amount of the annual interest rate. Give your answer to the nearest hundredth of a percentage.

Worked Solution
Create a strategy

Use the financial application on your calculator using the values from part (a).

Apply the idea

After pressing the enter on the empty field I\%, we should get 10.75\%.\text{Interest rate }=10.75\% \text{ per annum}

Idea summary

When using the financial application:

  • Always write down the value of each variable - this is your working out.

  • If you are investing money then PV is negative and FV is positive. Hint: think of investing as 'giving' your money to the bank so from your point of view the money is negative.

  • If you are borrowing money PV is positive and FV is negative. Hint: think of borrowing as 'receiving' money from the bank so from your point of view the money is positive.

  • Payments (PMT) made to the bank for either investments or loans are negative, again we can think of this as 'giving' your money to the bank.

  • N is the total number of payments.

Spreadsheets to model an investment

Spreadsheets can also include payment details and are a useful tool for solving financial problems as the progression of the investment can be clearly seen as well as the effect of changing interest rates and payments.

Exploration

Let's explore this interactive compound interest spreadsheet. When we explore different options with a financial problem we call it "what if analysis".

We can change the amount invested (the blue cell) to any value we'd like to invest.

We can change the annual interest rate (the green cell) to any value.

We can change the number of compounding periods (the pink cell) to quarterly (4), monthly (12), weekly (52) or perhaps daily (365).

Investigate:

  • What happens as we increase the number of compounding periods?

  • What happens as we increase the annual interest rate?

  • How has the value in cell \text{C10} been calculated?

  • How has the value in \text{D12} been calculated?

Loading interactive...

As we increase the number of compounding periods the interest for each period decreases.

As we increase the annual interest rate the interest for each period increases.

The value in cell \text{C10} has been calculated by multiplying the balance in \text{B10} by the interest rate in \text{B2} divided by 100 and divided by the number of compounding periods in \text{B3} using the formula: =(\text{B10}*\text{B2}/100)/\text{B3}

The value in cell \text{D12} has been calculated by adding the balance in \text{B12} to the interest in \text{C12} using the formula: =\text{B12}+\text{C12}

Examples

Example 4

The spreadsheet below shows the first year of an investment with regular deposits:

ABCDE
1\text{Year}\text{Beginning Balance}\text{Interest}\text{Deposit}\text{End Balance}
2160006605007160
3
4
5
a

Calculate the annual interest rate for this investment.

Worked Solution
Create a strategy

Divide the interest by the beginning balance.

Apply the idea
\displaystyle \text{Annual interest rate}\displaystyle =\displaystyle \dfrac{660}{6000}Divide 660 by 6000
\displaystyle =\displaystyle 0.11Evaluate
\displaystyle =\displaystyle 11\%Write as a percentage
b

Write a formula for cell \text{B3}.

Worked Solution
Create a strategy

The balance at the beginning of a year is equal to the balance at the end of the previous year.

Apply the idea

The balance at the end of the previous year is in cell \text{E2}. So the formula should be:=\text{E2}

c

Write a formula for cell \text{C6} in terms of \text{B6}.

Worked Solution
Create a strategy

Multiply the balance by the interest rate.

Apply the idea

In part (a) we found that the interest rate was 0.11 and the balance is in \text{B6}. So the formula for \text{C6} is:=0.11 * \text{B6}

d

Write a formula for cell \text{E5} in terms of one or more other cells.

Worked Solution
Create a strategy

Add the balance at the beginning of the year, the interest, and the deposit.

Apply the idea

For \text{E5} the balance is in cell \text{B5} the interest is in cell \text{C5} and the deposit is in cell \text{D5}. So the formula would be:=\text{B5}+\text{C5}+\text{D5}

e

Using the spreadsheet facility on your calculator, reproduce this spreadsheet and determine the end balance for the 4th year.

Worked Solution
Create a strategy

Use the formulas from the previous three parts to create a spreadsheet that looks similar to the one above and fill it down until you reach year 4.

Apply the idea

We can also format the cells from \text{B2} to \text{E5} as Currency, so that they display with dollar signs and two decimal places as shown.

ABCDE
1\text{Year}\text{Beginning Balance}\text{Interest}\text{Deposit}\text{End Balance}
21\$ \, 6000.00\$ \, 660.00\$ \, 500.00\$ \, 7160.00
32\$ \, 7160.00\$ \, 787.60\$ \, 500.00\$ \, 8447.60
43\$ \, 8447.60\$ \, 929.24\$ \, 500.00\$ \, 9876.84
54\$ \, 9876.84\$ \, 1086.45\$ \, 500.00\$ \, 11463.29

The end balance for the 4th year is \$11\,463.29.

f

Calculate the total interest earned over the 4 years.

Worked Solution
Create a strategy

Subtract the initial value of the investment and the total deposits from the end balance.

Apply the idea

The end balance is made up of the initial value of the investment of \$6000, the 4 deposits of \$500, and the total interest. So to find the total interest we need to subtract the initial value and deposits.

\displaystyle \text{Total interest}\displaystyle =\displaystyle 11\,463.29 - 6000 - 4 \times 500Subtract the initial value and deposits
\displaystyle =\displaystyle \$3463.29Evaluate
Idea summary

Spreadsheets can also include payment details and are a useful tool for solving financial problems.

Every spreadsheet formula starts with an equals (=) sign.

For multiplication in formulas we use *, for division we use (/).

The \$ signs in the cell references makes the reference absolute. That means the cell name, e.g. \text{\$A\$2}, will not change as the formula is copied down the column.

Outcomes

U3.AoS2.8

use a table to investigate and analyse on a step–by-step basis the amortisation of a reducing balance loan or an annuity, and interpret amortisation tables

U3.AoS2.4

the use of first-order linear recurrence relations to model compound interest investments and loans, and the flat rate, unit cost and reducing balance methods for depreciating assets, reducing balance loans, annuities, perpetuities and annuity investments

U3.AoS2.9

use technology with financial mathematics capabilities, to solve practical problems associated with compound interest investments and loans, reducing balance loans, annuities and perpetuities, and annuity investments

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