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6. Reducing Balance Loans and Annuities
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VCE 12 General 2023

6.03 Annuities

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Introduction

An annuity is a type of investment from which an individual will typically withdraw a regular amount at regular times, until the value of the annuity is \$0. Interest is generated on an annuity and added to the balance of the account.

Annuity with an amortisation table

An amortisation table for an annuity is very similar to an amortisation table for a loan.

Examples

Example 1

Consider the following table that shows an initial investment of \$8500 invested at an interest rate of4.5\% per annum, compounded monthly for four months. The payment made from this annuity is\$2140 per month.

PeriodStart balanceInterest earnedPaymentReduction in principalFinish balance
000008500
18500 21402108.126391.88
26391.8823.972140 4275.85
34275.8516.0321402123.97
4 8.07 2151.880
a

Calculate the interest for period 1 in the table.

Worked Solution
Create a strategy

Multiply the previous balance by the compound interest rate per payment.

Apply the idea
\displaystyle \text{Interest}\displaystyle =\displaystyle 8500 \times \dfrac{0.045}{12}Use the opening balance, \$8500
\displaystyle =\displaystyle \$31.88Evaluate using a calculator
b

Calculate the reduction in principal for period 2 in the table.

Worked Solution
Create a strategy

Subtract the ending balance from the previous balance.

Apply the idea
\displaystyle \text{Reduction in principal}\displaystyle =\displaystyle 6391.88-4275.84Subtract the balances
\displaystyle =\displaystyle \$2116.03Evaluate
Reflect and check

Notice that this is the same as the difference between the payment and the interest earned, since they are the only two factors affecting the balance, and one is an increase while the other is a decrease.

c

Calculate the ending balance for period 3 in the table.

Worked Solution
Create a strategy

Subtract the total reductions from the previous balance.

Apply the idea
\displaystyle \text{Balance}\displaystyle =\displaystyle 4275.85-2123.97Subtract the values
\displaystyle =\displaystyle \$2151.88Evaluate
Reflect and check

Notice that this is the same as the reduction in principal for the following period, since the account reaches \$0 at the end of period 4.

d

Calculate the repayment made in the final period in the table.

Worked Solution
Create a strategy

Add the previous balance to the interest generated that month.

Apply the idea

The amount left to pay at the end of the second last period (period 3) is \$2151.88.

\displaystyle \text{Final repayment}\displaystyle =\displaystyle 2151.88+8.07
\displaystyle =\displaystyle \$2159.95

So the final payment that comes out of the annuity will be \$2159.95.

Idea summary

An amortisation table displays how an annuity changes over time on a step by step basis.

The following steps are made with each payment:

  1. Interest needs to be calculated: compound interest rate per payment times the previous balance

  2. Payment: previous balance plus the interest generated that month.

  3. Reduction in principal: payment made minus the interest

  4. Balance of the loan: balance owing minus the reduction in principal

Annuity with a recurrence relation

An annuity with compound interest and regular withdrawals will results in the same type of situation as a reducing balance loan. As such, the same type of recurrence relation can be used.

Interest on an annuity can be modelled using the following recurrence relation: A_{n+1}=RA_n-d,\,A_0=P where A_{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 subtracted per time period, and P is the initial value of the investment or loan (the principal value).

Examples

Example 2

Tahlia received an inheritance of \$250\,000. and decides to invest the entire amount in an annuity earning 7.2\% per annum, compounded monthly. At the end of each month, after the interest has been paid into her account, Tahlia withdraws \$2000 to help pay for living expenses.

a

Write a recursive rule that gives the closing balance, A_{n+1}, at the end of month n+1. Write both parts of the rule (including for A_0) on the same line, separated by a comma.

Worked Solution
Create a strategy

Use the recursive rule A_{n+1}=R \times A_{n}-d, where A_{0} = P, and find R,\,d,\, and P.

Apply the idea

Tahlia monthly withdraws the amount of d=\$9000. The original investment was P=\$100\,000.

The balance is increased by 7.2\% =0.072 each year. To find the rate per month, we divide by 12 to get: R=1+\dfrac{0.072}{12}=1.006. So the recursive rule becomes:

\displaystyle A_{n+1}\displaystyle =\displaystyle R\times A_{n} - d,\,A_{0}=PWrite the recursive rule
\displaystyle =\displaystyle 1.006 \times A_{n} - 2000,\,A_{0}=250\,000Substitute R, \,d, \,P
\displaystyle A_{n+1}\displaystyle =\displaystyle 1.006A_{n} - 2000,\,A_{0}=250\,000Simplify
b

During which year and month will Tahlia's annuity end?

Worked Solution
Create a strategy

Enter the recursive rule and initial investment we found in part (a) into your calculator and find when the annuity first drops below \$0.

Apply the idea

Using your calculator you should get that the annuity drops below 0 at N=232. This means after 232 months. To find the number of years, we divide by 12.

\displaystyle 232\div 12\displaystyle \approx\displaystyle 19.33333\ldotsDivide the number of months by 12
\displaystyle =\displaystyle 19\dfrac{4}{12}Write as a mixed numeral

This tells us that 232 months is equal to 19 years and 4 months.

So Tahlia's annuity will last for 19 years and a further 4 months.

c

Graph the balance of the loan, A, against the number of months, n.

Worked Solution
Create a strategy

Enter the recursive rule and initial investment we found in part (a) into your calculator, then use the graph function on the calculator to graph the recurrence relation.

Apply the idea
A graphing calculator showing a decreasing concave down curve. Ask your teacher for more information.

We can see that the curve hits the horizontal axis shortly after month 220 and before month 240 which agrees with the finding that the annuity will last for 233 months . The curve hits the vertical axis at 250\,000, which agrees with the initial investment being \$250\,000. And the values on the curve are decreasing over time which agrees with the investment decreasing as Tahlia takes out money each month for living expenses.

Idea summary

Interest on an annuity can be modelled using the following recurrence relation:

\displaystyle A_{n+1}=R A_n -d,\,A_0=P
\bm{A_{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 subtracted per time period
\bm{P}
is the initial value of the investment or loan (the principal value)

Annuity with the financial application

As with a reducing balance loan, an annuity investment can be modelled using a CAS financial application. However, as an annuity is now an investment and not a loan, there is a subtle difference that occurs in the signs of some of the values.

Examples

Example 3

Dave opens a savings account at the start of year and will have invested \$8000 at the end of each year for 3 years. The account pays 6\% p.a. with interest compounded annually.

a

When the account is opened, what is the present value of his first deposit correct to the nearest cent?

Worked Solution
Create a strategy

Use the present value formula: PV=\dfrac{A}{(1+r)^n}

Apply the idea
\displaystyle PV\displaystyle =\displaystyle \dfrac{8000}{\left(1+\frac{6}{100}\right)^1}Substitute A=8000,\,r=\dfrac{6}{100},\,n=1
\displaystyle =\displaystyle \$7547.17Evaluate using your calculator
b

What is the present value of his second deposit correct to the nearest cent?

Worked Solution
Apply the idea
\displaystyle PV\displaystyle =\displaystyle \dfrac{8000}{\left(1+\frac{6}{100}\right)^2}Substitute A=8000,\,r=\dfrac{6}{100},\,n=2
\displaystyle =\displaystyle \$7119.97Evaluate using your calculator
c

What is the present value of his third deposit correct to the nearest cent?

Worked Solution
Apply the idea
\displaystyle PV\displaystyle =\displaystyle \dfrac{8000}{\left(1+\frac{6}{100}\right)^3}Substitute A=8000,\,r=\dfrac{6}{100},\,n=3
\displaystyle =\displaystyle \$7119.97Evaluate using your calculator
d

What is the present value of the annuity correct to the nearest cent?

Worked Solution
Create a strategy

Add all the present values found in the previous parts.

Apply the idea

The present values of the three deposits were \$7547.17,\,\$7119.97, and \$6716.95.

\displaystyle \text{Sum}\displaystyle =\displaystyle 7547.17+7119.97+6716.95Add the three present values
\displaystyle =\displaystyle \$21\,384.09Evaluate using your calculator

Example 4

Here are two ways of investing \$45\,000 for 25 years:

Lump-sum depositRateTime
\$45\,0007.5\% \text{ compounded annually}25 \text{ years}
Periodic deposit (yearly)RateTime
\$18007.5\% \text{ compounded annually}25 \text{ years}
a

After 25 years, how much will the lump-sum investment be worth? Write the answer correct to two decimal places.

Worked Solution
Create a strategy

For the lump sum investment, use the compound interest formula: A=P\left(1+r\right)^t

Apply the idea
\displaystyle A\displaystyle =\displaystyle 45\,000\left(1+\dfrac{7.5}{100}\right)^{25}Substitute P=45\,000,\,r=\dfrac{7.5}{100},\,t=25
\displaystyle =\displaystyle \$274\,425.28Evaluate using your calculator
b

After 25 years, how much will the annuity be worth? Write the answer correct to two decimal places.

Worked Solution
Create a strategy

For the periodic deposit, use the annuity formula: A=\dfrac{P\left((1+r)^t-1\right)}{r}

Apply the idea
\displaystyle A\displaystyle =\displaystyle \dfrac{1800\left((1+\frac{7.5}{100})^{25}-1\right)}{\frac{7.5}{100}}Substitute P=1800,\,r=\dfrac{7.5}{100},\,t=25
\displaystyle =\displaystyle \$122\,360.15Evaluate using your calculator
c

After 25 years, how much more will you have from the lump-sum investment than from the annuity? Write the answer correct to two decimal places.

Worked Solution
Create a strategy

Subtract the annuity found in part (b) from the annuity found in part (a).

Apply the idea
\displaystyle \text{Difference}\displaystyle =\displaystyle 274\,425.28-122\,360.15Subtract the annuities
\displaystyle =\displaystyle \$152\,065.13Evaluate using your calculator

Example 5

Iain invests \$190\,000 at a rate of 7\% per annum compounded monthly. At the end of each year he withdraws \$14\,300 from the investment after the interest is paid and the balance is reinvested in the account.

We will use the financial solver on our CAS calculator to determine how long the annuity lasts.

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

I\% is our annual interest rate which is 7\%.

PV is the present value of his investment which is initially \$190\,000. Since he is investing his money, he no longer has it, so we can think of it as he is losing it. So we should make it negative: -\$190\,000.

PMT is the amount Iain withdraws each time period which is \$14\,300. It represented by a positive value, as it functions as income for Iain.

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 1 since he makes a payment every year.

C/Y is the number of compounding periods each year which is also 1.

Value
NX
I\%7\%
PV-190\,000
PMT14\,300
FV0
P/Y1
C/Y1
b

At the end of which year will the annuity have run out?

Worked Solution
Create a strategy

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

Apply the idea

By pressing enter next to N in the financial application, we should get: N=40

The annuity lasts 40 years.

Reflect and check

If your calculator does not have dedicated financial solving, use the formula: N=\dfrac{\log\left(\frac{PMT}{PMT+PV \times I\%}\right)}{\log(1+I\%)} Don't forget that PV is negative. Finally, be careful with your rounding. If N=5.4, for instance, this means it happens during the 6th or at the end of the 6th time period.

Idea summary

When using a financial application for an annuity:

N is the number of instalment periods, or how long the annuity lasts.

I\% is our annual interest rate.

PV is the present value of our investment. This should be represented by a negative value, as the investment represents money has set aside.

PMT is the amount withdrawal each time period.

FV is the future value of our investment which should be 0.

P/Y is the number of payments or withdrawals made each year.

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

Outcomes

U3.AoS2.3

the concepts of financial mathematics including simple and compound interest, nominal and effective interest rates, the present and future value of an investment, loan or asset, amortisation of a reducing balance loan or annuity and amortisation tables

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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