Often an investor will make a regular payment to contribute to the growth of the investment. Tables and recursive rules are useful for more complex finance problems, such as a compound interest investment with regular deposits, or a reducing balance loan with regular payments. Investments involving compound interest are often displayed in a table of values so that the growth in the value of the investment can be clearly seen.

For a principal investment/loan, P, at the compound interest rate of r per period and payment d per period, the sequence of the value of the investment over time forms a first order linear recurrence .

The sequence which generates the value, V_n of the investment/loan at the end of each installment period is:

Recursive sequence:

\displaystyle V_n

\displaystyle =

\displaystyle V_{n-1} \times (1 + r)+ d,

where V_0=P

The sequence which generates the value, V_n of the investment/loan at the beginning of each instalment period is:

Recursive sequence:

\displaystyle V_n

\displaystyle =

\displaystyle V_{n-1} \times (1 + r)+ d,

where V_1=P

Examples

Example 1

Jenny is saving for a European holiday which will cost \$15\,000. She puts the \$5000 she has already saved in a savings account which offers interest compounded quarterly. She also makes a quarterly contribution of \$350. The progression of the investment is shown in the table below.

\text{Quarter}	\text{Account balance at} \\ \text{start of quarter } (\$)	\text{Interest } (\$)	\text{Payment } (\$)	\text{Account balance at} \\ \text{end of quarter } (\$)
1	5000	46.88	350	5396.88
2	5396.88	50.60	350	5797.48
3	5797.48	A	350	B

Find the interest rate per quarter for this account.

Worked Solution

Create a strategy

The interest rate per quarter can be found by dividing the interest by the account balance at the start of the quarter.

Apply the idea

Let's consider the first quarter with interest \$46.88 and starting balance of \$5000.

\displaystyle \text{Quarterly interest rate}	\displaystyle =	\displaystyle \dfrac{46.88}{5000}	Substitute the values
	\displaystyle =	\displaystyle 0.0093\,76	Evaluate
	\displaystyle =	\displaystyle 0.9376\%	Convert to a percentage

Calculate the value of A in dollars.

Worked Solution

Create a strategy

Multiply the interest rate per quarter found in part (a) to the account balance at the start of quarter 3.

Apply the idea

To find the interest for any quarter we need to find 0.9376\% of the starting balance, which for quarter 3 was \$5797.48.

\displaystyle A	\displaystyle =	\displaystyle 0.9376\% \text{ of }5797.48	Find the percentage of the balance
	\displaystyle =	\displaystyle 0.009\,376 \times 5797.48	Convert rate to a decimal
	\displaystyle =	\displaystyle \$54.36	Evaluate

Calculate the value of B in dollars.

Worked Solution

Create a strategy

Add the account balance at the start of the third quarter, the interest, and the payment.

Apply the idea

The account balance at the end of each quarter is the sum of the account balance at the start of the quarter, the interest, and the payment.

\displaystyle B	\displaystyle =	\displaystyle 5797.48 + 54.36 + 350	Add the balance, interest, and payment
	\displaystyle =	\displaystyle \$6201.84	Evaluate

Write a recursive rule, T_{n+1}, in terms of T_n that gives the balance in the account at the beginning of the (n+1)th quarter.

T_{n+1}= ⬚ \times T_n + ⬚, \, \, T_1=⬚

Worked Solution

Create a strategy

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

Apply the idea

The balance is increased by r=0.9376\%=0.009\,376 each quarter. Each quarter the a payment of d=350 is added. The original investment was P=\$5000.

\displaystyle T_{n+1}	\displaystyle =	\displaystyle (1+r)\times T_n+d, \, T_1=P	Write the recursive ule
	\displaystyle =	\displaystyle (1+0.009\,376) \times T_n+350, \, T_1=5000	Substitute r, \,d, \,P
\displaystyle T_{n+1}	\displaystyle =	\displaystyle 1.009\,376 \times T_n+350, \, T_1=5000	Simplify

Using your calculator, how many whole quarters will it take to until she has enough money to go on the holiday?

Worked Solution

Create a strategy

Enter the recursive rule and initial value into your calculator and look for the first value that is equal to or greater than \$15\,000.

Apply the idea

Using your calculator you should get that the 24th value is equal to or greater than \$15\,000.\text{Number of quarters} = 24

If she changes the payment to \$400 per quarter, how much sooner, in whole quarters, will she be able to go on holidays?

Worked Solution

Create a strategy

Change the value of d to \$400. Then enter the recursive rule and initial value into your calculator and look for the first value that is equal to or greater than \$15\,000.

Apply the idea

The new recursive rule with d=400 is:

\displaystyle T_{n+1}

\displaystyle =

\displaystyle 1.009\,376 \times T_n+400, \, T_1=5000

Using your calculator you should get that the 22nd value is equal to or greater than \$15\,000. So the number of quarters sooner which she will be able to go on holidays is:

\displaystyle \text{Number of quarters}	\displaystyle =	\displaystyle 24-22	Subtract the quarters
	\displaystyle =	\displaystyle 2	Evaluate

Idea summary

The first order linear recursive sequence which generates the value, V_n of the investment/loan at the end of each installment period is:

\displaystyle V_n

\displaystyle =

\displaystyle V_{n-1} \times (1 + r)+ d,

where V_0=P

The first order linear recursive sequence which generates the value, V_n of the investment/loan at the beginning of each instalment period is:

\displaystyle V_n

\displaystyle =

\displaystyle V_{n-1} \times (1 + r)+ d,

where V_1=P

Financial application for investments

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

Examples

Example 2

Valerie is saving for a European holiday which will cost \$20\,000. She puts the \$8000 she has already saved in a savings account which offers 3.75\% p.a compounded quarterly. She also makes a quarterly contribution of \$400. Valerie is interested in finding how long it will take her to save up for her holiday.

If N is the number of payments, complete the table of values showing the variables required to use a financial solver to solve this problem.

Variable	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 PV and Pmt should be opposite in sign to FV.

Apply the idea

I(\%) is the annual interest rate which is 3.75\%.

PV is the present value of what is in her savings account which is initially \$8000. Since she is giving this money to the bank, she no longer has it, so we can think of it as she is losing it. So we should make it negative: -\$8000.

PMT is the regular contribution of \$400 that she is giving to the bank. So since she no longer has it, we should make this negative: -\$400.

FV is the future value of the savings account which she wants to be \$20\,000. Valerie will take this from the bank and keep it, so she is gaining it. So it should stay positive.

P/Y is the number of contributions per year which is 4 since she makes a payment every quarter.

C/Y is the number of compounding periods each year which is also 4.

Variable	Value
N	-
I(\%)	3.75\%
PV	-8000
Pmt	-400
FV	20\,000
P/Y	4
C/Y	4

Determine the number of whole quarters it will take for Valerie to save the \$20\,000 required for the holiday.

Worked Solution

Create a strategy

Use a CAS calculator, select menu, then finance solver, and enter all the value from part (a). Go to the empty field on N and press enter.

Apply the idea

After pressing the enter on the empty field N, we have 22.7793 which we must round up to the nearest whole number:\text{Number of quarters}=23

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 3

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

	A	B	C	D	E
1	\text{Year}	\text{Beginning Balance}	\text{Interest}	\text{Deposit}	\text{End Balance}
2	1	6000	660	500	7160
3
4
5

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 0.11	Evaluate
	\displaystyle =	\displaystyle 11\%	Write as a percentage

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}

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}

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}

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.

	A	B	C	D	E
1	\text{Year}	\text{Beginning Balance}	\text{Interest}	\text{Deposit}	\text{End Balance}
2	1	\$ \, 6000.00	\$ \, 660.00	\$ \, 500.00	\$ \, 7160.00
3	2	\$ \, 7160.00	\$ \, 787.60	\$ \, 500.00	\$ \, 8447.60
4	3	\$ \, 8447.60	\$ \, 929.24	\$ \, 500.00	\$ \, 9876.84
5	4	\$ \, 9876.84	\$ \, 1086.45	\$ \, 500.00	\$ \, 11463.29

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

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 500	Subtract the initial value and deposits
	\displaystyle =	\displaystyle \$3463.29	Evaluate

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

ACMGM099

use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity

ACMGM100

with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount

6.04 Analysing investments with periodic payments

Ideas

Tables and sequences for investments

Examples

Example 1

Financial application for investments

Examples

Example 2

Spreadsheets to model an investment

Exploration

Examples

Example 3

Outcomes

ACMGM099

ACMGM100

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