Up until this chapter, our deposits or withdrawals have always been at the same frequency as the compounding of the interest for our investment (or loan).
In real life this is not always the case and we often make payments more frequently than the interest is compounded. We'll examine modelling some simple cases here.
Modelling different frequencies using a recurrence relation
example
Tayla takes out a personal loan to buy a car. She is charged interest which is compounded monthly and the conditions of her loan allow her to make quarterly repayments.
The balance of her loan can be defined recursively as:
$L_{n+1}=\left(1.006\right)^3L_n-400;L_0=18000$Ln+1=(1.006)3Ln−400;L0=18000
(a) How much did she borrow?
Think: The amount borrowed will be the initial amount.
Do: $L_0=18000$L0=18000, so $\$18000$$18000
(b) What annual interest rate is she charged?
Think: The monthly interest rate is shown as part of $1.006$1.006. Notice this is to the power of $3$3 since it is compounded for $3$3 months within each quarter.
Do: So the monthly interest rate is $0.6%$0.6% and thus the annual interest rate is $7.2%$7.2%
(c) How much does she owe after the first year?
Think: We'll enter the recurrence relation in our calculator.
Do:
So the balance at the end of the first year (after four quarters) is $\$17695.65$$17695.65
(d) Use to sequence facility on your calculator to determine during which quarter she will have paid off half her loan.
Think: We'll scroll down on our calculator and look for when the balance first falls below $\$9000$$9000
Do:
Modelling different frequencies using the CAS financial application
example
Alan opens a savings account after selling his car and makes weekly deposits of $\$50$$50.
He earns $4.2%$4.2% per annum, compounded monthly.
How much did Alan need to deposit initially if he'd like to save $\$30000$$30000 after $5$5 years?
Think: Let's review what each of the variables in our calculator represent.
N is the number of installment periods
I is our annual interest rate
PV is the present or initial value of our investment and will be negative because from our point of view we have given our money to the investment. This is what we want to find.
PMT is the payment Alan will make each month
FV is the future value of our investment
P/Y is the number of payments or withdrawals made each year
C/Y is the number of compounding periods each year
Do:
:
Solving we get:
So Alan needed to initially deposit $\$12603.36$$12603.36
Worked Examples
Question 1
Jenny opens a high-interest savings account where interest of $6.48%$6.48% per annum is compounded monthly. Her initial deposit is $\$12000$$12000 and she makes monthly deposits of $\$300$$300.
Complete the table below, rounding each answer to the nearest cent, and using the rounded answer to calculate the amounts for the following month.
Month
Balance at beginning of month ($\$$$) Interest ($\$$$) Deposit ($\$$$) Balance at end of month ($\$$$) 1 $12000$12000 $64.80$64.80 $300$300 $12364.80$12364.80 2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ For many investment accounts, interest is calculated daily, but paid into the account on a monthly basis. Choose the most accurate statement.
The interest earned over a year would be more since compounding more regularly results in faster exponential growth.
AThe interest earned over a year would be less since the daily interest rate would be a lot smaller.
BThe interest earned over a year would be the same.
C
Question 2
Iain opens a savings account which earns interest of $12%$12% compounded quarterly. He also adds an additional deposit to his account each year. The balance of the investment, in dollars, at the end of each year where interest is compounded quarterly is given by$B_n=\left(1+0.03\right)^4\times B_{n-1}+4000$Bn=(1+0.03)4×Bn−1+4000, where $B_0=22000$B0=22000.
How much did Iain initially invest?
How much does he deposit each year?
Use the sequence facility in your calculator to determine how much is in his account at the end of six years, after he makes his deposit. Give your answer correct to the nearest cent.
Use the sequence facility of your calculator to determine during which year the value of his savings will double.
Rewrite the recursive rule to reflect the interest being compounded monthly instead of quarterly.
Question 3
Uther opens a savings account that earns $4.7%$4.7% per annum compounded monthly. He initially deposits $\$2500$$2500 when he opens the account at the beginning of the month, and then deposits $\$100$$100 at the end of every week. Assume for this question that there are $52$52 weeks in a year.
We will use the finance facility on our calculator to determine how much money Uther has saved after $5$5 years.
Fill in the value for each of the following, and type an $X$X next to the variable we wish to solve for.
$N$N $\editable{}$ $I$I$%$% $\editable{}$ $PV$PV $\editable{}$ $PMT$PMT $\editable{}$ $FV$FV $\editable{}$ $P$P$/$/$Y$Y $\editable{}$ $C$C$/$/$Y$Y $\editable{}$ Use your calculator to determine the value of his savings after $5$5 years, correct to the nearest cent.
We will now use the financial facility on your calculator to determine how much Uther needs to deposit initially if he would like to have $\$35000$$35000 saved after $5$5 years.
Fill in the value for each of the following, leaving an $X$X for the variable you wish to solve for.
$N$N $\editable{}$ $I$I$%$% $\editable{}$ $PV$PV $\editable{}$ $PMT$PMT $\editable{}$ $FV$FV $\editable{}$ $P$P$/$/$Y$Y $\editable{}$ $C$C$/$/$Y$Y $\editable{}$ Use your calculator to determine the value of the initial investment, correct to the nearest cent.