Modelling a Loan
We're now familiar with the concept of a reducible balance loan and how we can analyse them using a spreadsheet and the financial capability of our CAS calculators.
Now we want to be able to model this type of loan with a recurrence relation and use the power of the sequence facility on our calculator to answer questions about the loan.
Modelling a Reducible Balance Loan with a Recurrence Relation
example
Tim is starting up his own small business. He has saved $\$15000$$15000 to buy equipment and borrows another $\$50000$$50000 from the bank. He is charged interest at a rate of $4.5%$4.5% per annum, compounded monthly and he makes monthly repayments of $\$400$$400
(a) How much does Tim owe at the end of the first month?
Think: We'll take the amount Tim has borrowed, add the interest on and then take away his repayment.
Do: $15000+50000\times\frac{0.045}{12}-400$15000+50000×0.04512−400
$\$49787.50$$49787.50
(b) Write a recurrence relation which gives the balance at the end of the month, $B_{n+1}$Bn+1
Think: The calculation we used in part (a) will help us a lot with this part. Just remember, that when we want to add on the interest we can do it in two parts as shown above, or simply multiply by $1.00375$1.00375.
Do: $B_{n+1}=1.00375\times B_n-400;B_0=50000$Bn+1=1.00375×Bn−400;B0=50000
Notice we use $B_0$B0 as the amount borrowed. If we wanted to use $B_1$B1, we'd need to use the amount owing at the end of month $1$1.
(c) Use the sequence facility on your calculator to determine how many months it will take Tim to pay off the loan.
Think: We'll now enter our recurrence relation into the sequence facility of our calculator and scroll down until we see the balance become negative.
Do:
So we can see Tim pays off the loan in month $169$169.
(d) Calculate the amount of his final repayment
Think: Using the balance of the loan for the month prior to Tim paying it off, we can add the interest on to the balance and this will be the amount of his final repayment.
Do: Final repayment = $1.00375\times394.03$1.00375×394.03
$\$395.51$$395.51
(e) Hence determine the total amount he paid for the equipment
Think: Tim has made $168$168 repayments of $\$400$$400each and a final repayment of $\$395.51$$395.51
Do: Total amount paid for equipment = $15000+168\times400+395.51$15000+168×400+395.51
Total amount = $\$82595.51$$82595.51
Worked Examples
question 1
Bart borrows $\$61000$$61000 from a banking institution. He is charged $6.6%$6.6% per annum interest, compounded monthly. At the beginning of each month, before interest is charged, he makes a repayment of $\$400$$400.
Fill in the missing values in the table. Give all values correct to the nearest cent and use your rounded answers for all subsequent calculations in the table.
Month Opening Balance Repayment Interest Closing Balance 1 $61000$61000 $400$400 $333.30$333.30 $[60933.29\le60933.3\le60933.31]$[60933.29≤60933.3≤60933.31] 2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Write a recursive rule that gives the closing balance, $B_n$Bn, at the end of month $n$n.
Write both parts of the rule (including for $B_0$B0) on the same line, separated by a comma.
Use your calculator to determine how much is owing on the loan after $4$4 years.
Give your answer to the nearest cent.
At the end of which year and month will the loan have been repaid?
$\editable{}$ year(s) $\editable{}$ month(s)
question 2
Xavier takes out a mortgage to purchase an apartment. A portion of his payments and balances are shown in the table below. At the beginning of each month, after interest is charged, he makes a repayment of $\$2900$$2900.
| Month | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| $1$1 | $X$X | $2900$2900 | $408740$408740 | |
| $2$2 | $408740$408740 | $1634.96$1634.96 | $2900$2900 | |
| $3$3 | $Y$Y | |||
| $4$4 |
Calculate the monthly interest rate, $r$r , charged on this loan. Give your exact answer as a percentage or decimal.
Calculate the value of $X$X in the table.
Calculate the value of $Y$Y in the table, correct to two decimal places.
Write a recursive rule that gives the closing balance, $A_n$An, of the loan after $n$n months and state the initial balance $A_0$A0.
Write both parts of the rule on the same line separated by a comma.
Use the sequence facility on your calculator to calculate the value of the final repayment.
Hence calculate the total repayments made.
How much interest does Xavier pay on this loan?
question 3
Tara takes out a personal loan to go on a holiday. A portion of her payments and balances are shown in the table below. At the beginning of each quarter, after interest is charged, she makes a repayment of $\$350$$350.
| Quarter | Opening Balance | Interest | Repayment | Closing Balance |
|---|---|---|---|---|
| $1$1 | $X$X | $10831.50$10831.50 | ||
| $2$2 | $10831.50$10831.50 | $178.72$178.72 | $350$350 | |
| $3$3 | $Y$Y |
Calculate the quarterly interest rate charged on this loan.
Give your final answer as a percentage to 2 decimal places.
Calculate the value of $X$X in the table.
Give your answer correct to two decimal places.
Calculate the value of $Y$Y in the table.
Give your answer correct to two decimal places.
Write a recursive rule that gives the opening balance, $A_n$An, of the loan at the beginning of $n$n quarters.
Use the sequence facility on your calculator to calculate the value of the final repayment.
Hence calculate the total repayments made.
If Tara had been offered half the rate of interest and everything else remained equal, select the most appropriate statement about the interest charged on the loan over its lifetime.
The interest would be half as much.
AThe interest would be less than half as much.
BThe interest would be more than half as much.
C