Taking out a loan from a bank or a financial institution is a common way for people to buy expensive items such as cars or houses. If we borrow money, we will receive our desired amount (called the principal) from the lender (usually a bank), and we will sign a contract outlining specific details such as the interest rate and the length of the loan. The rate at which we repay the debt will determine the overall amount of interest we end up paying.
If we make regular, equal deposits on our loan then the repayments form an annuity. We can use tables of future values to calculate interest rates and instalments for these loans.
In the case of an investment, we can also use tables of future values to calculate the regular instalments needed to reach a certain savings target.
How much will a loan cost?
Each bank will have different conditions for offering you a loan, so it is important to consider the loan with the best repayment strategy, as often repaying more early on can save you a lot of money in the long run!
To calculate the total amount of the loan, we need to multiply the repayment amount, the length of the loan and how often the repayment is to be made (e.g. weekly, monthly) all together.
$\text{Total repayments }=\text{Individual repayment }\times\text{Number of time periods each year }\times\text{Number of years }$Total repayments =Individual repayment ×Number of time periods each year ×Number of years
For example, if you have to make weekly repayments of $\$55.50$$55.50 for $2$2 years, then:
$\text{Individual repayment }=\$55.50$Individual repayment =$55.50
$\text{Number of time periods each year }=52$Number of time periods each year =52 (we assume $52$52 weeks in a year)
$\text{Number of years }=2$Number of years =2
So the total repayments would be $55.50\times52\times2=\$5772$55.50×52×2=$5772
Interest tables with loans
Banks offer different interest rates depending on size of your repayments and the length of your loan. From Table $1$1, you can see that for a $20$20-year loan at an annual interest rate of $6%$6%, you would need a monthly repayment of $\$7.16$$7.16 for each $\$1000$$1000 that is borrowed in order to pay off the loan in $20$20 years.
Table 1: Monthly repayments on a $\$1000$$1000 loan
| Term of loan of $\$1000$$1000 (years) | |||||
| Annual interest rate | $10$10 | $15$15 | $20$20 | $25$25 | $30$30 |
| $3%$3% | $9.66$9.66 | $6.91$6.91 | $5.55$5.55 | $4.74$4.74 | $4.22$4.22 |
| $4%$4% | $10.12$10.12 | $7.40$7.40 | $6.06$6.06 | $5.28$5.28 | $4.77$4.77 |
| $5%$5% | $10.61$10.61 | $7.91$7.91 | $6.60$6.60 | $5.85$5.85 | $5.37$5.37 |
| $6%$6% | $11.10$11.10 | $8.44$8.44 | $7.16$7.16 | $6.44$6.44 | $6.00$6.00 |
| $7%$7% | $11.61$11.61 | $8.99$8.99 | $7.75$7.75 | $7.07$7.07 | $6.65$6.65 |
| $8%$8% | $12.13$12.13 | $9.56$9.56 | $8.36$8.36 | $7.72$7.72 | $7.34$7.34 |
If we borrow $\$15000$$15000 at $6%$6% per annum over $20$20 years, we would be borrowing $15$15 groups of $\$1000$$1000 and the monthly repayments would be:
$15\times\$7.16=\$107.40$15×$7.16=$107.40
The total repayments are given by:
$\$7.16\times\text{12 months }\times\text{20 years }\times\text{15 lots }=\$25776$$7.16×12 months ×20 years ×15 lots =$25776
So over $20$20 years, we'd actually end up paying $\$10776$$10776 interest on top of the $\$15000$$15000 repayment of the loan.
Interest tables with savings
We can also use interest tables to determine how long it will take us to save a certain amount. The numbers in them will be different to loans, but they work in the same way.
Table $2$2 shows us how much we need to save each month to reach $\$1000$$1000 after a certain number of years.
Table 2: Monthly instalments required to save $\$1000$$1000
| Time to save $\$1000$$1000 (years) | |||||
| Annual interest rate | $5$5 | $10$10 | $15$15 | $20$20 | $25$25 |
| $1%$1% | $16.26$16.26 | $7.93$7.93 | $5.15$5.15 | $3.77$3.77 | $2.94$2.94 |
| $2%$2% | $15.86$15.86 | $7.53$7.53 | $4.77$4.77 | $3.39$3.39 | $2.57$2.57 |
| $3%$3% | $15.47$15.47 | $7.16$7.16 | $4.41$4.41 | $3.05$3.05 | $2.24$2.24 |
| $4%$4% | $15.08$15.08 | $6.79$6.79 | $4.06$4.06 | $2.73$2.73 | $1.95$1.95 |
| $5%$5% | $14.70$14.70 | $6.44$6.44 | $3.74$3.74 | $2.43$2.43 | $1.68$1.68 |
| $6%$6% | $14.33$14.33 | $6.10$6.10 | $3.44$3.44 | $2.16$2.16 | $1.44$1.44 |
For example, if we save each month for $10$10 years at $2%$2% per annum compounded monthly, we would need to save $\$7.53$$7.53 each month to get to $\$1000$$1000 by the end of $10$10 years.
We can calculate the monthly instalments required to save $\$30000$$30000 in $10$10 years if the savings account earns $2%$2% interest per annum, compounded monthly.
We want to save a total of $\$\left(30\times1000\right)$$(30×1000)
| $\text{Instalment }$Instalment | $=$= | $30\times\$7.53$30×$7.53 |
| $=$= | $\$225.90$$225.90 |
Worked examples
Question 1
Using Table $1$1, calculate the monthly instalments required to pay off a $15$15-year loan of $\$200000$$200000 at $7%$7% p.a. monthly reducible interest.
Question 2
Petra has a $15$15-year loan for $\$10000$$10000 at $4%$4% p.a. monthly reducible interest. Using Table $1$1, calculate each of the following:
The monthly instalments required to repay the loan.
The total amount she will repay
The interest she will be charged over the term of the loan.
The total interest expressed as a percentage of the principal loan.
Question 3
Jacob wants to save $\$30000$$30000 over $10$10 years in a savings account that earns $3%$3% interest p.a., compounded monthly. Using Table $2$2, calculate:
The monthly instalments required to save the target amount.
The total amount she will invest herself.
The amount earned through interest over the saving period.
Question 4
Trey and Greta each borrow $\$5000$$5000. Trey's loan is at $5%$5% p.a. reducing interest for $15$15 years. Greta's loan is at $6%$6% p.a. interest reducing interest for $10$10 years.
Using Table $1$1, calculate the difference between the total repayments for each loan. Who pays the least amount?