Lesson

Taking a loan from a bank or a financial institution is the way most people afford to buy expensive items, such as cars or houses. The borrower receives a full amount of money, called the principal, from a lender, and signs a contract outlining specific conditions, such as the interest rate and duration of the loan. This determines the amount of the borrower's repayment.

Although banks and financial institutions offer lots of different types of loans with lots of different conditions, we are going to focus on **amortized loans**, where both the principal and the interest are paid back gradually over an agreed period of time.

Terminology

**Deposit:** an initial, partial payment for the cost of the loan.

**Balance:** The amount owing after interest has been incurred or repayments have been made.

**Repayment:** a fixed amount paid at regular time periods to repay the balance and any interest incurred, i.e. $\frac{\text{balance + interest }}{\text{number of time periods }}=\text{repayment }$balance + interest number of time periods =repayment .

**Interest:** an amount charged for loaning you the amount of the balance. Usually calculated as a percentage of the balancing owning using the compound interest formula.

**Interest rate:** the amount of interest charged expressed as a percentage.

Remember, each bank will have different conditions for offering you a loan. So it is important to consider the total amount the loan will cost you because it could end up saving you *heaps* of cash! 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. In other words:

$\text{total loan amount }=\text{repayment }\times\text{number of time periods }$total loan amount =repayment ×number of time periods

Remember: we need to express the number of time periods relative to the length of the loan. For example, if I had to made weekly repayments of $\$5.55$$5.55 for $2$2 years:

$\text{Repayment }=\$5.55$Repayment =$5.55

$\text{Number of time periods }=104$Number of time periods =104 (because there are $52$52 weeks in a year and $52\times2=104$52×2=104)

So the total would be $5.55\times104=\$577.20$5.55×104=$577.20

If you didn't work out the number of time periods first, you could work it out in one step as $5.55\times52\times2$5.55×52×2 and you will get the same answer of $\$577.20$$577.20.

Banks offer different interest rates depending on the interest rate you pay and the length of your loan. For example, in the table below you can see that a $20$20-year loan at an annual 6% interest, will incur a monthly repayment of $\$7.16$$7.16 for each $\$1000$$1000 that is borrowed.

That means that if we borrowed $\$15000$$15000, we would be borrowing $15$15 lots of $\$1000$$1000. So to work out the total amount of the loan, we would calculate $\$7.16\times\text{12 months }\times\text{20 years }\times\text{15 lots }$$7.16×12 months ×20 years ×15 lots . More simply, $7.16\times12\times20\times15$7.16×12×20×15, which equals $\$25776$$25776. So we'd actually end up paying $\$10776$$10776 interest over the $20$20 years.

Monthly repayments on a $1000 loan | |||||
---|---|---|---|---|---|

Term of Loan in 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 |

When we were studying compound interest, we looked at situations where an amount of money compounded more than once a year (e.g. monthly). Even though we divided the interest rate to match the time period, you would have noticed that the more time interest is compounded in a year, the more interest you will earn. For example, let's compare 1) the value of $\$1000$$1000 invested in an account at an interest rate of $5%$5% p.a. for $3$3 years with 2) the value of $\$1000$$1000 in an account at an interest rate of $5%$5% p.a. that compounds monthly for $3$3 years:

1)

$A$A |
$=$= | $P\left(1+r\right)^n$P(1+r)n |

$=$= | $1000\times1.05^3$1000×1.053 | |

$=$= | $\$1157.63$$1157.63 |

2)

$A$A |
$=$= | $P\left(1+r\right)^n$P(1+r)n |

$=$= | $1000\times1.004167^{36}$1000×1.00416736 | |

$=$= | $\$1161.49$$1161.49 |

See how the amount that compounds monthly earns more interest than the one that compounds annually? This difference will get bigger as time passes by. So how do we compare different loans if they compound at different time periods? We compare the **effective interest rates**. The effective interest rate tells us the *actual* amount of annual interest that your loan will incur.

We calculate the effective annual interest rate using the formula:

Effective Interest Rates

$\text{effective annual interest rate }=\left(1+\frac{r}{n}\right)^n-1$effective annual interest rate =(1+`r``n`)`n`−1

Using the financial table, calculate the monthly instalments required to pay off a $25$25-year loan of $\$1000$$1000 at $4%$4% p.a. monthly reducible interest.

Monthly repayments on a $\$1000$$1000 loan

Term of Loan in 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 |

Term of Loan in 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 |

Apply everyday compounding rates

Apply numeric reasoning in solving problems