# Long term loans

A loan is usually repaid in instalments.

A loan instalment consists of two parts: the loan repayment and the loan interest.

Customer can usually choose between two different repayment methods.

## Loan with a fixed amortisation schedule

In a loan with a fixed amortisation schedule , the loan is repaid at regular intervals, always in equal principal repayments. With this interest is paid with every payment.

**Example 1.**

The annual interest rate on a loan with a fixed amortisation schedule taken for the purchase of a summer house is 4,8%. The loan is repaid every half a year for two years. The amount of the loan is 30 000 euros.

Prepare a loan repayment table showing the instalments, interest, instalment amount, and loan balance.

## Annuity loan

In an annuity loan, the installments remain the same throughout the loan period. The amount of one installment, ie annuity, is calculated from the formula

A is an annuity

K is the size of the loan, ie the loan principal

q is the interest rate factor

n is the number of payments

**Example 2.**

The amount of an annuity loan taken for the purchase of an apartment is €120 000. The loan period has been agreed to be ten years and the interest rate of the loan is 5,2%. The loan is repaid monthly.

a) Calculate the size of the annuity.

b) How much interest is paid on the loan in total?

Solution

a) Calculate the annuity using the formula

Loan capital K = €120 000.

When the loan is repaid every month for ten years, the instalments become total

10 ∙ 12 = 120 n = 120.

For the interest factor, the interest rate must be calculated per month, in which case the annual interest rate of 5,2% is divided by 12. The interest rate is then 0,433% and the interest rate factor q = 1,00433.

Substitute into the formula

Be careful with your calculator that all the parts are correctly in place!

The monthly installment of the loan is 1284,31 €.

b) The monthly installment is therefore 1284,31 € and the number of instalments during the loan period is 120. The total loan repayment is €154 117,20.

Interest paid in total

When you want to calculate how much of the loan is left after a certain instalment, the following formula is used

V_{k} is the amount of the remaining loan after the loan has been repaid k times

K is the size of the loan, i.e. the loan principal

q is the interest rate factor

k is the number of instalments made

A is the annuity, i.e. the amount of the instalment

**Example 3.**

** **Returning to the previous example. How much loan was left after five years of repayments?

The loan capital K was 120 000 €.

The interest rate factor q was 1,00433.

The annuity A was calculated to be 1284,31 €.

The number of repayments made k, which after five years is 60 (this is in the middle of the loan repayment).

Substitute into the formula

Be careful again with your calculator!

**Turn on the subtitles if needed.**

### Exercises

1. A person took out an annuity loan of €80 000 for the purchase of an apartment. The loan period was agreed for ten years with an interest rate of 6,00%. The loan is repaid monthly.

a) How many instalments are there in total?

b) Calculate the interest factor.

c) Calculate the annuity, i.e. the amount of the instalment.

2. A person took out an annuity loan of €80 000 for the purchase of an apartment. The loan period was agreed for ten years with an interest rate of 6,00%. The loan is repaid semi-annually.

a) Calculate the interest factor.

b) Calculate the size of the annuity.

c) How much of the loan is left after eight years of payments?

3. A loan of €6000 is taken out for the purchase of a boat. The loan is repaid once a year with a fixed amortisation schedule. The loan period is for three years and the interest rate is 9,00%.

a) How many instalments are there on the loan?

b) What is the amount of the instalment?

c) What is the first instalment?

d) What is the final instalment?

e) How much interest is paid in total for the loan?

(You can make a loan calculation, i.e. a table for loan repayment)