Question

A four year loan of $5000 is repaid by equal annual payments at the end of each year. Compute the annual payment on the basis of an interest rate of 6% p.a. and draw up a loan
schedule, showing the interest component of every payment and the outstanding balance.

Answer 1

solution

Loan amount $A=5000$

Time $n=4$

Interest $I=6\%$

Payments $P=?$

$$A=P * \frac{1-(1+I)^{-n}}{I}$$

$$5000=P * \frac{1-(1.06)^{-4}}{0.06}$$

$$5000=P*3.4651$$

$$P=1442.9575$$

the annual payment amount is $1442.96

Each annual payment pays interest on the outstanding loan balance and a portion of the loan balance.

Interest paid is calculated by multiplying the interest rate by the loan balance from the previous period.

$$Interest\ paid = Interest (I)*Loan\ Balance$$

Principle repaid is the portion of the annual payment that covers part of the loan amount.

$$Principle\ repaid= payment\ amount-Interest \ paid$$

Loan balance for a given year is the difference between the loan balance from the previous year and the portion of the loan amount repaid during the year

$$Loan\ balance_{year\ t} = Loan\ balance_{year\ t-1} - principle\ repaid_{year\ t}$$

Note that at the beginning of the loan term, loan balance is equal to the loan amount.

Loan schedule:

year 1:

payment amount = $1442.96

interest paid

$=5000*0.6$

= $300

principle repaid

$=1442.96-300$

= $1142.96

Loan balance

$=5000- 1142.96$

= $3857.04

year 2:

payment amount = $1442.96

interest paid

$=3857.04*0.06$

= $231.42

principle repaid

$=1442.96-231.42$

= $1211.54

Loan balance

$=3857.04-1211.54$

= $ 2645.50

year 3:

payment amount = $1442.96

interest paid

$=2645.50*0.06$

= $158.73

principle repaid

$=1442.96-158.73$

= $1284.23

Loan balance

$=2645.50-1284.23$

= $1361.27

year 4:

payment amount = $1442.96

interest paid

$=1361.27*0.06$

= $81.68

principle repaid

$=1442.96-81.68$

= $1361.28

Loan balance = 0.00