A loan schedule shows the interest and the repayment component of each payment. It also shows the outstanding balance (OB) after each payment.
Consider a loan of $£y$ over $n$ years with an annual payment of $£x$ every year. The present value of this annuity paid in arrears is equal to the value of the loan $(£y)$: \begin{align} &xa_{n|}=y\\ &x=\frac{y}{a}_{n|} \end{align}
This is the amount that must be paid every year for $n$ years.
If the loan is borrowed at a rate of $i\%$, then the interest component is a fraction of the previous year’s outstanding balance. \[\text{Interest component of year } k= i\%\times(\text{OB of year }k-1)\]
The repayment component of the current year is the interest component subtracted from the annual payment.
\[\text{Repayment component of year } k=x-(\text{interest component of year }k)\]
The outstanding balance of the current year is the outstanding balance of the previous year minus the repayment component of the current year:
\[\text{OB of year }k= (\text{OB of year }k-1)-(\text{repayment component of year }k)\]
The outstanding balance at the end of the loan period must equal $0$.
Draw up a loan schedule for a loan of $£1000$ which it to be repaid over five years with equal annual payments at the end of each year. The interest rate is $7.5\%$ per annum.
\[xa_{5\vert}=1,000\] \[\Rightarrow x=\frac{1000}{a_{5\vert}~} =1000 \frac{0.075}{(1-(1+0.075)^{-5}~}\approx £247.16.\]
|
Year |
Payment |
Interest Component |
Repayment Component |
Outstanding Balance |
|---|---|---|---|---|
|
$0$ |
$0$ |
$0$ |
$0$ |
$£1,000$ |
|
$1$ |
$£247.16$ |
$£75$ |
$£172.16$ |
$£827.84$ |
|
$2$ |
$£247.16$ |
$£62.09$ |
$£185.07$ |
$£642.77$ |
|
$3$ |
$£247.16$ |
$£48.21$ |
$£198.95$ |
$£443.82$ |
|
$4$ |
$£247.16$ |
$£33.29$ |
$£213.87$ |
$£229.95$ |
|
$5$ |
$£247.16$ |
$£17.25$ |
$£229.91$ |
$£0.04$ |
The final outstanding balance does not equal $0$ as there are rounding errors. The annual payment is rounded to two decimal places as is the interest and repayment components and the outstanding balance. Had we used the full figure, the outstanding balance at the end would equal $0$.
Dr Graham Murphy solving a loan schedule problem.
If a loan is to be paid in arrears every $p$ times a year, then the payment for each period should be calculated using the $a_{n|}^{(p)}$ formula. This would be the amount that would be paid each period for $np$ periods.
The interest component for each period should be calculated using the interest rate per period: \[\text{Interest component of year }k= i_{[p]}\times(\text{OB of year }k-1)\] The repayment component of the current year is the interest component subtracted from the annual payment: \[\text{Repayment component of year }k= x-(\text{interest component of year} k)\] The outstanding balance of the current year is the outstanding balance of the previous year minus the repayment component of the current year: \[\text{OB of year }k= (\text{OB of year }k-1)-(\text{repayment component of year }k)\]
Like before, the outstanding balance at the end of the loan period must equal $0$.
A loan of $£1000$ is to be repaid by an annuity payable quarterly in arrears for two years at a rate of $6\%$ per annum. Draw up a loan schedule for this loan.
\begin{align} &i^{(4)}=0.05869538467\\ &i_{[4]} =\frac{~i^{(4)}~}{4}=0.01467384617\\ &4xa_{2\vert}^{(4)}=1,000 \end{align} \[\Rightarrow x=\frac{1000}{~4a_{2\vert}^{(4)}~}=1000\frac{0.058695384617}{4\left(1-(1+0.06)^{-2}\right)}\approx£133.39.\]
|
Month |
Payment |
Interest Component |
Repayment Component |
Outstanding Balance |
|---|---|---|---|---|
|
$0$ |
$0$ |
$0$ |
$0$ |
$£1,000$ |
|
$3$ |
$133.39$ |
$£14.67$ |
$£118.72$ |
$£881.28$ |
|
$6$ |
$£133.39$ |
$£12.93$ |
$£120.46$ |
$£760.82$ |
|
$9$ |
$133.39$ |
$£11.16$ |
$£122.23$ |
$£638.59$ |
|
$12$ |
$133.39$ |
$£9.37$ |
$£124.02$ |
$£514.57$ |
|
$15$ |
$133.39$ |
$£7.55$ |
$£125.84$ |
$£388.73$ |
|
$18$ |
$133.39$ |
$£5.70$ |
$£127.69$ |
$£261.04$ |
|
$21$ |
$133.39$ |
$£3.83$ |
$£129.56$ |
$£131.48$ |
|
$24$ |
$133.39$ |
$£1.93$ |
$£131.46$ |
$£0.02$ |
Due to rounding errors, the final outstanding balance does not equal 0.