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An investor takes out a 15 -year loan. The loan is repaid by monthly payments in arrear with the first repayment being $\mathrm{R} 1,000$ and the repayments increasing by R200 each month until the repayments reach R10,000. The repayments then remain at $\mathrm{R} 10,000$ for the rest of the term. The annual effective rate of interest is $8.5 \%$ per annum.
i. Show that the amount of the loan is R848,200.
[7]
ii. Calculate, using the retrospective method, the amount of capital outstanding at the end of the third year, just after the repayment made at that time.
[5]
[4]
iv. Calculate the capital component and the interest component of the $37^{\text {th }}$ payment.
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(i)
$10000=1000+200(t-1) \Rightarrow t=46$ months
\begin{aligned} &800 \times a_{461}+200 \times(I a)_{\overline{4} 6}+10,000 \times v^{46} \times a_{134} \quad @ \frac{i^{(12)}}{12}=0.0068214933 \\ &=31,494.3014483263+175,610.2038747990+641,080.1491508610 \\ &=848,184.654 \approx 848200 \end{aligned}
(ii)
$848,200 \times(1.085)^{3}-800 \times s_{36 i \frac{(12)}{12}}-200 \times(I a)_{36 i^{121}} \times(1.085)^{3}=906,433.21$
(iii)
- Loan outstanding is larger than original loan
- Payments at start of loan term was not large enough to cover the interest
- Interest on loan was R57,859.90673 and repayment was R 1,000
- Interest not covered was added to outstanding loan
(iv)
$\begin{array}{ll}\text { Payment: } & 8,200 \\ \text { Interest component: } & 906,433.210189080 * 0.0068214933=6,183.23 \\ \text { Capital component: } & 8,200-6,183.22813=2,016.77\end{array}$
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