# Outstanding loan balance

A loan of ${$1000}$is being repaid with annual payments over 10 years. The size of the payment in the first five years is${$k}$. It is found that the payments in the last five years are five times the payments in the first five years. Let the annual rate $r=0.08$, calculate

(i) The value of $K$

(ii)The outstanding loan balance after making the $4th$ payment

(iii) The outstanding loan balance after making the $5th$, hence the principal amortised in the $5th$ payment.

$$\begin{array}{c|c|c} t=0 & ...t=5 & t=6 ... t=10 \\ \hline \\ K... &K & (5K)... \\ \end{array}$$

(i) In order to find $K$:

$L=K a_{5|0.08}+(5K)(a_{5|0.08}).(1+i)^{-5}$

$1000=K[\frac{1-1.08^{-5}}{0.08}]+5K[\frac{1-1.08^{-5}}{0.08}](1.08)^{-5} \rightarrow K=56.88$

(ii)Using the retrospective method to find $B_4^r$

$B^r_4=1000(1.08)^4-56.88[\frac{1.08^4-1}{0.08}]=1104.18$

(iii)I have tried some different thing. I doubt it works.

$B^5_r=1104.18-56.88[\frac{1.08-1}{0.08}]$=1047.30

My answers are absurd. I get to play more than the loan itself.

• (i) and (ii) are fine. But I don´t understand the third question. Maybe you can explain it in your own words. Apr 24, 2016 at 13:59
• Well that question was from a past test. According to me, I need to find the outstanding balance after 5years, I also need to calculate the Interest generated after these 5 years. Subtracting these 2 values shall give the principal amortised.
– Tosh
Apr 24, 2016 at 14:17
• If you interpretation is right then your result is right. $\checkmark$ Apr 24, 2016 at 14:25
• I have been able to calculate the interest generated for 5yrs as $1000(1-(1.08)^{-5}=319.416.$ Subtracting from the above I get $727.88$. Does this mean that the outstanding balance in all 3 cases is higher than the loan of $1000$.Does it make a little sense?
– Tosh
Apr 24, 2016 at 15:10

1. In order to find $K$ $$L=K\,a_{\overline{5}|i}+5K\,v^5\,a_{\overline{5}|i}=K\,a_{\overline{5}|i}(1+5v^5)$$ and then $$K=\frac{L}{a_{\overline{5}|i}(1+5v^5)}=56.88422\approx 56.88$$
2. The outstanding balance at $t=4$ $$B_4=K\,a_{\overline{5-4}|i}+5K\,v^5\,a_{\overline{5}|i}=Kv+5K\,v^5\,a_{\overline{5}|i}= 825.55$$
3. The outstanding balance at $t=5$ $$B_5=5K\,v^5\,a_{\overline{5}|i}= 772.88$$
 t    Paymet   Principal   Interest     Debt
0                                    1,000.00
1    56.88       38.71      18.17      961.29
2    56.88       41.81      15.07      919.47
3    56.88       45.16      11.73      874.32
4    56.88       48.77       8.12      825.55
5    56.88       52.67       4.21      772.88
6   284.42      131.74     152.68      641.14
7   284.42      142.28     142.14      498.85
8   284.42      153.66     130.76      345.19
9   284.42      165.96     118.46      179.23
10   284.42      179.23     105.19        0.00

• I appreciate the effort for the table. I have done this table in previous questions. But I do not grasp something here, The interest at $t=1$ is calculated $i$X$a_{n|i}=0.08$X$1000=80$. How come it is $38.71$ in the table.
– Tosh
Apr 24, 2016 at 16:40
• Actually you have two different payments and you cannot apply the rule you know with only one payment. So for the first 5 years you have an annuity with payment $K$ and with present value $227.12$, so $8\%\times 227.12=18.17$. Apr 24, 2016 at 16:49