I was not quite sure what the question was asking and I would like to have some input.
a), Is it asking us to actually calculate knowing what each principal paid? or
b), Working from almost complete scratch from the given information?
A loan of present value $L$ is made with payments starting 1 quarter after the loan was made. Each quarter has an effective interest rate of $i=0.2$. Each payments consists of $\$250$ plus the interest accumulated from each of the outstanding balance. If the payment is complete right after the 12th payment, what is $L$?
This is rather a special problem where in an example, it is known that $L=3000$ and each interest added for the payment periods are $I_1=60, I_2=55, \cdots , I_{12}=5$ which means that each payments are $K_1=310, K_2=305, \cdots, K_{12}=255$.
So, the value of $L$ can be "confirmed" as
$$\begin{align} L& =310v+305v^2+\cdots 255v^{12}\\ &= (310v+310v^2+\cdots 310v^{12})-(5v^2+10v^3+\cdots 55v^{11})\\ &=310a_{\overline {12} \rceil .02}-5v((Ia)_{\overline {11} \rceil .02})\\ & \approx 3278.36-278.36 \end{align}$$
But I'm not sure if the question meant to have us do this.
What if I did not know what $L$ was in the first place? Would that not be a more natural problem?
If so, all I know is that
$$L = (250+Li)v+(250+OB_1i)v^2+ \cdots +(250+OB_{12}iv^{12})$$
where each $OB_j$ is the outstanding balance.
I know that $OB_{t}=OB_{t+1}-PR_{t+1}$ where $PR_j$ is the principal paid and each $OB$ will depend on $L$ and it seems solvable, but I haven't succeeded so far.
May I ask for some help?
Thanks
You have a loan amortization with equal principal payments $P=250$ for $n=12$ periods at an interest rate $i=2\%$. This is known aslso as Italian amortization plan.
Thus $L=nP=3000$ and for the other quantities: $$ \begin{align*} D_k&=D_{k-1}-P=\frac{n-k}{n}S & &\text{debt at time }k \\ I_k&=D_{k-1}i=(n-k+1)\frac{S}{n}i & &\text{interest at time }k\\ R_k&=I_k+P_k=(n-k+1)\frac{S}{n}i+\frac{S}{n} &&\text{payment at time }k \end{align*} $$
n R P I D 0 L=3,000.00 1 310.00 250.00 60.00 2,750.00 2 305.00 250.00 55.00 2,500.00 3 300.00 250.00 50.00 2,250.00 4 295.00 250.00 45.00 2,000.00 5 290.00 250.00 40.00 1,750.00 6 285.00 250.00 35.00 1,500.00 7 280.00 250.00 30.00 1,250.00 8 275.00 250.00 25.00 1,000.00 9 270.00 250.00 20.00 750.00 10 265.00 250.00 15.00 500.00 11 260.00 250.00 10.00 250.00 12 255.00 250.00 5.00 - 3,000.00 390.00