I have a problem that asks me how to calculate the beginning loan balance based only on the monthly payment of debt.
The Coverage ratio is $1.25$ meaning the profit must be $1.25\times$ greater than the loan payment. The income is $200$ per month so when applying the coverage ratio of $1.25$ to $200$ we get a payment of $160$ because the income must be $1.25\times$ greater than the payment.
\begin{align*}
\text{Profit} &= 200\\[5pt]
\text{Payment} &= 160\\[5pt]
\end{align*}
From this I can find what the total loan and interest paid will be by taking the payment ($P$) and multiplying it by the term of the loan ($T$) which is $300$ (months) so
\begin{align*}
P \cdot T &= 48,000\\[5pt]
(300 \cdot 160 &= 48,000)\\[5pt]
\end{align*}
Then from this, I should be able to tell how much of this total amount was interest by doing the following:
$$\begin{align*}
\text{Interest} &= 12\%\ \text{per year or}\ 1\%\ \text{monthly (compounded monthly)}\\[5pt]
\frac{48,000}{(1+.01)^{300}} &= \text{total interest paid}\\[5pt]
\end{align*}$$
I know this is wrong but what am I not understanding?
Best Answer
The amount of the loan is the present value of the payments.
Let $v=\frac1{1.01}$ be the per-period discount rate. We have $$P=160(v+v^2+v^3+\cdots v^{300})$$ Can you take it from here?