# Calculate beginning balance of loan based on payment

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?

• Welcome to MSE. In order for MathJax commands to be effective, they must be enclosed in$ signs. For example, $x^2$ shows up as$x^2$. Commented Jun 4, 2021 at 23:28 ## 1 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? • Thank you for the response. I see now how this is done. This means then there is no comprehensive formula that can be used to compute this in excel outside of using the built in PV function? Commented Jun 4, 2021 at 23:44 • I don't know what's available in Excel, but there isn't a simple analytic formula for$v\$. You have to compute it numerically. I guess you could find it in Excel with the present value function by trial and error. Commented Jun 4, 2021 at 23:50
• Understood, thank you very much. Commented Jun 5, 2021 at 0:53