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\( B_0\) Principle or initial loan balance
\( r \) Interest rate
\( t_\text{term} \) Loan term during repayment
\( t_\text{deferred} \) Period of no payments
\( rt_\text{term} \) Loan product
\( P \) Repayment rate (dollars per time)
\( \frac{V}{B_0} \) Overpay ratio

Deferred payments create a piecewise integration problem

In many ads the offer is made "No payments for 3 years" or as a part of negotiations a distressed borrower is told, "do not worry about it, we will defer your payments." In either case, the loan is now in two distinct phases. In the first phase, the borrower makes no payments and in the second phase, the payments begin. Here \(t_\text{term}\) refers to the time over which payments are made and \(t_\text{deferred}\) is the time the loan payments are deferred.

$$ \frac{d B}{dt}= \begin{cases} rB & 0\geq t < t_\text{deferred} \\ rB-P & t_\text{deferred} \geq t < t_\text{deferred}+t_\text{term} \end{cases} $$

Start by finding the value of \(B(t=t_\text{deferred})\).

$$ B_\text{deferred} = B_0 e^{rt_\text{deferred}} $$

The continuous solution for the overpay ratio applies to the balance at \(B(t=t_\text{deferred})\).

$$ \frac{V}{B_\text{deferred}} = \frac{rt_\text{term}}{1- e^{-rt_\text{term}}} $$

Now substitute in \(B_\text{deferred}\) to solve for the overpay ratio.

$$ \frac{V}{B_0} = \frac{rt_\text{term} e^{rt_\text{deferred}}}{1- e^{-rt_\text{term}}} $$

Similarly, the loan repayment rate where the initial balance is \(B_\text{deferred}\) is

$$ \frac{P}{B_\text{deferred}} = \frac{r}{1- e^{-rt_\text{term}}} $$

Substitute in \(B_\text{deferred}\) to solve for the repayment rate.

$$ \frac{P}{B_0} = \frac{r e^{r t_\text{deferred}}}{1- e^{-rt_\text{term}}} $$

Avoid deferring payments at all costs. For a given repayment time, \(t_\text{term}\), the payment rate and the overpay ratio are exponential functions of \(t_\text{deferred}\).

© MC Byington