$$B_0$$ principal or initial loan balance
$$r$$ Interest rate
$$t_\text{term}$$ Loan term
$$rt_\text{term}$$ Loan product, important parameter which fully specifies a loan
$$P$$ Repayment rate (dollars per time)
$$P_\text{balloon}$$ Final payment due at the end of the loan term
$$\frac{V}{B_0}$$ Overpay ratio

# Balloon payments change the limits on the integral

Some loans end on balloon payments where the borrower is expected to pay a large final payment at the end of the loan term, $$P_\text{balloon}$$. This changes the limits on the solution to the loan equation a bit but the general continuous solution is still valid.

$$B = (B_0-\frac{P}{r})e^{rt}+\frac{P}{r}$$

Solving for $$t_\text{term}$$ requires setting $$B=P_\text{balloon}$$ instead of $$B=0$$.

$$t_\text{term} = \frac{1}{r} \log \left[ \frac{r P_\text{balloon}-P}{rB_0 -P} \right]$$

Solve for $$P$$ as a function of $$B_0$$, $$r$$, $$t_\text{term}$$, and $$P_\text{balloon}$$. Notice that as the balloon payment grows, the repayment rate, $$P$$, drops. As $$rt_\text{term}$$ increases, the effect of increasing the balloon payment drops considerably. This is because at large $$rt_\text{term}$$ the loan repayment is mostly covering interest, not principal.

$$P = \frac{r \left(B_0-P_\text{balloon} e^{-rt_\text{term}}\right)}{1-e^{-rt_\text{term}}}$$

Next, take the equation for overpay and solve as a function of $$P$$, $$B_0$$, $$P_\text{balloon}$$, and $$r$$.

$$\frac{V}{B_0} = \frac{Pt_\text{term}}{B_0} + \frac{P_\text{balloon}}{B_0}$$ $$\frac{V}{B_0} = \frac{P}{rB_0} \log\left[ \frac{r P_\text{balloon}-P}{rB_0 -P} \right] + \frac{P_\text{balloon}}{B_0}$$

Or solve for the overpay ratio as a function of $$t_\text{term}$$, $$\frac{P_\text{balloon}}{B_0}$$, and $$r$$. Increasing the ratio of the balloon payment to initial balance increases the interest when $$rt_\text{term}$$ is small because there is more money accumulating interest for a longer period. At large $$rt_\text{term}$$, the balloon payment matters less because, for the majority of the loan term, the entire principal is earning interest.

$$\frac{V}{B_0} = \frac{rt_\text{term}\left( 1- \frac{P_\text{balloon}}{B_0}e^{-rt_\text{term}} \right) }{ 1-e^{-rt_\text{term}}} + \frac{P_\text{balloon}}{B_0}$$