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\( B_0\) 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

Photo credit to Ritu Arya

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.

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