principal or initial loan balance | |
---|---|

Interest rate | |

Loan term | |

Loan product, important parameter which fully specifies a loan | |

Repayment rate (dollars per time) | |

Final payment due at the end of the loan term | |

Overpay ratio |

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.

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

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.

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

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.