The Master Formula (Payoff Time)
$$n = \frac{\ln\left(\frac{E}{E - Pr}\right)}{\ln(1 + r)}$$
Where:
- n: Number of months to payoff
- E: Monthly Payment (EMI + Extra)
- P: Current Loan Balance
- r: Monthly Interest Rate (decimal)
Dimensional Analysis
The formula involves the ratio of two logarithms. Logarithms themselves are dimensionless, and their arguments ($E / (E-Pr)$) are ratios of currency ($M/M$), which are also dimensionless. Thus, $n$ is a pure number representing the count of periods, which is dimensionally correct.
Variations: Impact of Extra Payments
If you add an extra amount $x$ to your payment, the new time $n'$ is:$$n' = \frac{\ln(E+x) - \ln(E+x-Pr)}{\ln(1+r)}$$Observe how the $\ln$ function causes a non-linear (exponential) decrease in time for a linear increase in $x$. This is the "Magic of Prepayment."
Shortcuts & Mnemonics
- The "One-Third" Rule: For many standard 20-year mortgages, paying just 1/12th extra every month (one extra EMI a year) reduces the tenure by nearly 5 years.
- Mnemonic: "Log of (Payment over Payment minus Interest) divided by Log of Growth."
Edge Cases
- $E = Pr$: The denominator of the log argument becomes zero. Payoff time $n = \infty$. You are in an "Interest-Only" trap.
- Continuous Compounding: The formula simplifies to $T = \frac{1}{r} \ln(\frac{k}{k-rD_0})$, where $k$ is the continuous payment rate. This is the "Physics Version" of the formula.
- Small $r$: Using $\ln(1+x) \approx x$, the formula approximates to $n \approx \frac{\ln(E/(E-Pr))}{r}$.
