The Master Formula (Payoff Months)
$$N = \frac{-\ln(1 - \frac{rB}{P})}{\ln(1 + r)}$$
Where:
- N: Number of months to zero balance
- B: Current Balance
- P: Monthly Payment
- r: Monthly Interest Rate (APR / 12)
Dimensional Analysis
The term $(rB/P)$ is $\frac{(\text{Interest/Time}) \times \text{Principal}}{\text{Payment/Time}}$. The "Time" units cancel out, and "Principal/Payment" (Currency/Currency) cancels out. The entire argument is dimensionless. The result $N$ is a count of cycles, consistent with the definition of time in a discrete series.
Variations: The Cost of Delay
Total Interest Paid ($I_{total}$) is given by:$$I_{total} = (N \times P) - B$$This shows that total interest is a linear function of time ($N$). Since $N$ increases exponentially as $P$ approaches $rB$, the interest paid can become several times the original balance.
Shortcuts & Mnemonics
- The "1% Rule": To make any real progress, your payment should be at least 1% of the principal plus the monthly interest.
- Snowball Method: Mathematically, paying off the highest interest rate first (Avalanche) is the most efficient, but "Snowball" (smallest balance first) helps with psychological momentum—similar to solving easy questions first in JEE to build confidence.
Edge Cases
- $P < rB$: The logarithm argument becomes negative. In the real world, this is Negative Amortization—your debt grows even though you are paying every month.
- $r = 0$: During a promotional 0% APR period, $N = B/P$. The exponential decay becomes a simple linear decrease.
- Daily vs Monthly: While the calculator uses monthly $r$ for simplicity, the actual cost is slightly higher due to the $EAR$ mentioned in the Method article.
