Loan Amortization

16 formulas

Loan amortization formulas break down loan payments into their interest and principal components and track the remaining balance over time. These are critical for understanding mortgages, auto loans, and any installment debt — showing exactly how much of each payment goes toward interest versus reducing the balance.

Balance & Schedule

Remaining Loan Balance

Outstanding loan balance after p payments. The final payment is capped at the amount owed, and the balance stays zero after payoff. At zero interest, balance is max(0, PV − PMT × p).

B_p = \max\left(0, PV \times (1+k)^p - PMT \times \frac{(1+k)^p - 1}{k}\right)

Showing balance, annual

2 compounding methods

Interest Portion of Payment

Calculates how much of a specific loan payment goes toward interest. Takes the original loan setup plus the payment number, computes the balance immediately before that payment, and multiplies by the per-period rate.

I_p = B_{p-1}\frac{k}{m},\quad B_{p-1}\geq 0

Principal Portion of Payment

Calculates how much of a specific loan payment goes toward reducing the principal balance. Equals the total payment minus that payment's interest portion.

P_p = \min\left(PMT,B_{p-1}(1+k/m)\right)-B_{p-1}k/m

Cumulative Interest Paid

Total interest paid over the first p payments of an amortizing loan, summed payment by payment from the schedule. Each payment is capped at the balance plus interest, so the total stops growing once the loan is paid off.

I_{1..p} = \sum_{t=1}^{p} B_{t-1} \times \frac{k}{m}

Cumulative Principal Paid

Total principal repaid over the first p payments: the original loan less the balance still outstanding after payment p.

P_{1..p} = PV - B_p

Total Interest Cost of a Loan

All interest paid over the life of an amortizing loan, from the first payment until the balance reaches zero, summed from the schedule so the final capped payment is handled exactly.

I_{\text{total}} = \sum_{t=1}^{T} B_{t-1} \times \frac{k}{m}, \quad B_T = 0

Payment Structures

Interest-Only Payment

The periodic payment on an interest-only loan: the balance times the periodic rate. Nothing is repaid, so the balance is unchanged at the end of the interest-only period.

PMT_{IO} = PV \times \frac{k}{m}

Balloon Payment

The lump sum due at the end of a balloon loan: the balance remaining after n years of regular payments that were sized for a longer amortization.

B_{nm} = PV(1+k/m)^{nm} - PMT \times \frac{(1+k/m)^{nm} - 1}{k/m}

Payoff Time with Extra Payments

How many years it takes to retire a loan when a fixed extra amount is added to every payment. The combined payment is treated as the annuity payment and the term solved for.

n = -\frac{\ln\left(1 - \frac{PV \cdot k/m}{PMT + E}\right)}{m \ln(1 + k/m)}

Mortgage Analysis

Loan-to-Value Ratio

The loan amount as a fraction of the property value. Lenders use it to price risk: higher LTV means a smaller equity cushion if the property must be sold.

LTV = \frac{\text{Loan}}{\text{Value}}

Down Payment

The cash paid up front on a purchase: the price times the down-payment percentage. The loan finances the rest.

\text{Down} = \text{Value} \times d

Mortgage Points Break-Even

Months of payments needed for the monthly saving from a lower rate to repay the up-front cost of the points that bought it. Points cost a percentage of the loan; the saving is the difference between the two amortizing payments.

\text{BE} = \frac{\text{Points} \times \text{Loan}}{PMT(k_{\text{without}}) - PMT(k_{\text{with}})}

Refinance Break-Even

Months of the lower new payment needed to recover the closing costs of refinancing.

\text{BE} = \frac{\text{Closing Costs}}{PMT_{\text{old}} - PMT_{\text{new}}}

Cost of Borrowing

Effective Borrowing Rate with Fees

The annual rate a borrower actually pays once up-front fees are counted: the rate that makes the scheduled payments worth the loan net of fees. This is the idea behind the APR quoted on loan disclosures.

PV - \text{Fees} = PMT \times \frac{1 - (1 + r/m)^{-nm}}{r/m}

Debt Capacity & Affordability

Debt Service Coverage Ratio (DSCR)

Net operating income divided by total debt service (principal plus interest due in the period). Above 1 the income covers the payments; below 1 the borrower is paying debt out of reserves. Lenders on income property and business loans typically require 1.20 to 1.35.

\text{DSCR} = \frac{\text{NOI}}{\text{Debt Service}}

Maximum Affordable Loan

The largest loan whose payment fits within a debt-to-income limit: income times the allowed DTI ratio per period, less existing debt payments, is the payment capacity; the loan is that capacity converted to a principal at the rate and term.

\text{Loan} = \left(\frac{\text{Income} \times \text{DTI}}{m} - \text{Other}\right) \times \frac{1 - (1 + k/m)^{-nm}}{k/m}