Calc FYI

Remaining Loan Balance

$$B_k = P\frac{(1+r)^n - (1+r)^k}{(1+r)^n - 1}$$

Variables

Symbol Name Unit Description
$B_k$ Remaining loan balance currency Outstanding principal after k payments have been made.
$P$ Original principal currency Total loan amount at origination.
$r$ Monthly interest rate decimal Annual rate divided by 12.
$n$ Total instalments months Full loan tenure in months.
$k$ Payments made months Number of EMI payments already made.

Remaining Loan Balance Formula

The outstanding principal remaining after making k monthly payments on a standard amortising loan is:

$$B_k = P \times \frac{(1+r)^n - (1+r)^k}{(1+r)^n - 1}$$

This formula tells you exactly how much you still owe at any point in the loan's life.

Why the Balance Decreases Slowly at First

In an amortising loan, early payments are mostly interest. The principal paydown accelerates over time:

For a $200,000 home loan at 8%, 20 years (240 months), EMI ≈ $1,672:

After payment # Balance % Paid off
12 (1 year) $196,350 1.8%
60 (5 years) $180,736 9.6%
120 (10 years) $152,143 23.9%
180 (15 years) $106,303 46.8%
240 (20 years) $0 100%

After paying for half the loan term (10 of 20 years), only 23.9% of the principal has been repaid. This "back-loading" of principal repayment is why mortgage refinancing in the early years saves less than borrowers expect.

Use Cases

  • Refinancing decisions: Know your balance before applying for a new loan
  • Early closure calculations: Determine prepayment amount
  • Home equity: Outstanding balance = current value minus equity
  • Investment property analysis: Calculate debt service coverage

Derivation & History

The formula is derived from two separate compound interest calculations:

  1. The unpaid principal grows as P(1+r)^k (as if no payments were made)
  2. Each EMI payment reduces future obligations; the k payments made are worth EMI × [(1+r)^k − 1] / r in future value

Subtracting gives B_k = P(1+r)^k − EMI × [(1+r)^k − 1] / r

Substituting the EMI formula EMI = P × r(1+r)^n / [(1+r)^n − 1] and simplifying yields the canonical form shown above.

Worked Examples

After 5 years of a 20-year home loan

  1. (1+r)^n = (1.006667)^240 = 4.9268
  2. (1+r)^k = (1.006667)^60 = 1.4898
  3. B_60 = 200,000 × (4.9268 − 1.4898) / (4.9268 − 1)
  4. B_60 = 200,000 × 3.4370 / 3.9268
  5. B_60 = 200,000 × 0.87528 = $175,056

Result: Balance after 5 years = $175,056 (only $24,944 principal repaid)

Halfway through a car loan

  1. (1.00583)^60 = 1.4176; (1.00583)^30 = 1.1907
  2. B_30 = 25,000 × (1.4176 − 1.1907) / (1.4176 − 1)
  3. B_30 = 25,000 × 0.2269 / 0.4176 = 25,000 × 0.5432 = $13,580

Result: Balance at 30 months = $13,580 (54.3% of original loan remains)

Edge Cases & Limitations

k = 0: B_0 = P (no payments made — balance equals original principal).

k = n: B_n = 0 (all payments made — loan fully repaid). Verify numerically as a sanity check.

Prepayment: If lump-sum payments are made, the formula is invalid; each prepayment changes the amortisation schedule.

Rate changes (variable loans): Requires separate B_k calculation for each rate period.

Real-World Applications

Banks use this formula to generate monthly loan statements showing outstanding balance. Mortgage borrowers use it to calculate how much they need to pay to close a loan early. Property valuers compare outstanding mortgage balance against property value to compute loan-to-value (LTV) ratios for refinancing eligibility. Bankruptcy courts use outstanding balances to determine secured versus unsecured debt. Accounting departments use it to record the principal and interest components of each loan payment.

Try This Formula

Loan EMI Calculator