FinanceLoans & AmortizationA-Level

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Remaining Loan Balance

Balance after k payments on an amortizing loan.

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B_{k} = P V (1 + r)^{k} - P M T \frac{( 1 + r ) ^{k} - 1}{r}

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Core idea

Overview

The Remaining Loan Balance formula, also known as the retrospective method, calculates the outstanding principal on a loan after a specific number of payments have been made. It determines the balance by finding the difference between the future value of the original loan amount and the future value of the series of payments made up to that point.

When to use: Use this formula when you need to determine the payoff amount for a fixed-rate amortizing loan at any specific point in its lifecycle. It is applicable in scenarios where the interest rate and payment amount remain constant, such as standard mortgages or auto loans. It assumes payments are made at the end of each period.

Why it matters: Understanding the remaining balance is crucial for financial planning, allowing borrowers to calculate their current equity in an asset. It provides the necessary information for making decisions about refinancing, early loan payoff, or selling property. In accounting, it ensures the accurate reporting of liabilities on a balance sheet.

Symbols

Variables

B_k = Remaining Balance, PV = Principal, r = Rate per Period, PMT = Payment, k = Payments Made

B_{k}

Remaining Balance

P V

Principal

r

Rate per Period

V a r iab l e

P M T

Payment

k

Payments Made

V a r iab l e

Walkthrough

Derivation

Derivation of Remaining Loan Balance After k Payments

The remaining balance equals the compounded principal minus the compounded value of payments already made.

Payments are equal and made at the end of each period.
Interest rate per period is constant r.
k payments have been made (0 ≤ k ≤ n).

Compound the original principal forward k periods:

If you made no payments, the balance after k periods would be the principal grown by (1+r)^k.

P V (1 + r)^{k}

Compute the future value of the k payments made:

k end-of-period payments form a geometric series; this is their value at time k.

P M T \frac{( 1 + r ) ^{k} - 1}{r}

Subtract payments-from-principal to get remaining balance:

This gives the outstanding balance immediately after the k-th payment. If r=0, it reduces to $B_{k}$ = PV − PMT·k.

B_{k} = P V (1 + r)^{k} - P M T \frac{( 1 + r ) ^{k} - 1}{r}

Result

B_{k} = P V (1 + r)^{k} - P M T \frac{( 1 + r ) ^{k} - 1}{r}

Visual intuition

Graph

The graph of the remaining balance (B) against the number of payments (k) follows an exponential decay curve. This shape occurs because the balance grows exponentially due to interest while being reduced by a constant payment amount, causing the remaining value to decrease at an accelerating rate until it reaches zero.

Graph type: exponential

Why it behaves this way

Intuition

Imagine two financial streams on a timeline: the initial loan amount growing exponentially due to interest, and a separate stream of all your payments also growing exponentially as if they were invested.

B_{k}

The outstanding principal amount of the loan at the end of period k.

This is the amount you still owe the lender after making 'k' payments, reflecting the portion of the original loan that has not yet been repaid.

The initial principal amount borrowed or the present value of the loan.

This is the starting amount of debt before any interest accrues or payments are made.

The periodic interest rate, expressed as a decimal.

This represents the cost of borrowing for each payment period. A higher 'r' means the loan grows faster due to interest, requiring more of each payment to service the interest.

The number of payments that have been made.

This is a counter for how many payment periods have passed. It determines how much the initial loan has grown and how many payments have contributed to reducing the balance.

PMT

The fixed payment amount made at the end of each period.

This is the regular amount paid by the borrower, which covers both the interest accrued during the period and a portion of the principal.

Signs and relationships

-: The negative sign indicates that the future value of the payments made (the second term) reduces the future value of the original loan amount (the first term).

Free study cues

Insight

Canonical usage

All monetary values (remaining balance, present value, payment amount) must be expressed in the same currency unit. Interest rates and payment counts are dimensionless.

Common confusion

A frequent mistake is using the annual interest rate directly in the formula without converting it to the periodic rate that matches the payment frequency, or using the percentage value (e.g., 5)

Dimension note

The interest rate 'r' is a dimensionless ratio representing the cost of borrowing per period. The number of payments 'k' is a dimensionless count of periods.

Unit systems

$B_{k}$ currency unit · The remaining loan balance after 'k' payments, expressed in a consistent currency.

$P V$ currency unit · The initial principal or present value of the loan, expressed in the same currency as B_k and PMT.

$P M T$ currency unit · The fixed payment amount per period, expressed in the same currency as B_k and PV.

$r$ dimensionless · The interest rate per period, expressed as a decimal (e.g., 0.05 for 5%). It must be consistent with the payment frequency (e.g., monthly rate for monthly payments).

$k$ dimensionless · The number of payment periods that have occurred. This count must be consistent with the period used for the interest rate 'r'.

One free problem

Practice Problem

A homeowner has a mortgage with an initial loan amount of 200,000. The monthly interest rate is 0.005 and they make monthly payments of 1,200. What is the remaining balance after 5 years (60 months)?

Principal200000 $

Rate per Period0.005

Payment1200 $

Payments Made60

Solve for: $B$

Hint: First calculate (1+r)^k, then find the future value of the principal and subtract the future value of the payments.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Remaining mortgage balance after 5 years.

Study smarter

Tips

Ensure the periodic interest rate (r) matches the frequency of the payments (k).
The value for 'k' should represent the number of payment periods that have already elapsed.
A negative result suggests the loan has been paid off and a surplus exists.
The formula is retrospective, meaning it does not require knowledge of the total loan term.

Avoid these traps

Common Mistakes

Using total n instead of k (payments made).
Using annual rate when payments are monthly.

Common questions

Frequently Asked Questions

The remaining balance equals the compounded principal minus the compounded value of payments already made.

Use this formula when you need to determine the payoff amount for a fixed-rate amortizing loan at any specific point in its lifecycle. It is applicable in scenarios where the interest rate and payment amount remain constant, such as standard mortgages or auto loans. It assumes payments are made at the end of each period.

Understanding the remaining balance is crucial for financial planning, allowing borrowers to calculate their current equity in an asset. It provides the necessary information for making decisions about refinancing, early loan payoff, or selling property. In accounting, it ensures the accurate reporting of liabilities on a balance sheet.

Using total n instead of k (payments made). Using annual rate when payments are monthly.