Loan Amortization & Outstanding Balance

Under the amortization method, a loan is repaid by regular payments, and the outstanding balance after any payment can be found two ways that always agree.

By the end you'll be able to compute a loan's outstanding balance both prospectively (present value of what's left) and retrospectively (loan minus payments made), and explain why the two never disagree.

Predict: does the outstanding balance fall in a straight line down to zero, or does it curve? Step the k slider through the payments and check.

This is a $10,000 loan repaid by n level payments at rate i. Drag k to see the balance just after payment k — the prospective value $R\cdot a_{n-k}$ (present value of what's left) always matches the retrospective value $L(1+i)^k - R\cdot s_k$ (accumulated loan minus accumulated payments), read out below in matching colors. Hover the marker or the curve for exact values.

Rate i = 8.0%

Term n = 5

Payment k = 2

R = L/a₅ = 10,000/3.99271 = 2,504.56 · Prospective B₂ = R·a₃ = 6,454.51 · Retrospective B₂ = L(1.08)²−R·s₂ = 6,454.51

Outstanding balance B_t after each payment, for t = 0…n

balance Bₜ selected payment k

Every payment on an amortizing loan does two jobs at once: it covers the interest that accrued since the last payment, then whatever is left over chips away at the principal still owed — so the balance shrinks with every payment, all the way down to zero at the last one.

Intuitive

If a loan of L is repaid by n level payments of R at rate i, the loan is exactly the present value of that payment stream, so $R = L/a_n$ — the same level annuity-immediate formula from the previous lesson. Once payments start, some of the loan is gone and some is still owed — that's the outstanding balance, and there are two equally valid ways to pin it down.

Formal

Let $B_t$ be the balance immediately after the t-th payment. Looking forward (prospective method), $B_t$ is the present value of the payments still to come: $B_t = R\cdot a_{n-t}$. Looking backward (retrospective method), $B_t$ is the original loan accumulated to time t minus the payments accumulated to time t: $B_t = L(1+i)^t - R\cdot s_t$. Substituting $L = R\cdot a_n$ into the retrospective formula and simplifying the annuity identities shows the two always reduce to the same value — it's a theorem, not a coincidence, because both are just "loan minus what's been paid," measured from opposite directions.

Applied

The prospective method is the natural one when you know the remaining payment stream (e.g., pricing a mortgage you're about to buy on the secondary market — you only care what's left). The retrospective method is the natural one when you know the history of past payments but the future stream might be about to change, such as right before a refinance.

Worked example

A $10,000 loan is repaid by 5 annual payments at $i = 8\%$. $a_5 = (1-1.08^{-5})/0.08 = 3.99271$, so $R = 10000/3.99271 = 2{,}504.56$. Balance right after the 2nd payment, prospectively: $B_2 = R\cdot a_3 = 2{,}504.5645\times 2.57710 = 6{,}454.51$. Retrospectively: $B_2 = 10000(1.08)^2 - 2{,}504.5645\cdot s_2 = 11{,}664 - 5{,}209.49 = $ 6,454.51 — both methods agree.

Your turn

Same idea, different numbers: a $12,000 loan repaid by 6 annual payments at $i = 5\%$. $a_6 = (1-1.05^{-6})/0.05 = 5.07569$, so $R = 12000/5.07569 = 2{,}364.21$. Balance right after the 3rd payment, prospectively: $B_3 = R\cdot a_3 = 2{,}364.21 \times 2.72324 = 6{,}438.31$. Now finish it retrospectively: $s_3 = ((1.05)^3-1)/0.05 = 3.1525$, so $B_3 = 12000(1.05)^3 - 2{,}364.21\times 3.1525 = 13{,}891.50 - $ ____

Reveal the answer

$2{,}364.21\times 3.1525 = 7{,}453.17$, so $B_3 = 13{,}891.50 - 7{,}453.17 =$ 6,438.33 — matching the prospective value above (small rounding differences come from truncating intermediate factors to 5 decimals). Set n = 6, i = 5%, k = 3 on the slider above and check it against the marker's tooltip.

More info — why the shape curves, not a line

Each payment's interest piece is $i$ times the previous balance, and the leftover principal piece grows geometrically as $(1+i)$ with every payment (see the next lesson on the principal/interest split). Early on, most of a level payment covers interest and only a little reduces principal, so the balance drops slowly. Later, the principal piece has grown to dominate, so the balance drops faster — a curve that's shallow at first and steep near the end, not a straight line. See the Finan study guide in Dive deeper for the full derivation.

Check your understanding

Question 1 of 3

An $8{,}000$ loan is repaid by 4 level annual payments at $i=6\%$. Find the outstanding balance immediately after the 1st payment.

Question 2 of 3

Why do the prospective and retrospective outstanding-balance formulas always give the same value at any time t?

Question 3 of 3

A $12{,}000$ loan is repaid by 6 level annual payments at $i=7\%$, so $R = L/a_6$ — the level-annuity-immediate present-value formula. Which expression gives the outstanding balance immediately after the 4th payment?

Recap

Amortization repays a loan of L with level payments $R = L/a_n$.
Prospective balance (payments remaining): $B_t = R\cdot a_{n-t}$.
Retrospective balance (loan minus payments made): $B_t = L(1+i)^t - R\cdot s_t$.
The two formulas always agree — they measure the same quantity from opposite directions.

Dive deeper

Amortization schedule — Wikipedia Explain how an amortizing loan's balance declines each period.
Marcel Finan — A Basic Course in the Theory of Interest (Exam FM study guide, PDF) Derive prospective and retrospective outstanding-balance formulas.

Sources

The Amortization Method — Prospective and Retrospective Balances