Splitting Each Payment into Interest and Principal
Every payment on an amortizing loan does two jobs: it pays the interest that accrued since the last payment, and whatever is left over chips away at the principal. On a level-payment loan that split shifts steadily — interest shrinks, principal repaid grows — in a strikingly regular geometric pattern.
By the end you'll be able to split any payment into its interest and principal pieces, and use the fact that principal repaid grows by a factor of \((1+i)\) each period to jump straight to any payment's split without rebuilding the whole schedule.
Predict: as the loan progresses, does the interest part of each payment grow or shrink — and what about the principal part? Step through k and check.
Each bar below is one level payment R on a 10,000 loan, split into its interest portion (bottom) and principal-repaid portion (top). Every bar is the same total height R, but the mix inside it shifts: interest is \(i\) times the balance carried into that period, and it shrinks every period as the outstanding balance shrinks. Drag k to pin one payment and read its exact split; drag i and n to reshape the whole schedule.
Payment 2 of 5: interest \(I_2\) = 663.64 · principal \(P_2\) = 1840.92 · growth ratio \(P_2/P_1 = 1+i =\) 1.0800
Think of each payment as two envelopes. The first envelope pays the bank for letting you keep the money one more period — that's interest, and it's proportional to how much you still owe. Whatever's left over after filling that envelope goes into the second envelope, chipping away at the debt itself — that's principal. Because the debt keeps shrinking, the first envelope needs less and less each time, so the second envelope keeps growing to make up the difference.
For a payment R made at time t, with \(B_{t-1}\) the balance just after the previous payment: \(I_t = i \cdot B_{t-1}\) and \(P_t = R - I_t\), so \(B_t = B_{t-1} - P_t\). Writing the prospective balance as \(B_{t-1} = R \cdot a_{n-t+1}\) gives \(P_t = R - i\cdot R\cdot a_{n-t+1} = R\cdot(1 - i\cdot a_{n-t+1}) = R\cdot v^{n-t+1}\) — a plain geometric sequence in t. Consecutive principal payments satisfy \(P_{t+1}/P_t = v^{n-t}/v^{n-t+1} = 1/v = 1+i\): principal repaid grows by exactly the factor \((1+i)\) every period, and correspondingly \(I_t = R\cdot(1-v^{n-t+1})\) shrinks.
A homeowner comparing year-1 and year-20 mortgage statements notices almost the whole payment goes to interest at first and almost all of it goes to principal near the end — the same level payment, but a completely different split. That matters for tax planning (mortgage interest is often deductible, so early years shelter more income) and for deciding whether an early payoff saves much interest at all: late in the loan there's very little interest left to save.
10,000 loan, 5 annual payments, \(i = 8\%\): \(R = 2504.56\). \(I_1 = 0.08 \times 10000 = \) 800.00, so \(P_1 = 2504.56 - 800.00 = \) 1704.56 and \(B_1 = 8295.44\). Next period: \(I_2 = 0.08 \times 8295.44 = \) 663.64, so \(P_2 = 2504.56 - 663.64 = \) 1840.92. Check the ratio: \(P_2/P_1 = 1840.92/1704.56 \approx\) 1.08 \(= 1+i\), confirming the geometric growth.
Same idea, different numbers: 10,000 loan, 4 annual payments, \(i = 6\%\). \(a_4 = (1-1.06^{-4})/0.06 = 3.465106\), so \(R = 10000/3.465106 = 2885.91\). \(I_1 = 0.06 \times 10000 = 600.00\), so \(P_1 = 2885.91 - 600.00 = 2285.91\) and \(B_1 = 7714.09\). Now finish it: \(I_2 = 0.06 \times 7714.09 = 462.85\), so \(P_2 = 2885.91 - 462.85 = \) ____
Reveal the answer
\(P_2 = 2885.91 - 462.85 = \) 2423.06. Check the growth ratio: \(P_2/P_1 = 2423.06/2285.91 \approx 1.06 = 1+i\), exactly as the formula predicts. Set i = 6% and n = 4 on the sliders above and step k from 1 to 2 to see these same numbers on the chart.
More info — how the two columns sum up
Every period the two envelopes add up to the same payment: \(I_t + P_t = R\), so summing across all n periods, \(\sum I_t + \sum P_t = nR\). The principal column has to repay the whole loan by the last payment, so \(\sum P_t = L\) exactly — and that forces \(\sum I_t = nR - L\), the loan's total interest cost. This is exactly the bookkeeping an amortization schedule tabulates period by period: payment, interest, principal, and the resulting balance, all four numbers reconciling at every row. See the Corporate Finance Institute link in Dive deeper below for a fully tabulated schedule.
Check your understanding
A 10000 loan is repaid by 5 level annual payments at \(i = 8\%\) (\(R = 2504.56\)). After the first payment the balance is \(B_1 = 8295.44\). What is the interest portion \(I_2\) of the second payment?
Under a level-payment loan, as the loan progresses from the first payment to the last, what happens to the interest portion and the principal portion of each payment?
Why does the interest portion of a payment equal i times the balance right after the previous payment, rather than i times the original loan amount?
For the 10000/8%/5-year loan, \(P_3 = 1988.20\). Using the geometric growth rule \(P_{t+1} = P_t \cdot (1+i)\), what is \(P_4\)?
Recap
- Each payment splits into interest \(I_t = i\cdot B_{t-1}\) (charged on the balance carried IN) and principal repaid \(P_t = R - I_t\).
- On a level-payment loan, \(P_t = R\cdot v^{n-t+1}\): principal repaid grows geometrically, \(P_{t+1}/P_t = 1+i\), while interest \(I_t = R\cdot(1-v^{n-t+1})\) shrinks by the same factor.
- Every payment splits the same total R differently — early payments are mostly interest, late payments are mostly principal.
- An amortization schedule tabulates R, \(I_t\), \(P_t\), and \(B_t\) each period; the principal column sums to the loan L and the interest column sums to \(nR - L\).
Dive deeper
- Amortization Schedule — Corporate Finance Institute Show how each payment splits into shrinking interest and growing principal.
- Amortization schedule — Wikipedia Tabulate interest, principal, and balance for each period.
Sources
- Splitting Each Payment into Interest and Principal