Annuity Loan – Equal Instalments
About this app
Same payment, shifting split: interest and principal per year, balance Rₙ = S qⁿ − A(qⁿ − 1)/(q − 1), interest line S(q − 1), annuity and term with formulas, in arrears and in advance, linear repayment for comparison.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Why the same payment repays more every year
With an annuity loan you pay the same amount A every year. Each payment first settles the interest on the outstanding balance, Z = R · (q − 1) with q = 1 + p/100; only the rest repays the debt. As the balance falls, the interest falls and the principal share grows geometrically by the factor q. The balance after n years is Rₙ = S · qⁿ − A · (qⁿ − 1)/(q − 1). This gives the annuity A = S · q^N (q − 1)/(q^N − 1) and the term N = ln(A/(A − S(q − 1)))/ln q.
What you see
The top card shows one bar of height A per year, with the principal part at the bottom and the interest part on top. The dividing line moves up year by year, and a dotted curve traces the geometric growth of the principal. Bottom left, the balance falls to the end point N. Bottom right, the debt grows like a deposit with compound interest and the payments grow like a savings plan: the gap is the balance, the crossing is the end of repayment. The Working panel puts every number into the formulas.
Try it yourself
Set debt, interest rate and term and read off the annuity, or set the payment and let the app compute the term. Drag the top edge of the bars down: if the payment is at or below the interest line S · (q − 1), nothing is ever repaid and the balance stays or grows. Switch to payments in advance, or show linear repayment with a fixed principal S/N for comparison. "Year by year" runs through the repayment schedule.
Frequently asked questions
- How do you calculate the annuity of a loan?
- The yearly annuity is A = S · q^N · (q − 1)/(q^N − 1) with debt S, term N in years and q = 1 + p/100. For €100,000 at 5 % over 10 years the payment is €12,950.46 per year. For payments in advance at the start of each year, replace A by A · q.
- Why does the principal share of an annuity grow?
- Each payment first covers the interest on the current balance. Since every repayment reduces the balance, next year's interest is smaller and more of the same payment goes towards the principal. The principal share grows by exactly the factor q = 1 + p/100 each year.
- Does doubling the term halve the payment?
- No. For €100,000 at 5 % the payment falls from €12,950.46 over 10 years only to €8,024.26 over 20 years, about 62 percent. Interest runs twice as long on a debt that is barely reduced at first; total interest rises from €29,504.57 to €60,485.17.
- What can I adjust in the app?
- You set debt, interest rate and either the term or the annuity, drag the payment directly at the bars and the end point N on the balance, choose payments in arrears or in advance, and show linear repayment for comparison. The app uses yearly payments and shows every formula with the numbers filled in.
Subject: Finance
More from Finance
- Savings Formula – Fixed Instalments
- Arithmetic vs. Geometric
- Loan types compared
- Perpetuity & Drawdown
- Geometric Series – the Proof
- Sub-annual Payments & Equivalent Annual Amount
- Compounding Frequency & Effective Rate
A VisuApp by heyprof: interactive, free in your browser, no sign-up. What is a VisuApp? · All apps