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Sub-annual Payments & Equivalent Annual Amount

About this app

Monthly payments with annual interest credit: interest strips as a Gauss staircase, equivalent annual deposit and remaining balance for withdrawals over n years.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

Sub-annual payments, annual interest

If the amount u is paid in m times a year and interest is credited only at year end, interest within the year is pro rata: a payment that sat on the account for k/m of a year earns u · k/m · p % interest. With payments in arrears the payments sit for (m−1)/m, (m−2)/m, …, 0 years. Their sum is the Gauss sum, (m−1)m/2 sub-periods. This gives the equivalent annual deposit K₁ = u · (m + p(m−1)/200): the single year-end amount with the same effect as the m payments. With it the annuity formula Kₙ = K₁ · (qⁿ − 1)/(q − 1) with q = 1 + p/100 holds over n years.

What you see

The left card shows one year: one row per payment with the payment and an interest strip until year end. The strips form a staircase. A rotated copy completes it to a rectangle of m · (m−1) cells; the staircase is exactly half of it. Next to it the payments stack into a column with the interest piece on top. The right card shows the years: every bar is made of equal blocks K₁ plus the interest. In withdrawal mode the amount a is taken at the start of every sub-period; then (m+1) replaces (m−1) and the equivalent annuity is a · (m + (m+1)p/200).

Try it

Play the year and watch the payments arrive and their strips grow. Click a row to see the interest of that single payment, and drag the top edge of the interest piece to change the interest rate. Switch between 2, 4 and 12 sub-periods and compare the staircases. Change the term: the number of blocks grows, their size stays the same. The working panel puts all numbers into the formulas and compares with monthly compounding.

Frequently asked questions

What is an equivalent annual payment for sub-annual deposits?
It is the single amount at year end that has the same effect as m equal sub-annual payments u with pro rata interest. For payments in arrears K₁ = u · (m + p(m−1)/200). With 100 € per month at 6 % this is 1,233 €, not 1,200 €.
Why does (m−1)/2 appear in the formula for sub-annual deposits?
The payments sit on the account for (m−1)/m, (m−2)/m, …, 0 years. By the Gauss sum, 1 + 2 + … + (m−1) = (m−1)m/2. Multiplied by u · p/(m · 100) this gives the interest Z₁ = u · p(m−1)/200.
How do you compute the balance with monthly withdrawals?
If the amount a is withdrawn at the start of each of the m sub-periods, the balance after one year is R₁ = K · q − a · (m + (m+1)p/200), and after n years Rₙ = Kqⁿ − a · (m + (m+1)p/200) · (qⁿ − 1)/(q − 1). The term a · (m + (m+1)p/200) is the equivalent annuity.
What can I adjust in this app?
You choose deposit (in arrears) or withdrawal (in advance), the amount per sub-period, the initial capital, the number of sub-periods from 1 to 12, the annual interest rate and the term up to 30 years. The app applies pro rata interest within the year and credits it at year end.

Subject: Finance

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