Compounding Frequency & Effective Rate
About this app
What happens when the bank credits interest several times a year? The balance grows as a staircase of interest pieces, each slightly larger than the last. An interest stack compares m = 1 with m periods and shows the small extra. The second card plots the effective annual rate p_eff = 100(1 + p/(m·100))^m − 100 against m: it rises but levels off at the continuous-compounding limit 100(e^{p/100} − 1). Slide example (€1000, 6 %, 10 years) and Bernoulli's limit e. Refutes "monthly gives 12 times as much" and "p_eff grows without bound".
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Interest several times a year
With compounding m times a year, the annual rate p is spread over m dates: every 1/m year, p/m % is credited and earns interest from then on. After n years Kₙ = K · (1 + p/(m · 100))^(m · n), with m = 4 for quarterly and m = 12 for monthly compounding. The effective annual rate p_eff is the rate that gives the same result with a single credit per year: 1 + p_eff/100 = (1 + p/(m · 100))^m. It exceeds p because every later interest piece is computed on a balance that has already grown.
What you see
The left card shows the balance as a staircase: each step is one interest piece, next to it the annual staircase and the continuous limit curve K · e^(pt/100). An interest stack puts the pieces on top of each other and highlights the extra over annual compounding. The right card plots p_eff against m on a logarithmic axis, with dots at m = 1, 2, 4, 12, 52 and 365, the nominal line p and the asymptote 100 · (e^(p/100) − 1). The Working panel lists q, Kₙ and p_eff with the numbers filled in.
Try it
Choose m with the pills 1, 2, 4, 12, 365 and ∞ or drag the dot in the right card. "Double m" refines the staircase step by step; it ends a little higher each time but stays below the e-curve. Drag the end point of the staircase to change the rate. The "10 years" example compounds €1000 at 6 %: annually €1790.85, monthly €1819.40, continuously €1822.12. The Bernoulli example with €1 at 100 % leads to Euler's number e = 2.718…
Frequently asked questions
- How do you calculate the effective annual rate for monthly compounding?
- With nominal annual rate p and m = 12 credits, p_eff = 100 · (1 + p/1200)^12 − 100. For p = 6 % this gives 6.168 %. It is larger than p because the interest credited each month earns interest for the rest of the year.
- Does monthly compounding give twelve times as much interest?
- No. Each credit is only p/12 %, so together they give about p % plus interest on interest. With €1000 at 6 % the first year yields €61.68 instead of €60, i.e. €1.68 more.
- What is continuous compounding?
- Continuous compounding is the limit of infinitely many compounding dates. Since (1 + p/(m · 100))^m tends to e^(p/100) as m → ∞, capital grows as K · e^(pt/100). The effective annual rate then reaches its upper bound 100 · (e^(p/100) − 1), which is 6.184 % at 6 %.
- What can I set in the app?
- You set principal K, rate p, term n and the number of compounding periods m (1 to 1000 or continuous) with sliders, pills or by dragging directly in the cards. The app uses Kₙ = K · (1 + p/(m · 100))^(m · n) without fees or taxes.
Subject: Finance
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