From finite frequency to a limit
Let P be initial amount, j nominal annual rate, m equal annual postings and t years. With no flows or costs, final value is P×(1+j/m)^(m×t). As m rises, each posting rate falls while the count rises. For fixed j and t the limit is P×exp(j×t). This is a mathematical limit, not an extra bank premium.
Use a separate continuous rate δ: F=P×exp(δ×t), and present value P=F×exp(−δ×t). Equivalent one-year effective rate is exp(δ)−1; inverse conversion is δ=ln(1+r), requiring 1+r>0. The same symbol in two sources does not guarantee the same definition or time unit.
Four results for a conditional 8%
For 100,000 roubles over one year at 8%: one posting gives 108,000; twelve equal postings about 108,299.95; 365 about 108,327.76; continuous limit about 108,328.71. Figures are fictional and unrounded internally. The daily-to-limit gap is small; “continuous” does not mean unlimited growth at fixed rate and term.
If exactly 8% effective annual growth is required, equivalent continuous rate is about 7.6961%, not 8%. Continuous 8% corresponds to effective growth about 8.3287%. These forms describe one final coefficient. Comparing printed percentages without identifying the measure is wrong.
Why reverse-check
Recover the final amount from the effective rate using P×(1+r) and compare it with P×exp(δ) for one year. Use consistent years for multiple periods. At δ=0 the factor is one. A negative continuous rate reduces the amount but exp remains positive. These cases expose sign and unit errors.
Where the limit stops answering the question
The continuous model omits weekends, posting dates, minimum balances, rounding and contribution timing. Each deposit must receive its own remaining-term factor; do not apply a full-year exponential to money deposited at year-end. With changing rates, constant δ is only a scenario. For a real contract, read the accrual procedure first. A theoretical limit does not remove access restrictions.
Common comparison errors
Do not use continuous interest as a synonym for daily payment. Do not call nominal rate effective without conversion. Do not claim a benefit from frequency when effective outcomes are equal. Do not transfer a mathematical limit to uncertain market return. This extends [effective rate](/guides/effective-rate/) and complements [capitalisation calendar](/guides/capitalization-calendar/).