Annuity calculator
Annuities-certain with the notation the exams use. Set the effective rate, term and payment, and every standard value updates live: in arrears, due, continuous and increasing, present and accumulated, plus the five equivalent ways of quoting the same interest rate.
Present values at time 0
Accumulated values at time n
Accumulated values are the present values rolled up: s = a × (1+i)ⁿ.
The same rate, quoted every way
All five describe identical growth: 1 invested for a year becomes 1 + i under any of them. δ compounds continuously; i⁽ᵖ⁾/p is applied p times a year.
The formulas behind the numbers
Everything reduces to the discount factor v = 1/(1+i). An annuity paying 1 at the end of each year for n years is worth the geometric sum of discounted payments:
Moving the payments to the start of each year swaps i for the discount rate d in the denominator; paying continuously swaps it for the force of interest δ. Accumulated values are the same quantities rolled forward n years at (1+i)ⁿ. The increasing annuity (Ia) pays 1, then 2, up to n, and is the workhorse behind premium and benefit escalation questions.
Why five versions of one interest rate?
Because payments arrive at different frequencies. The effective rate i compounds once a year; a mortgage quoted at i⁽¹²⁾ applies a twelfth of the nominal rate each month; bonds discount with d; and anything in continuous time uses δ. They all describe the same growth, and converting between them is the first skill CM1 tests. A handy sense check: for any positive rate, d < d⁽ᵖ⁾ < δ < i⁽ᵖ⁾ < i.
Studying CM1? Memori is a flashcard app built by actuarial students; the shop carries a ready-made CM1 notation and formulas set, and the app renders all of this notation natively. See the notation cheat sheet or join the beta.
Annuities-certain only: these values involve no mortality. For education, not financial advice.