Annuity-Due
A level annuity-due pays 1 at the start of each of n periods instead of the end. Rent, leases, and insurance premiums work this way — you pay to use the asset for the coming period, not the one just finished.
By the end you'll be able to compute \(\ddot{a}_n\) and \(\ddot{s}_n\), and explain why an annuity-due is always worth \((1+i)\) times its annuity-immediate counterpart.
Predict: for the same n and i, which is larger — annuity-immediate or annuity-due? Toggle the switch below and check the ratio readout.
The timeline below plots the same n payments of 1. Toggle to slide every payment one period earlier — from the end of its period (annuity-immediate) to the start (annuity-due) — and watch \(a_n\) become \(\ddot{a}_n\), always larger by exactly the factor \((1+i)\).
a₆ = 4.917324 · ä₆ = 5.212363 · ratio ä₆/a₆ = 1.0600 = (1+i)
An annuity-immediate pays at the end of each period; an annuity-due pays at the start. Slide every payment one period earlier and each one sits closer to today, so it's discounted less — the whole stream is worth more. That's the entire idea: same n payments, one uniform timing shift, more present value.
Due payments fall at times \(0, 1, \ldots, n-1\), so \(\ddot{a}_n = 1 + v + \cdots + v^{n-1} = (1-v^n)/(1-v)\). Using d = i·v = i/(1+i), this simplifies to \(\boxed{\ddot{a}_n = (1-v^n)/d}\) — the one change from \(a_n = (1-v^n)/i\) is dividing by d instead of i. Valuing at time n instead of time 0 gives the accumulated value \(\ddot{s}_n = ((1+i)^n-1)/d\). Because \((1+i)\cdot(1-v^n)/i = (1-v^n)/(i\cdot v) = (1-v^n)/d\), the two families are tied together by \(\boxed{\ddot{a}_n = (1+i)\cdot a_n}\) and \(\ddot{s}_n = (1+i)\cdot s_n\).
This is exactly how rent and insurance premiums work: you pay at the start of the month or year for the coverage or occupancy you're about to receive, not the period you just finished. Whenever a contract's payment schedule says "in advance," reach for \(\ddot{a}_n\) or \(\ddot{s}_n\), not the plain \(a_n\)/\(s_n\) formulas.
Value a 5-payment annuity-due of 100 per year at \(i = 6\%\). From the matching annuity-immediate, \(a_5 = 4.212364\), so \(\ddot{a}_5 = 1.06 \times 4.212364 = \) 4.465106, giving PV = 100 × 4.465106 = 446.51 — 25.27 more than the annuity-immediate's 421.24, which is exactly one year's interest on that 421.24 (6% × 421.24 ≈ 25.27).
Rent of 1000 is due at the start of each month for 12 months, at an effective monthly rate of 0.5%. First find \(a_{12}\) at 0.5%: it comes out to \(a_{12} \approx 11.619\). Now finish it: \(\ddot{a}_{12} = 1.005 \times 11.619 = \) ____, so PV = 1000 × \(\ddot{a}_{12}\) ≈ ____
Reveal the answer
\(\ddot{a}_{12} = 1.005 \times 11.619 \approx\) 11.677, so PV = 1000 × 11.677 ≈ 11,677.03. Set n = 12 and i = 0.5% on the slider above (nudge the rate slider as close as it goes) and compare the ratio readout to 1.005.
More info — two shortcuts that skip the (1+i) multiplication
Two identities fall straight out of the payment diagrams without multiplying by \((1+i)\) at all: \(\ddot{a}_n = 1 + a_{n-1}\) — the first due payment (at time 0) is worth exactly 1 today, and the remaining \(n-1\) payments (at times 1 through \(n-1\)) are just an \((n-1)\)-term annuity-immediate. Symmetrically, \(\ddot{s}_n = s_{n+1} - 1\) — value an \((n+1)\)-term immediate annuity up to time \(n+1\), then remove the extra payment that would have landed there. Both are handy sanity checks against the \((1+i)\) formula. See the Corporate Finance Institute link in Dive deeper below for more worked examples.
Check your understanding
A level annuity pays 1 at the start of each of 10 periods at \(i = 5\%\). Given \(a_{10} = 7.721735\), what is \(\ddot{a}_{10}\)?
For the same n and the same effective rate \(i > 0\), which is always at least as large: \(\ddot{a}_n\) or \(a_n\)?
An annuity-immediate pays 1 at the end of each of 6 years at \(i = 6\%\), with \(a_6 = 4.917324\). What is \(\ddot{a}_6\) for an annuity-due with the same n and i?
Recap
- An annuity-due pays 1 at the start of each of n periods (times 0 through n−1), instead of the end.
- \(\ddot{a}_n = (1-v^n)/d\) and \(\ddot{s}_n = ((1+i)^n-1)/d\) — the same numerators as \(a_n\) and \(s_n\), but divided by the discount rate d instead of i.
- \(\ddot{a}_n = (1+i)\cdot a_n\) and \(\ddot{s}_n = (1+i)\cdot s_n\) — shifting every payment one period earlier is always worth exactly one extra period of interest.
- Shortcuts: \(\ddot{a}_n = 1 + a_{n-1}\) and \(\ddot{s}_n = s_{n+1} - 1\).
Dive deeper
- Corporate Finance Institute — Annuity Due Explain beginning-of-period timing and the (1+i) uplift over an ordinary annuity.
- Marcel Finan — Exam FM study guide (PDF) Prove äₙ=(1−v^n)/d and the identity äₙ=(1+i)·aₙ.
Sources
- The Level Annuity-Due