Annuity Due: Payments at the Start of Each Period

Ordinary vs. due

The timing of payments changes everything:

• Ordinary annuity: payment at the end of each period. Common for loans, mortgages, most savings plans.
• Annuity due: payment at the beginning of each period. Common for rent, insurance premiums, leases.

Every deposit in an annuity due earns interest for one extra period compared with the ordinary version. So: $FV_{\text{due}} = FV_{\text{ordinary}} \cdot (1 + r)$

Formulas

Future value (annuity due): $FV = PMT \cdot \dfrac{(1 + r)^n - 1}{r} \cdot (1 + r)$

Present value (annuity due): $PV = PMT \cdot \dfrac{1 - (1 + r)^{-n}}{r} \cdot (1 + r)$

The $(1 + r)$ factor is the timing premium — you earn (or avoid) one extra period of interest.

Step-by-step solution

Setup: $PMT = 200$/month, $r = 0.005$/month, $n = 240$ payments.

Step 1 — Ordinary-annuity factor: From the previous problem, $\dfrac{(1.005)^{240} - 1}{0.005} \approx 462.04$

Step 2 — Ordinary-annuity FV: $FV_{\text{ord}} = 200 \cdot 462.04 = 92{,}408$

Step 3 — Multiply by $(1 + r) = 1.005$: $FV_{\text{due}} = 92{,}408 \cdot 1.005 \approx \boxed{92{,}870}$

Step 4 — Difference vs. ordinary: $\Delta = 92{,}870 - 92{,}408 = 462$

That's one extra period of interest on the whole annuity — and it's identical to the FV factor times the monthly rate times one more month's deposit.

Why the extra period matters

Picture the first deposit: in an ordinary annuity it's invested for $n - 1$ months; in an annuity due it's invested for all $n$ months. Same for every subsequent deposit — they all get one extra month of compounding.

When each form shows up

• Annuity due: rent, leases, insurance premiums, upfront subscription plans, pension drawdowns paid on day 1.
• Ordinary annuity: loan amortization, bond coupons, end-of-period paychecks, typical retirement contributions.

If you're unsure which model applies, ask: "Do you pay on day 1, or on day 30?"

Common mistakes

• Applying $(1 + r)$ to $PMT$ instead of the whole FV. The correct form multiplies the ordinary factor, not the payment, by $(1 + r)$.
• Using annual $(1 + R)$ when payments are monthly. The multiplier must match the per-period rate.
• Mixing cash-flow sign conventions. Bank calculators often treat deposits as negative and withdrawals as positive — stay consistent.

Try it in the visualization

A side-by-side timeline shows deposits at month start vs. month end. Bars labelled "ordinary" and "due" run in parallel; the due bars pull ahead each period and the gap widens smoothly over time.