What is an annuity due?
– Definition: An annuity due is a sequence of equal payments made at the beginning of each period (month, quarter, year). Because payments occur at the start of each interval, each cash flow is effectively invested one period earlier than with an ordinary annuity (payments at the end of each period).
Why timing matters
– If you receive payments, an annuity due is usually better because you can use or invest each payment immediately.
– If you make payments, an annuity due is less favorable because you must pay sooner and lose the use of the funds during the period.
– The difference in timing changes present-value (PV) and future-value (FV) calculations: annuity-due values equal ordinary-annuity values multiplied by (1 + r), where r is the per-period interest rate.
Key definitions
– Ordinary annuity: equal payments made at the end of each period.
– Immediate annuity: a contract that begins making periodic payments soon after a lump-sum purchase (typically within one payment period); contrast with a deferred annuity, which starts later.
– Present value (PV): the current worth of a future cash-flow stream discounted at rate r.
– Future value (FV): the accumulated value of a series of payments at a future date, grown at rate r.
– Time value of money: principle that a sum today is worth more than the same sum in the future because of its earning potential.
Formulas (level payments, constant rate, payments at period start)
– Present value of an annuity due:
PV = Pmt × [1 − (1 + r)^(-n)] / r × (1 + r)
– Future value of an annuity due:
FV = Pmt × [(1 + r)^n − 1] / r × (1 + r)
Variables:
– Pmt = payment per period
– r = interest rate per period (decimal)
– n = number of periods
Step-by-step checklist for valuing an annuity due
1. Confirm payments are equal and occur at the beginning of each period.
2. Choose the correct per-period interest rate. If annual nominal rate is given but payments are monthly, divide rate by 12.
3. Ensure number of periods n matches the rate’s period (years × periods per year).
4. Use the annuity-due formulas above (or compute ordinary-annuity value and multiply by 1 + r).
5. Check units and round consistently.
Worked numeric example
– Problem: You will receive $1,000 at the start of each year for 10 years. The annual interest rate is 3% (r = 0.03). What are the PV and FV of these payments?
– Step 1: Compute the ordinary-annuity PV factor: [1 − (1 + r)^(-n)] / r
(1 + r)^n = 1.03^10 ≈ 1.3439164 → (1 + r)^(-n) ≈ 0.744093
Factor = (1 − 0.744093) / 0.03 ≈ 0.255907 / 0.03 ≈ 8.530235
– Step 2: PV of ordinary annuity = Pmt × factor = 1,000 × 8.530235 ≈ $8,530.24
– Step 3: Convert to annuity due: PV_due = PV_ordinary × (1 + r) = 8,530.24 × 1.03 ≈ $8,786.14
– For FV:
– Ordinary FV factor = [(1 + r)^n − 1] / r = (1.3439164 − 1) / 0.03 ≈ 11.463879
– FV ordinary = 1,000 × 11.463879 ≈ $11,463.88
– FV due = FV ordinary × (1 + r) ≈ 11,463.88 × 1.03 ≈ $11,807.80
– Result: PV ≈ $8,786.14; FV ≈ $11,807.80 (rounded).
Which is better: annuity due or ordinary annuity?
– For the recipient: annuity due typically has a higher PV because you receive funds earlier.
– For the payer: ordinary annuity is usually preferred because payments are delayed until the
…end of the period, reducing the present value of payments (for a given nominal amount).
When annuity-due timing matters
– Accounting and pricing: You must recognize whether cash flows occur at the beginning or end of periods when valuing leases, pensions, rent, or installment contracts.
– Cash-flow planning: Receiving payments at the beginning increases liquidity sooner; paying at the beginning increases financing needs.
– Comparative analysis: To compare two offers, convert both to the same timing convention (PV or FV) before judging which is better.
Key formulas (assume constant rate r per period, n periods, payment PMT at start of each period)
– Present value of an annuity due:
PV_due = PMT × [1 − (1 + r)^−n] / r × (1 + r)
– Future value of an annuity due:
FV_due = PMT × [(1 + r)^n − 1] / r × (1 + r)
Note: The factor (1 + r) converts the ordinary-annuity formula (payments at period end) into the annuity-due equivalent.
Worked example (different from earlier)
– Problem: You will receive $500 at the start of each year for 5 years. Interest rate = 4% per year. Find PV.
– Step 1 — ordinary PV factor: [1 − (1 + r)^−n] / r
(1.04)^5 = 1.2166529 → (1.04)^−5 ≈ 0.821927
1 − 0.821927 = 0.178073
0.178073 / 0.04 = 4.4518275
– Step 2 — PV ordinary = 500 × 4.4518275 = $2,225.91
– Step 3 — convert to annuity due: PV_due = PV_ordinary × (1 + r) = 2,225.91 × 1.04 ≈ $2,314.95
How to calculate in common tools
– Excel / Google Sheets:
PV function: =PV(rate, nper, pmt, [fv], [type])
Use type = 1 for beginning-of-period (annuity due). Example: =PV(0.04,5,-500,0,1) → returns ≈ 2314.95
FV function: =FV(rate, nper, pmt, [pv], [type]) with type = 1 for annuity due.
– Financial calculator (TI BA II Plus / similar):
1) Set to BGN mode (2nd + PMT → BGN → 2nd + ENTER → 2nd + CPT to exit
2) Enter N (number of periods) — press N, type the periods, ENTER.
3) Enter I/Y (interest rate per period) — press I/Y, type the periodic rate (not percent sign), ENTER.
4) Enter PMT (payment) — press PMT, type the payment amount. Use a negative sign for cash outflows (payments you make) and positive for cash inflows, ENTER.
5) Enter FV (future value) — press FV, type the future value
6) Compute PV (present value) — press CPT (compute), then PV. The calculator will display the present value for an annuity due using the entries and BGN mode you set.
7) Exit BGN (return to END mode) when finished — press 2nd + PMT, then 2nd + ENTER (this toggles BGN/END), then 2nd + CPT to exit the settings menu. Confirm the display shows “END” if you want ordinary-annuity behavior again.
Worked numeric example (TI BA II Plus)
– Inputs: N = 5, I/Y = 4, PMT = -500 (you pay 500 at the start of each period), FV = 0, BGN mode on.
– After entering values, press CPT then PV. The calculator returns ≈ 2314.95, which equals the present value of a 5-period annuity due with 4% per period and 500 payment at the beginning of each period.
Formulas (clear and reusable)
– Present value of an annuity due:
PVannuity_due = PMT * [1 − (1 + r)^−n] / r * (1 + r)
where r = interest rate per period, n = number of periods, PMT = payment each period (positive for receipts, negative for payments).
– Future value of an annuity due:
FVannuity_due = PMT * [ (1 + r)^n − 1 ] / r * (1 + r)
Numeric check (from the worked example above)
– Ordinary-annuity PV factor for r = 4%, n = 5: (1 − 1.04^−5) / 0.04 ≈ 4.45182
– Multiply by PMT = 500 → PVordinary ≈ 2,225.91
– Convert to annuity due by multiplying (1 + r) = 1.04 → PVannuity_due ≈ 2,314.95
Excel / Google Sheets quick reference
– PV function (annuity due): =PV(rate, nper, pmt, [fv], 1) — the final argument “type = 1” indicates payments at the beginning of each period.
– FV function (annuity due): =FV(rate, nper, pmt, [pv], 1)
– Sign convention: Use negative for cash outflows (payments you make) and positive for cash inflows (payments you receive) to avoid sign-confusion in results.
Checklist before you compute
– Determine whether payments are at the beginning (annuity due) or end (ordinary annuity) of each period. Use type = 1 (or BGN mode) only for annuity due.
– Convert APR to the periodic rate: r = APR / periods_per_year (e.g., monthly rate = APR/12).
– Use consistent units for rate and nper (both monthly, both quarterly, etc.).
– Decide on sign convention for cash flows and be consistent.
Common mistakes to avoid
– Forgetting to set BGN mode or type = 1 for annuity due — that will understate PV/FV.
– Mixing annual rates with monthly periods without converting the rate.
– Entering PMT with wrong sign relative to PV/FV — this flips the sign of the result and can be confusing.
Short alternative: multiplying ordinary annuity result
– You can compute PV (