What is an annuity table?
An annuity table is a ready-made lookup chart that converts a stream of equal periodic payments into a single present-value factor. Instead of computing the full formula each time, you find the cell that matches your discount (interest) rate and number of periods, then multiply that factor by the payment amount to get the present value.
Key definitions
– Annuity: A series of equal cash flows paid at regular intervals (for example, $5,000 each year).
– Present value (PV): The current equivalent value of future cash flows after discounting for the time value of money.
– Discount rate (r): The rate used to convert future dollars into today’s dollars; it reflects opportunity cost and risk.
– Ordinary annuity: Payments occur at the end of each period (e.g., year-end).
– Annuity due: Payments occur at the beginning of each period (e.g., start of each year).
– Annuity table: A table of precomputed factors for (1 − (1 + r)^−n) / r for various r and n (for ordinary annuities); an annuity-due factor is the ordinary factor multiplied by (1 + r).
Why use an annuity table
– Speed: avoids repeated calculation of powers and fractions.
– Simplicity: good for hand calculations, teaching, and quick comparisons.
– Transparency: you can see how PV changes with rate and term.
Core formula (ordinary annuity)
PV = PMT × [(1 − (1 + r)^−n) / r]
where:
– PV = present value of the annuity
– PMT = payment each period
– r = discount rate per period (decimal)
– n = number of periods
For an annuity due, multiply the ordinary-annuity PV by (1 + r):
PV_due = PV_ordinary × (1 + r)
Step-by-step checklist for using an annuity table
1. Confirm payment frequency and period count (n).
2. Determine the appropriate discount rate per period (r).
3. Decide whether payments are at period end (ordinary) or start (due).
4. Find the intersection cell for r and n in the annuity table to get the factor.
5. If using an annuity-due and the table gives ordinary factors, multiply the factor by (1 + r).
6. Multiply the factor by PMT to get PV.
7. Compare PV to any lump-sum offered; account for taxes, fees, or other considerations separately.
Worked numeric example (step-by-step)
Scenario: You can take either
– $50,000 per year for 25 years (payments at year-end), or
– a lump sum of $650,000 today.
Assume a discount rate of 6% (r = 0.06) and ordinary annuity.
1. Compute (or look up) the ordinary-annuity factor for r = 6% and n = 25.
– Factor ≈ 12.7834 (this equals (1 − (1.06)^−25) / 0.06, rounded).
2. Multiply factor by PMT:
– PV = $50,000 × 12.7834 ≈ $639,170 (rounded
3. Compare PV to the lump sum and interpret the result.
– PV (ordinary annuity) ≈ $639,170 (from step 2).
– Lump-sum offer today = $650,000.
– Difference = $650,000 − $639,170 = $10,830. The lump sum is larger by about $10.8k under the assumed 6% discount rate and year‑end payments.
Interpretation: if you believe you can earn 6% on money and accept year‑end payments, the lump sum is slightly preferable in pure present‑value terms. That conclusion depends entirely on the 6% assumption and ignores taxes, fees, and personal circumstances.
4. Sensitivity checks (quick variations).
– If your discount rate is 5% instead of 6%:
– Ordinary annuity factor ≈ 14.093 → PV = $50,000 × 14.093 ≈ $704,650 (annuity preferred).
– If your discount rate is 7%:
– Ordinary annuity factor ≈ 11.654 → PV = $50,000 × 11.654 ≈ $582,700 (lump sum preferred).
– Breakeven (approximate internal rate where PV = $650,000): factor must equal 13 (because 650,000/50,000 = 13). Solving numerically gives r ≈ 5.82% (approx). If your expected discount/return is below ~5.82%, the annuity wins; above it, the lump sum wins.
5. Check the “annuity-due” variant (payments at period start).
– If payments are at the beginning of each year (annuity-due), multiply the ordinary factor by (1 + r).
– With r = 6%: annuity-due factor ≈ 12.7834 × 1.06 = 13.5504 → PV = $50,000 × 13.5504 ≈
≈ $677,520.
Interpretation (annuity-due vs. ordinary)
– Because annuity-due payments arrive at the period start, the PV is larger by the factor (1 + r). At r = 6% the ordinary annuity PV was about $639,170; the annuity-due PV is $677,520. That flips the choice versus a $650,000 lump sum: ordinary annuity (payments at year-end) loses at 6%, annuity-due (payments at year-start) wins.
Formulas and how to compute
1. Ordinary annuity (payments at period end)
PV = PMT × [1 − (1 + r)^−n] / r
– PMT = payment per period (positive for receipts)
– r = discount/interest rate per period (decimal)
– n = number of periods
2. Annuity-due (payments at period start)
PV_due = PV_ordinary × (1 + r)
Worked numeric recap (assumptions)
– PMT = $50,000, n =