How annuity savings is calculated
Convert the annual effective return to the payment-period rate. Future value = opening balance × (1+r)^n + payment × ((1+r)^n−1) ÷ r. Beginning-of-period deposits multiply the payment term by 1+r. At zero return the payment factor is n.
Worked example
$100 initially plus $10 monthly for two years at 0% grows to $340. Changing payment timing has no effect when the return is zero.
Deferral and payout phase
After the saving phase you can model a payout. Deferral years let the saved value keep growing with no new payments: value at payout = FV × (1 + i)^(payments per year × deferral years). The payout then either spreads that value over a set period, payment = value ÷ {[1 − (1 + i)^−n] ÷ i}, or shows how long a chosen withdrawal lasts. Payouts use the same return, frequency and timing as the saving phase.
Worked example: $100,000 today with no further payments at a 5% effective return, 1 year of saving and 1 year of deferral: 100,000 × 1.05 × 1.05 = $110,250. Paid out over 20 yearly payments at the end of each year, the annuity factor is 12.4622, so each payment is 110,250 ÷ 12.4622 = $8,846.75. Withdrawing $10,000 a year from $100,000 at 5% instead lasts ln 2 ÷ ln 1.05 = 14.2 years.