Immediate annuity: what will it pay?

How it works

The premium buys a level payout A, made m times a year (12, 4, or 1), with the first payout one period after you buy. The calculator sets the premium equal to the expected present value of all payouts:

premium = A × Σ v^t × S(t), so A = premium ÷ Σ v^t × S(t)

• t = j ÷ m years for payout j = 1, 2, 3, …
• v = 1 ÷ (1 + rate), where the rate is a yearly effective rate (compounded once a year), so v^t discounts a payout t years away.
• S(t) is the weight of payout j: 1 while t is within the guaranteed years (t ≤ n), and otherwise the chance you are alive at t.
• The sum runs over m × (the larger of 120 − age and n) payouts.

Chance of being alive. From the Social Security 2023 period life table, q(a) is the chance that a person of exact age a dies within a year, for men or for women. For a whole number of years k after buying at age x:

S(k) = (1 − q(x)) × (1 − q(x + 1)) × … × (1 − q(x + k − 1))

Within a year of age, deaths are spread evenly, so for a fraction s of a year (0 < s < 1):

S(k + s) = S(k) × (1 − s × q(x + k))

Nobody lives past 120: q(119) is the last row, and the chance of being alive at 120 or later is 0.

Other results.

• Income per year = A × m; payout rate = income per year ÷ premium.
• Expected total payouts = Σ A × S(t), not discounted; expected years of payouts = that ÷ income per year.
• Present value of expected payouts = Σ A × v^t × S(t), which equals the premium.
• Premium back: the number of payouts k = premium ÷ A, rounded up, is reached at t = k ÷ m. The calculator shows age + t and S(t), the chance payouts are still made then.
• The table has one row per year after you buy: your age at the end of the year, S at the end of the year as a percentage, what is paid that year if payouts continue, the expected payouts, and their present value.

Assumptions

• This is a fair-price estimate. It leaves out the insurer’s costs and profit, state premium taxes, and income tax, so real quotes are usually lower.
• Survival is from the general population table, not an insurer’s annuitant table.
• The rate is the same for every year; the payout is level, with no cost-of-living increase.
• This is an estimate for planning, not financial advice.

Rules

• Premium: $1,000 to $100,000,000. Age: a whole number from 50 to 90. Rate: 0% to 15%. Guaranteed years: a whole number from 0 to 40.
• Money is shown to the cent, rounded half up; the calculation keeps full precision.

When the data is out of date

The life table is the SSA 2023 period table (2026 Trustees Report). When SSA publishes a newer table, the page keeps using this one until it is updated; survival changes little from one table to the next.

Worked examples by hand

$96,000 at age 90, 40 years guaranteed, 0%, monthly. 40 years from 90 run to age 130, past the end of the table, so every one of the 40 × 12 = 480 payouts is guaranteed, with weight 1 and no discount. A = 96,000 ÷ 480 = $200.00 a month. Payouts reach $96,000 after all 480, at age 90 + 40 = 130 (to your heirs), with certainty.

$100,000 at 65, man, 0%, monthly. With no interest, v = 1, so the premium equals A × (the expected number of payouts). The expected number is Σ S(j ÷ 12), from the life table: 216.9 payouts, about 18.07 years. A = 100,000 ÷ 216.9 = $461.05 a month, and the expected total of payouts is exactly $100,000.

$100,000 at 65, man, 4.5%, monthly. The same sum with each weight times 1.045^−t gives 139.8, so A = 100,000 ÷ 139.8 = $715.24 a month, $8,582.89 a year (an 8.58% payout rate). The payouts add up to $100,000 after 140 payouts, at age 65 + 140 ÷ 12 = 76.7; the chance of being alive then is 74.8%.

The same for a woman. Her weights sum to 153.5, so A = $651.44 a month.

$200,000 at 70, man, 5%, yearly, 10 years guaranteed. The first 10 payouts have weight 1; the rest use the life table. A = $19,353.28 a year. The premium comes back after 11 payouts, at age 81, with a 66.7% chance of being alive then.