Hong Life CalculatorsUnderstanding the lottery annuity vs lump sum calculator — concept and principle
Big US lottery jackpots are advertised as the total of an annuity paid over many years, but winners can instead take a smaller cash lump sum. This calculator finds the yearly payments, their present value at your discount rate and the break-even rate at which both choices are worth the same.
How it is calculated
- With N payments growing by g a year and summing to the jackpot J, the first payment = J × g ÷ ((1 + g)^N − 1), or J ÷ N if g = 0.
- Payment k = first × (1 + g)^(k − 1).
- The first payment is made at once and the rest on each anniversary, so present value PV = Σ payment_k ÷ (1 + d)^(k − 1).
- Break-even rate: the d at which PV equals the cash option, found by bisection.
The principle
Money received later is worth less than money today, because today's money can be invested. Discounting each annuity payment back to today makes it comparable with the lump sum. If the annuity's present value at your rate is higher than the cash option, the annuity offers more value. If you could earn more than the break-even rate on the cash, the lump sum comes out ahead. Taxes are not included; both are taxed as income when received.
Worked example
Default inputs: jackpot 500,000,000 USD, cash 230,000,000 USD, 30 payments, 5% yearly increase, discount rate 4%.
- First payment = 500,000,000 × 0.05 ÷ (1.05^30 − 1) ≈ 7,525,718 USD.
- Last payment = 7,525,718 × 1.05^29 ≈ 30,976,874 USD.
- Discounting all 30 payments at 4% gives PV ≈ 260,267,110 USD.
- PV − cash = 30,267,110 USD, so the annuity is worth more at 4%.
- Break-even discount rate ≈ 4.87%; cash is 46% of the jackpot.
| Advertised jackpot (sum of all annuity payments) | Present value of the annuity |
|---|---|
| 1.25e+08 | $65,066,778 |
| 2.5e+08 | $130,133,555 |
| 3.75e+08 | $195,200,333 |
| 5e+08 | $260,267,110 |
| 6.25e+08 | $325,333,888 |
| 7.5e+08 | $390,400,665 |
| 1e+09 | $520,534,220 |
How to read the result
If you expect to earn more than the break-even rate on the lump sum, after tax and risk, cash is better in pure money terms. If you value guaranteed income, protection from overspending or spreading tax over many years, the annuity has advantages beyond this number. The ratio of cash to jackpot tracks current interest rates.
Common mistakes
- Comparing the advertised jackpot directly with the cash option.
- Using an optimistic stock-market return as a safe discount rate.
- Ignoring that both options are taxed, and the tax timing differs.
- Assuming payments are equal when the lottery uses graduated payments.
Frequently asked questions
Is the lump sum or annuity better?
It depends on the return you can achieve, taxes and personal preferences. The break-even rate gives a numeric reference point.
Why do annuity payments increase each year?
Powerball and Mega Millions annuities use 30 graduated payments rising 5% a year to offset inflation.
What happens to annuity payments if the winner dies?
Under typical US multistate lottery rules remaining payments go to the estate; check the specific game rules.
Sources: Powerball official game rules (Multi-State Lottery Association), annuity payment description · Brealey, Myers & Allen, Principles of Corporate Finance (growing annuities, present value)