Lump Sum vs Annuity: Present-Value Crossover

The Choice

Lump sum today $1,000,000

Annual payment $120,000

Discount rate 7.0%

Years of payments 10

Live Result

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PV of annuity

Lump sum

Difference (annuity − lump sum)

Break-even discount rate

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How it works

A future payment is worth less than the same amount today, because money today can earn interest. Discounting a payment PMT received t years from now at rate r gives its present value:

PV_t = PMT / (1 + r)^t

Each colored slab in the right-hand stack is one year's discounted payment, stacked bottom-to-top from year 1 to year n — its height shrinks the farther out it sits. The full annuity is their sum, the standard ordinary-annuity formula:

PV_annuity = PMT × [1 − (1+r)^(−n)] / r

The left column is simply the lump sum — it needs no discounting because it already sits at t = 0. Whichever column is taller is the better deal at that discount rate.

Discount rate slider — raise it and every future payment gets discounted harder, so the annuity stack shrinks while the lump-sum column stays fixed.
Break-even rate — the single discount rate at which PV_annuity exactly equals the lump sum; below it the annuity wins, above it the lump sum wins. Found here by bisection since the annuity formula has no closed-form inverse for r.
This is exactly the "$1,000,000 today or $X/year for N years" lottery/settlement decision, and the same math prices bonds, pensions, and loan payoffs.

Lump sum (today) Annuity PV, stacked by year

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