Reference

The two-phase formula, derived.

Retirement is two distinct math problems back-to-back: an accumulation phase that builds the nest egg, and a drawdown phase that depletes it. Each has a closed-form expression. Together they determine whether your plan works.

Phase 1: Accumulation

You start with savings S₀, contribute a fixed amount PMT per period, earn rate r per period, for n periods. The balance at retirement is the future-value-of-an-annuity-with-initial-balance:

S_n = S₀(1+r)ⁿ + PMT · ((1+r)ⁿ − 1) / r

The same expression as the rifinance.xyz compound-interest engine, expressed in months: r = annual rate ÷ 12 and n = years × 12.

Phase 2: Drawdown

You start retirement with balance B₀. Each period you withdraw W and earn rate r_d on the residual. We want to know how many periods k the balance lasts. The closed form is the present-value-of-annuity equation rearranged for periods:

k = − ln(1 − B₀ · r_d / W) / ln(1 + r_d)

The expression is undefined (gives a negative under-the-log) if the withdrawal rate is too small to deplete the balance — in that case, the balance lasts indefinitely. The boundary condition is W = B₀ · r_d: withdrawing exactly the interest each period leaves principal intact forever.

The joint problem: sustainability

Plug the output of Phase 1 (the projected nest egg S_n) into Phase 2 as B₀, set W to the user's target monthly spending, and use a more conservative drawdown return r_d. The number of months the balance lasts is the answer to the question retirement calculators should actually answer: does this plan reach the user's life expectancy?

Our calculator uses r_d = 0.6 × r to reflect a typical glidepath de-risking from equities to bonds. The glidepath page walks through alternative assumptions.

Worked example: the sample case

Inputs: age 35, retire at 65, life expectancy 90, savings €50,000, contribution €500/month, real return 5%, target spending €3,500/month.
Phase 1: 30-year accumulation. S_n ≈ 50,000 × 4.467 + 500 · 832.4 ≈ 223,400 + 416,200 = €639,600.
Phase 2: drawdown at €3,500/month, real return 3% (60% of accumulation). The closed form gives k ≈ 296 months ≈ 24.7 years.
Life expectancy gap: 25 years needed, 24.7 years projected. Marginal shortfall — either lift contributions, delay retirement by one year, or reduce target spending by ~3 %.

Sanity checks

Zero return. Both phases reduce to simple division. Accumulation: S_n = S₀ + PMT · n. Drawdown: k = B₀ / W.
No contributions. Phase 1 reduces to S_n = S₀(1+r)ⁿ — just compound interest.
Withdrawal ≤ interest. If W ≤ B₀ · r_d, the balance is self-sustaining and the calculator reports “indefinitely.”