Core answer: An annuity converts a lump sum into a payment stream (or vice versa). Present value of ¥2,000/month for 20 years at 3%: PV = PMT × (1 − (1+i)^-n) ÷ i = 2,000 × 180.3 ≈ ¥360,600. That's the fair lump-sum price of the pension stream. Future value of saving ¥2,000/month for 20 years at 3%: FV = 2,000 × ((1.0025²⁴⁰ − 1) ÷ 0.0025) ≈ ¥656,600.

The two core formulas (ordinary annuity, end-of-period)

  • PV = PMT × [1 − (1+i)⁻ⁿ] ÷ i — what a payment stream is worth today
  • FV = PMT × [(1+i)ⁿ − 1] ÷ i — what regular saving grows to

Annuity-due (payments at period START, like rent) multiplies by (1+i).

Worked examples

Example 1 — Lottery choice. ¥5M jackpot as 20 annual payments of ¥250k vs ¥3.2M lump sum, at 4% discount rate: PV of the annuity = 250,000 × [1 − 1.04⁻²⁰] ÷ 0.04 = 250,000 × 13.59 = ¥3.4M. The annuity is worth ¥200k more on paper — before taxes and your own investment skill enter the debate.

Example 2 — Pension adequacy. You want ¥5,000/month for 25 years of retirement at 3%: PV needed = 5,000 × [1 − 1.0025⁻³⁰⁰] ÷ 0.0025 = 5,000 × 211 ≈ ¥1,055,000 at retirement day. That's the target the savings-goal math then works backward from.

Example 3 — Car lease vs buy. Lease: ¥3,500/month × 36 + ¥20k down ≈ ¥146k total outflow. Buy: ¥150k − resale ¥85k after 3 years = ¥65k net cost + interest. Annuity math on the payment streams makes the real comparison visible (buying usually wins on cash terms; leasing wins on cash flow).

Example 4 — Reverse check. A commercial annuity product: pay ¥100k/year × 10 years, receive ¥2,100/month for life from 60. Solving the implied rate: ≈ 2.2–2.8% — below long-bond alternatives at times; the product sells longevity insurance, not yield. Price the insurance value honestly.

Common mistakes and myths

  1. Ignoring the discount rate — the same stream is worth ¥360k at 3% and ¥298k at 5%; the rate assumption IS the answer, so justify it.
  2. Start vs end of period — rent-style annuities-due are worth (1+i)× more; mixing conventions shifts results by a full period.
  3. Forgetting inflation — a fixed ¥5,000/month in 2045 buys ~¥3,700 of today's goods at 2% inflation; real annuities (inflation-indexed) cost more for a reason.
  4. "Guaranteed income" without credit check — an annuity is only as good as the insurer's solvency; spread very large amounts across institutions where protection schemes cap out.
  5. Comparing total received vs total paid — "pay ¥1M, receive ¥1.5M" sounds +50% but over 25 years is ~2%/yr; always convert to a rate.