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Capitalized worth; deposit for perpetual withdrawals

Short answer

Capitalized worth is the present sum that yields a perpetual series of payments. Using 12% nominal compounded monthly (effective annual 12.683%), required deposit = 3000/0.01 + 20000/0.12683 + 50000(A/F,12.683%,5)/0.12683 ≈ Rs 5,18,920.

Define capitalized worth. How much money should Mr X deposit now in a bank which gives 12% interest annually, so that he can draw Rs 3000 per month plus Rs. 20,000 annually and Rs. 50,000 in every five years for infinite period?

Capitalized worth

Capitalized worth (CW) is the present worth of a series of cash flows that continues forever (infinite life). For a uniform perpetual amount A it is CW = A / i. For a perpetual series that repeats every k years, convert each payment to an equivalent end-of-year amount first, then divide by i. It is used for bridges, dams, endowments and scholarships.

Numerical

Assumption: 12% per year is the nominal rate compounded monthly, so monthly i = 1% and effective annual i = (1.01)^12 − 1 = 12.683%.

 Monthly : 3,000 every month, forever
 Annual  : 20,000 every year, forever
 5-yearly: 50,000 at years 5, 10, 15, ... forever

1. Monthly Rs 3,000 P₁ = 3,000 / 0.01 = Rs 300,000

2. Annual Rs 20,000 P₂ = 20,000 / 0.12683 = Rs 157,698

3. Rs 50,000 every 5 years

  • Equivalent annual amount A = 50,000 (A/F, 12.683%, 5) = 50,000 × 0.15529 = Rs 7,764.51
  • P₃ = 7,764.51 / 0.12683 = Rs 61,222

Total deposit P = P₁ + P₂ + P₃ = 300,000 + 157,698 + 61,222

Answer: Mr X should deposit about Rs 518,920 (≈ Rs 5.19 lakh).

(If 12% is taken as the effective annual rate, the monthly rate is 0.9489% and the total becomes about Rs 548,417.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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