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Deposit now to draw growing monthly amount

Short answer

With nominal 12% compounded monthly, i=1%. Use geometric gradient present worth: P = 5,000 × [1 - (1.02/1.01)^120] / (0.01 - 0.02) = Rs 11,30,894.

How many rupees should you deposit now so that you will be able to draw Rs 5000 at the end of this month which increases by 2 percent per month for 10 years. Bank interest rate is 12% per year.

Data and assumption

First withdrawal A₁ = Rs 5,000 at the end of month 1, growing by g = 2% per month for n = 120 months. The interest of 12% per year is taken as nominal, compounded monthly: i = 1% per month.

Formula (geometric gradient present worth)

For i ≠ g: P = A₁ [1 − ((1 + g)/(1 + i))ⁿ] / (i − g)

 A1=5000  5100  5202 ...  (x1.02 each month)
   ^       ^     ^           ^
 --+-------+-----+--- ... ---+--> month
 0 1       2     3           120
 P = ?

Calculation

(1 + g)/(1 + i) = 1.02/1.01 = 1.009901 ((1.02)/(1.01))¹²⁰ = 3.26179 i − g = 0.01 − 0.02 = −0.01

P = 5,000 × (1 − 3.26179) / (−0.01) = 5,000 × 226.179 = Rs 1,130,894

Answer: Deposit now about Rs 11.31 lakh. (If 12% were an effective annual rate, i would be 0.9489% per month and the answer would be a little larger.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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