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Nominal vs effective; FW with monthly compounding

Short answer

Effective annual rate = (1+0.10/12)^12 -1 = 10.4713%. Annuity-due of 10 payments gives FV at year 9, then compound 6 years: FW = 50,000 × 16.302 × 1.8175 = Rs 14,81,529.

What is nominal and effective interest rate? Evaluate FW at the end of 15 years with 10% interest compounded monthly of a cash flow of Rs. 50,000 at the beginning of each year for first 10 years.

Nominal and effective interest rate

The nominal rate is the stated yearly rate, ignoring the effect of compounding within the year (here 10% per year). The effective rate is the actual yearly rate after compounding m times a year: i = (1 + r/m)ᵐ − 1.

Effective annual rate

i = (1 + 0.10/12)¹² − 1 = 10.4713% per year

Cash flow diagram

 Rs 50,000 at t = 0, 1, ..., 9 (beginning of years 1-10)
 ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
 |  |  |  |  |  |  |  |  |  |
 +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--> year
 0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15
                                              FW = ?

FW at the end of year 15

The payments at t = 0 to 9 form an annuity-due. The (F/A, i, 10) factor gives their value at t = 9, so this is then moved to t = 15 (6 years):

FW = 50,000 (F/A, 10.4713%, 10) (F/P, 10.4713%, 6)

  • (F/A, 10.4713%, 10) = [(1.104713)¹⁰ − 1] / 0.104713 = 16.302
  • (F/P, 10.4713%, 6) = (1.104713)⁶ = 1.8175

FW = 50,000 × 16.302 × 1.8175 = Rs 1,481,529

Answer: FW at the end of 15 years = about Rs 14.8 lakh.

✓ Answer checked by Claude Opus 5.5 (high effort)

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