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Nominal vs effective; FW with continuous compounding

Short answer

Continuous compounding effective rate = e^0.08 -1 = 8.3287%. Annuity-due of 5 payments yields value at year4; compound 6 years using e^(0.08×6)=1.6161: FW = 500 × 5.9052 × 1.6161 = Rs 4,772.

What is effective and nominal interest rate? Evaluate FW at the end of 10 years with 8% interest rate compounded continuously of a cash flow of Rs. 500 at the beginning of each year for first 5 years.

Effective and nominal rate

Nominal rate r is the stated annual rate (8%). Effective rate is the actual annual growth after compounding within the year. For continuous compounding, m → ∞: i = e^r − 1

i = e^0.08 − 1 = 8.3287% per year

Cash flow diagram

 Rs 500 at t = 0,1,2,3,4  (beginning of years 1-5)
 ^   ^   ^   ^   ^
 0   1   2   3   4   5  6  7  8  9  10
                                    FW = ?

FW at the end of year 10

The five payments (t = 0 to 4) form an annuity-due; (F/A, i, 5) gives the value at t = 4, which is then moved 6 years to t = 10.

FW = 500 (F/A, 8.3287%, 5) (F/P, 8.3287%, 6)

  • (F/A, 8.3287%, 5) = [(1.083287)⁵ − 1] / 0.083287 = 5.9052
  • (F/P, 8.3287%, 6) = e^(0.08×6) = e^0.48 = 1.6161

FW = 500 × 5.9052 × 1.6161 = Rs 4,772

Answer: FW at the end of 10 years = about Rs 4,772.

✓ Answer checked by Claude Opus 5.5 (high effort)

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