The effective annual rate of interest corresponding to a nominal rate of 8%…

Concept: When a nominal annual rate is compounded more than once a year, the effective annual rate is the actual percentage growth of the principal over one year, and it is always higher than the nominal rate whenever there is more than one compounding period. If the nominal rate is r% per annum compounded n times a year, each period earns r/n% interest, and the effective annual rate equals (1 + r/100n)n − 1, expressed as a percentage — because interest earned in an earlier period itself earns interest in the later period(s) within the same year.

Application:

1. Nominal rate r = 8% per annum, compounded half-yearly, so there are n = 2 compounding periods in the year.

2. Half-yearly interest rate = 8% ÷ 2 = 4%.

3. Take principal = Rs. 100 (any principal works, since only the rate is needed).

4. Amount after 1 year = 100 × (1 + 4/100)2 = 100 × 1.0816 = Rs. 108.16.

5. Compound interest earned in the year = 108.16 − 100 = Rs. 8.16.

6. Effective annual rate = 8.16%.

Cross-check: Interest for the first half-year = 4% of Rs. 100 = Rs. 4. Interest for the second half-year = 4% of (100 + 4) = Rs. 4.16, since the second half-year also earns interest on the first half-year's interest of Rs. 4. Total interest for the year = 4 + 4.16 = Rs. 8.16 — the same effective annual rate obtained above, confirming the result.

So the effective annual rate of interest is 8.16% per annum.