The nominal rate of interest per annum, compounded half-yearly, that gives the same effective rate as 8% per annum compounded quarterly is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8.08%
Two rates are equivalent when they give the same effective rate, that is, the same growth of Rs. 1 in one year.
At 8% compounded quarterly, the rate per quarter is 2%:
\( \text{Growth in a year} = (1.02)^4 \)
Let the nominal rate be r per annum compounded half-yearly:
\( \displaystyle \left(1 + \frac{r}{2}\right)^2 = (1.02)^4 \)
\( \displaystyle 1 + \frac{r}{2} = (1.02)^2 = 1.0404 \)
\( \displaystyle \frac{r}{2} = 0.0404 \)
\( r = 0.0808 \)
Why the other options are incorrect
Option (1) 8% — Incorrect. 8% compounded half-yearly gives (1.04)2 = 1.0816, which is less than (1.02)4 = 1.0824; with fewer compoundings a higher nominal rate is needed.
Option (3) 8.24% — Incorrect. This is the effective rate of 8% compounded quarterly, not a half-yearly nominal rate.
Option (4) 4.04% — Incorrect. This is the rate for one half-year; the nominal rate per annum is twice it.