Which of the following statements are correct?
(A) The rate of interest announced (stated) by a bank is called the effective rate of interest.
(B) 6% per annum compounded half-yearly gives the same interest in a year as 6.09% per annum compounded annually.
(C) If a nominal rate r is compounded continuously, the effective rate is \(e^r - 1\).
(D) For a nominal rate of 8% per annum compounded quarterly, the effective rate is more than 8%.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
The announced rate is the nominal rate. The effective rate is the rate at which money actually grows in a year:
\( \displaystyle r_{\text{eff}} = \left(1 + \frac{r}{m}\right)^m - 1 \), and \( e^r - 1 \) for continuous compounding.
6% half-yearly: \( (1.03)^2 - 1 = 0.0609 \)
8% quarterly: \( (1.02)^4 - 1 = 0.0824 \)
So the stated rate is the nominal rate, 6% half-yearly is 6.09% effective, and 8% quarterly is 8.24% effective.
Checking the statements
(A) The rate of interest announced (stated) by a bank is called the effective rate of interest. — Incorrect. The announced rate is the nominal rate.
(B) 6% per annum compounded half-yearly gives the same interest in a year as 6.09% per annum compounded annually. — Correct. (1.03)2 = 1.0609.
(C) If a nominal rate r is compounded continuously, the effective rate is \(e^r - 1\). — Correct. This is the continuous compounding formula.
(D) For a nominal rate of 8% per annum compounded quarterly, the effective rate is more than 8%. — Correct. It is 8.24%.