Sameer can invest his money either in Plan P at 10% per annum compounded semi-annually, or in Plan Q at 9.8% per annum compounded quarterly. Which of the following are correct? (Given \((1.0245)^4 = 1.1017\))
(A) The effective rate of Plan P is 10.25%.
(B) The effective rate of Plan Q is 10.17%.
(C) Plan Q is the better investment.
(D) The effective rate of Plan Q is less than its nominal rate.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
Two rates with different compounding are compared through their effective rates:
\( \displaystyle r_{\mathrm{eff}} = \left(1 + \frac{r}{m}\right)^{m} - 1 \)
Plan P: r = 0.10, m = 2
\(r_{\mathrm{eff}} = (1.05)^2 - 1 = 1.1025 - 1 = 10.25\%\)
Plan Q: r = 0.098, m = 4, so the rate per quarter is 0.0245
\(r_{\mathrm{eff}} = (1.0245)^4 - 1 = 1.1017 - 1 = 10.17\%\)
So the effective rate of P is 10.25% and that of Q is 10.17%; P gives more.
Checking the statements
(A) The effective rate of Plan P is 10.25%. — Correct. \((1.05)^2 - 1 = 0.1025\).
(B) The effective rate of Plan Q is 10.17%. — Correct. \((1.0245)^4 - 1 = 0.1017\).
(C) Plan Q is the better investment. — Incorrect. Q gives 10.17%, less than the 10.25% of P.
(D) The effective rate of Plan Q is less than its nominal rate. — Incorrect. 10.17% is more than 9.8%; compounding makes the effective rate larger.