Mathematics for FinancePYQ Dec 23Question 1472 of 512
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Calculate the present value of 2,000\displaystyle 2,000 to be required after 10\displaystyle 10 years compounded annually at 5%\displaystyle 5\% per annum given (1.05)10=1.62889\displaystyle (1.05)^{10} = 1.62889

Options

A1,227.82\displaystyle 1,227.82
B1,282.48\displaystyle 1,282.48
C1,328.35\displaystyle 1,328.35
D1,822.65\displaystyle 1,822.65
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Correct Answer

✅ Option c — 1,328.35\displaystyle 1,328.35

All Options:

  • A1,227.82\displaystyle 1,227.82
  • B1,282.48\displaystyle 1,282.48
  • C1,328.35\displaystyle 1,328.35
  • D1,822.65\displaystyle 1,822.65

Detailed Solution & Explanation

Let the principal amount be P=Rs. 100\displaystyle P = \text{Rs. }100. Given parameters: * Nominal Rate (r\displaystyle r) = 4.5%\displaystyle 4.5\% p.a. =0.045\displaystyle = 0.045 Let's calculate the effective annual rate under the two cases: 1. **Compounded quarterly (m=4\displaystyle m=4):** Eq=(1+r4)4−1E_q = \left(1 + \frac{r}{4}\right)^4 - 1 Eq=(1+0.0454)4−1E_q = \left(1 + \frac{0.045}{4}\right)^4 - 1 Eq=(1.01125)4−1E_q = (1.01125)^4 - 1 First, let's calculate (1.01125)4\displaystyle (1.01125)^4: (1.01125)2≈1.02262656(1.01125)^2 \approx 1.02262656 (1.01125)4=(1.02262656)2≈1.045765(1.01125)^4 = (1.02262656)^2 \approx 1.045765 So: Eq≈4.5765%E_q \approx 4.5765\% 2. **Compounded annually (m=1\displaystyle m=1):** Ea=r=4.5%E_a = r = 4.5\% The gain per Rs. 100\displaystyle \text{Rs. }100 when compounded quarterly over compounding annually is the difference between their effective rates: Gain per Rs. 100=Eq−Ea=4.5765%−4.5%=0.0765%\text{Gain per Rs. 100} = E_q - E_a = 4.5765\% - 4.5\% = 0.0765\% This corresponds to Rs. 0.076\displaystyle \text{Rs. }0.076 (or 0.0765\displaystyle 0.0765 rupees) per Rs. 100\displaystyle \text{Rs. }100. Hence, **Option C** is the correct answer.

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