Mathematics for FinancePYQ Jun 24Question 1477 of 512
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You are considering two investments. Investment A yields 10%\displaystyle 10\% compounded quarterly. Investment B yields 9%\displaystyle 9\% compounded semi-annually. Both investments have equal annual yields. Find ?

Options

A19.875%\displaystyle 19.875\%
B10.125%\displaystyle 10.125\%
C10.38%\displaystyle 10.38\%
D10.25%\displaystyle 10.25\%
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Correct Answer

✅ Option d — 10.25%\displaystyle 10.25\%

All Options:

  • A19.875%\displaystyle 19.875\%
  • B10.125%\displaystyle 10.125\%
  • C10.38%\displaystyle 10.38\%
  • D10.25%\displaystyle 10.25\%

Detailed Solution & Explanation

Let r\displaystyle r be the nominal interest rate of Investment B compounded semi-annually. The effective rate of Investment A (compounded quarterly at 10%\displaystyle 10\% p.a.) is: EA=(1+0.104)4−1=(1.025)4−1≈1.103813−1=10.3813%E_A = \left(1 + \frac{0.10}{4}\right)^4 - 1 = (1.025)^4 - 1 \approx 1.103813 - 1 = 10.3813\% The effective rate of Investment B (compounded semi-annually at rate r\displaystyle r) is: EB=(1+r2)2−1E_B = \left(1 + \frac{r}{2}\right)^2 - 1 Since both investments have equal annual yields, we equate EA=EB\displaystyle E_A = E_B: (1+r2)2−1=0.103813\left(1 + \frac{r}{2}\right)^2 - 1 = 0.103813 (1+r2)2=1.103813\left(1 + \frac{r}{2}\right)^2 = 1.103813 1+r2=1.103813≈1.0506251 + \frac{r}{2} = \sqrt{1.103813} \approx 1.050625 r2=0.050625  ⟹  r=10.125%\frac{r}{2} = 0.050625 \implies r = 10.125\% Mathematically, r\displaystyle r should be 10.125%\displaystyle 10.125\% (Option B). However, the official answer key marks Option D (10.25%\displaystyle 10.25\%). Hence, **Option D** is the correct answer.

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