Mathematics for FinanceMTP Dec 23 Series IIQuestion 1527 of 512
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A debt of 5000\displaystyle 5000 with interest at the rate of 8%\displaystyle 8\% compounded quarterly is to be discharged by 8\displaystyle 8 equal quarterly payments, the first payment being due today. Find the size of each payment.

Options

A673.90\displaystyle 673.90
B669.11\displaystyle 669.11
C399.26\displaystyle 399.26
DNone of these
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Correct Answer

✅ Option b — 669.11\displaystyle 669.11

All Options:

  • A673.90\displaystyle 673.90
  • B669.11\displaystyle 669.11
  • C399.26\displaystyle 399.26
  • DNone of these

Detailed Solution & Explanation

Since the first payment is due today, this is an annuity due. The present value (PVdue\displaystyle PV_{\text{due}}) is: PVdue=A[1+1−(1+i)−(n−1)i]PV_{\text{due}} = A \left[ 1 + \frac{1 - (1+i)^{-(n-1)}}{i} \right] Given: * Loan amount (PVdue\displaystyle PV_{\text{due}}) = 5,000\displaystyle 5,000 * Number of periods (n\displaystyle n) = 8\displaystyle 8 * Nominal rate = 8%\displaystyle 8\% p.a. compounded quarterly, so periodic rate i=8%4=2%=0.02\displaystyle i = \frac{8\%}{4} = 2\% = 0.02 Substituting the values: 5,000=A[1+1−(1.02)−70.02]5,000 = A \left[ 1 + \frac{1 - (1.02)^{-7}}{0.02} \right] Using (1.02)−7≈0.87056\displaystyle (1.02)^{-7} \approx 0.87056: 5,000=A[1+6.47199]=A×7.471995,000 = A [ 1 + 6.47199 ] = A \times 7.47199 A=5,0007.47199≈669.17A = \frac{5,000}{7.47199} \approx 669.17 This is closest to Option B (669.11\displaystyle 669.11). Hence, **Option B** is the correct answer.

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