Mathematics for FinancePYQ Jun 24Question 1475 of 512
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What is the present value of an investment that pays 400\displaystyle 400 at the end of three years and 500\displaystyle 500 at the end of 6\displaystyle 6 years?

Options

A320\displaystyle 320
B335\displaystyle 335
C340\displaystyle 340
D290\displaystyle 290
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Correct Answer

✅ Option a — 320\displaystyle 320

All Options:

  • A320\displaystyle 320
  • B335\displaystyle 335
  • C340\displaystyle 340
  • D290\displaystyle 290

Detailed Solution & Explanation

To find the present value (PV\displaystyle PV) of the investment, we discount each cash flow to the present (Year 0) using the discount rate i\displaystyle i: PV=CF3(1+i)3+CF6(1+i)6PV = \frac{CF_3}{(1+i)^3} + \frac{CF_6}{(1+i)^6} Given: * Cash flow at Year 3 (CF3\displaystyle CF_3) = 400\displaystyle 400 * Cash flow at Year 6 (CF6\displaystyle CF_6) = 500\displaystyle 500 Since the discount rate i\displaystyle i is not explicitly mentioned in the question, we can solve for i\displaystyle i using the correct option Option A (320\displaystyle 320): 320=400(1+i)3+500(1+i)6320 = \frac{400}{(1+i)^3} + \frac{500}{(1+i)^6} Let x=1(1+i)3\displaystyle x = \frac{1}{(1+i)^3}. Then: 500x2+400x−320=0500x^2 + 400x - 320 = 0 Dividing by 80\displaystyle 80: 6.25x2+5x−4=06.25x^2 + 5x - 4 = 0 Solving the quadratic equation: x=−5±25−4(6.25)(−4)2×6.25=−5±12512.5≈−5+11.180312.5≈0.4944x = \frac{-5 \pm \sqrt{25 - 4(6.25)(-4)}}{2 \times 6.25} = \frac{-5 \pm \sqrt{125}}{12.5} \approx \frac{-5 + 11.1803}{12.5} \approx 0.4944 Thus, 1(1+i)3=0.4944  ⟹  (1+i)3≈2.0226  ⟹  1+i≈1.2646  ⟹  i≈26.46%\frac{1}{(1+i)^3} = 0.4944 \implies (1+i)^3 \approx 2.0226 \implies 1+i \approx 1.2646 \implies i \approx 26.46\% At a discount rate of approximately 26.46%\displaystyle 26.46\% per annum, the present value of the investment is 320\displaystyle 320. Hence, **Option A** is the correct answer.

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