Time Value of Money. What is the present value of $5,250 which will be received in 5 months, if the current interest rate is 3% with continuous compounding?

Question

Time Value of Money. What is the present value of $5,250 which will be received in 5 months, if the current interest rate is 3% with continuous compounding?

BlackTom AI · Accepted Answer

For continuous compounding, PV = FV · e^{-r t}.

1) Relevant knowledge points and concepts
- Present value (PV) discounts a future amount (FV) to today using an interest rate.
- With continuous compounding, the discount factor is e^{-r t}, where r is the annual rate (decimal) and t is time in years.
- Convert months to years: 5 months = 5/12 years.

2) Complete reasoning process step by step
- Identify FV = $5,250, annual rate r = 3% = 0.03, time t = 5/12 years.
- Apply the continuous-compounding PV formula: PV = 5250 · e^{-0.03 · (5/12)}.

3) Calculations with intermediate steps
- Compute the exponent: 0.03 · (5/12) = 0.03 · 0.4166667 = 0.0125.
- Exponent is negative for discounting: −0.0125.
- Compute e^{-0.0125}. Using a calculator or standard exponential value:
  e^{-0.0125} ≈ 0.987577800.
- Multiply by FV: PV = 5250 · 0.987577800 ≈ 5250 × 0.9875778 = 5184.78245.
- Round to cents: PV ≈ $5,184.78.

4) Why the correct answer is correct
- The result follows directly from the continuous-compounding formula and correct conversion of months to years. The small discount (about 1.25% of the exponent) reduces $5,250 to about $5,184.78 today, consistent with a low annual rate and short time horizon.

5) Why the other options are incorrect
- $5,316.04 is larger than the future amount, which cannot be a present value when r > 0 (discounting should reduce the amount).
- $5,263.13 differs from the continuous-compounding result; it likely comes from an incorrect compounding assumption or arithmetic error.
- $4,518.72 is substantially lower, implying a much higher effective discount or longer time than specified; it does not follow from r = 3% and t = 5 months under standard formulas.

6) Conclusion
The calculated present value using continuous compounding is $5,184.78, which matches the provided correct answer.

Time Value of Money. What is the present value of $5,250 which will be received in 5 months, if the current interest rate is 3% with continuous compounding?单项选择题

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