You will receive $631 in 6 years time. If the discount rate is 12% per annum continuously compounded, what is the present value of this future cashflow? Enter your answer with 2 decimal places. Do not enter the dollar sign "$".

Question

BlackTom AI · Accepted Answer

The present value for continuous compounding uses the formula PV = FV · e^(−r·t).

1. Relevant knowledge points and concepts  
- Continuous compounding: PV = FV · e^(−r t), where FV is the future value, r is the continuously compounded rate, and t is time in years.  
- Here: FV = 631, r = 0.12 (12%), t = 6.

2. Step-by-step reasoning  
- Substitute the known values into the continuous discount formula and evaluate the exponential factor.

3. Calculations with intermediate steps  
- Compute the exponent: −r·t = −0.12 × 6 = −0.72.  
- Compute e^(−0.72). Using a calculator: e^(−0.72) ≈ 0.486752.  
- Multiply by the future value: PV = 631 × 0.486752 ≈ 307.140512.  
- Round to two decimal places: 307.14.

4. Why this answer is correct  
- Continuous compounding discounts continuously at rate r, so the correct discount factor is e^(−r t). The arithmetic above applies that exact factor to the specified FV and yields 307.14 when rounded to two decimals.

5. Common incorrect approaches and why they are wrong  
- Using annual discrete compounding: PV = 631 / (1 + 0.12)^6 ≈ 631 / 1.9743 ≈ 319.50 (different because discrete compounding assumes periodic, not continuous, compounding).  
- Using simple discount: PV = 631 / (1 + 0.12·6) = 631 / 1.72 ≈ 366.63 (incorrect because it ignores compounding).  
- These alternatives produce different results because they apply different interest/discount conventions; the problem explicitly states continuous compounding, so those are not applicable.

6. Conclusion  
The correctly computed present value under continuous compounding is 307.14, which matches the provided answer.

You will receive $631 in 6 years time. If the discount rate is 12% per annum continuously compounded, what is the present value of this future cashflow? Enter your answer with 2 decimal places. Do not enter the dollar sign "$".数值题

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