What is the NPV of the project that has two separate investments (cost) of: $5,000 at the time T(0) and $5,000 in 6 months and pays $14,000 in 1 year? Assume the interest rate is 4% (continuously compounded).

Question

BlackTom AI · Accepted Answer

With continuous compounding, PV = FV · e^{−rt}, so discount each cash flow to time 0 and sum inflows minus outflows.

1) Relevant concepts
- Continuous compounding discount formula: PV = FV · e^{−r t}.
- NPV = sum of present values of cash inflows minus sum of present values of cash outflows.
- Cash at t = 0 is already at present value; no discounting needed.

2) Step-by-step reasoning
- There is one cash inflow: $14,000 at t = 1 year.
- Two cash outflows (costs): $5,000 at t = 0 and $5,000 at t = 0.5 year.
- Discount the t = 1 and t = 0.5 cash flows to t = 0 using r = 0.04 (4%) continuous.

3) Calculations (all intermediate steps)
- Discount factor for t = 1: e^{−0.04·1} = e^{−0.04} ≈ 0.9607894391523232.
  PV(inflow) = 14,000 × 0.9607894391523232 = 14,000 × 0.960789439 ≈ 13,451.052147.
- t = 0 outflow: PV(cost1) = 5,000 (no discounting).
- Discount factor for t = 0.5: e^{−0.04·0.5} = e^{−0.02} ≈ 0.9801986733067553.
  PV(cost2) = 5,000 × 0.9801986733067553 ≈ 4,900.9933665.
- NPV = PV(inflows) − PV(outflows) = 13,451.052147 − (5,000 + 4,900.9933665)
       = 13,451.052147 − 9,900.9933665
       = 3,550.0587805 ≈ $3,550.06.

4) Why this is correct
Continuous compounding requires the exponential discount factors used above. Each cash flow was discounted to time 0 correctly and signs were handled as inflow vs outflow; the arithmetic yields the NPV computed.

5) Why the other options are incorrect
- $3,559.58 and $3,647.10 are close but incorrect due to slight misapplication of discounting (e.g., using discrete discounting or rounding errors).
- $4,099.01 is much larger, likely arising from failing to discount the 6-month outflow or misplacing a cost as an inflow. None of these match correct continuous-compounding discount factors used properly.

6) Conclusion
The correct NPV, computed with continuous compounding at 4%, is $3,550.06, which matches the provided answer.

What is the NPV of the project that has two separate investments (cost) of: $5,000 at the time T(0) and $5,000 in 6 months and pays $14,000 in 1 year? Assume the interest rate is 4% (continuously compounded).单项选择题

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