Use this graph to answer the question below. The owner of a hair salon spends $1,000,000 to renovate its premises, estimating that this will increase her cash flow by $220,000 per year. She constructs the above graph, which shows the net present value (NPV) as a function of the discount rate. If her discount rate is 6%, should she accept the project?

Question

Use this graph to answer the question below. The owner of a hair salon spends​ $1,000,000 to renovate its​ premises, estimating that this will increase her cash flow by​ $220,000 per year. She constructs the above​ graph, which shows the net present value​ (NPV) as a function of the discount rate. If her discount rate is​ 6%, should she accept the​ project?

BlackTom AI · Accepted Answer

NPV is positive only when the discounted present value (PV) of the future cash flows exceeds the initial cost; the decision rule is: accept if NPV > 0, reject if NPV < 0.

1) Relevant knowledge points and concepts
- NPV = PV(cash inflows) − initial cost.
- PV of an n‑year level cash flow (annuity) = CF × [1 − (1 + r)^(−n)] / r.
- If NPV < 0 at the chosen discount rate, the project should be rejected.

2) Complete reasoning process step by step
The graph shows the NPV line starting at +100,000 when discount rate = 0 and crossing the horizontal axis (NPV = 0) between 3% and 4%. That starting value (+100,000) equals the undiscounted net cash over the project, which implies the project pays 220,000 per year for 5 years (220,000 × 5 = 1,100,000; minus the 1,000,000 cost = +100,000). Thus we treat this as a 5‑year annuity of 220,000.

3) Calculations (intermediate steps shown)
Compute PV at r = 6%:
- (1 + r)^(−n) = 1.06^(−5) ≈ 0.747258
- Annuity factor = [1 − 0.747258] / 0.06 = 0.252742 / 0.06 ≈ 4.21236
- PV(cash inflows) = 220,000 × 4.21236 ≈ 926,719.20
- NPV = 926,719.20 − 1,000,000 ≈ −73,280.80 (negative)

4) Why the correct answer is correct
Since the calculated NPV at a 6% discount rate is negative (about −$73,281), the project destroys value at that discount rate, so it should be rejected.

5) Why the other options are incorrect
- a. Incorrect: it claims NPV is positive at 6%, but both the graph and the calculation show NPV < 0.
- c. Incorrect: the graph and the given cash flows and cost provide enough information to determine the sign of NPV.
- d. Incorrect and contradictory: it says “No, because the NPV is positive,” which is self-contradictory and factually wrong here.

6) Conclusion confirming the final answer
Therefore, at a 6% discount rate the NPV is negative and she should not accept the project. This confirms answer b.

类似问题

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