Project S has a pattern of high cash flows in its early life, while Project L has a longer life, with large cash flows late in its life. Neither has negative cash flows after Year 0, and at the current cost of capital, the two projects have identical NPVs. Now suppose interest rates and money costs decline. Other things held constant, this change will cause L to become preferred to S.

Question

BlackTom AI · Accepted Answer

Cash flows received later are more sensitive to changes in the discount rate than earlier cash flows, so a decline in interest rates (cost of capital) benefits projects with late payoffs more.

1) Relevant knowledge points and concepts
- Present value (PV) of a cash flow at time t: PV = CF / (1 + r)^t.
- Sensitivity of PV to the discount rate: d(PV)/dr = -t * CF / (1 + r)^(t+1). The negative sign shows PV rises when r falls; the magnitude is proportional to t.
- Projects with greater weight in later years have greater duration and thus gain more when r decreases.

2) Step-by-step reasoning
- Both projects have equal NPV at the current discount rate r0 by hypothesis.
- If r declines to r1 < r0, each cash flow’s PV increases, but the increase is larger for cash flows occurring at larger t because of the t multiplier in the derivative above.
- Since Project L concentrates cash flows late, its total PV increases by more than Project S’s total PV when r falls. Thus L’s NPV rises relative to S’s.

3) Illustrative calculation (intermediate steps shown)
- Suppose an initial outlay of -100 at t=0 for both and r0 = 10%.
- To make NPVs equal at r0 = 10%: for S with CF1 at t=1: -100 + CF1/(1.1) = 0 ⇒ CF1 = 110. For L with CF5 at t=5: -100 + CF5/(1.1)^5 = 0 ⇒ CF5 = 100*(1.1)^5 = 161.051.
- Now lower r to 5% (r1 = 5%): NPV_S = -100 + 110/1.05 = -100 + 104.7619 = 4.7619. NPV_L = -100 + 161.051/1.05^5; 1.05^5 = 1.2762816 ⇒ PV = 126.169 ⇒ NPV_L = 26.169.
- L’s NPV increased much more than S’s, so L becomes preferred.

4) Why the correct answer is correct
- The mathematics and example show later cash flows gain more PV when rates fall; therefore the long-lived project (L) benefits more and becomes preferred.

5) Why the other option (False) is incorrect
- Claiming S would be preferred contradicts the PV sensitivity result: earlier cash flows (S) gain less from rate declines, so S cannot overtake L when L’s later cash flows rise more in PV.

6) Conclusion
- Therefore the statement is True, matching the provided answer.

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