Hanse, Incorporated, has the following two mutually exclusive projects available.

Year Project R Project S
0 −$ 77,000 −$ 97,600
1 26,800 24,400
2 25,800 24,400
3 23,800 39,400
4 17,800 34,400
5 11,600 13,400

What is the crossover rate for these two projects?

Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

What is the NPV of each project at the crossover rate?

Note: Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.

Question

Hanse, Incorporated, has the following two mutually exclusive projects available.

Year	Project R	Project S
0	−$ 77,000	−$ 97,600
1	26,800	24,400
2	25,800	24,400
3	23,800	39,400
4	17,800	34,400
5	11,600	13,400

What is the crossover rate for these two projects?

Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

What is the NPV of each project at the crossover rate?

Note: Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.

BlackTom AI · Accepted Answer

We start by clearly restating the data for both mutually exclusive projects to set up the crossover-rate calculation.

- Project R cash flows: Year 0 = -77,000; Year 1 = 26,800; Year 2 = 25,800; Year 3 = 23,800; Year 4 = 17,800; Year 5 = 11,600.
- Project S cash flows: Year 0 = -97,600; Year 1 = 24,400; Year 2 = 24,400; Year 3 = 39,400; Year 4 = 34,400; Year 5 = 13,400.

The crossover rate is the discount rate r at which the NPVs of the two projects are equal, i.e., NPV_R(r) = NPV_S(r). This requires forming the NPVs as functions of r and solving for r.

Compute NPV expressions as functions of r:
- NPV_R(r) = -77,000 + 26,800/(1+r) + 25,800/(1+r)^2 + 23,800/(1+r)^3 + 17,800/(1+r)^4 + 11,600/(1+r)^5.
- NPV_S(r) = -97,600 + 24,400/(1+r) + 24,400/(1+r)^2 + 39,400/(1+r)^3 + 34,400/(1+r)^4 + 13,400/(1+r)^5.

Set them equal and solve for r. This is generally done numerically (trial-and-error, interpolation, or using a financial calculator or software). You look for r such that the difference NPV_R(r) − NPV_S(r) = 0.

As you test values of r, you adjust until the two net present values converge. In this problem, the numeric solution yields a crossover rate around 10.32%.

Next, once r_cross is found, compute the NPVs of each project at that rate: NPV_R(r_cross) and NPV_S(r_cross). By construction, these should be equal since r_cross is defined by NPV_R(r_cross) = NPV_S(r_cross).

If you plug r = 0.1032 (i.e., 10.32%) into the NPV formulas above, you obtain approximately:
- NPV_R(0.1032) ≈ 5,329.92
- NPV_S(0.1032) ≈ 5,329.92

Thus, the crossover rate is approximately 10.32%, and the NPVs of Projects R and S at that rate are both about $5,329.92.

Notes on the process and common pitfalls:
- The crossover rate is not the IRR of either project individually; it is the rate where their NPVs are equal, which is why you solve NPV_R(r) = NPV_S(r).
- Be careful with rounding: do not round intermediate calculations during the solving step, but final NPVs at the crossover rate are typically reported to two decimals.
- If you attempted to use a single IRR or compared NPVs at a fixed rate without finding the equality point, you would miss the true crossover rate.

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