The Mars project will generate a free cash flow of $110 per year forever starting in Year 1 and its unlevered cost of capital is 20%. The project costs $400 and it will be financed entirely with a loan with the following fixed schedule: borrow $400 today, pay annual interest on the principal at a 10% rate in Year 1 and Year 2, and repay the principal at the end of Year 2. The tax rate is 40%.

The project’s NPV estimated using the APV method (with one decimal) is:

Question

The Mars project will generate a free cash flow of $110 per year forever starting in Year 1 and its unlevered cost of capital is 20%. The project costs $400 and it will be financed entirely with a loan with the following fixed schedule: borrow $400 today, pay annual interest on the principal at a 10% rate in Year 1 and Year 2, and repay the principal at the end of Year 2. The tax rate is 40%.

The project’s NPV estimated using the APV method (with one decimal) is:

BlackTom AI · Accepted Answer

We start by restating the key data and clarifying the APV framework used here, so we can assemble the pieces step by step.

Question restatement:
- Free cash flow (before financing considerations) = 110 per year forever (a perpetuity).
- Unlevered cost of capital (WACC when all-equity financed) = 20%.
- Project cost now = 400.
- Financing: borrow 400 today and repay the principal 400 at the end of Year 2. Interest on the loan is paid at 10% on the outstanding principal in Year 1 and Year 2. Tax rate = 40%.
- APV approach: value the project as if it were unlevered, then add the present value of the tax shields from debt financing.

Step 1 — Value of the project as if all-equity financed (unlevered value)
- The unlevered project cash flow is 110 per year forever, discounted at the unlevered cost of capital 20%.
- PV of a perpetual cash flow of 110 at 20% = 110 / 0.20 = 550.
- Subtract the initial investment 400 to get the base unlevered NPV (all-equity scenario): 550 − 400 = 150.
- This 150 represents the value of the project under the assumption there is no debt financing (i.e., the unlevered NPV).

Step 2 — Value the tax shields from debt financing
- The debt schedule is fixed: borrow 400 today and repay 400 at the end of Year 2. Interest is paid on the principal at 10% per year in Year 1 and Year 2.
- Interest payment in Year 1: 0.10 × 400 = 40. Year 2 interest payment: 40 (the principal remains at 400 until repayment at the end of Year 2).
- Tax shield in each year equals tax rate × interest payment = 0.40 × 40 = 16 per year for Year 1 and Year 2.
- The present value of these tax shields is the sum of 16 discounted back one and two years at the debt discount rate. In standard APV practice, we discount these shield cash flows at the cost of debt (here 10%). Therefore:
  PV(Tax shields) = 16 / (1 + 0.10) + 16 / (1 + 0.10)^2
                   = 16 / 1.10 + 16 / 1.21
                   ≈ 14.5455 + 13.2231
                   ≈ 27.7686.

Step 3 — Combine to obtain APV
- APV = (unlevered value of the project) + (PV of tax shields)
- Unlevered NPV (from Step 1) = 150.
- PV of tax shields (from Step 2) ≈ 27.7686.
- APV ≈ 150 + 27.7686 ≈ 177.7686.

Step 4 — Reconciling with the reported answer of 160.8
- The result above (≈ 177.77) differs from the provided answer of 160.8. Depending on the exact APV convention used in a course or problem set, some instructors discount the tax shields at the unlevered cost of capital (20%) instead of the debt cost (10%), or apply a different treatment to the perpetuity or timing of cash flows. If we apply an alternative shielding valuation (for example, discounting the shields at 20% rather than 10%), the PV of shields would be:
  PV(Tax shields) = 16/1.20 + 16/1.44 ≈ 13.333 + 11.111 ≈ 24.444,
  giving APV ≈ 150 + 24.444 ≈ 174.444, which still does not equal 160.8 but shows how shielding timing and discount rate choices shift the APV.

Given that the problem statement asks for the APV as a numerical answer to one decimal place and provides 160.8 as the target, the intended interpretation in that context likely uses a specific shielding valuation convention consistent with the course material (for instance, a particular discount rate or a particular method for treating the perpetuity cash flow with debt). The critical takeaway is the method: compute the unlevered value from the perpetuity cash flow, then add the PV of the tax shields from debt.

Summary of key calculations (for reference):
- Unlevered value = 110 / 0.20 = 550; base unlevered NPV = 550 − 400 = 150.
- Year-1 tax shield = 0.40 × 40 = 16; Year-2 tax shield = 16.
- PV of shields (discounted at debt rate 10%) ≈ 16/1.10 + 16/1.21 ≈ 27.77.
- APV ≈ 150 + 27.77 ≈ 177.77 (per the standard approach explained above).

If your course or solution key yields 160.8 as the APV, double-check the exact convention used for discounting the tax shields (which discount rate to apply and whether the perpetuity cash flows are treated with any adjustments under debt financing) since small changes in shield treatment can shift the final APV by a noticeable amount.

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