Goodyear is thinking of divesting one of the plants. The plant will generate free cash flows (FCF) of $3.8 million at the end of the first year and the cash flows will grow at 3%. The plant is financed with a debt of 50 million which is expected to remain constant. Goodyear has an equity cost of capital of 10% and a debt cost of capital of 6% and a marginal tax rate of 40%. Personal tax rates for marginal equity and debt investors are 10% and 15%. There is a 10% chance that the firm will default in the next period. In case it defaults, the cost of default (after adjusting for appropriate discount rate) is 20 million. Further the present value of the net agency cost of the $50 million debt is estimated to be 5 million. If the plant has an average risk similar to the whole firm, value the plant using the APV method. Assume a leverage ratio of 0.5.

Question

Goodyear is thinking of divesting one of the plants. The plant will generate free cash flows (FCF) of $3.8 million at the end of the first year and the cash flows will grow at 3%. The plant is financed with a debt of 50 million which is expected to remain constant. Goodyear has an equity cost of capital of 10% and a debt cost of capital of 6% and a marginal tax rate of 40%. Personal tax rates for marginal equity and debt investors are 10% and 15%. There is a 10% chance that the firm will default in the next period. In case it defaults, the cost of default (after adjusting for appropriate discount rate) is 20 million. Further the present value of the net agency cost of the $50 million debt is estimated to be 5 million. If the plant has an average risk similar to the whole firm, value the plant using the APV method. Assume a leverage ratio of 0.5.

BlackTom AI · Accepted Answer

We begin by carefully restating what the problem is asking and what information is given, then we walk through the APV components step by step to build up the value.

Question restatement:
- Goodyear may divest a plant. The plant’s free cash flow (FCF) is 3.8 million at the end of year 1, and FCF grows at 3% thereafter.
- The plant is financed with 50 million of debt that is assumed to remain constant. The firm’s equity cost of capital is 10%, debt cost of capital is 6%, and the corporate tax rate is 40%. Personal tax rates are 10% on equity income and 15% on debt income. There is a 10% chance of default in the next period; if default occurs, the cost of default is 20 million (present value of this cost is given as 20m in the description, and we should account for it in the APV framework).
- The present value of net agency costs associated with the 50 million of debt is estimated at 5 million.
- The leverage ratio is 0.5 (i.e., D/V = 0.5). We are asked to value the plant using the APV method, assuming leverage similar to the whole firm.

Step 1: Compute the unlevered value of the project (V_U).
- In APV, we first value the project as if it were all-equity financed (no debt). The problem states the project risk is average with the same risk as the firm, so we use the firm’s equity cost of capital as the appropriate discount rate for the unlevered project cash flows, which is 10% (the given equity cost for the firm is used as the unlevered cost of capital here).
- With a perpetual growing FCF stream starting at FCF1 = 3.8 and growing at g = 3%, the value of the unlevered project is:
  V_U = FCF1 / (r_U − g) = 3.8 / (0.10 − 0.03) = 3.8 / 0.07 ≈ 54.2857 million.

Step 2: Determine the financing side effects to add in APV.
APV adds the present value of financing effects to V_U. The problem provides several financing-related items:
- PV of debt tax shield: With debt 50m and corporate tax rate τ_C = 40%, the gross tax shield is τ_C × D = 0.40 × 50 = 20 million. However, with personal taxes on debt income, the effective value of the tax shield is reduced because debt income is taxed at the personal level (15% in this problem). The after-personal-tax shield factor for debt is (1 − t_p) where t_p = 15%, so the adjusted PV of the debt tax shield is 20 × (1 − 0.15) = 20 × 0.85 = 17.0 million.
- PV of default costs: There is a 10% chance of default in the next period, and the cost of default (present value) is 20 million if it occurs. The APV approach would typically incorporate the expected present value of default costs as 0.10 × 20 = 2.0 million, discounted appropriately. For a one-period horizon implied by the statement, we would treat this as an immediate expected cost of 2.0 million to be subtracted from value.
- PV of net agency costs of debt: The problem provides this as a present value of 5 million, which is a cost to be subtracted from value in APV.
- Leverage ratio 0.5 (D/V = 0.5): This is consistent with the given debt level (D = 50m). If V is the total value after financing, D = 0.5 × V, which implies V = 100m total financing value under this target structure. In APV, we do not automatically recompute V from D and r; instead we use the stated leverage to interpret the scale of debt and the magnitude of tax shields and distress costs. Here we use the explicit numbers provided for the tax shield and distress costs as above.

Step 3: Assemble the APV value.
- Start with V_U ≈ 54.2857 million.
- Add the PV of the debt tax shield (after personal taxes): +17.0 million.
- Subtract the expected PV of default costs: −2.0 million.
- Subtract the PV of net agency costs: −5.0 million.
- Net APV value = 54.2857 + 17.0 − 2.0 − 5.0 ≈ 64.2857 million.

Step 4: Check for potential alternative interpretations that could yield a different result (and why they would differ).
- If one were to ignore personal taxes on debt income (i.e., treat the debt tax shield as 20 million without the 15% personal tax adjustment), the PV of the tax shield would be 20, and the APV would be 54.2857 + 20 − 2 − 5 = 67.2857 million, still well below the 87-million figure hinted by a provided answer in the data. This shows how sensitive the result can be to whether personal taxes are incorporated.
- If one incorrectly treated FCFs as a finite horizon rather than perpetual, or used a different discount rate for the unlevered cash flows, the value would differ as well.
- If the problem intended to fold the leverage effects into a levered cost of capital rather than explicitly adding a tax shield, the calculation path would change substantially. The APV method, as structured here, separates unlevered value from financing effects, so using the given APV components as described yields the value above.

Summary note on the numerical outcome:
- The unlevered project value (V_U) is about 54.29 million.
- The after-personal-tax debt tax shield adds about 17.0 million.
- Expected default costs subtract about 2.0 million.
- Net agency costs subtract about 5.0 million.
- The resulting APV (aggregate) value comes out to roughly 64.29 million.

Because the provided answer field contains 87 but the calculation above (using the standard APV treatment with personal taxes on debt, default costs, and agency costs as described) yields approximately 64.3 million, there is a discrepancy likely due to different assumptions or interpretations (e.g., not applying the personal tax adjustment to the debt tax shield, or treating the default cost differently, or using a finite horizon). If your instructor intended a different APV convention or different horizon, you may obtain a value closer to 87.

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