Bright Horizons Co. expects an EBIT of $12,500 every year in perpetuity. The firm currently has no debt, and its cost of equity is 12 percent. The company can borrow at an interest rate of 7 percent, and the corporate tax rate is 30 percent. What will the value of the firm be if it changes to a capital structure with 50 percent debt?

Question

BlackTom AI · Accepted Answer

Question restatement and options:
- The Bright Horizons Co. expects an EBIT of $12,500 every year in perpetuity.
- The firm currently has no debt, and its cost of equity is 12%.
- The company can borrow at an interest rate of 7%, and the corporate tax rate is 30%.
- What will the value of the firm be if it changes to a capital structure with 50% debt?
- Answer options: $83,854; $81,667; $76,885; $93,453

Step-by-step analysis of the underlying concepts and calculations:
- First, determine the value of the unlevered firm (no debt). When there is no debt, the firm’s cash flows are the EBIT after tax, i.e., EBIT*(1 - T). Here, EBIT = 12,500 and tax rate T = 30%, so after-tax operating cash flow is 12,500*(1 - 0.30) = 8,750 per year. For a perpetuity, the value of the unlevered firm, V_U, is these perpetual cash flows divided by the cost of equity (which equals the unlevered cost of capital when there is no debt): V_U = 8,750 / 0.12 = 72,916.67.
- Next, apply the classic tax shield reasoning for a levered firm with perpetual debt. The annual tax shield from debt is Tax_rate * Interest = T * (r_d * D). In the perpetuity model, the present value of the tax shields can be treated as an additional constant amount: PV(Tax shields) = T * D.
- Given the capital structure has 50% debt, D = 0.5 * V_L (the levered firm’s value). Therefore, V_L = V_U + PV(Tax shields) = V_U + T * D = V_U + 0.30 * (0.5 * V_L) = V_U + 0.15 * V_L.
- Solve for V_L: V_L - 0.15 * V_L = V_U ⇒ 0.85 * V_L = V_U ⇒ V_L = V_U / 0.85.
- Plugging in V_U = 72,916.67 gives V_L = 72,916.67 / 0.85 ≈ 85,781.96.
- Therefore, the calculated levered firm value under these assumptions is about $85,782 (rounded). Note that this relies on the standard, simplified perpetual debt tax shield model with no bankruptcy costs and constant debt, and uses the after-tax operating cash flow for the unlevered valuation.

Option-by-option evaluation:
- Option $83,854:
  - This value is somewhat close to the calculated $85,782 but somewhat lower. The discrepancy could arise from using a slightly different convention for the tax shield PV (for example, using PV(Tax shields) = 0.3 * D directly without the 0.5 scaling, or rounding in intermediate steps). In the strict derivation above, the result is about $85.8k, so $83,854 is not the exact result but is not wildly implausible if one uses alternative tax shield assumptions.
- Option $81,667:
  - This is noticeably lower than the computed $85.8k. It would require a larger deduction from the tax shield or a smaller unlevered value, which would come from either a lower after-tax cash flow realization or a different treatment of the tax shield, neither of which aligns with the standard perpetual debt tax shield model used here.
- Option $76,885:
  - This is considerably lower than the derived value. It would imply a substantially smaller V_U or a smaller PV of tax shields than the model yields, incompatible with the given perpetual EBIT and tax rate under the standard M&M framework used for perpetual debt with a fixed debt ratio.
- Option $93,453:
  - This is higher than the calculated $85.8k. Achieving this would require a larger PV of the tax shield or a larger unlevered firm value, which would necessitate either a higher EBIT, a lower tax rate, or a more favorable leverage effect than assumed. Under the specified inputs, this option overstates the levered value relative to the standard calculation.

Summary note:
- Using the conventional perpetual debt tax shield approach with D = 0.5 V_L yields V_L ≈ 85,782. The provided answer choices include values near this figure but do not match exactly; among them, $83,854 is the closest to the derived result, while the others diverge more significantly. Remember that small differences can stem from rounding conventions or alternative tax shield modeling assumptions.

FA25-BL-BUS-F305-1130 Final Exam

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