Serendipity Inc. is re-evaluating its debt level. Its current capital structure consists of 80% debt and 20% common equity, its beta is 1.60, and its tax rate is 25%. However, the CFO thinks the company has too much debt, and he is considering moving to a capital structure with 40% debt and 60% equity. The risk-free rate is 5.0% and the market risk premium is 6.0%. By how much would the capital structure shift change the firm's cost of equity?

Question

BlackTom AI · Accepted Answer

Use Hamada’s equation to remove the effect of leverage from the given beta, then reapply leverage for the new capital structure and compute equity costs with CAPM.

1) Relevant concepts
- Hamada’s formula links levered beta (βL) and unlevered (asset) beta (βU): βU = βL / [1 + (1 − Tc)·(D/E)] — this adjusts for the tax shield on debt.
- CAPM: cost of equity = rf + β·(market risk premium).

2) Step-by-step reasoning and calculations
- Given: βL (current) = 1.60, Tc = 25% = 0.25, current capital structure D/V = 0.80, E/V = 0.20 → current D/E = 0.80/0.20 = 4.
- Unlever the beta:
  βU = 1.60 / [1 + (1 − 0.25)·4] = 1.60 / [1 + 0.75·4] = 1.60 / [1 + 3] = 1.60 / 4 = 0.40.
- New capital structure: D/V = 0.40, E/V = 0.60 → new D/E = 0.40/0.60 = 2/3 ≈ 0.6666667.
- Re-lever to find new equity beta:
  β_new = βU · [1 + (1 − Tc)·(D/E)_new] = 0.40 · [1 + 0.75·(2/3)] = 0.40 · [1 + 0.5] = 0.40 · 1.5 = 0.60.
- Use CAPM (rf = 5.0%, market risk premium = 6.0%):
  Current cost of equity = 5.0% + 1.60·6.0% = 5.0% + 9.6% = 14.6%.
  New cost of equity = 5.0% + 0.60·6.0% = 5.0% + 3.6% = 8.6%.
- Change = New − Old = 8.6% − 14.6% = −6.0 percentage points.

3) Why this is correct
The decrease arises because lowering leverage reduces financial risk borne by equity holders; unlevering and relevering beta correctly accounts for the tax shield (1 − Tc) and the D/E ratio change, and CAPM translates the beta change into a cost-of-equity change.

4) Why other approaches are incorrect
Common mistakes: failing to unlever/re-lever (which would keep β at 1.60), using D/V instead of D/E inside Hamada’s formula, or ignoring taxes; any of these produce wrong beta and thus incorrect cost-of-equity changes.

Final confirmation: the firm’s cost of equity falls by 6.00 percentage points, matching the provided answer of −6.00%.

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