## Optimal Capital Structure and Advanced WACC Applications
### Optimal Capital Structure
The optimal mix minimises WACC. At each debt ratio:
- WACC = Debt weight × Kd(after-tax) + Equity weight × Ke
- Initially WACC falls as cheap debt replaces expensive equity
- Beyond the optimal point, financial distress raises both Kd and Ke, increasing WACC
- Identify the minimum WACC point — that is the optimal mix
### Reverse WACC — Solving for Ke
If WACC, debt cost, and D:E ratio are known:
WACC = [E/(D+E)] × Ke + [D/(D+E)] × Kd(1–t)
Rearrange to isolate Ke.
### Effect of Gearing on Equity Value
When a firm takes on debt to fund a project:
- Firm value = PV of all earnings / WACC
- Equity value = Firm value – Debt
- If WACC is unchanged (MM assumption): shareholders capture the full NPV of the project
- If Ke rises to exactly offset debt's benefit: WACC stays constant, equity value increases by project NPV
### WACC: Old vs New Capital Structure
When raising additional debt:
1. Compute new Ke (equity price falls, dividend rises → higher Ke)
2. Include new debt at its after-tax rate
3. Reweight using new total capital
4. Higher financial risk → higher WACC is common
### Example 1
Q58 — Optimal capital structure
| Debt % | Kd (after-tax) | Ke | WACC |
|---|
| 0% | 0 | 15.0 | 15.00% |
| 10% | 7.0 | 15.0 | 0.10×7+0.90×15 = 14.20% |
| 20% | 7.0 | 15.5 | 0.20×7+0.80×15.5 = 13.80% |
| 30% | 7.5 | 16.0 | 0.30×7.5+0.70×16 = 13.45% |
| 40% | 8.0 | 17.0 | 0.40×8+0.60×17 = 13.40% ← minimum |
| 50% | 8.5 | 19.0 | 0.50×8.5+0.50×19 = 13.75% |
| 60% | 9.5 | 20.0 | 0.60×9.5+0.40×20 = 13.70% |
Optimal mix: 40% debt : 60% equity (WACC = 13.40%)
### Example 2
Q59 — Reverse WACC: find Ke
D:E = 2:1 → Debt weight = 2/3, Equity weight = 1/3
WACC = 12%; Kd = 15%; Tax = 35% → After-tax Kd = 15%×0.65 = 9.75%
12% = (1/3)×Ke + (2/3)×9.75%
12% = Ke/3 + 6.50%
Ke/3 = 5.50%
Ke = 16.50%
### Example 3
Q50 — Gearing and equity value (Rtp)
Zeta Ltd: all-equity, MV = ₹6,00,000; Annual dividend = ₹1,20,000 (perpetual)
Ke = 1,20,000/6,00,000 = 20%
New project: Outlay ₹5,00,000; NCR = ₹1,05,000/year perpetual. Funded by 18% debentures.
Project NPV (at 20%) = 1,05,000/0.20 – 5,00,000 = 5,25,000 – 5,00,000 = ₹25,000
If Ke rises to 21.6% (equity holders bear increased risk):
Equity annual income = 1,20,000 + 1,05,000 – 90,000 (interest) = ₹1,35,000
Equity value = 1,35,000/0.216 = ₹6,25,000
Shareholder gain = 6,25,000 – 6,00,000 = ₹25,000 = project NPV ✓
Verify WACC unchanged:
Total firm value = 6,25,000 + 5,00,000 = ₹11,25,000
Total earnings = 1,20,000 + 1,05,000 = ₹2,25,000
WACC = 2,25,000/11,25,000 = 20% ✓ (unchanged)
Conclusion: Under MM (no taxes), gearing transfers risk to equity holders (Ke rises), but WACC is unchanged and shareholders capture the full project NPV.
### Example 4
Q47 — WACC before and after additional debt (Rtp, PYQ)
Existing: Equity 2L shares at ₹20; Ke = 2/20 + 5% = 15%; 11.5% Pref ₹10L; 10% Deb ₹30L → Kd = 6.5%
BV weights: Eq 40L, Pref 10L, Deb 30L. Total 80L.
WACC = (40/80)×15 + (10/80)×11.5 + (30/80)×6.5
= 7.50 + 1.44 + 2.44 = 11.375%
New: Additional ₹20L at 12% Deb; Ke rises (D₁ = ₹2.40, P₀ = ₹16):
New Ke = 2.40/16 + 5% = 15% + 5% = 20%; New Kd = 12%×0.65 = 7.8%
New total BV = 100L
New WACC = (40/100)×20 + (10/100)×11.5 + (30/100)×6.5 + (20/100)×7.8
= 8.00 + 1.15 + 1.95 + 1.56 = 12.66%
Additional gearing increased both financial risk (Ke rose) and WACC — the capital structure moved beyond its optimal point.