# Derivation: Modigliani & Miller's 2nd Proposition (With Taxes)

**Modigliani & Miller’s 2nd Proposition (With Taxes):**

M&M’s 2nd Proposition states that as leverage increases, expected return on equity increases. Since equity holders are paid after debt holders, it is the residual payment and therefore becomes riskier.

*(Note: As per the assumption, cost of leverage doesn't increase with increase in debt)*Even though extra leverage makes equity riskier, the proportion of equity in the capital structure reduces and the tax shield on debt reduces some of the overall risk.

Therefore in the presence of taxes WACC decreases as we add more leverage due to the additional tax shield.

The following two equations are proposed by M&M and will be derived through this exercise:

**re = ru + (1 – t) (D0 / E0) (ru – rd) - (1)****WACC = ru ( 1 – t (D0 / V0 ) )****- (2)**

**Derivation:**

Consider a firm with both debt and equity and a **life of 1 year**:

Value of the firm today: V0 = E0 + D0

Value of the firm after 1 year: V1 = E1 + D1

Value of a similar unlevered firm today: V0U

Value of a similar unlevered firm today: V1U

Proportion of equity in the capital structure: E

Proportion of debt in the capital structure: D

When there are no intermediate payments:

E1 = E0 (1 + re)

D1 = D0 (1 + rd)

X1 = Cash flow distributed to both debt and equity holders in period 1

Value of the levered firm next year on the basis of cash flow distributions can be written as: *(Note: Firm has a life of one year)*

V1 = (X1 – D1)(1 – t) + D1 = X1 (1 – t) + t D1 = **V1U + t D1**

V1 = E1 + D1 = **(1 + re)E0 + (1 + rd)D0**

V1U = (1 + ru) V0U and D1 = (1 + rd) D0

**Substituting for V1**

(1 + ru) V0U + t(1 + rd)D0 = (1 + re)E0 + (1 + rd)D0

**Rearranging the terms:**

(1 + re) = (1 + ru) ( V0U / E0) – (1 – t) (1 + rd) (D0 / E0)

V0U = V0 – t D0 (This equation is PV of V1 = V1U + t D1)

(1 + re) = (1 + ru) (V0 – t D0 ) / E0 - (1 – t) (1 + rd) (D0 / E0)

since V0 = E0 + D0

(1 + re) = (1 + ru) (E0 + D0 – t D0 ) / E0 - (1 – t) (1 + rd) (D0 / E0)

(1 + re) = (1 + ru) (E0 + D0 (1– t) ) / E0 - (1 – t) (1 + rd) (D0 / E0)

(1 + re) = (1 + ru) (E0) / (E0) + (1 + ru) (1– t) (D0 / E0) - (1 – t) (1 + rd) (D0 / E0)

(1 + re) = 1 + ru + (D0 / E0) (1– t) (1 + ru - 1 - rd)

re = 1 + ru + (D0 / E0) (1– t) (ru - rd) - 1

We can rewrite:

**re = ru + (1 – t) (D0 / E0) (ru** **– rd)** **[ Equation (1) derived]**

If this value of re is inserted in the WACC equation:

WACC = [(E0 / (D0 + E0)] re + [(D0 / (D0 + E0)] rd (1 - t)

Remember V0 = E0 + D0

= E0 / V0 [**ru + (1 – t) (D0 / E0) (ru** **– rd)] **+ D0 / V0 ((rd (1 – t))

= E0 / V0 (ru) + E0 / V0 (1 – t) (D0 / E0) ru - E0 / V0 (1 – t) (D0 / E0) rd + D0 / V0 ((rd (1 – t))

Cancelling out terms in the numerator and denominator

= (ru) [E0 / V0 + D0 / V0 (1 – t) ]

= (ru) [E0 / V0 + D0 / V0 - D0 / V0 (t) ]

Remember V0 = E0 + D0

**WACC = (ru) [ 1 - D0** **/ V0 ** **(t) ] [Equation (2) derived]**

Assumptions of M&M'a 2nd Proposition:

No transaction cost

Perfect information & homogeneous expectations

Investors care only about their wealth

Financing decisions do not affect investment outcomes.