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.