MODIGLIANI-MILLER PROPOSTIONS



MODIGLIANI-MILLER PROPOSITIONS

M-M Proposition 1: In competitive, transaction costless, information efficient markets, with no taxes, the market value of the firm (i.e., market value of all of its securities) is independent of the firm’s capital structure. That is, [pic] = [pic] (see definition below) [Brealey, Myers and Allen, Chapter 17]

The proof of this proposition is based on the following arbitrage property of perfect markets.

Arbitrage Property: Two identical assets must have the same market price. Two assets are identical if either can be converted into the other.

Logic of M-M Proof: Let firm U be an unlevered firm and let firm L be an identical firm if levered (or be the same firm if levered); U and L differ only in that L is levered. [pic] is the market value of firm U (therefore [pic] is also the market value of the equity of firm U since it is unlevered); [pic] and [pic]are the market values, respectively, of the equity and debt of firm L, and firm value [pic]= [pic] + [pic].

Compare buying percent P of firm L with the following: buying percent P of the equity (shares) of firm U plus personal borrowing of amount P [pic] (using the shares of firm U as collateral). Exhibit 1 shows that the incomes of the two “strategies” are identical.

Exhibit 1. Convert U into L using [2]

|Strategy |Net Investment of the Strategy |Income |

|[1] Buy percent P of equity of firm L |P [pic] |P [Profit ( Interest] |

|[2] Buy percent P of equity of firm U and borrow |P [pic] ( P [pic] |P Profit ( P Interest = |

|amount [P [pic]] | |= P [Profit ( Interest] |

Since we can use strategy [2] to create the stock under strategy [1], strategy [2] must be at least as good as strategy [1], and therefore we must be willing to pay at least as much for strategy [2] as for strategy [2]. That is,

P [pic] ( P [pic]( P [pic],

which implies that:

[pic] ( [pic] + [pic] =[pic] (1)

Now we will show that we can convert L into U. Compare two strategies: [1(] Buy P percent of firm U; [2(] Buy P percent of the shares of firm L, and buy percent P of the debt of firm L. Exhibit 2 shows that the incomes of the two strategies are identical.

Exhibit 2. Convert L into U [using [2(])

|Strategy |Net Investment of the strategy |Income |

|[1(] Buy percent P of firm U equity |P [pic] |P Profit |

|[2(] Buy percent P of firm L debt and |P [pic] + P [pic]= P [pic] |P [Profit ( Interest] + P Interest |

|percent P of firm L equity | |= P Profit |

Since we can use strategy [2(] to create the stock under strategy [1(], strategy [2(] must be at least as good as strategy [1(], and therefore we must be willing to pay at least as much for strategy [2(] as for strategy [2(]. That is,

P [pic] ( P [pic]

which implies that:

[pic] ( [pic] (2)

The only way for (1) and (2) to hold is for:

[pic] = [pic] (3)

which is M-M Proposition 1.

Brealey, Myers and Allen Proposition 1 illustration (page 449-451): Assume that M-M Proposition 1 holds. Then we will see that the same investment is required to generate the same income whether or not the firm is levered. One share of the levered firm generates the same income as two shares of the unlevered firm with personal borrowing of $10.

One share of the levered firm:

Investment = cost of one shares of L = $10

Income is one of: $0 $1 $2 or $3 [see BM&A Table 17.2 on page 450]

Two shares of the unlevered firm with borrowing of $10:

Investment = cost of two shares of U ( amount borrowed = $20 ( $10 = $10

Income = firm income on two share ( interest on personal debt

= one of: $0 $1 $2 or $3 [see BM&A Table on page 451]

M-M Proposition 2: In competitive, transaction costless, information efficient markets, with corporate tax-deductibility of interest, the market value of the firm (i.e., market value of all of its securities) equals:

[pic] = [pic] + T [pic]

where [pic] is the value of the firm if it has debt, [pic] is the value of the firm if it has no debt, T is the corporate tax savings per dollar of debt, and [pic] is the market value of the firm’s debt.

In the above relationship, T is equal to the firm’s tax rate if all debt interest is tax deductible. However, if some or all of the interest is not tax deductible, T is not the marginal tax rate. For example, if 80 percent of the interest were tax deductible and the corporate tax rate were 34 percent, then, in the above equations, T = .8 ( .34 = .272.

The proof M-M Proposition 2 is similar to the proof with no taxes and so will not be shown here.

To illustrate, suppose that Royal Corporation is all-equity financed and worth $400 million ([pic] = $400 million). It plans to change its financial structure (but not its assets) by borrowing $100 million and using the $100 million for a share repurchase. Royal’s tax rate is 34 percent and all of the interest on it debt is tax deductible. Proposition 2 states that it can expect its new value to be:

[pic] = [pic] + T [pic]= $400,000,000 + .34($100,000,000) = $434,000,000

7/5/2005

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