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Modigliani and Miller's Propositions
in Financial Economics

Modigliani and Miller's Propositions Concerning
the Equity Value of a Corporation

At the time that Franco Modigliani and Merton Miller (M&M) did their analysis there were four schools of thought as to what determines the value of a corporation:

Modigliani and Miller established that, properly interpreted, each of these approaches will give the same total equity value for the corporation. "Properly interpreted" means that dividends must be taken as "net dividends;" i.e., dividends paid out minus funds raised from the sale of new stock. Earnings must be taken as net earnings, earnings minus an imputed interest on cumulative investment. Cash flow must take into account the cash outflow of investment (free cash flow).

In addition to the four approaches that M&M brought together, there is a fifth more fundamental approach. The value of a corporations is the sum of the net present values of all its worthwhile projects.

It is important to note that their result is for the total equity in the corporation, as opposed to the value of a single share.

The cash flow of a corporation, which is defined as

After-tax profits plus Depreciation,

can go for dividends or for investment. Investment could also be covered by the sale of new stock. This means that

Cash Flow + Sale of New Stock = Dividends + Investment.

Therefore

Dividends - Sale of New Stock = Cash Flow - Investment.

Cash Flow minus Investment is called Free Cash Flow, and Dividends - Sale of New Stock is called Net Dividends.

Thus Free Cash Flow is the same as Net Dividends and therefore the value of the equity in a corporation is equal to the present value of future free cash flows and also the present value of net dividends.

M&M established that if one computes the cumulative investment of a corporation and deducts an interest from earnings based on this cumulative investment the result, which they label Net Earnings, is equal to both Net Dividends and Free Cash Flow. Therefore the equity value of a corporation is equal to the present value of all future Net Earnings.

M&M also established that if the present value of growth opportunities is calculated as the present value of of the earnings of investment projects which are in excess of the rate of discount r then the capitalized value of the current earnings plus the present value of the growth opportunities is equal to the other three methods of determining the equity value of a corporation. In the form of an equation, this says that the equity value of the corporation, P, is equal to

P = E/r + PVGO.

So M&M's analysis revealed that there is no conflict between the four schools of thought on the valuation of the equity in a corporation. The relationship also applies for a single share of the corporation.

The present value of the free cash flows is the totalling up of the cash flows for the separate projects and then computing the present values. Since net dividends are identically equal to free cash flows, it follows that the present value of the net dividends must be equal to the value of the corporation.

There is a fifth view of the valuation of the equity in a corporation; i.e., the sum of the net present values of all of its investments. This is compatible with the other four valuations.

Thus, the total value of the equity in a corporation should be equal to:


A Mathematical Derivation of M&M's Results

M&M analysis starts from the proposition that the stock price at any time t, pt, is equal to the present value of the next dividend payment and the price of the stock at the time of that payment; i.e.,

(1) pt = (dt+1 + pt+1)/(1+r)

This is equivalent to the condition that the price of the stock is equal to the present value of all future dividends.

In this analysis a lower case letter will denote the quantity per share and the upper case letter the quantity for the whole corporation. Let Nt be the number of shares outstanding at time t. The value of the equity in the corporation at time, Pt, is just Ntpt. Thus if Equation (1) is multiplied by Nt one obtains

(2) Pt = Ntpt = Nt(dt+1+ Ntpt+1)/(1+r) = (Dt+1 + Nt pt+1)/(1+r).

The quantity Ntpt+1 can be expressed as

Ntpt+1 = Ntpt+1 -Nt+1pt+1 +Nt+1pt+1 = -pt+1(Nt+1-Nt)+Pt+1.

The term pt+1(Nt+1-Nt) is the number of new shares sold between time t and time t+1 valued at the price of the stock at t+1. This may be taken to be the value of new stock sold in year t+1, St+1. Thus equation (2) reduces to

(3) Pt = ((Dt+1 - St+1) + Pt+1)/(1+r)

This equation implies also that

Pt+1 = ((Dt+2 - St+2) + Pt+2)/(1+r)

When this value for Pt+1 is substituted into the equation for t the result is:

Pt = (Dt+1 - St+1)/(1+r) + ((Dt+2 - St+2) + Pt+2)/(1+r)2

If this process is continued the result is

Pt = (Dt+1 - St+1)/(1+r) + (Dt+2 - St+2)/(1+r)2 + (Dt+3 - St+3)/(1+r)3 + ....

This means that the value of the corporation is equal to the present value of all future Net Dividends, dividends paid out less the funds brought in by the sale of new stock.

So M&M's analysis revealed that there is no conflict between the four schools of thought on the valuation of the equity in a corporation.


For more on Modigliani and Miller's propositions see M&M2.


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