# Proof of Miller and Modigliani Hypothesis

Definition: Miller and Modigliani Hypothesis or MM Approach supports the “dividend irrelevance theory”, stating that the dividends are irrelevant and has no effect on the firm’s share value. Also, it is believed that it is the investment policy that increases the value of the shares and hence should be given more importance than the payouts to the shareholders.

To justify his argument Miller and Modigliani have given proof in the form of equations, through which it is very clear that dividends play no role in determining the share price.

## Proof of Miller and Modigliani Hypothesis

Step 1: The market price of a share, in the beginning, is equal to the present value of dividends received at the end of the period plus the market price of a share at the end, is represented as:

P0 = [1/(1+ Ke)] * (D1 +P1)

Where, P0 = market price of a share in the beginning of the period
Ke=cost of equity capital
D1= Dividends received at the end of the period
P1= market price of a share at the end of the period

Step 2: It is assumed that no external financing is raised, thus the total capitalized value of the firm would be the number of shares (n) times the price of the share P0.

nP0 = [1/(1+Ke)] * (nD1+nP1)

Step 3: If the retained earnings fall short to finance the investment opportunity then, ∆n is the number of new shares issued at the end of year 1 at price P1.

nP0 = [1/(1+Ke)] * [(nD1 + (n+ n ) P1 – ∆P1)]

Where, n= no of shares outstanding at the beginning

Step 4: If the firm finances all its investments, the total amount raised through new shares is given as:

nP1 = I – (E – nD1)

Where, ∆nP1= amount received from the sale of new shares to finance capital budget

I = requirement of capital budget
E= earnings
nD1 = Dividends
E-nD1 = Retained Earnings

Step 5: If we substitute the equation of step 4 in step 3, we get the following equation:

nP0 = [1/(1+Ke)] * [(nD1 + (n+ ∆n ) P1 – ∆P1) – (I –E + nD1)

OR

nP0 = [nD1+ (n+ ∆n) P1 – I + E – nD1] / (1+Ke)

The negative nD1 and the positive nD1 get cancelled and then, we get the final equation:

nP0 = [(n + n) P1 – I + E] / (1+Ke)

Thus, we found out that there is no dividend in the equation above, and hence it is proved from the Miller and Modigliani Hypothesis that dividends are irrelevant and has no effect on the firm’s share price.