The residuals theory of dividends tends to imply that the dividends are irrelevant and the value of the firm is independent of its dividend policy. The irrelevance of dividend policy for a valuation of the firm has been most comprehensively presented by Modigliani and Miller. They have argued that the market price of a share is affected by the earnings of the firm and not influenced by the pattern of income distribution. What matters, on the other hand, is the investment decisions which determine the earnings of the firm and thus affect the value of the firm. They argue that subject to a number of assumptions, the way a firm splits its earnings between dividends and retained earnings has no effect on the value of the firm.

The MM approach to irrelevance of dividend is based on the following assumptions:

- The capital markets are perfect and the investors behave rationally.
- All information is freely available to all the investors.
- There is no transaction cost.
- Securities are divisible and can be split into any fraction. No investor can affect the market price.
- There are no taxes and no flotation cost.
- The firm has a defined investment policy and the future profits are known with certainty. The implication is that the investment decisions are unaffected by the dividend decision and the operating cash flows are same no matter which dividend policy is adopted.

The model

Under the assumptions stated above, MM argue that neither the firm paying dividends nor the shareholders receiving the dividends will be adversely affected by firms paying either too little or too much dividends. They have used the arbitrage process to show that the division of profits between dividends and retained earnings is irrelevant from the point of view of the shareholders. They have shown that given the investment opportunities, a firm will finance these either by ploughing back profits of if pays dividends, then will raise an equal amount of new share capital externally by selling new shares. The amount of dividends paid to existing shareholders will be replaced by new share capital raised externally.

In order to satisfy their model, MM has started with the following valuation model.

**P0= 1* (D1+P1)/ (1+ke)**

Where,

P0 = Present market price of the share

Ke = Cost of equity share capital

D1 = Expected dividend at the end of year 1

P1 = Expected market price of the share at the end of year 1

With the help of this valuation model we will create a arbitrage process, i.e., replacement of amount paid as dividend by the issue of fresh capital. The arbitrage process involves two simultaneous actions. With reference to dividend policy the two actions are:

- Payment of dividend by the firm
- Rising of fresh capital.

With the help of arbitrage process, MM have shown that the dividend payment will not have any effect on the value of the firm. Even if the firm pays dividends, resulting in a increase in market value of the share, the effect on the value of the firm will be neutralised by the decrease in terminal value of the share. The working of the arbitrage process is substantiated as follows:

Suppose a firm has 100000 shares outstanding and is planning to declare a dividend of $5 at the end of current financial year. The present market price of the share is $100. The cost of equity capital, ke, may be taken at 10%. The expected market price at the end of the year 1 may be found under two options:

- If dividend of $5 is paid
- If dividend is not paid

When Dividend of $5 is paid (the value of D1 is 5) :

P0 = (D1 + P1)/ (1+ ke)

P0 (1+ ke) = D1 + P1

P1 = P0 (1+ ke) – D1

= 100(1.10) – 5

= 105

So, the market price is expected to be $105, if the firm pays dividend of $5

When, Dividend of $5 is not pain (the value of D1 is 0):

P0 = (D1 + P1)/ (1+ ke)

P0 (1+ ke) = D1 + P1

P1 = P0 (1+ ke) – D1

= 100(1.1)

= 110

So, the market price of the share is expected to be $110, if the firm does not pay any dividend.

However, in both the cases, the position of the shareholders would be the same. A shareholder having for example 1 share will be having same worth of his holding if the firm pays dividend or not. In case, the dividend of $5 is paid, he will receive $5 from the firm as dividend and the market price of the share would be $105, giving a total worth of $110. In case, the dividend is not paid then the market price of the share or the worth of the shareholder would be still $110. So, the shareholder would be indifferent if dividend is paid or not to him. The same example can be extended further to analyze the effect of arbitrage employed by the firm.

Say, the firm has total profits of $1000000 during the year 1 and is planning to make an investment of $2000000 at the end of the year 1. The arbitrage process and value of the firm may be explained as follows:

**If dividend of $5 is paid by the firm at the end of year 1:**

Total Earnings $1000000

Dividends Paid (100000 * $5) $500000

Retained Earnings $500000

Total funds required for investment $2000000

Therefore, fresh capital to be issued $1500000

Market price at the end of year 1 $105

Number of shares to be issued (1500000/105) 14285.71

Total number of shares (100000+14285.71) 114285.71

Applying the formula, the value of the firm, nP0 is

**nP0 = [(n + m)P1 – I +E] / (1+ ke)**

** = **[(114285.71)105 – 2000000 + 1000000] / (1.10)

= $10000000

**If dividend of $5 is not paid by the firm at the end of the year 1:**

Total earnings $1000000

Dividends Paid -

Retained Earnings $1000000

Total funds required for investment $2000000

Therefore, fresh capital to be issued $1000000

Market price at the end of year 1 $110

Number of share to be issued (1000000/110) 9090.9

Total number of shares (100000+9090.0) 109090.9

Applying the formula, the value of the firm, nP0 is

**nP0 = [(n + m) P1 – I +E] / (1+ ke)**

= [(109090.9)110 – 2000000 + 1000000]

= $10000000

So, the value of the firm remains same at $10000000 if the dividend is paid or not. With the help of above process, it can be showed that dividend policy is irrelevant for the valuation of the firm. Dividend payment does not affect the value of the firm.

It should be noted that, the formula used above, gives the current market value of the firm. The MM model shows that if dividend is paid or not at the end of current year, the present market value of the firm remains same at $10000000. The same example can be applied to find out the expected market value of the firm at the end of current year as follows:

**If dividend of $5 is paid**

Total number of shares 114285.71

Market price $105

Total market value $12000000

**If dividend of $5 is not paid**

Total number of shares 109090.90

Market price $110

Total market value $12000000

Thus, the expected market value remains same at $12000000, whether the firm pays dividend of $5 or not. The MM Model shows therefore, that the current market value or the expected market value of the firm, both are unaffected by the dividend decision of the firm.

Under the assumptions set by MM, this model testifies that dividend is irrelevant and the investors are indifferent between the current dividends and the future capital gains. Given these assumptions, the effect of a dividend decision may be stated as: that there is not relationship between dividend policy and value of the share. One dividend policy is as good as other. Investors are concerned only with the total returns and are indifferent if these returns are from dividend income or capital gains. That is, to finance the growth, the firm may choose to issue shares and thereby allowing profits to be used to pay dividend, or may use internally generated funds for financing the growth and thereby paying less in dividends and not issuing any shares.

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