## Cost of equity

In a similar way to the cost of debt the cost of equity represents the equilibrium or minimum rate of return required by the firm's common shareholders. These funds can be obtained in two ways: internally, from retained earnings, and externally, from issuing new stock. These two sources are now discussed.

a. Internal sources

The cost of retained earnings represents an opportunity cost to investors. These earnings could be paid out in the form of dividends to investors, who could then reinvest the funds in other shares, bonds or property. Unless the retained earnings can earn at least the same return within the firm as they can outside the firm, assuming the same degree of risk, it is not profitable to use them as a source of funds.

There is no simple way to calculate this internal cost; there are several alternative approaches, which are not mutually exclusive. Two main approaches will be discussed here: the capital asset pricing model (CAPM), and the dividend valuation model (DVM).

1. The capital asset pricing model (CAPM). The general nature of this model was discussed in the previous section. It was seen that the variability, or volatility, in returns to an individual security can be divided into two components: that part which is related to the corporate risk of the firm, sometimes called unsystematic risk, and that part which affects the market as a whole, the systematic risk. Rational investors will diversify their portfolios so as to eliminate unsystematic risk; such risk carries no benefit in terms of additional return, since investors can obtain the same returns through holding a diversified portfolio of securities with similar corporate risk, but with reduced market risk.

The cost of equity, in terms of retained earnings, can now be estimated using the equation of the SML in (11.6). Thus, assuming kM =10 per cent, kRF =6 per cent and a beta coefficient of 2.0, the cost of equity would be:

While the CAPM may appear to be an objective and precise method for estimating the cost of capital it is subject to a number of drawbacks, which arise from the nature of the assumptions underlying the model. One of the most important ofthese is the use ofbeta coefficients based on historical data. In practice this is the only objective method for estimating such coefficients, but conceptually the cost of capital should be based on a model involving expected beta coefficients for the future. It is beyond the scope of this text to examine these assumptions and drawbacks in more detail, but they are discussed in most texts on financial management.

2. The dividend valuation model (DVM). This model is also known as the DCF

model since it involves the now familiar present-value formula from Chapter 2, and is similar to expression (11.7). The value of a shareholder's wealth is the sum of expected future returns, discounted by their required rate of return. These returns come in the form of dividends and an increase in the market value of the firm's shares. Thus the present value of the share is given by:

where Dt is the dividend paid by the firm in period t. Since the future value of the share, Vn, is in turn determined by the sum of expected future dividends, equation (11.9) can be rewritten as the sum to infinity of all expected future dividends:

We now make the assumption that the dividends of the firm grow at a constant rate of g per year. Thus the value of the shares is given by:

where D1 is the dividend that is expected to be paid in the following period. This is a geometric series which can be summed to infinity as long as the terms become smaller, in other words as long as g<k. The sum is given by:

D1 n

This equation can be rearranged to solve for the cost of equity as follows:

For example, if a firm has a current share price of £20, the dividend next year is expected to be £1.20 and dividends have been growing on average at 4 per cent per year, then the cost of equity is given by:

1 20

Just as the CAPM has its problems, so does the DVM also have drawbacks. Its main failing is essentially the same as with the CAPM: it looks backwards rather than forwards. While historical data can be used to estimate the average growth rate of dividends over the last ten or twenty years say, such information is not a reliable indicator of future dividend growth rates. Furthermore, current share prices can be highly volatile, and many firms do not pay dividends at all if management believes that the funds can be more profitably reinvested in the firm than returned to shareholders. It can therefore readily be seen that many fast-growing high-tech firms would on this basis have very uncertain estimates of their cost of capital.

b. External sources

It is more expensive for a firm to raise equity externally rather than internally for two reasons:

1 There are flotation costs, as discussed with the cost of debt.

2 New shares have to be sold at a price lower than the current market price, in order to attract buyers. The reason for this is that the current price normally represents an equilibrium between existing demand and supply. A new issue involves an increase in supply, thus reducing the equilibrium price.

The result of these factors is that equation (11.13) has to be modified in order to provide an estimate of the cost of external equity as follows:

where kne represents the cost of new equity, and V* is the net proceeds to the firm from the new issue (per share) after deducting flotation costs.

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