# Capital Growth and the Cost of Equity

Because dividend growth increases price, we can reformulate Equations (14) and (15) by focusing on the capital gain impact on equity value and the corporate cut-off rate. If share price grows at a constant annual rate G = (P1 - P0) / P0 then next year's price:

From Equation (14) we also know that the current price based on dividend growth (g):

So, logically share price one year from now must equal:

Because the same share cannot sell at different prices, it follows from Equation (16) that the dividend growth rate (g) must equal (G) the annual growth in share price (capital gain). Equations (16) and (17) can therefore be redefined as follows:

A comparison of Equations (16) and (18) reveals that if share price grows at a rate G, this must equal g, the annual growth in dividends. If we substitute G for g into Equation (14), this produces a dividend-capital gain model equivalent to the Gordon growth model.

The current ex div share price is determined by capitalizing next year's dividend at the amount by which the desired rate of return on equity exceeds the percentage capital gain.

Activity 3

If a company's forecast dividend is 20 pence per share, price is expected to grow at five percent per annum, and the equity capitalisation rate is 10 per cent:

Use the Gordon growth model to confirm that the current ex-div price still equals £4.00

Turning now to an equity capitalisation rate, which incorporates capital gains as a managerial investment criterion, we can substitute G for g into Equation (14) and rearrange terms so that:

This equation states that the total cost of equity comprises a dividend yield one year hence (D1/P0), plus a capital gain yield [G = (P1 - P0) / P0] equivalent to the growth in dividends (g).

Activity 4

If a company currently trading at £4.00 per share with a forecast 20 pence dividend is expected to grow at five percent per annum, confirm that the equity capitalisation rate is 10 per cent using the appropriate dividend and capital gain models.

# Growth Estimates and the Cut-Off Rate

So far so good, but if management wish to finance future projects by retaining profits in their quest for shareholder wealth, how do they calculate the growth rate?

Obviously, dividend and capital gains are rarely constant, which gives rise to complex valuation models that are beyond the scope of this text. But even if they are uniform, management still need annual growth estimators. Since the future is so uncertain, a simple solution favored by management is to assume that the past and future are interdependent. Without information to the contrary, Gordon (op cit) believed that a company's anticipated growth should be determined from its financial history. Consider the following data:

Year Dividend per Share (pence)

2005 20

2006 22

2007 24.2

2008 26.62

2009 29.28

Using the formula (Dt-Dt 1)/Dt 1 we can determine annual dividend growth rates

The average periodic growth rate, as an estimator of g, is therefore given by: g = 0.4 / 4 = 10%

Alternatively, we can calculate dividend growth by solving for g in the following equation and rearranging terms. 20 pence (1+g)4 = 29.28 pence.

Activity 5

Using the previous data and the appropriate equations, confirm that the forecast dividend for 2010 should be 32.21 pence. If shares are currently priced at £2.68 and dividends are expected to grow at ten percent per annum beyond 2010, confirm that the equity capitalisation rate (managerial cut-off rate for new investment) is 22 per cent.

# Earnings Valuation and the Cut-Off Rate

Whether or not growth is incorporated into the model, there is still no consensus as to whether dividends alone determine a share's value and hence the firm's cut-off rate for investment.

As long ago as 1961, the Nobel economic prize winners, Franco Modigliani and Merton Miller (MM) argued that given the problems of estimating retention-financed dividend growth, why not assume that dividends and retentions are perfect economic substitutes? Because if so; a company's share price and capitalisation rate can be determined by its overall earnings, rather than dividend policy. Since the future is uncertain, they also recommended a one period model.

According to MM, the current exit share price (P0) equals the anticipated earnings per share (E1) plus the ex div price (P1) at the end of the year, discounted at the shareholders' rate of return (K ).Algebraically, their single-period earnings model is:

Of course, earnings (like dividend) proponents confident with their forecasts need not restrict themselves to one period, or zero- growth. Assuming the cost of equity Ke is constant, the current ex-div price of a share held for any finite number of years (n) and then sold ex-div for Pn equals the finite-period earnings model

If n tends to infinity, then the general earnings valuation model is given by

If annual earnings Et are constant in perpetuity, Equation (23) simplifies to the constant earnings valuation model:

We can also incorporate growth into the previous equation to derive a constant earnings growth model analogous to the Gordon dividend model such that:

Review Activity

The only apparent difference between Equations (21) to (25) and our earlier dividend valuation models is the substitution of an earnings term (E) for dividends (D) in a parallel series of equations. However, because the same share cannot trade at two prices, the reformulation of corresponding P0 equations to derive the cost of equity (Ke) may have important consequences for the managerial cut-off rate. Can you explain why?

If a company adopts a full distribution policy, where dividend per share equals earnings per share, then substituting Et for Dt into either valuation models has no effect on the cost of equity (Ke).For example, reformulating the constant valuation model that solves for P0:

But what if a company adopts a partial distribution policy (where Dt < Et,).

Because the same share cannot trade at two prices, the equity return (Ke) must differ in the corresponding dividend and earnings equations if P0 is to remain the same. Mathematically:

Moreover, if P0 is identical throughout both series of dividend and earnings value equations, outlined earlier, then not only must the equity yield for dividends and earnings (Ke) differ, but a unique relationship must also exist between the two.

For example, if a dividend yield equals 10 percent per annum in response to a dividend of £1.00, the current share price should be

But if we now assume the dividend-payout ratio is 50 per cent and substitute the annual earnings per share of £2.00 into the previous equation, then subject to the law of one price (where P0 still equals £10.00) we produce the following equation with one unknown.

Rearranging terms, we can therefore define the earnings yield as an alternative to dividends as a managerial cut-off (discount) rate for new investment.

And solving for the earnings yield, we observe a difference to the dividend yield

Not only do the two yields differ but note they exhibit an inverse relationship defined by the dividend payout (earnings retention) ratio. Because the same share cannot sell at different prices and the dividend per share is half the earnings per share, then the earnings yield must be twice the dividend yield.

# Summary and Conclusions

We began our study of strategic financial management way back in Part One with an explanation of how companies employ their overall cost of capital as an investment criterion designed to maximise shareholder wealth. You will recall that under conditions of reasonably perfect markets, certainty and equilibrium, the correct cost is defined as the minimum return required by investors from an alternative investment of equivalent risk (The Separation Theorem of Fisher). So, if an all equity company undertakes a capital project using the marginal cost of equity as its discount rate, the total market value of ordinary shares should increase by the project's NPV.

In this Chapter we therefore addressed the crucial issue of equity valuation and the derivation of its associated capital cost as a discount rate, from both a dividend and earnings perspective under growth and non-growth conditions. We concluded that an equity capitalisation rate based on earnings, rather than dividends, should be management's preferred cut-off rate for new investment. But what if fund sources other than share capital are available to management. How do these affect project discount rates in our newly leveraged firm?

# Selected References

1. Fisher, I., The Theory of Interest, Macmillan (New York), 1930.

2. Gordon, M.J., The Investment, Financing and Valuation of a Corporation, Irwin, 1962.

3. Miller, M.H. and Modigliani, F., "Dividend policy, growth and the valuation of shares", The Journal of Business of the University of Chicago, Vol. XXXIV, No. 4 October 1961.