﻿ The Cost of Equity Capital
San José State University

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 The Cost of Equity Capital

Corporations can raise additional capital by the sale of debt or the sale of their common stock. The determination of the cost of capital from debt is straight-forward. If R is the interest rate on debt and t is the tax rate on corporate profits then the cost of capital from debt is R(1-t). The adjustment for the tax rate on profit comes because the interest payments on debt are tax-deductible.

The definition and determination of the cost of capital from equity is more complicated. The Gordon Model of stock price offers some insights . The Gordon Model says that if a corporation has dividends that will be growing at a constant rate of ΔV forever and d1 is the next dividend to be paid then the price of a share of the stock should sell for

#### p = d1/(r-g)

where r is the discount rate the market places on the future dividends of the stock. The discount rate r is the effective cost of capital from equity. If the price of the stock is known and the discount rate r is unknown it can be found by solving the Gordon formula; i.e.,

#### r = d1/p + g.

The ratio of dividend to stock price d1/p is known as the yield rate for the stock. Thus the Gordon Model implies that the cost of capital from equity is the sum of the yield rate plus the dividend growth rate.

But the Gordon Model is for a rather special class of corporations. There is uncertainty about the relevance of the above formula for the general class of corporations. What the analysis below shows is that properly interpreted the above formula for the cost of equity capital applies to the general case.

## The Definition and Determination of the Cost of Equity Capital

The first thing to note is that the cost of equity capital means the cost of the sale of new equity for the present stockholders of the corporation. The second thing is that the determination of costs and benefits are separated. That means that the costs are calculated presuming no benefits and that the benefits are calculated presuming no costs. The relevant quantity is the net benefits which are computed by subtracting the costs from the benefits. The preliminary separation of costs and benefits greatly simplifies the analysis.

Let V be the total value of the corporation, N the number of shares outstanding and D the total funds available for the payment of dividends. Now suppose ΔN shares are issued but there are no benefits. The old stockholders would receive N/(N+ΔN) share of the dividends and also of the capital gain of the corporation. Let ΔV be the increase in value of the corporation. If there were no issue of new stock the old stockholders would have received (D+ΔV), but with the issue of ΔN new shares the old stockholder will receive only (D+ΔV)N/(N+ΔN). The cost to the old stockholders of the issue of new stock is then

#### (D+ΔV)[1 - N/(N+ΔN)] = (D+ΔV)ΔN/(N+ΔN)

If the number of new shares sold is relatively small so the stock price p is unaffected then the amount of capital raised is pΔN. If the above cost of the new issue to the old stockholders is divided by the amount of capital raised the result is the cost of equity capital

#### req = [(D+ΔV)ΔN/(N+ΔN)]/pΔN = [(D+ΔV)/p](1/(N+ΔN))

By multiplying and dividing by N the above formula can be rearranged to

#### req = [(D/N + ΔV/N)/p][N/(N+ΔN)]

D/N is the dividends per share and (D/N)/p is the yield rate. ΔV/Np is the same as ΔV/V the growth rate in the value of the corporation g. Since for ΔN relatively small compared to N the ratio N/(N+ΔN) is approximately one

#### req = yield rate + growth rate

In the Gordon Model the growth rate for dividends is the same as the growth rate for earnings and the growth rate for the value of the corporation so the above formula is the same as that derived from the Gordon Model.