# 7.2: Valuing Corporate Equities

- Page ID
- 584

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- How are corporate equities valued?

A corporate equity, or stock, is sometimes called a share because it is just that, a share in the ownership of a joint-stock corporation. Ownership entitles investors to a say in how the corporation is run. Today that usually means one vote per share in corporate elections for the board of directors, a group of people who direct, oversee, and monitor the corporation’s professional managers. *Ownership also means that investors are residual claimants, entitling them to a proportionate share of the corporation’s net earnings (profits), its cash flows, and its assets once all other claims against it have been settled*.

In exchange for their investment, **stockholders** may receive a flow of cash payments, usually made quarterly, called **dividends**. Dividends differ from bond coupons in important ways. Unlike coupons, they are not fixed. They may go up or down over time. Also, if a company fails to pay dividends on its stock, it is not considered in default. (We speak here of common stock. Another type of financial instrument, a preferred share [preference shares in the United Kingdom], promises to pay a fixed dividend. Such instruments are a type of equity-debt hybrid and are priced more like coupon bonds.) In fact, many corporations today do not pay any dividends, and for good reasons. Small, rapidly growing companies, it is widely believed, should plow their profits back into their businesses rather than return money to shareholders. *That is not cheating the stockholders, because profits left with the company instead of paid out as dividends raise the share price*. The company has more cash than it otherwise would, after all, and stockholders own the profits whether they are left with the company or put into their pockets. Plus, it is generally thought that growing companies put the money to more profitable use than stockholders could.

There is a tax benefit to retaining earnings, too. Taxes on dividends, which the **Internal Revenue Service (IRS)** considers income, are usually higher than taxes on share appreciation, which the IRS considers capital gains.^{[1]} Also, dividends are taxed in the year they are paid, which may be inconvenient for stockholders, but capital gains taxes are incurred only when the stockholders sell their shares, so they have more control over their tax liabilities. Similarly, companies that have stopped growing will sometimes buy their own stock in the market rather than pay dividends. Fewer shares outstanding means that each share is worth more (the price per share equals the total value of the company divided by the number of shares, so as the denominator declines, the price per share increases), so stockholders are “paid” with a higher stock price. Nevertheless, some corporations continue to pay dividends. *The point here is that what really matters when valuing corporate equities is earnings or profits because, as noted above, they belong to the stockholders whether they are divided, kept as cash, or used to repurchase shares*.

*The simplest stock valuation method, the one-period valuation model, simply calculates the discounted present value of dividends and selling price over a one-year holding period*:

\[P = \dfrac{E}{ 1 + k} + \dfrac{P_1}{ 1 + k}\nonumber\]

where:

- P = price now
- E = yearly dividends
*k*= required rate of return- P₁ = expected price at year’s end

So if a company is expected to pay no dividends, its share price is expected to be $75 at the end of the year, and the **required rate of return (or k)** (a sort of risk-adjusted interest rate) on investments in its risk class is 10 percent, an investor would buy the stock if its market price was at or below P = 0/1.10 + 75/1.10 = $68.18. Another investor might also require a 10 percent return but think the stock will be worth $104 at the end of the year. He’d pay P = 0/1.10 + 104/1.1 = $94.55 for the stock today! A third investor might agree with the first that the stock will be worth $75 in a year, but she might need a 12 percent return. She’d pay only up to P = 0/1.12 + 75/1.12 = $66.96 per share. Yet another investor might also require a 12 percent return to hold the stock and think $75 a reasonable price a year from now, but he might also think earnings of $1 per share is in the offing. He’d pay P = 1/1.12 + 75/1.12 = .89 + 66.96 = $67.85 per share.

*For longer holding periods, one can use the generalized dividend valuation model, which discounts expected future earnings to their present value*. That can be done mechanically, as we did for coupon bonds in Chapter 4, or with a little fancier math:

\[P = t = 1 ∞ \dfrac{E t}{1 + k t}\]

That sideways 8 means infinity. So this equation basically says that the price of a share now is the sum (σ) of the discounted present values of the expected earnings between now and infinity. The neat thing about this equation is that the expected future sales price of the stock drops out of the equation because the present value of any sum at any decent required rate of return quickly becomes negligible. (For example, the present value of an asset expected to be worth $10 in 20 years at 15 percent interest is only PV = 10/(1.15)^{20} = $0.61 today.) So for all intents and purposes in this model, called the Gordon Growth model, *a corporate equity is worth the discounted present value of its expected future earnings stream*.

\[P = \dfrac{E ( 1 + g )}{k - g} \]

where

- P= price today
- E = most recent earnings
*k*= required return*g*= constant growth rate

So the price of a stock today that recently earned $1 per share and has expected earnings growth of 5 percent would be $21.00 if the required return was 10 percent (P = 1.05/.05). If another investor estimates either *k* or *g* differently, perhaps because he knows more (or less) about a country, industry, or company’s future prospects, P will of course change, perhaps radically. For a little practice, complete the following exercises now.

- Use the one-period valuation model P = E/(1 +
*k*) + P1/(1 +*k*) to price the following stocks (remember to decimalize percentages).Dividends (E = $) Required return ( *k*= %)Expected price next year (P1 = $) Answer: price today (P = $) 1.00 10 20 19.10 1.00 15 20 18.26 1.00 20 20 17.50 0 5 20 19.05 0 5 30 28.57 0 5 40 38.10 1.00 10 50 46.36 1.50 10 50 46.82 2.00 10 50 47.27 0 10 1 0.91 - Use the Gordon growth model P = E × (1 +
*g*)/(*k*−*g*) to value the following stocks (remember to decimalize percentages).Earnings (E = $) Required return ( *k*= %)Expected earnings growth rate ( *g*= %)Answer: price today (P = $) 1 10 5 20.00 1 15 5 10.00 1 20 5 6.67 1 10 5 20.00 2 10 5 40.00 3 10 5 60.00 1 30 5 4.00 1 30 10 10.00 1 30 15 20.00 100 20 10 1,000.00

Stock prices plummeted after the terrorist attacks on 9/11. Use the Gordon growth model to explain why.

Stock prices plummeted after 9/11 because risks increased, raising *k*, and because expectations of corporate profits dropped, decreasing *g*. So the numerator of the Gordon growth model decreased and the denominator increased, both of which caused P to decrease.

- In general, corporate equities are valued the same way that any financial security is, by discounting expected future cash flows.
- With stocks, corporate earnings replace actual cash payments because shareholders own profits, whether they receive them as cash dividends or not.
- The formula for valuing a stock to be held one year, called the one-period valuation model, is P = E/(1 +
*k*) + P_{1}/(1 +*k*), where E is dividends, P_{1}is the expected sales price of the stock next year, and*k*is the return required to hold the stock given its risk and liquidity characteristics. - In the Gordon growth model, earnings are assumed to grow at a constant rate forever, so stock values can be estimated without guessing the future sales price by using the following formula: P = E(1 +
*g*)/(*k*–*g*), where E = the most recent earnings,*g*= the rate of earnings growth,*k*= the required return where*k*>*g*.

[1] __www.irs.gov/__