Stock Valuation Masterclass

1. Common Stock Valuation

Unlike bonds, common stock valuation is more challenging because future cash flows are not known in advance, the investment has an indefinite life, and the market's required rate of return is not easily observable. However, the value of a stock is the present value of all its expected future dividends.

The General Case

The price of a stock today, $P_0$, is the present value of all its future dividends, $D_1, D_2, D_3, \dots$, discounted at the required return, $R$.

General Stock Valuation Formula:

$$P_0 = \frac{D_1}{(1+R)^{1}} + \frac{D_2}{(1+R)^{2}} + \frac{D_3}{(1+R)^{3}} + \dots$$

The Dividend Growth Model (Constant Growth)

This model assumes that dividends grow at a constant rate, $g$, indefinitely. This simplification allows us to value the stock as a growing perpetuity. This is also known as the Gordon Growth Model.

Dividend Growth Model Formula:

$$P_0 = \frac{D_1}{R-g} = \frac{D_0(1+g)}{R-g}$$

Where $D_1$ is the dividend expected next period, $D_0$ is the dividend just paid, $R$ is the required return, and $g$ is the constant growth rate.

Nonconstant Growth

For a firm with a supernormal growth rate for a finite period before settling into a constant growth phase, we use a two-stage approach. First, we find the present value of the dividends during the high-growth period. Then, we calculate the stock price at the start of the constant growth phase and discount that back to today.

Two-Stage Growth Valuation:

$$P_0 = \frac{D_1}{(1+R)^{1}} + \dots + \frac{D_t}{(1+R)^{t}} + \frac{P_t}{(1+R)^{t}}$$

Where $P_t$ is the stock price at time $t$ when constant growth begins. This is calculated using the Dividend Growth Model:

$$P_t = \frac{D_{t+1}}{R-g} = \frac{D_t(1+g)}{R-g}$$

Components of the Required Return

The total required return, $R$, can be broken down into two components: the dividend yield and the capital gains yield.

Total Required Return Formula:

$$R = \frac{D_1}{P_0} + g$$

Where $\frac{D_1}{P_0}$ is the dividend yield and $g$ is the capital gains yield (equal to the constant growth rate).

Stock Valuation Using Multiples

For companies that don't pay dividends, valuation can be based on multiples of earnings or sales. The most common is the Price-to-Earnings (PE) ratio.

PE Ratio Valuation:

$$P_t = \text{Benchmark PE Ratio} \times \text{EPS}_t$$

Textbook Concepts Review & Critical Thinking

1. Why does the value of a share of stock depend on dividends?

The value of a share of stock is the present value of the expected future dividends. For a long-term investor, the only cash flows received from a stock are the dividends. The final selling price of the stock is also determined by the present value of its future dividends at that point in time.

2. A substantial percentage of the companies listed on the NYSE and Nasdaq don't pay dividends, but investors are nonetheless willing to buy shares in them. How is this possible given your answer to the previous question?

The value of these stocks is based on the expectation that they will start paying dividends at some point in the future. The company is likely reinvesting all its earnings back into the business to fuel growth. Investors believe this growth will eventually lead to larger dividends (or a higher future selling price) than if the company paid dividends today.

3. Under what circumstances might a company choose not to pay dividends?

A company might choose not to pay dividends if it has profitable growth opportunities that require reinvesting all its earnings. This strategy is often employed by young, fast-growing companies that need capital for expansion and research and development.

4. Under what two assumptions can we use the dividend growth model presented in the chapter to determine the value of a share of stock? Comment on the reasonableness of these assumptions.

The two assumptions are:
1. Dividends are expected to grow at a constant rate, $g$, forever.
2. The constant growth rate, $g$, is less than the required return, $R$.
The constant growth assumption is more reasonable for mature, stable companies like utilities. For younger, rapidly growing companies, it is not a realistic assumption for the immediate future. The second assumption is a mathematical requirement; if $g \geq R$, the model produces a non-sensical or infinite stock price.

5. Suppose a company has a preferred stock issue and a common stock issue. Both have just paid a \$2 dividend. Which do you think will have a higher price, a share of the preferred or a share of the common?

The preferred stock will likely have a higher price. Preferred stock has a fixed dividend in perpetuity, similar to a zero-growth common stock. However, preferred stock holders have preference over common stock holders in dividend payments and liquidation, which makes their cash flows more secure. As a result, the required return on preferred stock is typically lower than on common stock, leading to a higher present value (price) for the same dividend amount.

6. Based on the dividend growth model, what are the two components of the total return on a share of stock? Which do you think is typically larger?

The two components are the dividend yield ($D_1/P_0$) and the capital gains yield ($g$). The capital gains yield is typically larger for most stocks, especially for growth companies that reinvest a large portion of their earnings and pay small or no dividends.

7. In the context of the dividend growth model, is it true that the growth rate in dividends and the growth rate in the price of the stock are identical?

Yes, it is true. The dividend growth model implicitly assumes that the stock price will grow at the same constant rate as the dividends. This is because the valuation formula is a function of the next period's dividend, which is constantly increasing at a rate of $g$.

8. When it comes to voting in elections, what are the differences between U.S. political democracy and U.S. corporate democracy?

In a political democracy, it's generally "one person, one vote." In corporate democracy, it's "one share, one vote." This means that in a corporation, an individual's voting power is proportional to the number of shares they own, not their status as an individual person.

9. Is it unfair or unethical for corporations to create classes of stock with unequal voting rights?

This is a highly debated topic. From one perspective, it is not unfair because investors are fully aware of the voting rights (or lack thereof) when they purchase the stock. From another perspective, it can be seen as unfair as it allows a small group of insiders to maintain control without a proportionate financial stake, which can make management less accountable to outside investors.

10. Some companies, such as Alphabet, have created classes of stock with no voting rights at all. Why would investors buy such stock?

Investors buy non-voting stock primarily for the capital appreciation and dividends, not for the voting rights. They are betting on the company's future growth and profitability, which they expect will increase the stock price. The founders or insiders who retain the voting control are often trusted by these investors to make sound decisions for the long-term benefit of the company.

Textbook Problems & Solutions

1. Stock Values (Basic): The RLX Co. just paid a dividend of \$3.20 per share on its stock. The dividends are expected to grow at a constant rate of 4 percent per year indefinitely. If investors require a return of 10.5 percent on the company's stock, what is the current price? What will the price be in 3 years? In 15 years?

Given: $D_0 = \$3.20$, $g = 4\% = 0.04$, $R = 10.5\% = 0.105$.

Current Price ($P_0$):

$$P_0 = \frac{D_0(1+g)}{R-g} = \frac{\$3.20(1.04)}{0.105-0.04} = \frac{\$3.328}{0.065} = \textbf{\$51.20}$$

Price in 3 years ($P_3$):

$$P_3 = P_0(1+g)^3 = \$51.20(1.04)^3 = \$51.20(1.124864) = \textbf{\$57.59}$$
Alternatively, using the dividend at time 4: $$D_4 = D_0(1+g)^4 = \$3.20(1.04)^4 = \$3.7445$$ $$P_3 = \frac{D_4}{R-g} = \frac{\$3.7445}{0.105-0.04} = \frac{\$3.7445}{0.065} = \textbf{\$57.61}$$

Note: The slight difference is due to rounding in the intermediate steps. Both methods are correct.

Price in 15 years ($P_{15}$):

$$P_{15} = P_0(1+g)^{15} = \$51.20(1.04)^{15} = \$51.20(1.8009) = \textbf{\$92.20}$$

2. Stock Values (Basic): The next dividend payment by Im, Inc., will be \$1.87 per share. The dividends are anticipated to maintain a growth rate of 4.3 percent forever. If the stock currently sells for \$37 per share, what is the required return?

Given: $D_1 = \$1.87$, $g = 4.3\% = 0.043$, $P_0 = \$37$.

Required Return ($R$):

$$R = \frac{D_1}{P_0} + g = \frac{\$1.87}{\$37} + 0.043 = 0.05054 + 0.043 = 0.09354 \text{ or } \textbf{9.35\%}$$

3. Stock Values (Basic): For the company in the previous problem, what is the dividend yield? What is the expected capital gains yield?

Dividend Yield:

$$\text{Dividend Yield} = \frac{D_1}{P_0} = \frac{\$1.87}{\$37} = 0.05054 \text{ or } \textbf{5.05\%}$$

Capital Gains Yield:

$$\text{Capital Gains Yield} = g = \textbf{4.3\%}$$

4. Stock Values (Basic): Five Star Corporation will pay a dividend of \$3.04 per share next year. The company pledges to increase its dividend by 3.75 percent per year indefinitely. If you require a return of 11 percent on your investment, how much will you pay for the company's stock today?

Given: $D_1 = \$3.04$, $g = 3.75\% = 0.0375$, $R = 11\% = 0.11$.

Current Price ($P_0$):

$$P_0 = \frac{D_1}{R-g} = \frac{\$3.04}{0.11 - 0.0375} = \frac{\$3.04}{0.0725} = \textbf{\$41.93}$$

5. Stock Valuation (Basic): Caccamise Co. is expected to maintain a constant 3.4 percent growth rate in its dividends indefinitely. If the company has a dividend yield of 5.3 percent, what is the required return on the company's stock?

Given: $g = 3.4\% = 0.034$, Dividend Yield = $5.3\% = 0.053$.

Required Return ($R$):

$$R = \text{Dividend Yield} + g = 0.053 + 0.034 = 0.087 \text{ or } \textbf{8.7\%}$$

6. Stock Valuation (Basic): Suppose you know that a company's stock currently sells for \$78 per share and the required return on the stock is 10.9 percent. You also know that the total return on the stock is evenly divided between a capital gains yield and a dividend yield. If it's the company's policy to always maintain a constant growth rate in its dividends, what is the current dividend per share?

Given: $P_0 = \$78$, $R = 10.9\% = 0.109$. Total return is split evenly between capital gains yield ($g$) and dividend yield ($D_1/P_0$).

Step 1: Find the dividend yield and capital gains yield.

Since $R = \text{Dividend Yield} + \text{Capital Gains Yield}$ and they are equal, each is $0.109 / 2 = 0.0545$. So, $g = 5.45\%$ and $D_1/P_0 = 5.45\%$.

Step 2: Find the next dividend ($D_1$).

$$D_1 = \text{Dividend Yield} \times P_0 = 0.0545 \times \$78 = \$4.251$$

Step 3: Find the current dividend ($D_0$).

We know that $D_1 = D_0(1+g)$, so $D_0 = D_1 / (1+g)$.

$$D_0 = \frac{\$4.251}{1.0545} = \textbf{\$4.03}$$

7. Stock Valuation (Basic): Hailey Corp. pays a constant \$9.45 dividend on its stock. The company will maintain this dividend for the next 13 years and will then cease paying dividends forever. If the required return on this stock is 10.7 percent, what is the current share price?

Given: Constant dividend of $D = \$9.45$ for 13 years, $R = 10.7\% = 0.107$.

This is a an annuity problem. We need to find the present value of a 13-year annuity with a payment of \$9.45 per year.

$$\text{PV} = C \times \left[ \frac{1 - \frac{1}{(1+R)^t}}{R} \right]$$ $$\text{PV} = \$9.45 \times \left[ \frac{1 - \frac{1}{(1.107)^{13}}}{0.107} \right]$$ $$\text{PV} = \$9.45 \times \left[ \frac{1 - 0.2831}{0.107} \right] = \$9.45 \times \frac{0.7169}{0.107} = \$9.45 \times 6.700 = \textbf{\$63.32}$$

8. Valuing Preferred Stock (Basic): Fegley, Inc., has an issue of preferred stock outstanding that pays a \$3.80 dividend every year in perpetuity. If this issue currently sells for \$93 per share, what is the required return?

Given: Dividend = $D = \$3.80$, Price = $P_0 = \$93$.

Preferred stock is a perpetuity. The formula for the value of a perpetuity is $P_0 = D/R$. We can rearrange this to solve for $R$.

$$R = \frac{D}{P_0} = \frac{\$3.80}{\$93} = 0.04086 \text{ or } \textbf{4.09\%}$$

9. Stock Valuation and Required Return (Basic): Red, Inc., Yellow Corp., and Blue Company each will pay a dividend of \$4.15 next year. The growth rate in dividends for all three companies is 4 percent. The required return for each company's stock is 8 percent, 11 percent, and 14 percent, respectively. What is the stock price for each company? What do you conclude about the relationship between the required return and the stock price?

Given: $D_1 = \$4.15$, $g = 4\% = 0.04$.

Red, Inc.: $R = 8\% = 0.08$

$$P_0 = \frac{D_1}{R-g} = \frac{\$4.15}{0.08-0.04} = \frac{\$4.15}{0.04} = \textbf{\$103.75}$$

Yellow Corp.: $R = 11\% = 0.11$

$$P_0 = \frac{D_1}{R-g} = \frac{\$4.15}{0.11-0.04} = \frac{\$4.15}{0.07} = \textbf{\$59.29}$$

Blue Company: $R = 14\% = 0.14$

$$P_0 = \frac{D_1}{R-g} = \frac{\$4.15}{0.14-0.04} = \frac{\$4.15}{0.10} = \textbf{\$41.50}$$

Conclusion: As the required return ($R$) increases, the stock price ($P_0$) decreases, assuming all other variables remain constant. This is an inverse relationship.

10. Voting Rights (Basic): After successfully completing your corporate finance class, you feel the next challenge ahead is to serve on the board of directors of Cornwall Enterprises. Unfortunately, you will be the only person voting for you. If the company has 720,000 shares outstanding, and the stock currently sells for \$48, how much will it cost you to buy a seat if the company uses straight voting?

Given: Total shares outstanding = 720,000, Price per share = \$48.

With straight voting, a simple majority of votes is required to elect a director, which means you need to own 50% + 1 share of the company's stock to guarantee a seat. Since you are the only one voting for yourself, you need to own enough shares to have a majority of all outstanding shares.

Shares needed = $(720,000 / 2) + 1 = 360,001$

Cost = $360,001 \times \$48 = \textbf{\$17,280,048}$

11. Voting Rights (Basic): In the previous problem, assume that the company uses cumulative voting, and there are four seats in the current election. How much will it cost you to buy a seat now?

Given: Total shares outstanding = 720,000, Price per share = \$48, Number of seats = 4.

With cumulative voting, the number of shares required to guarantee a seat is calculated as: $1 / (N+1)$ of the total shares outstanding, plus one share, where $N$ is the number of directors up for election.

$$\text{Shares needed} = \left( \frac{\text{Total Shares}}{N+1} \right) + 1$$ $$\text{Shares needed} = \left( \frac{720,000}{4+1} \right) + 1 = \left( \frac{720,000}{5} \right) + 1 = 144,000 + 1 = 144,001$$

Cost = $144,001 \times \$48 = \textbf{\$6,912,048}$

12. Stock Valuation and PE (Basic): The Dahlia Flower Co. has earnings of \$3.64 per share. The benchmark PE for the company is 18. What stock price would you consider appropriate? What if the benchmark PE were 21?

Given: EPS = \$3.64.

Using the formula $P = \text{PE} \times \text{EPS}$.

Case 1: Benchmark PE = 18

$$P = 18 \times \$3.64 = \textbf{\$65.52}$$

Case 2: Benchmark PE = 21

$$P = 21 \times \$3.64 = \textbf{\$76.44}$$

13. Stock Valuation and PS (Basic): Z Space, Inc., is a new company and currently has negative earnings. The company's sales are \$2.7 million and there are 175,000 shares outstanding. If the benchmark price-sales ratio is 4.3, what is your estimate of an appropriate stock price? What if the price-sales ratio were 3.6?

Given: Total sales = \$2.7 million, Shares outstanding = 175,000.

Step 1: Calculate Sales per Share.

$$\text{Sales per Share} = \frac{\text{Total Sales}}{\text{Shares Outstanding}} = \frac{\$2,700,000}{175,000} = \$15.43$$

Step 2: Calculate stock price using Price-Sales Ratio.

Using the formula $P = \text{Price-Sales Ratio} \times \text{Sales per Share}$.

Case 1: Price-Sales Ratio = 4.3

$$P = 4.3 \times \$15.43 = \textbf{\$66.35}$$

Case 2: Price-Sales Ratio = 3.6

$$P = 3.6 \times \$15.43 = \textbf{\$55.55}$$

18. Supernormal Growth (Intermediate): Synovec Co. is growing quickly. Dividends are expected to grow at a rate of 30 percent for the next three years, with the growth rate falling off to a constant 4 percent thereafter. If the required return is 10 percent, and the company just paid a dividend of \$2.65, what is the current share price?

Given: $D_0 = \$2.65$, $g_1 = 30\% = 0.30$ (for 3 years), $g_2 = 4\% = 0.04$ (indefinitely), $R = 10\% = 0.10$.

Step 1: Calculate dividends for the supernormal growth period.

$$D_1 = D_0(1+g_1) = \$2.65(1.30) = \$3.445$$ $$D_2 = D_1(1+g_1) = \$3.445(1.30) = \$4.4785$$ $$D_3 = D_2(1+g_1) = \$4.4785(1.30) = \$5.82205$$

Step 2: Calculate the stock price at the end of the supernormal growth period ($P_3$).

This is when the constant growth rate begins. We use the dividend at time 4 ($D_4$) and the constant growth formula.

$$D_4 = D_3(1+g_2) = \$5.82205(1.04) = \$6.054932$$ $$P_3 = \frac{D_4}{R-g_2} = \frac{\$6.054932}{0.10 - 0.04} = \frac{\$6.054932}{0.06} = \$100.9155$$

Step 3: Calculate the present value of the dividends and the future stock price.

$$P_0 = \frac{D_1}{(1+R)^1} + \frac{D_2}{(1+R)^2} + \frac{D_3}{(1+R)^3} + \frac{P_3}{(1+R)^3}$$ $$P_0 = \frac{\$3.445}{1.10} + \frac{\$4.4785}{(1.10)^2} + \frac{\$5.82205}{(1.10)^3} + \frac{\$100.9155}{(1.10)^3}$$ $$P_0 = \$3.1318 + \$3.7012 + \$4.3739 + \$75.8360 = \textbf{\$87.04}$$