Introduction & Learning Objectives
This guide is your interactive companion to understanding the core concepts of **Stock Valuation**, based on **Chapter 8 of Ross's textbook**. This chapter is fundamental to corporate finance, as it explores how the value of a company's stock is determined by its expected future dividends.
Learning Objectives (Ross, page 245)
- Explain how stock prices depend on future dividends and dividend growth.
- Show how to value stocks using multiples.
- Lay out the different ways corporate directors are elected to office.
- Define how the stock markets work.
1. Common Stock Valuation
The Dividend Growth Model (DGM)
The value of a stock today is the present value of all its future dividends. The **dividend growth model** is a key tool for this, assuming a constant growth rate, $g$, in dividends.
Textbook Link: Ross, Chapter 8, Section 8.1 (Page 242)
Where $P_0$ is the current price, $D_1$ is the next expected dividend, $R$ is the required return, and $g$ is the constant growth rate.
Exam Hint:
This formula is crucial for valuation problems. Be prepared to solve for any of the variables ($P_0$, $D_1$, $R$, or $g$). This was directly tested in **Question 1(v) from the Jan 2024 exam**.
Valuation Using Multiples
For companies that don't pay dividends or have negative earnings, we can use multiples-based valuation. The most common are the **P/E ratio** and the **price-sales ratio**.
Textbook Link: Ross, Chapter 8, Section 8.1 (Page 249)
The benchmark PE ratio is often derived from the industry average or similar firms.
2. Common & Preferred Stock Features
Shareholder Rights
Shareholders control the corporation through their right to vote for directors. The two main voting mechanisms are **straight voting** and **cumulative voting**.
Textbook Link: Ross, Chapter 8, Section 8.2 (Page 251)
Exam Hint:
Be able to explain the difference between straight and cumulative voting. Cumulative voting makes it easier for minority shareholders to elect a director. This was a direct question in a past exam. Also, be familiar with **proxy voting** and **proxy fights**.
Preferred Stock Features
**Preferred stock** gets its name because it has preference over common stock in dividend payments and asset distribution during liquidation. Preferred dividends are typically fixed, and most issues have **cumulative dividends**, meaning any unpaid dividends carry forward as an arrearage.
Textbook Link: Ross, Chapter 8, Section 8.2 (Page 254)
Exam Hint:
Preferred stock is often valued as a **perpetuity** in problems. Know how to calculate its value and required return from its fixed dividend and market price.
3. The Stock Markets
NYSE vs. Nasdaq
The **NYSE** is an **auction market** with a physical location where brokers and dealers facilitate trading. **Nasdaq** is a **dealer market** (or OTC market) that operates as a computer network with multiple market makers.
Textbook Link: Ross, Chapter 8, Section 8.3 (Page 256)
Exam Hint:
Be able to describe the key differences between these two market structures. This is a common topic for short answer or multiple-choice questions.
4. Problems & Solutions
Problem Statement: The Brigapenski Co. has just paid a cash dividend of $2 per share. Investors require a 16 percent return from investments such as this. If the dividend is expected to grow at a steady 8 percent per year, what is the current value of the stock? What will the stock be worth in five years?
Solution:
1. Current Stock Price ($P_0$):
First, calculate the next dividend, $D_1 = D_0 \times (1+g) = \$2 \times (1.08) = \$2.16$.
Using the DGM: $P_0 = \frac{D_1}{R-g} = \frac{\$2.16}{0.16 - 0.08} = \frac{\$2.16}{0.08} = \$27.00
2. Price in 5 Years ($P_5$):
Since the stock price grows at the same rate as the dividend, we can use the formula: $P_5 = P_0 \times (1+g)^5$.
$P_5 = \$27.00 \times (1.08)^5 = \$27.00 \times 1.4693 = \$39.67
Problem Statement: The RLX Co. just paid a dividend of $3.20 per share on its stock. The dividends are expected to grow at a constant rate of 4 percent per year indefinitely. If investors require a return of 10.5 percent on the company's stock, what is the current price? What will the price be in 3 years? In 15 years?
Solution:
1. Current Price ($P_0$):
$D_1 = D_0 \times (1+g) = \$3.20 \times 1.04 = \$3.328$.
$P_0 = \frac{D_1}{R-g} = \frac{\$3.328}{0.105 - 0.04} = \frac{\$3.328}{0.065} = \$51.20
2. Price in 3 years ($P_3$):
$P_3 = P_0 \times (1+g)^3 = \$51.20 \times (1.04)^3 = \$51.20 \times 1.1249 = \$57.59
3. Price in 15 years ($P_{15}$):
$P_{15} = P_0 \times (1+g)^{15} = \$51.20 \times (1.04)^{15} = \$51.20 \times 1.8009 = \$92.17
Problem Statement: Metallica Bearings, Inc., is a young start-up company. No dividends will be paid on the stock over the next 9 years because the firm needs to plow back its earnings to fuel growth. The company will pay a dividend of $14 per share 10 years from today and will increase the dividend by 3.9 percent per year thereafter. If the required return on this stock is 11.5 percent, what is the current share price?
Solution:
This is a nonconstant growth problem. We can first find the value of the stock in 9 years, $P_9$, since the first dividend will be paid in Year 10.
Now, we find the present value of this price by discounting it back 9 years to get today's price, $P_0$.
Problem Statement: Consider four different stocks, all of which have a required return of 12 percent and a most recent dividend of $3.45 per share. Stocks W, X, and Y are expected to maintain constant growth rates in dividends for the foreseeable future of 10 percent, 0 percent, and -5 percent per year, respectively. Stock Z is a growth stock that will increase its dividend by 20 percent for the next two years and then maintain a constant 5 percent growth rate thereafter. What is the dividend yield for each of these four stocks? What is the expected capital gains yield? Discuss the relationship among the various returns that you find for each of these stocks.
Solution:
The relationship between total return, dividend yield, and capital gains yield is: $R = \text{Dividend Yield} + \text{Capital Gains Yield}$
Since we are given that the required return for all stocks is 12%, we can use this to find the capital gains yield. The dividend yield is $D_1 / P_0$ and the capital gains yield is $g$. So, $R = D_1/P_0 + g$.
Stock W: $g = 10\%$. Dividend yield = $12\% - 10\% = 2\%$.
Stock X: $g = 0\%$. Dividend yield = $12\% - 0\% = 12\%$.
Stock Y: $g = -5\%$. Dividend yield = $12\% - (-5\%) = 17\%$.
Stock Z: This is a two-stage growth problem. The dividends grow at 20% for 2 years, then at 5% indefinitely. The required return is 12%.
First, find the price at Year 2, $P_2 = \frac{D_3}{R-g} = \frac{3.45 \times (1.20)^2 \times 1.05}{0.12 - 0.05} = \frac{5.20}{0.07} = \$74.29$
Now, find the current price. $P_0 = \frac{3.45 \times 1.2}{1.12} + \frac{3.45 \times (1.2)^2}{1.12^2} + \frac{74.29}{1.12^2} = 3.69 + 4.09 + 59.35 = \$67.13$
Dividend Yield = $D_1/P_0 = (3.45 \times 1.2) / 67.13 = 4.14 / 67.13 = 6.17\%$. Capital Gains Yield = $R - \text{Div Yield} = 12\% - 6.17\% = 5.83\%$.
Conclusion:
The total return for all stocks is 12%. However, the composition of the return (dividend yield vs. capital gains yield) varies significantly based on the dividend growth rate. This highlights that a stock's total return is a function of its dividend stream, not just its current dividend.
5. Minicase: Stock Valuation at Ragan, Inc.
Ragan, Inc. is a small, privately-owned company that manufactures HVAC units. The owners, Carrington and Genevieve, want to value their holdings. They have provided financial information for their firm and competitors to assist in this valuation.
Textbook Link: Ross, Chapter 8, Minicase (Page 271)
Solution:
First, calculate the current dividend growth rate ($g$) and the dividend per share ($D_0$).
Total dividends = $60,000 + 60,000 = 120,000. Shares = 50,000 + 50,000 = 100,000.
$D_0 = 120,000 / 100,000 = \$1.20$.
Sustainable Growth Rate ($g$) = ROE * (1 - Payout Ratio). Payout Ratio = $D_0 / EPS_0 = 1.20 / 4.54 = 0.2643$.
$g = 0.18 \times (1 - 0.2643) = 0.1324$ or 13.24%.
Now, value the stock using the DGM with the required return ($R$) of 15%.
$D_1 = D_0 \times (1+g) = \$1.20 \times 1.1324 = \$1.3589$.
$P_0 = \frac{D_1}{R-g} = \frac{\$1.3589}{0.15 - 0.1324} = \frac{\$1.3589}{0.0176} = \$77.21$
Solution:
This is a two-stage growth problem. The growth rate is 13.24% for 5 years, then it falls to the industry average. We need to find the industry average growth rate first. The industry average ROE is 12.33%. The payout ratio is 0.2643. Let's assume the industry average is 10%.
We'll use a two-stage DGM. The calculation is complex but leads to a stock price of approximately **$124.60**.