Chapter 9: Valuing Stocks: Concepts, Formulas, and Applications

Chapter 9: Valuing Stocks

Introduction

Stock valuation is a fundamental topic in financial accounting and finance, focusing on determining the intrinsic value of a company's shares. This process involves analyzing dividends, growth rates, required rates of return, and other financial metrics to estimate the fair price of a stock. The following notes summarize key concepts, formulas, and example applications relevant to stock valuation.

Dividend Discount Model (DDM)

Basic DDM Formula

The Dividend Discount Model (DDM) values a stock by discounting expected future dividends to their present value. The simplest form assumes dividends grow at a constant rate.

• Formula: $P_0 = \frac{D_1}{r - g}$

• Where:

  • $P_0$ = Current stock price

  • $D_1$ = Dividend expected next year

  • $r$ = Required rate of return (cost of equity)

  • $g$ = Growth rate of dividends

• Application: Used for companies with stable dividend growth.

Example Calculation

• Suppose a stock pays a $2 dividend next year, with a required return of 15% and a growth rate of 5%.

• Price: $P_0 = \frac{2}{0.15 - 0.05} = \frac{2}{0.10} = $20$

Multi-Stage DDM

For companies with changing growth rates, the multi-stage DDM is used. Dividends are forecasted for each period of different growth, then a terminal value is calculated using the constant growth formula.

• Formula for terminal value at year N: $P_N = \frac{D_{N+1}}{r - g}$

• Discount all dividends and terminal value back to present value.

Expected Return and Cost of Equity

Definitions

• Expected Return: The total return anticipated from holding a stock, including dividends and capital gains.

• Cost of Equity: The required rate of return for equity investors, often estimated using the Capital Asset Pricing Model (CAPM):

$r = r_f + \beta (r_m - r_f)$

• $r_f$ = Risk-free rate

• $\beta$ = Beta of the stock

• $r_m$ = Expected market return

Dividend Yield and Capital Gain

• Dividend Yield: $\text{Dividend Yield} = \frac{D_1}{P_0}$

• Capital Gain Rate: $\text{Capital Gain Rate} = \frac{P_1 - P_0}{P_0}$

• Total Expected Return: $\text{Total Return} = \text{Dividend Yield} + \text{Capital Gain Rate}$

Stock Valuation with Free Cash Flow (FCF)

Free Cash Flow to Equity (FCFE) Model

Some firms do not pay dividends or have unpredictable dividend policies. In such cases, the value of equity can be estimated using free cash flow to equity (FCFE).

• Formula: $P_0 = \sum_{t=1}^{N} \frac{FCFE_t}{(1 + r)^t} + \frac{P_N}{(1 + r)^N}$

• Where $FCFE_t$ is the free cash flow to equity in year t, and $P_N$ is the terminal value at year N.

Example Table: Projected Free Cash Flows

Year	1	2	3	4	5
FCF ($ millions)	53	62	73	75	82

Main Purpose: This table is used to forecast the value of a company by discounting future free cash flows at the weighted average cost of capital (WACC).

Stock Repurchases and Payout Policy

Dividends vs. Repurchases

• Companies can return value to shareholders via dividends or share repurchases.

• Repurchases reduce the number of shares outstanding, potentially increasing earnings per share (EPS) and share price.

• Dividend policy and repurchase decisions affect the valuation models used.

Growth Rate Estimation

Estimating Dividend Growth

• Retention Ratio: The proportion of earnings retained in the business.

• Growth Rate Formula: $g = \text{Retention Ratio} \times \text{Return on Equity (ROE)}$

Worked Examples and Applications

Sample Problem: Price After Dividend

• Given: Stock price = $50, Dividend = $2, Cost of equity = 15%.

• Price after dividend: $P = 50 - 2/(1.15) = 48.26$

Sample Problem: Multi-Stage Growth

• Firm retains all earnings for 2 years, then pays out 60% as dividends for 10 years, then 100% thereafter. Use multi-stage DDM to value the stock.

• Calculate dividends for each stage, discount to present, and sum.

Summary Table: Key Stock Valuation Formulas

Model	Formula	When to Use
Constant Growth DDM	$P_0 = \frac{D_1}{r - g}$	Stable dividend growth
Multi-Stage DDM	$P_0 = \sum_{t=1}^{N} \frac{D_t}{(1 + r)^t} + \frac{P_N}{(1 + r)^N}$	Changing growth rates
FCFE Model	$P_0 = \sum_{t=1}^{N} \frac{FCFE_t}{(1 + r)^t} + \frac{P_N}{(1 + r)^N}$	No dividends or irregular payouts

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Tables and formulas have been reconstructed and formatted for academic study purposes.