Unit 9 asks: how much is a share of stock actually worth? Unlike bonds with fixed contractual cash flows, stocks represent ownership claims on uncertain future earnings and dividends. Valuing them requires connecting TVM (Unit 5), risk and required return (Units 6–8), and expectations about future growth. The Gordon Growth Model is the central tool — used by analysts at every major investment firm — and directly applies to BBYM cooperative investing, The Swanson Initiative equity portfolio, and evaluating whether a stock is over or underpriced before investing community savings.
Part 1 — Core Topics Explained
Every major concept tested on the Unit 9 assessment
📋 Learning Objectives
- Distinguish common stock from preferred stock and explain the rights and risks of each
- Explain the Dividend Discount Model (DDM) as the conceptual foundation for stock valuation
- Apply the Gordon Growth Model to value a constant-growth dividend stock
- Calculate a stock's expected total return as dividend yield plus capital gains yield
- Interpret P/E ratios and use them to assess relative stock valuation
- Distinguish growth stocks from value stocks and explain the tradeoffs of each
- Compare active vs. passive investing strategies and evaluate which is appropriate for BBYM community members
- Design a simulated diversified stock portfolio aligned with BBYM community wealth-building goals
1. Common Stock vs. Preferred Stock
A corporation can issue two types of equity: common stock (the standard ownership share) and preferred stock (a hybrid between common stock and bonds). Understanding the differences is essential for both investors and the corporations that issue them.
| Feature | Common Stock | Preferred Stock |
|---|---|---|
| Dividends | Variable — set by the board, can be cut or eliminated. No guaranteed payment. | Fixed stated rate — must be paid before any common dividends. Often cumulative. |
| Voting Rights | Yes — one vote per share on board elections and major corporate decisions | Generally none — preferred holders trade voting rights for dividend priority |
| Claim in Bankruptcy | Last — receives whatever remains after all creditors AND preferred stockholders are paid. Often zero. | Senior to common — receives par value before common stockholders get anything |
| Upside Potential | Unlimited — benefits fully from growth in company value | Limited — price stays near par value; rarely appreciates dramatically |
| Valuation Method | DDM / Gordon Growth Model / P/E multiples / DCF | Treated like a perpetuity: P = Dividend ÷ Required Return |
| Best For | Long-term growth investors comfortable with variability | Income-seeking investors wanting more stability than common stock |
Preferred Stock as a Perpetuity:
A preferred stock paying a fixed $5.00/year dividend with a 6% required return:
P = Dividend ÷ r = $5.00 ÷ 0.06 = $83.33
This is the Unit 5 perpetuity formula applied directly to equity. Preferred stock is essentially a corporate perpetuity — and this is why the Unit 5 TVM foundation matters for Unit 9. Every valuation model builds on discounted cash flows.
A preferred stock paying a fixed $5.00/year dividend with a 6% required return:
P = Dividend ÷ r = $5.00 ÷ 0.06 = $83.33
This is the Unit 5 perpetuity formula applied directly to equity. Preferred stock is essentially a corporate perpetuity — and this is why the Unit 5 TVM foundation matters for Unit 9. Every valuation model builds on discounted cash flows.
2. The Dividend Discount Model (DDM) — Conceptual Foundation
The Dividend Discount Model is the theoretical foundation of stock valuation. It states that a stock's intrinsic value equals the present value of ALL future dividends, discounted at the required return. This is the stock valuation equivalent of the bond pricing formula.
Dividend Discount Model (General Form)
P₀ = D₁/(1+rₛ) + D₂/(1+rₛ)² + D₃/(1+rₛ)³ + … + D∞/(1+rₛ)∞
P₀ = stock's intrinsic value today | Dᵀ = dividend in period t
rₛ = required return on the stock (from CAPM: rᵣᵓ + RPᵖ × β)
This infinite series simplifies when dividends grow at a constant rate g — giving the Gordon Growth Model
rₛ = required return on the stock (from CAPM: rᵣᵓ + RPᵖ × β)
This infinite series simplifies when dividends grow at a constant rate g — giving the Gordon Growth Model
Why Dividends — Even for Non-Dividend Stocks:
A student might ask: "Why use dividends if many growth companies (like Amazon, early) don't pay dividends?" The answer: even non-dividend-paying stocks eventually distribute cash to shareholders — either through future dividends or share repurchases when the company matures. The DDM captures the idea that all stock value ultimately comes from future cash distributions. For non-dividend stocks, analysts use the Discounted Free Cash Flow (DCF) model instead — which follows the same PV logic with free cash flows replacing dividends.
A student might ask: "Why use dividends if many growth companies (like Amazon, early) don't pay dividends?" The answer: even non-dividend-paying stocks eventually distribute cash to shareholders — either through future dividends or share repurchases when the company matures. The DDM captures the idea that all stock value ultimately comes from future cash distributions. For non-dividend stocks, analysts use the Discounted Free Cash Flow (DCF) model instead — which follows the same PV logic with free cash flows replacing dividends.
3. The Gordon Growth Model — Constant Dividend Growth
When dividends grow at a constant rate g forever, the infinite DDM series simplifies to a single elegant formula — the Gordon Growth Model (also called the constant growth model).
Gordon Growth Model — Stock Price
P₀ = D₁ ÷ (rₛ − g)
D₁ = next year's expected dividend | rₛ = required return | g = constant perpetual growth rate
Critical requirement: rₛ must be greater than g. If g ≥ rₛ, the formula breaks (denominator ≤ 0 implies infinite value).
Critical requirement: rₛ must be greater than g. If g ≥ rₛ, the formula breaks (denominator ≤ 0 implies infinite value).
Assessment Q9 — Worked Out:
D₁ = $3.00 | rₛ = 10% | g = 4%
P₀ = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
The stock's intrinsic value is $50.00. If the market price is exactly $50.00, the stock is fairly valued. If trading at $42, it is undervalued — buy. If trading at $58, it is overvalued — sell (or avoid).
D₁ = $3.00 | rₛ = 10% | g = 4%
P₀ = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
The stock's intrinsic value is $50.00. If the market price is exactly $50.00, the stock is fairly valued. If trading at $42, it is undervalued — buy. If trading at $58, it is overvalued — sell (or avoid).
Why g Must Be Less Than rₛ:
If g = rₛ, the denominator is zero — the formula implies infinite value, which makes no economic sense. If g > rₛ, the denominator is negative — a negative price, also nonsensical.
More fundamentally: no company can grow faster than the overall economy forever. A company growing at 15% while the economy grows at 3% would eventually own the entire economy — impossible. The Gordon Growth Model is valid only for mature, stable-growth companies where g reflects a sustainable long-run rate (typically 2–5%).
For high-growth companies, analysts use a multi-stage model: project high early growth for 5–10 years explicitly, then apply the Gordon Growth Model at the end of the high-growth period using a terminal growth rate.
If g = rₛ, the denominator is zero — the formula implies infinite value, which makes no economic sense. If g > rₛ, the denominator is negative — a negative price, also nonsensical.
More fundamentally: no company can grow faster than the overall economy forever. A company growing at 15% while the economy grows at 3% would eventually own the entire economy — impossible. The Gordon Growth Model is valid only for mature, stable-growth companies where g reflects a sustainable long-run rate (typically 2–5%).
For high-growth companies, analysts use a multi-stage model: project high early growth for 5–10 years explicitly, then apply the Gordon Growth Model at the end of the high-growth period using a terminal growth rate.
4. Expected Total Return on a Stock
Rearranging the Gordon Growth Model gives the expected total return — the two components of equity returns.
Expected Total Return
rₛ = D₁/P₀ + g
rₛ = Dividend Yield + Capital Gains Yield
D₁/P₀ = dividend yield (income component) | g = capital gains yield (growth component)
A stock growing dividends at g% per year sees its price grow at g% per year (capital appreciation)
A stock growing dividends at g% per year sees its price grow at g% per year (capital appreciation)
Decomposing Total Return — Two Stock Examples:
Income Stock (utility): P₀ = $40, D₁ = $3.20, g = 2%
Dividend Yield = $3.20 ÷ $40 = 8.0% | Capital Gains = 2.0% | Total Return = 10.0%
Growth Stock (tech): P₀ = $100, D₁ = $0.50, g = 9%
Dividend Yield = $0.50 ÷ $100 = 0.5% | Capital Gains = 9.0% | Total Return = 9.5%
The income stock offers most of its return as dividends (spendable income); the growth stock offers most through price appreciation. Neither is inherently better — the right choice depends on whether the investor needs current income or long-term capital accumulation.
Income Stock (utility): P₀ = $40, D₁ = $3.20, g = 2%
Dividend Yield = $3.20 ÷ $40 = 8.0% | Capital Gains = 2.0% | Total Return = 10.0%
Growth Stock (tech): P₀ = $100, D₁ = $0.50, g = 9%
Dividend Yield = $0.50 ÷ $100 = 0.5% | Capital Gains = 9.0% | Total Return = 9.5%
The income stock offers most of its return as dividends (spendable income); the growth stock offers most through price appreciation. Neither is inherently better — the right choice depends on whether the investor needs current income or long-term capital accumulation.
5. The P/E Ratio — Market Multiple Approach
The Price-to-Earnings (P/E) ratio is the most widely cited stock valuation metric in financial media. It tells you how much investors are paying per dollar of the company's earnings.
P/E Ratio
P/E = Stock Price ÷ Earnings Per Share (EPS)
Estimated Stock Price = EPS × Industry P/E
EPS = Net Income ÷ Shares Outstanding | Industry P/E = benchmark from comparable companies
Forward P/E uses next year's expected EPS; Trailing P/E uses last 12 months' actual EPS
Forward P/E uses next year's expected EPS; Trailing P/E uses last 12 months' actual EPS
| P/E Level | Interpretation | Typical Characteristics | BBYM Investing Implication |
|---|---|---|---|
| P/E < 15 | Potentially undervalued, or slow-growth / declining industry | Mature industries (utilities, banks, energy). Low growth expectations. | May represent a value opportunity — but check WHY it's cheap (low growth? problems?) |
| P/E 15–25 | Fair value for most established companies | S&P 500 historical average is ~16–18. Most large-cap stocks fall here. | Reasonable entry point for long-term investment in established companies |
| P/E 25–40 | Premium valuation — market pricing in above-average growth | Fast-growing companies with strong competitive positions | Only justified if the company can sustain high growth; risky if growth disappoints |
| P/E > 40 | High speculation — requires exceptional sustained growth to justify | Early-stage tech companies, hot growth stories, market bubbles | High risk — avoid with community savings; appropriate only for speculative portion of portfolio |
P/E Ratio Worked Example — Two Birmingham-Area Companies:
Company A (regional bank): EPS = $4.00, Stock Price = $48
P/E = $48 ÷ $4.00 = 12× — cheap relative to market average; typical for banks
Company B (tech firm): EPS = $2.00, Stock Price = $80
P/E = $80 ÷ $2.00 = 40× — market paying a premium for high expected growth
If the industry average P/E is 15× and Company A's earnings are $4.00:
Estimated fair value = $4.00 × 15 = $60 — Company A may be undervalued at $48.
Company A (regional bank): EPS = $4.00, Stock Price = $48
P/E = $48 ÷ $4.00 = 12× — cheap relative to market average; typical for banks
Company B (tech firm): EPS = $2.00, Stock Price = $80
P/E = $80 ÷ $2.00 = 40× — market paying a premium for high expected growth
If the industry average P/E is 15× and Company A's earnings are $4.00:
Estimated fair value = $4.00 × 15 = $60 — Company A may be undervalued at $48.
Part 2 — Valuation Models: Formulas & Worked Examples
All four valuation approaches with complete Birmingham-Bessemer worked examples
The Four Valuation Methods
Gordon Growth Model
P₀ = D₁ ÷ (rₛ − g)
Best for stable, mature dividend-paying companies. Requires constant growth assumption. Most commonly tested formula in Unit 9.
P/E Multiple
Price = EPS × Industry P/E
Best for earnings-generating firms when peer comparisons are available. Quick and intuitive but depends on quality of comparable companies.
P/B Multiple
Price = Book Value × P/B Ratio
Best for banks, insurance companies, and asset-heavy firms where balance sheet values are meaningful.
Discounted Cash Flow (DCF)
P₀ = PV of all future FCF
Most theoretically complete — works for any firm with projectable cash flows. Used for non-dividend-paying growth companies. Requires detailed financial modeling.
Gordon Growth Model — Five Worked Examples
Example 1 — Assessment Q9 (Exact):
D₁ = $3.00 | rₛ = 10% | g = 4%
P₀ = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
D₁ = $3.00 | rₛ = 10% | g = 4%
P₀ = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
Example 2 — Swanson Initiative Portfolio Stock:
A utility in the Swanson equity portfolio: D₁ = $2.40, rₛ = 8%, g = 3%
P₀ = $2.40 ÷ (0.08 − 0.03) = $2.40 ÷ 0.05 = $48.00
If the utility is currently trading at $41, it is undervalued — worth buying at $41 when intrinsic value is $48.
A utility in the Swanson equity portfolio: D₁ = $2.40, rₛ = 8%, g = 3%
P₀ = $2.40 ÷ (0.08 − 0.03) = $2.40 ÷ 0.05 = $48.00
If the utility is currently trading at $41, it is undervalued — worth buying at $41 when intrinsic value is $48.
Example 3 — Finding D₁ When D₀ Is Given:
A company just paid a dividend D₀ = $2.00. Dividends grow at g = 5%. rₛ = 11%.
Step 1: D₁ = D₀ × (1+g) = $2.00 × 1.05 = $2.10
Step 2: P₀ = $2.10 ÷ (0.11 − 0.05) = $2.10 ÷ 0.06 = $35.00
A company just paid a dividend D₀ = $2.00. Dividends grow at g = 5%. rₛ = 11%.
Step 1: D₁ = D₀ × (1+g) = $2.00 × 1.05 = $2.10
Step 2: P₀ = $2.10 ÷ (0.11 − 0.05) = $2.10 ÷ 0.06 = $35.00
Example 4 — Finding Required Return from Price:
Stock trades at $60. D₁ = $2.40. g = 5%. What is the market's implied required return?
rₛ = D₁/P₀ + g = $2.40/$60 + 0.05 = 0.04 + 0.05 = 9.0%
The market is pricing this stock as if the required return is 9.0%. If CAPM says the required return should be 10%, the stock is overpriced (market is accepting less return than the risk demands).
Stock trades at $60. D₁ = $2.40. g = 5%. What is the market's implied required return?
rₛ = D₁/P₀ + g = $2.40/$60 + 0.05 = 0.04 + 0.05 = 9.0%
The market is pricing this stock as if the required return is 9.0%. If CAPM says the required return should be 10%, the stock is overpriced (market is accepting less return than the risk demands).
Example 5 — Effect of Growth Rate on Price:
Same stock: D₁ = $2.00, rₛ = 10%. Compare g = 2% vs. g = 6%:
g = 2%: P₀ = $2.00 ÷ (0.10 − 0.02) = $2.00 ÷ 0.08 = $25.00
g = 6%: P₀ = $2.00 ÷ (0.10 − 0.06) = $2.00 ÷ 0.04 = $50.00
Doubling the growth rate from 2% to 6% doubles the stock price. This illustrates why growth expectations dominate stock valuations — small changes in perceived growth rate cause enormous price swings. It also explains why high-growth companies trade at very high P/E multiples: investors are paying for future growth potential, not current earnings.
Same stock: D₁ = $2.00, rₛ = 10%. Compare g = 2% vs. g = 6%:
g = 2%: P₀ = $2.00 ÷ (0.10 − 0.02) = $2.00 ÷ 0.08 = $25.00
g = 6%: P₀ = $2.00 ÷ (0.10 − 0.06) = $2.00 ÷ 0.04 = $50.00
Doubling the growth rate from 2% to 6% doubles the stock price. This illustrates why growth expectations dominate stock valuations — small changes in perceived growth rate cause enormous price swings. It also explains why high-growth companies trade at very high P/E multiples: investors are paying for future growth potential, not current earnings.
Sensitivity of Stock Price to Inputs
| Scenario | D₁ | rₛ | g | P₀ = D₁/(rₛ−g) | Change from Base |
|---|---|---|---|---|---|
| Base Case (Q9) | $3.00 | 10% | 4% | $50.00 | — |
| Higher required return | $3.00 | 12% | 4% | $37.50 | −25% |
| Lower required return | $3.00 | 8% | 4% | $75.00 | +50% |
| Higher growth rate | $3.00 | 10% | 6% | $75.00 | +50% |
| Lower growth rate | $3.00 | 10% | 2% | $37.50 | −25% |
| Higher dividend | $4.00 | 10% | 4% | $66.67 | +33% |
Stock prices are highly sensitive to both rₛ and g. A 2% rise in required return drops the price 25%. A 2% increase in growth raises it 50%. This is why interest rate announcements and quarterly earnings surprises move stock prices dramatically — they revise market estimates of rₛ and g.
Part 3 — Equity Markets, Investing Strategies & BBYM Application
How markets work, active vs. passive investing, and building community wealth through cooperative equity ownership
Growth Stocks vs. Value Stocks
| Feature | Growth Stocks | Value Stocks |
|---|---|---|
| P/E Ratio | High (25–60+) — market pays premium for expected growth | Low (8–15) — trading below perceived intrinsic value |
| Dividend Yield | Low or zero — profits reinvested for growth | Often higher — mature firms return cash to shareholders |
| Growth Rate (g) | High (10–25%+) but may not be sustainable | Low (2–5%) but stable and predictable |
| Examples | Amazon, Nvidia, early Tesla — high-growth tech and innovation companies | Johnson & Johnson, Procter & Gamble, utilities — mature stable businesses |
| Volatility | High — price swings dramatically on growth expectation revisions | Lower — stable earnings and dividends provide price support |
| Risk Profile | High — price depends entirely on future growth materializing | Lower — already generating earnings; less dependent on future promises |
| Best For | Young investors with long time horizons who can ride volatility | Income-seeking investors, conservative portfolios, near-retirement investors |
The BBYM Community Portfolio Approach:
Neither pure growth nor pure value investing is optimal for most community wealth-building goals. A blend is most practical:
• Core (60–70%): Broad market index funds (blend of both growth and value automatically)
• Value tilt (15–20%): Dividend-focused ETFs for income that can fund community programs
• Growth tilt (10–15%): Small allocation to growth for long-term capital appreciation
This approach captures the return benefits of both styles while avoiding concentration in either extreme.
Neither pure growth nor pure value investing is optimal for most community wealth-building goals. A blend is most practical:
• Core (60–70%): Broad market index funds (blend of both growth and value automatically)
• Value tilt (15–20%): Dividend-focused ETFs for income that can fund community programs
• Growth tilt (10–15%): Small allocation to growth for long-term capital appreciation
This approach captures the return benefits of both styles while avoiding concentration in either extreme.
ETFs and Index Funds — BBYM's Primary Investment Vehicle
| Feature | Index Fund (Mutual Fund) | ETF (Exchange-Traded Fund) | Actively Managed Fund |
|---|---|---|---|
| What it does | Tracks a market index (e.g., S&P 500) by holding all index stocks | Same as index fund but trades like a stock throughout the day | Fund manager selects stocks trying to beat the index |
| Annual cost (expense ratio) | 0.03–0.10% (very low) | 0.03–0.20% (very low) | 0.5–1.5%+ (high) |
| Typical performance vs. S&P 500 | Matches index (minus tiny fee) | Matches index (minus tiny fee) | ~80% underperform index over 15 years after fees |
| Diversification | Instant — holds all index components | Instant — same as index fund | Varies — depends on manager's strategy |
| Minimum investment | Often $1–$1,000 | Price of 1 share (can be $1 with fractional shares) | Often $1,000–$3,000 |
| Tax efficiency | High — low turnover means fewer taxable events | Highest — unique structure minimizes capital gains distributions | Lower — frequent trading creates taxable events |
The Index Fund Math — 40-Year Fee Impact on $500 Monthly Investment at 10%:
Index fund (0.05% fee) Net return: 9.95% Final value: $3,194,000
Active fund (1.00% fee) Net return: 9.00% Final value: $2,340,000
The 0.95% fee difference costs $854,000 over 40 years on a $500/month savings plan. This is money that belongs to BBYM families — paid instead to fund managers who, on average, do not beat the index. Fees are the most controllable variable in long-term investing.
Index fund (0.05% fee) Net return: 9.95% Final value: $3,194,000
Active fund (1.00% fee) Net return: 9.00% Final value: $2,340,000
The 0.95% fee difference costs $854,000 over 40 years on a $500/month savings plan. This is money that belongs to BBYM families — paid instead to fund managers who, on average, do not beat the index. Fees are the most controllable variable in long-term investing.
Active vs. Passive Investing — The BBYM Decision
| Dimension | Active Investing | Passive (Index) Investing |
|---|---|---|
| Philosophy | Skilled analysis can identify mispriced stocks and beat the market | Markets are efficient — consistent outperformance is not possible after fees |
| Evidence | ~20% of active funds outperform over 15 years (before fees); far fewer after fees | Low-cost index funds outperform most active funds over 10–15 year periods |
| Cost | High fees (0.5–1.5%), research costs, frequent trading taxes | Minimal fees (0.03–0.1%), low turnover, tax efficient |
| Time required | High — requires ongoing research, monitoring, and trading decisions | Low — "set and forget" — periodic rebalancing only |
| Appropriate for | Professional investors with edge, high-net-worth individuals, institutional funds | Most individual investors, retirement accounts, community trust funds |
| BBYM recommendation | Limit active positions to <10–15% of portfolio | Core 85–90% of community savings and endowment funds |
BBYM Cooperative Investing — Community Equity Ownership
The Cooperative Investing Model for Birmingham-Bessemer:
BBYM can structure community wealth-building through a cooperative investing framework where community members pool resources to:
1. Investment Club Model: Groups of 10–20 community members each contribute $50–$100/month, collectively building a diversified equity portfolio using index funds. Monthly meetings combine financial education with portfolio reviews. Each member owns proportional shares of the collective portfolio.
2. Community Development Investment Trust: The Swanson Initiative endowment holds a diversified equity portfolio (60% stocks / 40% bonds) using low-cost index funds. Annual returns above the 6% perpetuity threshold are reinvested to grow the endowment principal. Distributions fund community programs annually.
3. Cooperative Business Equity: Community members own equity stakes in BBYM-affiliated enterprises — applying stock valuation skills learned in this unit to real businesses in their own community. Gordon Growth Model and DCF methods help evaluate whether a proposed enterprise is financially viable before committing community capital.
This connects the academic content of Unit 9 directly to wealth-building activity in Birmingham-Bessemer.
BBYM can structure community wealth-building through a cooperative investing framework where community members pool resources to:
1. Investment Club Model: Groups of 10–20 community members each contribute $50–$100/month, collectively building a diversified equity portfolio using index funds. Monthly meetings combine financial education with portfolio reviews. Each member owns proportional shares of the collective portfolio.
2. Community Development Investment Trust: The Swanson Initiative endowment holds a diversified equity portfolio (60% stocks / 40% bonds) using low-cost index funds. Annual returns above the 6% perpetuity threshold are reinvested to grow the endowment principal. Distributions fund community programs annually.
3. Cooperative Business Equity: Community members own equity stakes in BBYM-affiliated enterprises — applying stock valuation skills learned in this unit to real businesses in their own community. Gordon Growth Model and DCF methods help evaluate whether a proposed enterprise is financially viable before committing community capital.
This connects the academic content of Unit 9 directly to wealth-building activity in Birmingham-Bessemer.
Part 4 — Key Terms Defined
Master these 15 terms for the Unit 9 assessment
Common Stock
The standard ownership share in a corporation. Holders receive variable dividends (if declared by the board), vote on major corporate decisions (one vote per share), and have the residual claim on assets in bankruptcy — meaning they receive whatever remains after all creditors and preferred stockholders are paid. Unlimited upside; last priority in liquidation.
Preferred Stock
A hybrid equity security with bond-like features. Pays a fixed stated dividend that must be paid before common dividends; typically does not carry voting rights; has priority over common stock in bankruptcy. Valued as a perpetuity: P = Dividend ÷ Required Return. Less upside than common stock but more predictable income and greater safety.
Dividend Discount Model (DDM)
A stock valuation framework stating that intrinsic value equals the present value of all future dividends discounted at the required return: P₀ = Σ Dᵀ/(1+rₛ)ᵀ. The conceptual foundation for all equity valuation — stock value comes entirely from future cash distributions to shareholders, regardless of whether those distributions are near-term or distant.
Gordon Growth Model (Constant Growth Model)
The simplified DDM for stocks with dividends growing at a constant rate g forever: P₀ = D₁ ÷ (rₛ − g). Requires rₛ > g. Best for mature, stable dividend-paying companies. Most widely taught and used stock valuation formula in introductory finance. The Assessment Q9 formula: $3.00 ÷ (10%−4%) = $50.00.
D₁ (Next Expected Dividend)
The dividend expected to be paid one period from now — the numerator in the Gordon Growth Model. If the most recent dividend D₀ is given instead, calculate D₁ = D₀ × (1+g). Critical: the Gordon Growth Model uses D₁ (next year's dividend), NOT D₀ (just paid). This is the most common calculation error on assessments.
Required Return on Stock (rₛ)
The minimum return an investor demands to hold a stock given its risk level. Determined by CAPM: rₛ = rᵣᵓ + RPᵖ × β. As required return increases (due to higher risk or rising rates), the Gordon Growth Model denominator grows, and the stock price falls. This links Unit 8 (CAPM) directly to Unit 9 (stock valuation).
Constant Growth Rate (g)
The perpetual rate at which dividends (and the stock price) are assumed to grow in the Gordon Growth Model. Must be less than rₛ for the model to work. In practice, g reflects the sustainable long-run growth rate — typically GDP growth (2–3%) for mature companies. Higher g assumptions dramatically increase estimated stock prices.
Dividend Yield
The annual dividend per share divided by the current stock price (D₁/P₀). The income component of total return. High-dividend-yield stocks (utilities, REITs, mature companies) appeal to income-seeking investors. Low-yield or zero-yield stocks (growth companies) offer return primarily through capital appreciation rather than current income.
Capital Gains Yield
The rate of price appreciation of a stock — equal to g in the Gordon Growth Model. If dividends grow at 4%/year, the stock price also grows at 4%/year (all else equal). The growth component of total return. Total return = Dividend Yield + Capital Gains Yield = D₁/P₀ + g.
Price-to-Earnings Ratio (P/E)
Stock price divided by earnings per share. The most widely cited valuation multiple in financial media. Reflects how much investors pay per dollar of current earnings — a higher P/E implies greater growth expectations or higher investor confidence. Industry comparisons make P/E meaningful: a 15× P/E for a bank (normal) and a 15× P/E for a tech company (very low) carry different implications.
Earnings Per Share (EPS)
Net income divided by the number of shares outstanding. The per-share profit of the company — the denominator in the P/E ratio and the foundation of earnings-based valuation. EPS growth drives both dividend growth and stock price appreciation over time. Diluted EPS accounts for all potential shares (options, convertibles); basic EPS uses only current outstanding shares.
Growth Stock
A stock in a company expected to grow revenues and earnings significantly faster than the market average. Typically has a high P/E ratio (market paying a premium for growth), low or zero dividend yield (profits reinvested), and high price volatility. Value depends heavily on future growth materializing. Examples: technology, biotech, and early-stage innovative companies.
Value Stock
A stock trading at a low valuation relative to its earnings, book value, or dividends — often because it's in a mature, slow-growth industry or temporarily out of favor with investors. Typically has low P/E, higher dividend yield, and less volatility than growth stocks. Value investing seeks to buy quality companies at discounts to their intrinsic value.
Exchange-Traded Fund (ETF)
A fund that holds a collection of securities (stocks, bonds, or other assets) and trades on an exchange like a stock throughout the day. Most ETFs passively track an index. Advantages: instant diversification, very low fees (0.03–0.2%), intraday liquidity, high tax efficiency. For BBYM investors, broad market ETFs (e.g., VTI, SPY) provide optimal diversification at minimal cost.
Intrinsic Value
The estimated fair value of a security based on fundamental analysis of its future cash flows, growth prospects, and required return — independent of current market price. Calculated using DDM, Gordon Growth Model, P/E multiples, or DCF. If intrinsic value exceeds market price, the stock is undervalued (buy); if below market price, it is overvalued (sell). The goal of fundamental analysis.
Part 5 — Practice Questions
Show all work — these mirror the Unit 9 assessment format exactly
Conceptual Questions
Q1A stock pays D₁ = $3.00, required return = 10%, constant growth = 4%. Estimated stock price is: A) $30.00 B) $50.00 C) $75.00 D) $43.00. (Unit 9 curriculum assessment question.)▼
Answer: B — $50.00
P₀ = D₁ ÷ (rₛ − g) = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
Common errors: A ($30) divides by r alone (3.00/0.10); C ($75) uses 4% denominator only; D ($43) is incorrect arithmetic. The key step is taking the difference (10% − 4% = 6%) as the denominator, not just one rate.
P₀ = D₁ ÷ (rₛ − g) = $3.00 ÷ (0.10 − 0.04) = $3.00 ÷ 0.06 = $50.00
Common errors: A ($30) divides by r alone (3.00/0.10); C ($75) uses 4% denominator only; D ($43) is incorrect arithmetic. The key step is taking the difference (10% − 4% = 6%) as the denominator, not just one rate.
Q2Explain why the Gordon Growth Model requires rₛ > g. What happens economically if a company tries to grow forever at a rate equal to or greater than the required return?▼
If g = rₛ: denominator = 0 → price is mathematically infinite. If g > rₛ: denominator is negative → price is negative. Both are economically impossible.
Economic reasoning: No company can sustain a growth rate at or above the overall required return forever. If a company grew at 10%/year while the economy grew at 3%, it would eventually dwarf the entire global economy — an absurdity. Competition, market saturation, and the law of large numbers all constrain growth to eventually converge toward the economy's growth rate (roughly 2–4%).
The model is valid only for the mature, stable phase of a company's life when growth has moderated to a sustainable long-run rate. For high-growth companies, analysts use multi-stage DDM: model the high-growth phase explicitly for 5–10 years, then apply the Gordon Growth Model at the end using a conservative terminal growth rate (2–4%).
Economic reasoning: No company can sustain a growth rate at or above the overall required return forever. If a company grew at 10%/year while the economy grew at 3%, it would eventually dwarf the entire global economy — an absurdity. Competition, market saturation, and the law of large numbers all constrain growth to eventually converge toward the economy's growth rate (roughly 2–4%).
The model is valid only for the mature, stable phase of a company's life when growth has moderated to a sustainable long-run rate. For high-growth companies, analysts use multi-stage DDM: model the high-growth phase explicitly for 5–10 years, then apply the Gordon Growth Model at the end using a conservative terminal growth rate (2–4%).
Q3A stock has a P/E of 35× while the industry average is 18×. What does this tell you about investor expectations for this company? Under what conditions is the premium P/E justified?▼
A P/E of 35× vs. industry average of 18× tells you that investors expect this company to grow significantly faster than its peers — they are paying nearly twice the industry multiple for each dollar of current earnings.
Specifically, investors believe:
• Future earnings growth will be much higher than industry peers
• The company has competitive advantages (brand, patents, network effects) that will sustain above-average profitability
• Current earnings understate the company's true earning power (e.g., heavy investment phase)
The premium is justified when:
(1) The company actually delivers on its growth expectations — high-growth periods sustained for 5–10+ years
(2) The company has durable competitive moats protecting future profits from competition
(3) Earnings quality is high and not inflated by accounting choices
The premium destroys value when: Growth disappoints. A stock priced for 20% growth that delivers 10% can fall 40–50% as the P/E compresses back toward the industry average. This is why growth stocks are more volatile — the price is highly sensitive to growth expectation revisions.
Specifically, investors believe:
• Future earnings growth will be much higher than industry peers
• The company has competitive advantages (brand, patents, network effects) that will sustain above-average profitability
• Current earnings understate the company's true earning power (e.g., heavy investment phase)
The premium is justified when:
(1) The company actually delivers on its growth expectations — high-growth periods sustained for 5–10+ years
(2) The company has durable competitive moats protecting future profits from competition
(3) Earnings quality is high and not inflated by accounting choices
The premium destroys value when: Growth disappoints. A stock priced for 20% growth that delivers 10% can fall 40–50% as the P/E compresses back toward the industry average. This is why growth stocks are more volatile — the price is highly sensitive to growth expectation revisions.
Q4What is the difference between dividend yield and capital gains yield? Which type of investor cares most about each, and why might a BBYM retiree prefer a high-dividend stock over a growth stock?▼
Dividend yield = D₁/P₀ — the income return from dividends as a percentage of the stock's current price. Actual cash received each year without selling any shares.
Capital gains yield = g — the price appreciation component. Return realized only when shares are sold.
Total return = both combined: rₛ = D₁/P₀ + g
Who cares about each:
Dividend yield matters most to income-seeking investors — retirees, endowments, anyone who needs regular cash distributions without selling holdings. Dividend income is predictable and doesn't require timing the market.
Capital gains yield matters most to long-term accumulators — young investors in the wealth-building phase who don't need current income and benefit from deferred taxation on unrealized gains.
A BBYM retiree prefers high-dividend stocks because:
• They need regular income to cover living expenses without selling shares (which would reduce principal)
• Dividend income is relatively predictable — established companies rarely cut dividends suddenly
• They avoid forced selling during market downturns to generate income — a growth stock investor must sell shares to generate cash, potentially at depressed prices
• Dividends provide a psychological floor — even if the stock price drops temporarily, the dividend income continues
Capital gains yield = g — the price appreciation component. Return realized only when shares are sold.
Total return = both combined: rₛ = D₁/P₀ + g
Who cares about each:
Dividend yield matters most to income-seeking investors — retirees, endowments, anyone who needs regular cash distributions without selling holdings. Dividend income is predictable and doesn't require timing the market.
Capital gains yield matters most to long-term accumulators — young investors in the wealth-building phase who don't need current income and benefit from deferred taxation on unrealized gains.
A BBYM retiree prefers high-dividend stocks because:
• They need regular income to cover living expenses without selling shares (which would reduce principal)
• Dividend income is relatively predictable — established companies rarely cut dividends suddenly
• They avoid forced selling during market downturns to generate income — a growth stock investor must sell shares to generate cash, potentially at depressed prices
• Dividends provide a psychological floor — even if the stock price drops temporarily, the dividend income continues
Calculation Questions
Q5A company just paid a dividend of D₀ = $2.50. Dividends grow at g = 6% per year. Required return rₛ = 12%. (a) Calculate D₁. (b) Estimate the stock's intrinsic value. (c) If the stock trades at $48, is it over or undervalued?▼
(a) D₁ = D₀ × (1+g) = $2.50 × 1.06 = $2.65
(b) P₀ = D₁ ÷ (rₛ − g) = $2.65 ÷ (0.12 − 0.06) = $2.65 ÷ 0.06 = $44.17
(c) Market price ($48) > Intrinsic value ($44.17) → Stock is overvalued by $3.83 per share.
A rational investor using the Gordon Growth Model would not pay $48 for a stock worth $44.17 — or would sell if they owned it. The stock will either need earnings/dividend growth to accelerate, or the price will need to fall toward $44 for the investment to be fairly priced at a 12% required return.
(b) P₀ = D₁ ÷ (rₛ − g) = $2.65 ÷ (0.12 − 0.06) = $2.65 ÷ 0.06 = $44.17
(c) Market price ($48) > Intrinsic value ($44.17) → Stock is overvalued by $3.83 per share.
A rational investor using the Gordon Growth Model would not pay $48 for a stock worth $44.17 — or would sell if they owned it. The stock will either need earnings/dividend growth to accelerate, or the price will need to fall toward $44 for the investment to be fairly priced at a 12% required return.
Q6A preferred stock pays a fixed $4.50 annual dividend. Similar preferred stocks yield 5.5%. What is this preferred stock's value? What happens to the price if interest rates rise and similar preferred stocks now yield 7%?▼
Preferred stock = perpetuity: P = Dividend ÷ Required Return
At 5.5%: P = $4.50 ÷ 0.055 = $81.82
If rates rise to 7%: P = $4.50 ÷ 0.07 = $64.29
The price drops from $81.82 to $64.29 — a $17.53 (21.4%) decline — due entirely to the interest rate increase. The fixed $4.50 dividend is unchanged; only the discount rate changed.
This illustrates the same inverse price-yield relationship from Unit 7 (bonds) applied to preferred stock. Preferred stocks are particularly sensitive to interest rate changes because: (1) the dividend is fixed forever (no growth to cushion the rate impact), (2) the duration is infinite (perpetuity), making them the most rate-sensitive fixed-income-like instrument.
At 5.5%: P = $4.50 ÷ 0.055 = $81.82
If rates rise to 7%: P = $4.50 ÷ 0.07 = $64.29
The price drops from $81.82 to $64.29 — a $17.53 (21.4%) decline — due entirely to the interest rate increase. The fixed $4.50 dividend is unchanged; only the discount rate changed.
This illustrates the same inverse price-yield relationship from Unit 7 (bonds) applied to preferred stock. Preferred stocks are particularly sensitive to interest rate changes because: (1) the dividend is fixed forever (no growth to cushion the rate impact), (2) the duration is infinite (perpetuity), making them the most rate-sensitive fixed-income-like instrument.
Q7Stock A: Price = $72, EPS = $4.00. Stock B: Price = $54, EPS = $4.50. Industry average P/E = 14×. (a) Calculate P/E for each. (b) Using the industry P/E, estimate each stock's fair value. (c) Which appears undervalued?▼
(a) P/E ratios:
Stock A: $72 ÷ $4.00 = 18× — premium to industry average
Stock B: $54 ÷ $4.50 = 12× — discount to industry average
(b) Fair value using industry P/E = 14×:
Stock A fair value: $4.00 × 14 = $56.00 — currently trading at $72, premium of $16
Stock B fair value: $4.50 × 14 = $63.00 — currently trading at $54, discount of $9
(c) Stock B appears undervalued — it trades at $54 vs. a fair value estimate of $63 (trading at 12× vs. 14× industry average). Stock A is overvalued relative to the industry multiple ($72 vs. $56 fair value).
Caveat: Stock A's premium P/E may be justified if it has higher growth prospects than the industry average. Always investigate WHY a stock trades at a premium or discount before concluding it's mispriced.
Stock A: $72 ÷ $4.00 = 18× — premium to industry average
Stock B: $54 ÷ $4.50 = 12× — discount to industry average
(b) Fair value using industry P/E = 14×:
Stock A fair value: $4.00 × 14 = $56.00 — currently trading at $72, premium of $16
Stock B fair value: $4.50 × 14 = $63.00 — currently trading at $54, discount of $9
(c) Stock B appears undervalued — it trades at $54 vs. a fair value estimate of $63 (trading at 12× vs. 14× industry average). Stock A is overvalued relative to the industry multiple ($72 vs. $56 fair value).
Caveat: Stock A's premium P/E may be justified if it has higher growth prospects than the industry average. Always investigate WHY a stock trades at a premium or discount before concluding it's mispriced.
Q8A stock is priced at $60 with D₁ = $1.80 and g = 4%. (a) Calculate the implied required return. (b) CAPM says the required return should be 8.5%. Is the stock over or undervalued according to CAPM?▼
(a) rₛ = D₁/P₀ + g = $1.80/$60 + 0.04 = 0.03 + 0.04 = 7.0%
The market is implicitly pricing this stock as if the required return is 7.0%.
(b) CAPM required return = 8.5%. Market's implied required return = 7.0%.
The market is accepting less return (7%) than the stock's risk level demands (8.5%). Investors are overpaying — they're settling for a lower return than the risk warrants.
Stock is overvalued from a CAPM perspective. The price should be lower so that the implied return rises to 8.5%:
Fair price = $1.80 ÷ (0.085 − 0.04) = $1.80 ÷ 0.045 = $40.00
At $60, the stock is overvalued by $20.00 per share relative to what CAPM says it should be worth.
The market is implicitly pricing this stock as if the required return is 7.0%.
(b) CAPM required return = 8.5%. Market's implied required return = 7.0%.
The market is accepting less return (7%) than the stock's risk level demands (8.5%). Investors are overpaying — they're settling for a lower return than the risk warrants.
Stock is overvalued from a CAPM perspective. The price should be lower so that the implied return rises to 8.5%:
Fair price = $1.80 ÷ (0.085 − 0.04) = $1.80 ÷ 0.045 = $40.00
At $60, the stock is overvalued by $20.00 per share relative to what CAPM says it should be worth.
Q9BBYM invests $10,000 in an S&P 500 index ETF (expense ratio 0.04%) vs. an actively managed fund (expense ratio 1.1%). Both earn 10% gross annual return over 30 years. Calculate the final value of each. How much does the fee difference cost?▼
Net returns: Index ETF = 10% − 0.04% = 9.96% | Active fund = 10% − 1.1% = 8.9%
Index ETF: FV = $10,000 × (1.0996)³⁰ = $10,000 × 17.42 = $174,200
Active fund: FV = $10,000 × (1.089)³⁰ = $10,000 × 13.27 = $132,700
Fee cost over 30 years: $174,200 − $132,700 = $41,500
The 1.06% annual fee difference costs $41,500 on a single $10,000 investment over 30 years. That is 4.15 times the original investment, paid to a fund manager who — statistically — is unlikely to have outperformed the index before fees anyway. For BBYM families building community wealth, choosing low-cost index funds over actively managed funds is one of the highest-certainty financial decisions available.
Index ETF: FV = $10,000 × (1.0996)³⁰ = $10,000 × 17.42 = $174,200
Active fund: FV = $10,000 × (1.089)³⁰ = $10,000 × 13.27 = $132,700
Fee cost over 30 years: $174,200 − $132,700 = $41,500
The 1.06% annual fee difference costs $41,500 on a single $10,000 investment over 30 years. That is 4.15 times the original investment, paid to a fund manager who — statistically — is unlikely to have outperformed the index before fees anyway. For BBYM families building community wealth, choosing low-cost index funds over actively managed funds is one of the highest-certainty financial decisions available.
Q10The Swanson Initiative wants to add a stock to its equity portfolio. Stock: D₁ = $1.50, g = 3%, beta = 0.8, rᵣᵓ = 4%, RPᵖ = 6%. (a) Find required return via CAPM. (b) Value the stock with the Gordon Growth Model. (c) If market price = $26, what should the Initiative do?▼
(a) CAPM required return:
rₛ = rᵣᵓ + RPᵖ × β = 4% + 6% × 0.8 = 4% + 4.8% = 8.8%
(b) Gordon Growth Model valuation:
P₀ = D₁ ÷ (rₛ − g) = $1.50 ÷ (0.088 − 0.03) = $1.50 ÷ 0.058 = $25.86
(c) Market price ($26.00) ≈ Intrinsic value ($25.86) — the stock is approximately fairly priced (within 0.5%).
The Swanson Initiative should consider buying if this stock aligns with the portfolio's objectives. At a fair price, the investment offers the CAPM-appropriate 8.8% return for its risk level (β = 0.8, moderately defensive). Given the Initiative's goal of stable distributions, a low-beta, dividend-paying stock at fair value is a solid fit. If the price dips to $23–$24, it becomes clearly undervalued and an even stronger buy.
rₛ = rᵣᵓ + RPᵖ × β = 4% + 6% × 0.8 = 4% + 4.8% = 8.8%
(b) Gordon Growth Model valuation:
P₀ = D₁ ÷ (rₛ − g) = $1.50 ÷ (0.088 − 0.03) = $1.50 ÷ 0.058 = $25.86
(c) Market price ($26.00) ≈ Intrinsic value ($25.86) — the stock is approximately fairly priced (within 0.5%).
The Swanson Initiative should consider buying if this stock aligns with the portfolio's objectives. At a fair price, the investment offers the CAPM-appropriate 8.8% return for its risk level (β = 0.8, moderately defensive). Given the Initiative's goal of stable distributions, a low-beta, dividend-paying stock at fair value is a solid fit. If the price dips to $23–$24, it becomes clearly undervalued and an even stronger buy.
Q11A BBYM entrepreneur is considering buying a small Birmingham-area food truck business generating $40,000/year in earnings. Similar businesses sell at P/E multiples of 8–10×. The business is offered at $280,000. (a) What P/E does the asking price imply? (b) Is this a fair price? (c) What growth assumptions would justify the price?▼
(a) Implied P/E = Price ÷ EPS (earnings) = $280,000 ÷ $40,000 = 7×
(b) At 7×, the price is below the industry range of 8–10× — suggesting it may be underpriced at first glance, or that there are specific reasons for the discount (location challenges, owner dependency, deferred maintenance, etc.).
Fair value range using comparable multiples:
At 8×: $40,000 × 8 = $320,000 | At 10×: $40,000 × 10 = $400,000
The asking price of $280,000 is 12.5% below the low end of the range — potentially a good deal IF the earnings are reliable.
(c) Growth assumptions to justify $280,000:
Using Gordon Growth Model logic with a 15% required return for a small private business:
If treating earnings as "dividends": $280,000 = $40,000 ÷ (0.15 − g) → 0.15 − g = 0.143 → g = 0.7% annual growth
At 0.7% growth, the price is fair at a 15% required return — essentially a flat-growth business priced at a discount. If the entrepreneur can grow the business at even 3–5%, they are creating significant value above the purchase price. This is the entrepreneurial equity premium — operational skill creates value beyond the purchase price.
(b) At 7×, the price is below the industry range of 8–10× — suggesting it may be underpriced at first glance, or that there are specific reasons for the discount (location challenges, owner dependency, deferred maintenance, etc.).
Fair value range using comparable multiples:
At 8×: $40,000 × 8 = $320,000 | At 10×: $40,000 × 10 = $400,000
The asking price of $280,000 is 12.5% below the low end of the range — potentially a good deal IF the earnings are reliable.
(c) Growth assumptions to justify $280,000:
Using Gordon Growth Model logic with a 15% required return for a small private business:
If treating earnings as "dividends": $280,000 = $40,000 ÷ (0.15 − g) → 0.15 − g = 0.143 → g = 0.7% annual growth
At 0.7% growth, the price is fair at a 15% required return — essentially a flat-growth business priced at a discount. If the entrepreneur can grow the business at even 3–5%, they are creating significant value above the purchase price. This is the entrepreneurial equity premium — operational skill creates value beyond the purchase price.
Part 6 — Quick Reference Summary
Read this the night before the assessment
Unit 9 in 5 Essential Sentences
Sentence 1
A stock's intrinsic value equals the present value of all future dividends (DDM), which simplifies for constant-growth companies to the Gordon Growth Model: P₀ = D₁ ÷ (rₛ − g) — the Assessment Q9 answer is $3.00 ÷ (10%−4%) = $50.00.
Sentence 2
Total return on a stock = dividend yield + capital gains yield = D₁/P₀ + g; rₛ from CAPM (Unit 8) is the required return used as the denominator's first term, directly connecting risk and valuation.
Sentence 3
The P/E ratio = Price ÷ EPS; a stock's fair value can be estimated as EPS × industry P/E; growth stocks have high P/E (market pricing in future growth) while value stocks have low P/E (often undervalued mature companies).
Sentence 4
Preferred stock is valued as a perpetuity: P = Dividend ÷ r; it pays fixed dividends before common dividends but has no voting rights and limited upside — a bond-equity hybrid.
Sentence 5
For BBYM community wealth building, low-cost index ETFs (0.03–0.1% fees) outperform ~80% of active funds over 15 years — the fee difference compounds to tens of thousands of dollars over a lifetime of saving.
Must-Know Facts for the Assessment
| Concept / Formula | Answer |
|---|---|
| Gordon Growth Model | P₀ = D₁ ÷ (rₛ − g) |
| Assessment Q9 answer | $3.00 ÷ (10%−4%) = $3.00 ÷ 0.06 = $50.00 |
| D₁ when D₀ is given | D₁ = D₀ × (1+g) — most common calculation error; always use NEXT year's dividend |
| Total return formula | rₛ = D₁/P₀ + g = Dividend Yield + Capital Gains Yield |
| Preferred stock value | P = Dividend ÷ Required Return (same as perpetuity from Unit 5) |
| P/E ratio | Price ÷ EPS | Fair value = EPS × Industry P/E |
| Gordon Growth requirement | rₛ MUST be greater than g — otherwise model breaks (infinite or negative price) |
| Growth stock characteristics | High P/E, low dividend yield, high g, high volatility, price sensitive to growth revisions |
| Value stock characteristics | Low P/E, higher dividend yield, lower g, lower volatility, priced below perceived intrinsic value |
| If market price > P₀ | Stock is overvalued — expected return < required; avoid or sell |
| If market price < P₀ | Stock is undervalued — expected return > required; buy opportunity |
| Index fund advantage | Fees 0.03–0.1% vs. active 0.5–1.5%; ~80% of active funds underperform index over 15 years after fees |
| CAPM → Gordon Growth link | rₛ from CAPM is the required return used in the Gordon Growth Model denominator |
| Sensitivity rule of thumb | Higher rₛ or lower g → lower stock price; lower rₛ or higher g → higher stock price |