Chapter 9 — Valuing Stocks
Core Question
Companies issue stocks to raise external capital when internal cash flow isn't enough. This chapter teaches you how to determine a fair price for those stocks using three rigorous valuation methods.
Why Do Companies Issue Shares?
If a company doesn't have enough profit/cash flow to invest in new projects (internal financing), it goes to the market to issue stock or take a loan (external financing).
Equity vs. Debt — Key Differences
| Debt | Equity |
|---|---|
| Fixed payments (contract) | Variable/irregular payments (dividends) |
| Interest + repayment | Upside — profits & dividends |
| No upside participation | Voting rights |
| Stated maturity date | Infinite maturity |
| Subject to default risk | Residual claimant |
| Lower risk | Higher risk |
Three Critical Principles
Price ≥ 0
Limited liability — shareholders can't lose more than invested.
Debtors First
Debt holders are paid before equity. Equity = residual claimant.
Bankruptcy ≠ Zero
Debt can trigger bankruptcy, but stock can fall without bankruptcy.
Three Valuation Approaches
| # | Present Value of… | Determines the… |
|---|---|---|
| 1 | Dividend Payments | Stock Price (per share) |
| 2 | Total Payouts (Dividends + Repurchases) | Equity Value |
| 3 | Free Cash Flow (all security holders) | Enterprise Value |
Navigate the Chapter
📊 9.1 — Stock Prices & Returns
One-period valuation · total return · dividend yield
💰 9.2 — Dividend-Discount Model
Constant growth · Gordon Growth · payout rate
🏭 9.3 — Total Payout & FCF
Buybacks · discounted FCF · WACC
📏 9.4 — Comparable Firms & P/E
Valuation multiples · P/E · V/EBITDA
ℹ️ 9.5 — Information & Markets
EMH · market mechanism · valuation triad
🎯 Exam Simulator — 15 Questions
Mixed theory + math · real exam style
9.1 Stock Prices, Returns & the Investment Horizon
Key Insight
Buying a stock gives you dividends while you hold it and the sale price when you sell. Both must be discounted at rE — the equity cost of capital — because the cash flows are risky.
One-Period Timeline
0
−P₀
pay today
1
Div₁ + P₁
receive
One-Period Stock Price
P₀ = (Div₁ + P₁) / (1 + rE)
Why rE? Cash flows are risky — can't use risk-free rate. The equity cost of capital is the return on other investments with comparable risk.
Multi-Period NPV
NPV over N periods
NPV = −P₀ + Σ Divt/(1+rE)t + PN/(1+rE)N
If NPV > 0 → buy the stock. In a competitive market, prices adjust so NPV ≈ 0 for all investors — you pay fair value.
Total Return
Total Return = Equity Cost of Capital
rE = Div₁/P₀ + (P₁−P₀)/P₀ = Dividend Yield + Capital Gain Rate
Dividend Yield= Div₁ / P₀
Cash income from holding the stock.
Capital Gain Rate= (P₁ − P₀) / P₀
Return from price appreciation.
✏️ Quick Check — Theory
What are the two components of total return?
Dividend yield (Div₁/P₀) = cash income. Capital gain rate ((P₁−P₀)/P₀) = price appreciation. Together = rE.
✏️ Quick Check — Calculation
P₀ = €50, Div₁ = €2, P₁ = €54. What is rE?
rE = (2 + 54 − 50)/50 = 12% (4% div yield + 8% capital gain)
🧮 Stock Price Calculator — One Period
Current Stock Price P₀
9.2 Applying the Dividend-Discount Model
Key Insight
If dividends grow at a constant rate g forever, an infinite series collapses into a simple formula. Elegant — but extremely sensitive to g.
Gordon Growth Model
P₀ = Div₁ / (rE − g)
Critical: rE must be strictly greater than g. If g ≥ rE, the formula produces nonsense.
Example: StudentKaf
Div = €0.445, rE = 8.5%, g = 1.5%.
P₀ = 0.445 / 0.07 = €6.36
Check: 7.0% div yield + 1.5% capital gain = 8.5% ✓
P₀ = 0.445 / 0.07 = €6.36
Check: 7.0% div yield + 1.5% capital gain = 8.5% ✓
Where Does Dividend Growth Come From?
Dividend Growth Rate
g = Retention Rate × Return on New Investment
Retention Rate = 1 − Payout Rate. Higher retention → more investment → higher growth but lower current dividends.
Three Ways to Increase Dividends per Share
1. Increase Profit
More earnings → more dividends
2. Raise Payout %
Pay a larger fraction of earnings
3. Reduce Shares
Buybacks → higher div per share
⚠️ Limitations of DDM:
• Difficult to forecast g
• Small changes in g → huge changes in P₀
• Only useful for dividend payers
• Difficult to forecast g
• Small changes in g → huge changes in P₀
• Only useful for dividend payers
✏️ Quick Check
Retention rate = 40%, return on new investment = 15%. What is g?
g = 0.40 × 15% = 6%
✏️ Quick Check
Div₁ = €3, g = 5%, rE = 10%. What is P₀?
P₀ = 3 / (0.10 − 0.05) = €60
🧮 Gordon Growth Calculator
Stock Price
9.3 Total Payout & Free Cash Flow Models
Key Insight
Firms return cash via share repurchases too — basic DDM misses this. DCF goes further: values the entire enterprise using cash flows to all investors.
1. Total Payout Model
Total Payout
P₀ = PV(Future Total Dividends + Repurchases) / Shares Outstanding
2. Discounted Free Cash Flow (DCF)
Enterprise Value
Enterprise Value = Market Value of Equity + Debt − Cash
Free Cash Flow
FCF = EBIT × (1−τc) + Depreciation − CapEx − ΔNWC
Why FCF? Earnings include non-cash charges (depreciation) and exclude real cash outflows (CapEx). FCF = real cash production.
WACC — Discounting FCF
WACC
rwacc = rE × E/(E+D) + rD × D/(E+D)
Terminal Value
VN = FCFN+1 / (rwacc − gFCF)
Stock Price from DCF
P₀ = (V₀ + Cash − Debt) / Shares Outstanding
Remember: Add Cash, subtract Debt to convert enterprise value → equity value → per-share price.
Model Comparison
| Model | Discounts | Best For | Rate |
|---|---|---|---|
| Dividend Discount | Dividends/share | Stable dividend payers | rE |
| Total Payout | Divs + buybacks | Firms with repurchases | rE |
| Discounted FCF | Free cash flows | Any firm, non-div payers | rwacc |
✏️ Quick Check
Why add Cash and subtract Debt when converting enterprise value to stock price?
V₀ includes all financing claims. Subtract Debt removes what's owed to bondholders. Add Cash because it belongs to equity but isn't in operating FCF.
✏️ Quick Check — Calc
EBIT = €100M, tax = 30%, Dep = €20M, CapEx = €30M, ΔNWC = €10M. FCF?
FCF = 100×0.70 + 20 − 30 − 10 = €50M
🧮 DCF Terminal Value Calculator
Terminal Enterprise Value
9.4 Valuation Based on Comparable Firms
Key Insight
The Law of One Price says comparable assets have comparable values. Since perfect matches don't exist, we use valuation multiples to scale.
Key Valuation Multiples
P/E Ratio= Share Price / EPS
Most common. Higher P/E = higher growth expectations.
V/EBITDA= Enterprise Value / EBITDA
Less sensitive to capital structure.
Steps: 1. Find comparables · 2. Average their P/E · 3. Multiply target's EPS · 4. Cross-check with DCF
Forward P/E
Forward P/E = P₀/EPS₀ = Payout Rate / (rE − g)
⚠️ Limitations: Doesn't reveal industry-wide mispricing. Doesn't adjust for growth/risk differences. No single technique is definitive.
Best Practice: Use multiple methods. Gain confidence when DCF and comparables converge.
✏️ Quick Check
Comp B: cap €175M, earnings €27.7M. Comp C: cap €302M, earnings €59.8M. Firm A has 10M shares, €37M earnings. Estimate A's share price.
P/E(B) = 6.32, P/E(C) = 5.05, avg = 5.69. EPS(A) = €3.70. Price = 5.69 × 3.70 ≈ €21
✏️ Quick Check
Payout = 40%, rE = 10%, g = 4%. Forward P/E?
0.40 / 0.06 = 6.67×
🧮 Forward P/E Calculator
Forward P/E
9.5 Information, Competition & Stock Prices
Key Insight
The Valuation Model links Share Value, Future Cash Flows, and Cost of Capital. In efficient markets, competition instantly incorporates all public information into prices.
The Market Mechanism
Price < True Value
→ Investors buy → price UP → NPV = 0
Price > True Value
→ Investors sell → price DOWN → NPV = 0
EMH: Competition among informed investors eliminates positive-NPV trades. Prices incorporate all public information instantly.
Key Takeaways
| Principle | What it means |
|---|---|
| Equity gives right to dividend | Expected return = capital gains + dividend yield |
| Use correct discount rate | rE for equity; rwacc for FCF |
| Payout policy trade-off | Lower dividend → higher growth (if NPV>0 exists) |
| 3 valuation techniques | DDM, DCF, Comparables |
| Markets price info quickly | EMH: instant pricing of public information |
✏️ Quick Check
According to EMH, what happens after positive news?
Investors buy instantly, driving price up until NPV = 0. Information is "priced in" — no arbitrage remains.
📐 Key Formulas — Chapter 9
One-Period Stock Price
P₀ = (Div₁ + P₁) / (1 + rE)
Total Return
rE = Div₁/P₀ + (P₁−P₀)/P₀
Dividend Growth Rate
g = Retention Rate × Return on New Investment
Gordon Growth Model
P₀ = Div₁ / (rE − g)
Free Cash Flow
FCF = EBIT(1−τ) + Depreciation − CapEx − ΔNWC
WACC
rwacc = rE·E/(E+D) + rD·D/(E+D)
DCF Terminal Value
VN = FCFN+1 / (rwacc − gFCF)
Stock Price from DCF
P₀ = (V₀ + Cash − Debt) / Shares Outstanding
Forward P/E
P/E = Payout / (rE − g)
🃏 Flashcards — Chapter 9
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Chapter 10 — Capital Markets & the Pricing of Risk
Core Question
Investors demand a higher return for taking more risk — but which kind of risk? This chapter quantifies risk, separates risk you get paid for from risk you don't, and builds the formula that prices every risky asset: the CAPM.
What You Will Learn
History
96 years of returns: stocks vs bonds vs bills
Measure
Expected return, variance, standard deviation
Diversify
Some risks vanish in a portfolio — others don't
The Big Picture
The chapter's punchline: Only systematic (market) risk earns a return premium. Firm-specific risk can be diversified away for free — so the market doesn't pay you for bearing it. The amount of systematic risk in a stock is measured by its beta (β), and the CAPM tells you the expected return: r = rf + β × (E[RMkt] − rf).
Historical Performance ($100 invested 1925 → 2021)
| Asset Class | Avg Annual Return | Volatility (SD) | $100 grew to… |
|---|---|---|---|
| Small Stocks | 18.6% | 38.6% | $93,939 |
| S&P 500 | 12.2% | 19.7% | $11,535 |
| Corporate Bonds | 6.5% | 6.3% | $716 |
| Treasury Bills | 3.3% | 3.1% | $24 |
| CPI (Inflation) | — | — | $16 |
Pattern: Higher historical return ↔ higher volatility. But this relationship only holds clearly for portfolios — not for individual stocks.
Navigate the Chapter
📜 10.1 — 96 Years of History
Big picture: stocks vs bonds vs bills
🎲 10.2 — Risk & Return Measures
E[R], Variance, Standard Deviation
📊 10.3 — Historical Returns
Realized return · sample variance · standard error
⚖️ 10.4 — Risk-Return Tradeoff
Excess return · portfolios vs individuals
🌪️ 10.5 — Common vs Independent
Theft vs earthquake insurance
🧺 10.6 — Diversification
Firm-specific vs systematic · no arbitrage
β 10.7 — Measuring Beta
Market portfolio · beta interpretation
🎯 10.8 — CAPM
The pricing formula for every risky asset
10.1 Risk and Return: Insights from 96 Years of History
Key Insight
Looking at investment performance from 1925 to 2021, we see a clear pattern: asset classes with higher average returns also had higher volatility. There's no free lunch — extra return requires bearing extra risk.
The Long-Run Picture
Imagine you invest $100 in 1925. By 2021, depending on what you chose:
• Small Stocks → $93,939 (massive growth, massive swings)
• S&P 500 → $11,535 (strong growth, big drawdowns)
• Corporate Bonds → $716 (moderate, stable)
• Treasury Bills → $24 (tiny, near risk-free)
• CPI Inflation → $16 (T-bills barely beat inflation!)
• Small Stocks → $93,939 (massive growth, massive swings)
• S&P 500 → $11,535 (strong growth, big drawdowns)
• Corporate Bonds → $716 (moderate, stable)
• Treasury Bills → $24 (tiny, near risk-free)
• CPI Inflation → $16 (T-bills barely beat inflation!)
Three Key Lessons
Stocks Win Long-Run
Over 96 years, stocks crushed bonds and bills.
Higher Return = Higher Volatility
Small stocks: 38.6% SD vs T-bills: 3.1% SD.
Time Smooths Risk
Over 20 years, even small stocks rarely lost money.
Real Returns (after inflation)
Treasury Bills barely beat inflation: $24 nominal vs $16 inflation — almost the same. If you hold only "safe" assets, you may preserve nominal capital but lose real purchasing power.
✏️ Quick Check — Theory
Looking at the 96-year data, what is the relationship between average return and volatility across asset classes?
Positive: assets with higher average returns also have higher volatility. Small stocks (highest return) have the highest SD (38.6%). T-bills (lowest return) have the lowest SD (3.1%). Risk and return move together at the asset-class level.
10.2 Common Measures of Risk and Return
Key Insight
To compare investments, we need numbers: a measure of typical return (expected value) and a measure of how much actual returns swing around that typical return (variance & standard deviation).
Probability Distribution
Future returns aren't known — they have possible outcomes, each with a probability. Example for stock BFI:
| Economy | Probability (pR) | Return (R) |
|---|---|---|
| Strong | 25% | +40% |
| Normal | 50% | +10% |
| Weak | 25% | −20% |
Expected Return
Expected Return
E[R] = Σ pR × R
BFI Example:
E[R] = 0.25(40%) + 0.50(10%) + 0.25(−20%) = 10% + 5% − 5% = 10%
E[R] = 0.25(40%) + 0.50(10%) + 0.25(−20%) = 10% + 5% − 5% = 10%
Variance & Standard Deviation
Variance
Var(R) = Σ pR × (R − E[R])²
Standard Deviation (Volatility)
SD(R) = √Var(R)
BFI Variance:
= 0.25(0.40−0.10)² + 0.50(0.10−0.10)² + 0.25(−0.20−0.10)²
= 0.25(0.09) + 0 + 0.25(0.09) = 0.045
SD = √0.045 = 21.2%
= 0.25(0.40−0.10)² + 0.50(0.10−0.10)² + 0.25(−0.20−0.10)²
= 0.25(0.09) + 0 + 0.25(0.09) = 0.045
SD = √0.045 = 21.2%
What SD Tells You
SD = how spread out outcomes are around E[R]. Roughly, ⅔ of outcomes fall within ±1 SD of the mean. For BFI: typical year ranges from 10% − 21.2% = −11% to 10% + 21.2% = +31%. SD is also called volatility.
✏️ Quick Check — Calculation
A stock has equal (50/50) probability of returning +45% or −25%. Compute E[R], Variance, and SD.
E[R] = 0.5(45%) + 0.5(−25%) = 22.5% − 12.5% = 10%
Var = 0.5(0.45−0.10)² + 0.5(−0.25−0.10)² = 0.5(0.1225) + 0.5(0.1225) = 0.1225
SD = √0.1225 = 35%
Var = 0.5(0.45−0.10)² + 0.5(−0.25−0.10)² = 0.5(0.1225) + 0.5(0.1225) = 0.1225
SD = √0.1225 = 35%
🧮 Expected Return & SD Calculator (3 scenarios)
Result
10.3 Historical Returns of Stocks and Bonds
Key Insight
In real life we rarely know the true probability distribution. Instead we estimate E[R] and SD from historical data — and recognize there's estimation error in any such estimate.
Realized Return (year by year)
Realized Return — Year t→t+1
Rt+1 = (Divt+1 + Pt+1) / Pt − 1 = Dividend Yield + Capital Gain
Example: Microsoft 2005: bought at $26.15, received $0.32 div, sold at $26.71.
R = (0.32 + 26.71)/26.15 − 1 = 3.49% (1.22% div yield + 2.14% capital gain)
R = (0.32 + 26.71)/26.15 − 1 = 3.49% (1.22% div yield + 2.14% capital gain)
Average Historical Return
Sample Average
R̄ = (1/T) × Σ Rt
This is our best estimate of the expected return E[R] going forward, assuming the distribution is stable.
Historical Variance & Standard Deviation
Sample Variance (Note: divides by T−1, not T)
Var(R) = [1/(T−1)] × Σ (Rt − R̄)²
Why T−1? We "use up" one degree of freedom estimating R̄ from the same data. Dividing by T−1 corrects for this bias and gives an unbiased estimate of variance.
Estimation Error: Can We Trust R̄?
Standard Error of the Mean
SE(R̄) = SD(R) / √T
95% Confidence Interval for E[R]
R̄ ± 2 × SE(R̄)
Real example: S&P 500 1926–2021: R̄ ≈ 12.2%, SD ≈ 19.7%, T = 96. SE = 19.7%/√96 = 2.0%. So 95% CI for E[RS&P] = 12.2% ± 4% = 8.2% to 16.2%. Even with 96 years of data, our estimate is rough!
✏️ Quick Check — Theory
Why do we divide by T−1 instead of T when computing historical variance?
Because we used the same data to compute R̄, which is itself an estimate. Dividing by T−1 ("degrees of freedom" correction) gives an unbiased estimate of the true variance. Dividing by T would systematically underestimate variance.
✏️ Quick Check — Math
A stock has SD = 25% based on 25 years of data. What is the standard error of the mean?
SE = 25% / √25 = 25% / 5 = 5%. The 95% CI for E[R] is therefore R̄ ± 10% — quite wide!
10.4 The Historical Tradeoff Between Risk and Return
Key Insight
For large portfolios, more volatility → more excess return. For individual stocks, this relationship breaks down. Why? Individual stocks carry firm-specific risk that the market doesn't compensate.
Excess Return
Excess Return
Excess Return = Average Realized Return − rf
The excess return is what investors earn above the risk-free rate as compensation for bearing risk.
Portfolios: Clear Tradeoff
| Portfolio | SD | Excess Return vs T-Bills |
|---|---|---|
| Small Stocks | 38.6% | 15.3% |
| S&P 500 | 19.7% | 8.9% |
| Corporate Bonds | 6.3% | 3.2% |
| Treasury Bills | 3.1% | 0% (baseline) |
Pattern is monotonic: as SD rises, excess return rises. Investors are rewarded for taking on more portfolio risk.
Individual Stocks: No Clear Tradeoff
The shock: if you plot individual stocks (volatility vs return) you see a scatter, not a line. Many individual stocks have higher volatility than the S&P 500 but lower average returns.
Why? Total volatility of a single stock includes firm-specific risk that disappears in a diversified portfolio. The market doesn't pay you to bear risk you could eliminate for free → the extra volatility of a single stock doesn't earn extra return.
✏️ Quick Check — Theory
Why does the positive risk-return tradeoff hold for large portfolios but NOT for individual stocks?
Large portfolios contain only systematic risk (firm-specific risk has been diversified away). Individual stocks contain both systematic AND firm-specific risk, but only systematic risk is rewarded. So a volatile single stock doesn't earn higher return — its volatility is mostly diversifiable noise.
10.5 Common Versus Independent Risk
Key Insight
Not all risks behave the same. Some cancel out in a large portfolio (independent risks). Others hit everything at once (common risks) and cannot be diversified.
The Insurance Analogy
Theft Insurance
Independent risk. One house being burgled is unrelated to another. With 100,000 policies, average payout is very predictable.
Earthquake Insurance
Common risk. One earthquake hits all houses in San Francisco at once. Risk does NOT shrink with more policies.
The Math: Law of Large Numbers
If you write N independent insurance policies, each with SD σ:
• Average payout has SD = σ / √N
• As N → ∞, the per-policy uncertainty disappears
But for common risks, all N policies move together → SD stays at σ no matter how big N gets.
• Average payout has SD = σ / √N
• As N → ∞, the per-policy uncertainty disappears
But for common risks, all N policies move together → SD stays at σ no matter how big N gets.
The Takeaway
| Type of Risk | Diversifiable? | Example |
|---|---|---|
| Independent | ✅ Yes | House fires, theft, single-firm news |
| Common | ❌ No | Earthquake, recession, interest-rate shock |
✏️ Quick Check — Theory
Why is theft insurance much cheaper per dollar of coverage than earthquake insurance in the same area?
Theft risk is independent across homes — with many policies, the insurer's average payout is very predictable, so they need only a small risk premium. Earthquake risk is common — one event triggers all claims at once, so the insurer faces huge variance even with many policies → much higher premium required.
10.6 Diversification in Stock Portfolios
Key Insight
Apply the insurance idea to stocks: each company has firm-specific news (independent) and economy-wide news (common). A diversified portfolio eliminates the first kind. The market doesn't pay you for risk you could have diversified away.
Two Types of Risk in a Stock
Firm-Specific Risk
CEO leaves, product recall, lawsuit, factory fire, patent expiry. Also called idiosyncratic, unsystematic, or diversifiable risk.
Systematic Risk
Recession, interest-rate change, inflation, oil shock, pandemic. Also called market or non-diversifiable risk.
How Diversification Works
Hold N stocks instead of 1:
• Firm-specific shocks partially cancel (some firms have good news, others bad)
• As N grows, firm-specific risk → 0
• Market-wide shocks don't cancel — they hit everyone
In the limit (~30+ well-chosen stocks), the only risk left is systematic.
• Firm-specific shocks partially cancel (some firms have good news, others bad)
• As N grows, firm-specific risk → 0
• Market-wide shocks don't cancel — they hit everyone
In the limit (~30+ well-chosen stocks), the only risk left is systematic.
The No-Arbitrage Argument: Why Diversifiable Risk Pays Zero Premium
Suppose firm-specific risk paid a premium. Then any investor could:
1. Build a diversified portfolio (eliminating firm-specific risk)
2. Earn the premium without bearing the risk
3. → Free money (arbitrage)
This can't happen in equilibrium. So only systematic risk is compensated. The risk premium for firm-specific risk = 0.
1. Build a diversified portfolio (eliminating firm-specific risk)
2. Earn the premium without bearing the risk
3. → Free money (arbitrage)
This can't happen in equilibrium. So only systematic risk is compensated. The risk premium for firm-specific risk = 0.
Implication for individual stocks: Total volatility of a single stock = firm-specific + systematic. Only the systematic portion earns a return. That's why high-volatility individual stocks don't reliably beat low-volatility ones.
✏️ Quick Check — Theory
Which of these risks is systematic (cannot be diversified): (a) a fire at one factory, (b) the Federal Reserve raising interest rates, (c) the CEO resigning, (d) a single product recall?
(b) Fed raising rates — this affects every firm in the economy (cost of borrowing, valuations, demand). The others are firm-specific and disappear in a large portfolio.
✏️ Quick Check — Theory
Why must the risk premium for firm-specific risk equal zero in equilibrium?
Because investors can eliminate firm-specific risk for free by diversifying. If it paid a premium, arbitrage would force the premium back to zero. Investors are only paid for risk they cannot eliminate.
10.7 Measuring Systematic Risk: Beta (β)
Key Insight
If only systematic risk matters, we need a way to measure how much systematic risk a stock has. The answer: beta — the sensitivity of a stock's return to the overall market.
The Market Portfolio
The market portfolio is the value-weighted combination of all risky assets. In practice we use a broad index like the S&P 500 as a proxy. It contains only systematic risk (firm-specific risks have been diversified away).
Definition of Beta
Beta
βi = % change in Ri for a 1% change in RMarket
More precisely: βi = Cov(Ri, RMkt) / Var(RMkt). Computed by regressing the stock's returns on market returns — the slope is β.
Interpreting Beta
| Beta | Interpretation | Example |
|---|---|---|
| β = 0 | No systematic risk — moves independently of market | Risk-free asset |
| 0 < β < 1 | Less volatile than market — defensive | Hormel (0.46), Coca-Cola (0.65) |
| β = 1 | Moves one-for-one with market | S&P 500 itself |
| β > 1 | Amplifies market moves — cyclical | Apple (1.21), Tesla (1.66), Avis (2.15) |
| β < 0 | Moves opposite to market | Rare — some hedges, gold miners |
Real Beta Examples (S&P 500, 2017–2022)
Defensive (β < 1)
Hormel 0.46 · Walmart 0.55 · Coca-Cola 0.65 · J&J 0.66 · McDonald's 0.60
Market-like (β ≈ 1)
Microsoft 0.92 · Pfizer 0.81 · Amgen 0.74
Cyclical (β > 1)
Apple 1.21 · Goldman Sachs 1.34 · Tesla 1.66 · Avis 2.15
✏️ Quick Check — Theory
A stock has β = 1.5. If the market rises 4% next month, what is the expected return on this stock (ignoring the risk-free rate)?
Beta says the stock moves 1.5% for every 1% move in the market. So expected move = 1.5 × 4% = +6%. (Note: this is the systematic component — actual return will also include firm-specific noise.)
✏️ Quick Check — Theory
Two stocks have equal total volatility (SD = 30%). Stock A has β = 1.5; Stock B has β = 0.5. Which one would you expect to have a higher expected return, and why?
Stock A. Both have the same total volatility, but A has more systematic risk (higher β) and less firm-specific risk. Only systematic risk earns a premium → A has a higher expected return. The "extra" volatility of B is mostly diversifiable noise.
10.8 Beta and the Cost of Capital — CAPM
Key Insight
The Capital Asset Pricing Model (CAPM) ties everything together: it tells you the expected return required on any asset, based solely on its systematic risk (beta).
Step 1: The Market Risk Premium
Market Risk Premium
MRP = E[RMkt] − rf
The extra return investors demand for holding the market portfolio over T-bills. Historically ≈ 5–7% per year. This is what β = 1 earns above rf.
Step 2: Scale by Beta
CAPM — The Single Most Important Formula in Finance
ri = rf + βi × (E[RMkt] − rf)
In words: Required return = risk-free rate + (amount of systematic risk) × (price of systematic risk).
Worked Example: AMD
AMD has β = 1.51. Risk-free rate rf = 3%. Market risk premium = 5%.
rAMD = 3% + 1.51 × 5% = 3% + 7.55% = 10.55%
This is what investors should demand to hold AMD. It's also the discount rate AMD's CFO should use when valuing equity-financed projects with similar risk.
rAMD = 3% + 1.51 × 5% = 3% + 7.55% = 10.55%
This is what investors should demand to hold AMD. It's also the discount rate AMD's CFO should use when valuing equity-financed projects with similar risk.
What CAPM Tells You
Linear in β
Plot E[R] vs β and you get a straight line — the Security Market Line.
Only β Matters
Two assets with the same β must have the same expected return.
Practical Tool
CFOs use CAPM to compute the cost of equity for WACC.
✏️ Quick Check — Calculation
rf = 4%, β = 1.5, market risk premium = 6%. What is the expected return on this stock?
r = 4% + 1.5 × 6% = 4% + 9% = 13%
✏️ Quick Check — Calculation
rf = 2%, E[RMkt] = 10%, β = 0.8. What is the expected return?
MRP = 10% − 2% = 8%. r = 2% + 0.8 × 8% = 2% + 6.4% = 8.4%
✏️ Quick Check — Theory
A stock has β = 0. What does CAPM say its expected return should be?
r = rf + 0 × MRP = rf. Even though the stock may have firm-specific volatility, it has no systematic risk → no risk premium → just earn the risk-free rate.
🧮 CAPM Calculator
CAPM Expected Return
📐 Key Formulas — Chapter 10
Expected Return
E[R] = Σ pR × R
Variance
Var(R) = Σ pR × (R − E[R])²
Standard Deviation
SD(R) = √Var(R)
Realized Return
Rt+1 = (Divt+1 + Pt+1)/Pt − 1
Historical Average
R̄ = (1/T) × Σ Rt
Sample Variance
Var = [1/(T−1)] × Σ (Rt − R̄)²
Standard Error of the Mean
SE = SD / √T
95% Confidence Interval
R̄ ± 2 × SE
Excess Return
Excess = R̄ − rf
Beta (Regression Definition)
βi = Cov(Ri, RMkt) / Var(RMkt)
Market Risk Premium
MRP = E[RMkt] − rf
CAPM
ri = rf + βi × (E[RMkt] − rf)
🃏 Flashcards — Chapter 10
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