Chapter 9 — Valuing Stocks
Core Question
Companies issue stocks to raise external capital. This chapter teaches you how to determine a fair price using three valuation methods.
Equity vs. Debt
| Debt | Equity |
|---|---|
| Fixed payments | Variable dividends |
| Stated maturity | Infinite maturity |
| Subject to default risk | Residual claimant |
| Lower risk | Higher risk |
Three Valuation Approaches
| PV of… | Determines the… |
|---|---|
| Dividend Payments | Stock Price per share |
| Total Payouts (Divs + Repurchases) | Equity Value |
| Free Cash Flow | Enterprise Value |
Navigate
📊 9.1 Stock Prices & Returns
One-period valuation · total return
💰 9.2 Dividend-Discount Model
Gordon Growth · payout · growth
🏭 9.3 Total Payout & FCF
Buybacks · DCF · WACC
📏 9.4 Comparable Firms
P/E · V/EBITDA · multiples
ℹ️ 9.5 Information & Markets
EMH · valuation triad
🎯 Exam Simulator (15)
Real exam-style mixed Q
9.1 Stock Prices, Returns & the Investment Horizon
Key Insight
Buying a stock gives you dividends while you hold and the sale price when you sell. Both must be discounted at rE.
One-Period Timeline
0
−P₀
1
Div₁ + P₁
One-Period Price
P₀ = (Div₁ + P₁) / (1 + rE)
Total Return
Total Return = Equity Cost of Capital
rE = Div₁/P₀ + (P₁−P₀)/P₀ = Dividend Yield + Capital Gain Rate
Market equilibrium: Competition forces price so that NPV = 0 for the marginal investor.
✏️ Quick Check
P₀=€50, Div₁=€2, P₁=€54. What is rE?
(2+54−50)/50 = 12% (4% div yield + 8% capital gain)
🧮 Stock Price Calculator
P₀
9.2 Dividend-Discount Model
Key Insight
If dividends grow at constant g forever, the infinite series collapses into a simple formula. Elegant but very sensitive to g.
Gordon Growth Model
P₀ = Div₁ / (rE − g)
Critical: rE must be > g, otherwise nonsense.
Dividend Growth Rate
g = Retention Rate × Return on New Investment
Retention = 1 − Payout. Higher retention → more growth, fewer current dividends. Trade-off.
↑ Profit
↑ Payout %
↓ Shares
⚠️ DDM Limitations: Hard to forecast g · Small Δg → huge ΔP₀ · Only for dividend payers
✏️ Quick Check
Retention=40%, return on new investment=15%. What is g?
g = 0.40 × 15% = 6%
✏️ Quick Check
Div₁=€3, g=5%, rE=10%. P₀?
3/(0.10−0.05) = €60
🧮 Gordon Growth Calculator
9.3 Total Payout & Free Cash Flow Models
Key Insight
Firms also return cash via buybacks. DCF goes further: values the whole enterprise using cash flows to all investors.
Total Payout Model
Total Payout
P₀ = PV(Future Dividends + Repurchases) / Shares
Discounted Free Cash Flow (DCF)
Enterprise Value
EV = Market Equity + Debt − Cash
Free Cash Flow
FCF = EBIT(1−τ) + Depreciation − CapEx − ΔNWC
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
Model Comparison
| Model | Discounts | Best For | Rate |
|---|---|---|---|
| DDM | Dividends | Stable dividend payers | rE |
| Total Payout | Divs + buybacks | Repurchasing firms | rE |
| DCF | FCF | Any firm | rwacc |
✏️ Quick Check
EBIT=€100M, tax=30%, Dep=€20M, CapEx=€30M, ΔNWC=€10M. FCF?
100×0.70 + 20 − 30 − 10 = €50M
🧮 DCF Terminal Value
9.4 Valuation Based on Comparable Firms
Key Insight
The Law of One Price says comparable assets have comparable values. Use multiples to scale.
Valuation Multiples
P/E Ratio= Price / EPS
Higher P/E = higher growth expectations.
V/EBITDA= Enterprise Value / EBITDA
Capital-structure neutral.
Forward P/E
P/E = Payout / (rE − g)
⚠️ Limits: Doesn't catch industry-wide mispricing · Ignores risk/growth differences
✏️ Quick Check
Comp B: cap €175M, earnings €27.7M. Comp C: cap €302M, earnings €59.8M. Firm A: 10M shares, €37M earnings. Estimate A's price.
P/E(B)=6.32, P/E(C)=5.05, avg=5.69. EPS(A)=€3.70. Price=5.69×3.70 ≈ €21
🧮 Forward P/E Calculator
9.5 Information, Competition & Stock Prices
Key Insight
Valuation Model links Share Value, Future Cash Flows, and Cost of Capital. EMH: prices instantly reflect all public information.
P < Value
Buy → price up → NPV=0
P > Value
Sell → price down → NPV=0
EMH: Competition eliminates positive-NPV trades. Prices reflect all public info instantly.
✏️ Quick Check
Per EMH, what happens after positive news?
Investors buy instantly, driving price up until NPV=0.
📐 Formulas — Chapter 9
One-Period Stock Price
P₀ = (Div₁ + P₁) / (1 + rE)
Total Return
rE = Div₁/P₀ + (P₁−P₀)/P₀
Dividend Growth
g = Retention × Return on New Investment
Gordon Growth
P₀ = Div₁ / (rE − g)
FCF
FCF = EBIT(1−τ) + Dep − CapEx − ΔNWC
WACC
rwacc = rE·E/(E+D) + rD·D/(E+D)
DCF Terminal
VN = FCFN+1 / (rwacc − g)
Stock from DCF
P₀ = (V₀ + Cash − Debt) / Shares
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 higher return for more risk — but which kind of risk? Only systematic risk earns a premium. The amount is measured by beta (β), priced by CAPM.
96-Year Performance ($100 invested 1925 → 2021)
| Asset | Avg Return | Volatility | $100 grew to |
|---|---|---|---|
| Small Stocks | 18.6% | 38.6% | $93,939 |
| S&P 500 | 12.2% | 19.7% | $11,535 |
| Corp Bonds | 6.5% | 6.3% | $716 |
| T-Bills | 3.3% | 3.1% | $24 |
| Inflation (CPI) | — | — | $16 |
Pattern: Higher avg return ↔ higher volatility, but only at portfolio level — not individual stocks.
📜 10.1 — 96 Years of History
🎲 10.2 — Risk & Return Measures
📊 10.3 — Historical Returns
⚖️ 10.4 — Risk-Return Tradeoff
🌪️ 10.5 — Common vs Independent
🧺 10.6 — Diversification
β 10.7 — Measuring Beta
🎯 10.8 — CAPM
10.1 Risk & Return: 96 Years of History
Key Insight
Over 1925–2021, asset classes that delivered higher average returns also had higher volatility. There is no free lunch — extra return is the reward for bearing extra risk.
The Growth of $100 (1925 → 2021)
The single most famous picture in investing: $100 invested in each asset class in 1925, plotted on a log scale (equal vertical distances = equal % growth). The ranking is dramatic.
Reading the chart: $100 → 2021 became Small Stocks $93,939, S&P 500 $11,535, Corp Bonds $716, T-Bills $24, Inflation (CPI) $16. The ordering of the curves is exactly the ordering of their volatilities — the riskiest asset (small stocks) is also the one that grew the most.
Three Lessons From the Data
Stocks Win Long-Run
Equities crushed bonds & bills over any long horizon.
↑ Return = ↑ Volatility
Small stocks: ~39% SD. T-bills: ~3% SD.
Time Smooths Risk
Stocks rarely lost money over 20-yr holding periods.
"Safe" is not safe in real terms: T-Bills grew $100 → $24 in nominal terms, but inflation alone turned $100 into $16 of 1925 purchasing power equivalent. After inflation, T-bills earned almost nothing — capital preservation in nominal dollars is real-terms erosion.
✏️ Quick Check
Across asset classes, is the relationship between average return and volatility positive or negative?
Positive. Higher average returns came with higher standard deviation historically — the risk-return tradeoff.
✏️ Quick Check
Why does the chart use a logarithmic vertical axis instead of a linear one?
On a log scale, equal vertical distances represent equal percentage growth. With 96 years of compounding, small stocks ($93,939) would dwarf everything else on a linear axis, making the smaller series invisible.
10.2 Common Measures of Risk and Return
Key Insight
Quantify return with E[R] and risk with Variance / SD.
| Economy | Probability | Return |
|---|---|---|
| Strong | 25% | +40% |
| Normal | 50% | +10% |
| Weak | 25% | −20% |
Expected Return
E[R] = Σ pR × R
BFI: 0.25(40) + 0.50(10) + 0.25(−20) = 10%
Variance
Var(R) = Σ pR × (R − E[R])²
Standard Deviation
SD(R) = √Var(R)
BFI Var = 0.25(0.30)² + 0 + 0.25(0.30)² = 0.045 → SD = 21.2%
Picturing the Probability Distribution
What the figure shows: the height of each bar is its probability. The dashed line at +10% is the expected return (probability-weighted mean). The two outer outcomes (−20% and +40%) are each 30 percentage points from the mean — their squared distances, weighted by 25% each, produce the variance of 0.045 and the 21.2% standard deviation. A wider, flatter distribution would mean higher SD.
✏️ Quick Check
50/50 chance of +45% or −25%. E[R], Var, SD?
E[R]=10%, Var=0.5(0.35)²+0.5(0.35)²=0.1225, SD=35%
✏️ Quick Check
If the +40% and −20% outcomes were replaced by +25% and −5% (same probabilities, same E[R]=10%), would SD rise or fall?
Fall. The outcomes are now closer to the mean (15 pts away instead of 30), so squared deviations — and therefore variance and SD — shrink.
🧮 E[R] & SD Calculator (3 scenarios)
10.3 Historical Returns of Stocks and Bonds
Key Insight
We estimate E[R] and SD from historical data — and recognize estimation error.
Realized Return
Rt+1 = (Div + Pt+1)/Pt − 1
Historical Average
R̄ = (1/T) × Σ Rt
Sample Variance
Var = [1/(T−1)] × Σ (Rt − R̄)²
Why T−1? Degrees-of-freedom correction since R̄ was estimated from the same data.
Standard Error
SE(R̄) = SD / √T
95% CI for E[R]
R̄ ± 2 × SE
S&P: R̄=12.2%, SD=19.7%, T=96 → SE=2.0% → 95% CI = 8.2% to 16.2%
✏️ Quick Check
SD=25%, T=25. SE?
25/√25 = 5%
10.4 Historical Risk-Return Tradeoff
Key Insight
Portfolios show clear tradeoff. Individual stocks don't — firm-specific risk isn't rewarded.
Excess Return
Excess = R̄ − rf
| Portfolio | SD | Excess vs T-Bills |
|---|---|---|
| Small Stocks | 38.6% | 15.3% |
| S&P 500 | 19.7% | 8.9% |
| Corp Bonds | 6.3% | 3.2% |
| T-Bills | 3.1% | 0% |
Portfolios Line Up — Individual Stocks Don't
What the figure shows: the four portfolios (colored dots) fall almost perfectly on an upward-sloping line — more risk, more reward. But hundreds of individual stocks (grey dots) form a cloud with no clear pattern: many high-volatility stocks earn lower returns than the S&P 500. The reason: a single stock's total volatility is mostly firm-specific risk, which the market does not reward. Only the systematic part is priced.
✏️ Quick Check
Why does the tradeoff hold for portfolios but not single stocks?
Well-diversified portfolios contain only systematic risk, which is rewarded. Single stocks also carry firm-specific risk that earns no premium, so their total volatility doesn't predict return.
✏️ Quick Check
Stock X has higher volatility than the S&P 500 but a lower average return. Does this violate "risk = reward"?
No. Most of X's volatility is diversifiable firm-specific noise, which earns nothing. "Risk = reward" applies to systematic risk, not total volatility.
10.5 Common vs Independent Risk
Key Insight
Independent risks cancel out in large groups. Common risks don't.
Theft (Independent)
Unrelated across homes → 100,000 policies give predictable payouts.
Earthquake (Common)
Hits all houses at once → risk doesn't shrink with N.
Law of Large Numbers: With N independent risks of SD σ, average has SD = σ/√N. For common risks: SD stays at σ.
| Type | Diversifiable? | Example |
|---|---|---|
| Independent | ✅ Yes | Fires, theft, firm news |
| Common | ❌ No | Recession, rates, pandemic |
✏️ Quick Check
Why is theft insurance cheaper than earthquake insurance?
Theft is independent → diversifiable. Earthquake is common → not diversifiable, demands big premium.
10.6 Diversification in Stock Portfolios
Key Insight
Diversification eliminates firm-specific risk but not systematic risk. The market pays you only for risk you can't avoid.
Firm-Specific
Idiosyncratic, unsystematic, diversifiable. CEO leaves, product recall, lawsuit.
Systematic
Market, non-diversifiable. Recession, inflation, rates.
No-Arbitrage: If firm-specific risk paid a premium, diversify and earn it free. So premium = 0. Only systematic risk earns return.
How Volatility Falls as You Add Stocks
What the figure shows: a single stock is very volatile. Adding more stocks rapidly cancels the firm-specific portion of risk — most of the benefit is captured within the first 20–30 names. But the curve flattens onto a floor it can never cross: the systematic (market) risk every stock shares. That floor is exactly the risk the CAPM prices.
✏️ Quick Check
Which is systematic: (a) factory fire, (b) Fed raises rates, (c) CEO resigns, (d) product recall?
(b) — interest-rate moves affect every firm. The others are firm-specific and diversify away.
✏️ Quick Check
Why does firm-specific risk earn zero premium?
Investors can eliminate it for free by diversifying. If it paid a premium, arbitrage would compete that premium away to zero.
✏️ Quick Check
Roughly how many stocks capture most of the diversification benefit?
About 20–30. Beyond that the curve is nearly flat — extra names barely lower risk because only the un-removable systematic floor remains.
10.7 Measuring Systematic Risk: Beta (β)
Key Insight
Beta measures a stock's sensitivity to overall market moves — its systematic risk.
Market Portfolio: Value-weighted combination of all risky assets. Proxy: S&P 500. Contains only systematic risk.
Beta
βi = Cov(Ri, RMkt) / Var(RMkt)
| β | Meaning | Example |
|---|---|---|
| 0 | No systematic risk | Risk-free |
| 0–1 | Defensive (dampens) | Hormel 0.46, Coca-Cola 0.65 |
| 1 | Moves with market | S&P 500 |
| >1 | Cyclical (amplifies) | Apple 1.21, Tesla 1.66 |
Beta = Slope of the Best-Fit Line
What the figure shows: plot the stock's return (y) against the market's return (x) for many periods. The slope of the best-fit line through that cloud is beta. Slope > 1 (drawn here) → the stock amplifies market moves (cyclical); slope < 1 → it dampens them (defensive); slope = 1 → it moves one-for-one with the market. The scatter around the line is firm-specific risk and does not affect β.
✏️ Quick Check
β=1.5. Market rises 4%. Stock's expected systematic move?
1.5 × 4% = +6%
✏️ Quick Check
Two stocks, SD=30% each. βA=1.5, βB=0.5. Higher expected return?
A. More systematic risk earns a premium. B's volatility is mostly diversifiable noise.
10.8 Beta and the Cost of Capital — CAPM
Key Insight
CAPM: expected return = risk-free rate + β × market risk premium.
Market Risk Premium
MRP = E[RMkt] − rf
CAPM
ri = rf + βi × (E[RMkt] − rf)
AMD example: β=1.51, rf=3%, MRP=5% → r = 3% + 1.51×5% = 10.55%
The Security Market Line (SML)
What the figure shows: the CAPM is a straight line — the SML. The intercept is the risk-free rate (β=0); the slope is the market risk premium. Every asset's required return is read off this line from its β alone. Total volatility never appears — only systematic risk (β) is priced.
✏️ Quick Check
rf=4%, β=1.5, MRP=6%. Expected return?
4% + 1.5×6% = 13%
✏️ Quick Check
rf=2%, E[RMkt]=10%, β=0.8. r?
2% + 0.8×8% = 8.4%
✏️ Quick Check
β=0. r?
r = rf. No systematic risk → no premium.
🧮 CAPM Calculator
📐 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 = (Div + Pt+1)/Pt − 1
Historical Average
R̄ = (1/T) Σ Rt
Sample Variance
Var = [1/(T−1)] Σ (Rt − R̄)²
Standard Error
SE = SD / √T
95% CI
R̄ ± 2·SE
Excess Return
R̄ − rf
Beta
βi = Cov(Ri, RMkt) / Var(RMkt)
CAPM
ri = rf + βi × (E[RMkt] − rf)
🃏 Flashcards — Chapter 10
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Chapter 11 — Optimal Portfolio Choice & the CAPM
Core Question
Given many risky assets, which combination should investors hold? Chapter 11 builds the efficient frontier, finds the tangent portfolio, and derives the CAPM from first principles.
The Storyline
Step 1: Compute portfolio expected return (easy — weighted average) and variance (harder — depends on covariances).
Step 2: Diversification works because returns are imperfectly correlated.
Step 3: Plot all possible portfolios → identify the efficient frontier.
Step 4: Add a risk-free asset → the best portfolio is the tangent portfolio, measured by the Sharpe ratio.
Step 5: If everyone holds the tangent portfolio, it must equal the market portfolio. CAPM is born.
Step 2: Diversification works because returns are imperfectly correlated.
Step 3: Plot all possible portfolios → identify the efficient frontier.
Step 4: Add a risk-free asset → the best portfolio is the tangent portfolio, measured by the Sharpe ratio.
Step 5: If everyone holds the tangent portfolio, it must equal the market portfolio. CAPM is born.
Navigate
💼 11.1 Portfolio Expected Return
Weights · weighted-average return
📈 11.2 Two-Stock Volatility
Covariance · correlation
🌐 11.3 Large Portfolios
Variance with many stocks
🏆 11.4 Efficient Frontier
Best risk-return combos
🏦 11.5 Risk-Free Borrowing
Tangent · Sharpe ratio
⚙️ 11.6 Required Returns
Beta of a portfolio
🎯 11.7 CAPM
Derivation · assumptions
📊 11.8 Security Market Line
CAPM in pictures
11.1 The Expected Return of a Portfolio
Key Insight
A portfolio's expected return is just the weighted average of the assets it holds. Easy.
Portfolio Weights
Weight of asset i
xi = Value of investment i / Total value of portfolio
Weights always sum to 1: Σ xi = 1. Positive = long, negative = short.
Portfolio Return
Realized Return
RP = Σ xi Ri
Expected Portfolio Return
E[RP] = Σ xi E[Ri]
Worked Example
200 shares of Dolby at $30 + 100 shares of Coca-Cola at $40.
Total = $6,000 + $4,000 = $10,000. Weights: xDol = 60%, xCoke = 40%.
If E[RDol] = 10% and E[RCoke] = 7%:
E[RP] = 0.60(10%) + 0.40(7%) = 8.8%
Total = $6,000 + $4,000 = $10,000. Weights: xDol = 60%, xCoke = 40%.
If E[RDol] = 10% and E[RCoke] = 7%:
E[RP] = 0.60(10%) + 0.40(7%) = 8.8%
✏️ Quick Check — Math
50% in A (E[R]=14%) and 50% in B (E[R]=8%). E[RP]?
0.5(14%) + 0.5(8%) = 11%
✏️ Quick Check — Theory
Why is portfolio expected return the weighted average, but variance is NOT?
Expectation is linear — averages of averages work. Variance involves squared deviations and covariances between assets, so it depends on how returns co-move, not just individual variances.
11.2 The Volatility of a Two-Stock Portfolio
Key Insight
Diversification reduces risk because stock returns aren't perfectly correlated. The math is covariance and correlation.
Covariance
Covariance
Cov(Ri, Rj) = E[(Ri − E[Ri])(Rj − E[Rj])]
Sign of covariance = direction of co-movement. Magnitude is in return-units² → hard to interpret.
Correlation (Standardized Covariance)
Correlation
Corr(Ri, Rj) = Cov(Ri, Rj) / [SD(Ri) · SD(Rj)]
Always in [−1, +1]:
• +1 = perfectly positively correlated (move together)
• 0 = uncorrelated
• −1 = perfectly negatively correlated (perfect hedge)
• +1 = perfectly positively correlated (move together)
• 0 = uncorrelated
• −1 = perfectly negatively correlated (perfect hedge)
Two-Stock Portfolio Variance
Var of 2-Stock Portfolio
Var(RP) = x₁² SD(R₁)² + x₂² SD(R₂)² + 2 x₁ x₂ Cov(R₁, R₂)
Using correlation: Var(RP) = x₁²σ₁² + x₂²σ₂² + 2 x₁ x₂ ρ σ₁ σ₂
Example: Correlation Matters
Two stocks, SD = 20% each, equal weights:
• ρ = +1.0 → Portfolio SD = 20% (no benefit)
• ρ = 0.0 → Portfolio SD = 14.1%
• ρ = −1.0 → Portfolio SD = 0% (perfect hedge!)
Lower correlation → bigger diversification gain.
• ρ = +1.0 → Portfolio SD = 20% (no benefit)
• ρ = 0.0 → Portfolio SD = 14.1%
• ρ = −1.0 → Portfolio SD = 0% (perfect hedge!)
Lower correlation → bigger diversification gain.
How Correlation Bends the Combination Curve
What the figure shows: each curve traces every mix of two stocks as the weight changes. At ρ=+1 the set is a straight line — no risk reduction. As correlation falls, the curve bows left, meaning you can reach the same expected return at lower volatility. At ρ=−1 the curve touches the vertical axis: a particular mix has zero volatility — a perfect hedge. This curvature is the whole engine of diversification.
✏️ Quick Check — Theory
What does Corr = +1 mean for diversification benefit?
No diversification benefit. Portfolio SD is just the weighted average of individual SDs. Returns move in lockstep.
✏️ Quick Check — Math
x₁=x₂=0.5, σ₁=σ₂=30%, ρ=0.3. Portfolio SD?
Var = 0.25(0.09) + 0.25(0.09) + 2(0.5)(0.5)(0.3)(0.3)(0.3) = 0.0225 + 0.0225 + 0.0135 = 0.0585. SD = 24.2%
🧮 Two-Stock Portfolio SD Calculator
11.3 Volatility of a Large Portfolio
Key Insight
As N grows, individual variances become negligible — only average covariance remains. That's the systematic risk floor.
Equally Weighted Portfolio of N Stocks
Variance Decomposition
Var(RP) = (1/N) × (Avg Variance) + (1 − 1/N) × (Avg Covariance)
As N → ∞:
• First term → 0 (firm-specific risk vanishes)
• Second term → Average Covariance (systematic floor)
The portfolio's volatility cannot fall below the systematic risk built into the market.
• First term → 0 (firm-specific risk vanishes)
• Second term → Average Covariance (systematic floor)
The portfolio's volatility cannot fall below the systematic risk built into the market.
Volatility vs Number of Stocks
What the figure shows: the variance formula has two terms. The (1/N)·Avg Variance term shrinks toward zero as you add stocks (diversifiable, firm-specific risk). The Avg Covariance term does not — it is the irreducible systematic floor the curve flattens onto. You can never diversify below √(Avg Cov).
What This Tells Us
Most Gains by N≈20
Bulk of diversification happens with first 20–30 stocks
Floor Exists
Can't diversify below systematic risk
Low Corr ↑ Benefit
Lower average correlation → lower floor
✏️ Quick Check — Theory
Why can't diversification eliminate all risk in a stock portfolio?
Stocks share common (systematic) risk — recessions, rates, inflation hit them all. Average covariance > 0, so as N grows, portfolio variance approaches that floor, not zero.
✏️ Quick Check — Math
N stocks, equal weights, each with SD=40%, all pairwise correlation = 0. Portfolio SD as N grows?
Avg cov = 0. Var = (1/N) × (0.16). As N→∞, Var → 0. SD → 0. Independent risks alone can be fully diversified.
11.4 Risk vs Return: Choosing an Efficient Portfolio
Key Insight
For each level of risk, one portfolio has the highest expected return. The set of these is the efficient frontier.
Definitions
Efficient Portfolio
No other portfolio has both higher E[R] and lower (or equal) SD.
Inefficient Portfolio
Dominated — you can find another with higher E[R] for the same or lower risk.
Building the Frontier (Two-Stock Example)
Vary the weight x₁ from 0 to 1 → trace a curve in (SD, E[R]) space. The upper part of that curve is efficient. Adding short sales (negative weights) extends it further.
✏️ Quick Check — Theory
An investor finds a portfolio with the same SD as theirs but a higher E[R]. What can we conclude?
Their current portfolio is inefficient — dominated by the alternative. They should switch.
✏️ Quick Check — Theory
If correlation between two stocks decreases, what happens to the efficient frontier?
It curves further left — more diversification benefit, lower minimum volatility achievable.
11.5 Risk-Free Saving and Borrowing — The Tangent Portfolio
Key Insight
Add a risk-free asset. The best risky portfolio is now the one with the highest Sharpe ratio — the tangent portfolio. Every investor should hold it.
The Sharpe Ratio
Sharpe Ratio
Sharpe = (E[RP] − rf) / SD(RP)
Excess return per unit of risk. The portfolio with the highest Sharpe is the one you'd combine with the risk-free asset.
Combining With Risk-Free
If you invest fraction y in a risky portfolio P (and 1−y in rf):
• E[Rcombo] = rf + y(E[RP] − rf)
• SD(Rcombo) = y × SD(RP)
→ A straight line from rf through P. Slope = Sharpe ratio of P. You want the steepest line.
• E[Rcombo] = rf + y(E[RP] − rf)
• SD(Rcombo) = y × SD(RP)
→ A straight line from rf through P. Slope = Sharpe ratio of P. You want the steepest line.
The Tangent Portfolio
The single risky portfolio that, when combined with rf, gives the steepest line is called the Tangent Portfolio. Every rational investor holds this same risky mix, scaled up or down by their risk tolerance using rf.
✏️ Quick Check — Math
Portfolio: E[R]=12%, SD=20%, rf=2%. Sharpe ratio?
(12 − 2) / 20 = 0.50
✏️ Quick Check — Theory
Why does every investor hold the same risky portfolio (the tangent) regardless of risk tolerance?
Because the tangent gives the maximum Sharpe ratio — best return per unit of risk. To adjust risk, you slide along the line using rf (lend or borrow), not by changing the risky mix.
11.6 The Efficient Portfolio and Required Returns
Key Insight
An asset's required return depends on how much it contributes to the risk of the efficient portfolio — captured by its beta with that portfolio.
Beta With Respect to a Portfolio P
Beta of stock i relative to portfolio P
βiP = Cov(Ri, RP) / Var(RP)
Measures how much risk asset i adds to portfolio P. High βP → adds a lot of risk → must earn high return to justify inclusion.
Required Return on Asset i
Required Return given efficient portfolio P
ri = rf + βiP × (E[RP] − rf)
Beta of a Portfolio
Portfolio Beta = Weighted Average of Betas
βP = Σ xi βi
Practical: The beta of a portfolio is just the weighted average of the betas of its components. Useful for computing the beta of an index fund or a mutual fund.
✏️ Quick Check — Math
Portfolio: 30% in A (β=0.8), 70% in B (β=1.4). Portfolio beta?
0.3(0.8) + 0.7(1.4) = 0.24 + 0.98 = 1.22
✏️ Quick Check — Theory
Why does a stock's expected return depend on its β with the efficient portfolio, rather than its total volatility?
Because once you hold the efficient portfolio, the only risk that matters is the marginal contribution — measured by covariance with that portfolio, i.e. βP. Volatility uncorrelated with P is diversified away.
11.7 The Capital Asset Pricing Model (CAPM)
Key Insight
If all investors hold the same tangent portfolio, in equilibrium it must equal the market portfolio. Then the formula for required return becomes the famous CAPM.
CAPM's Three Assumptions
| # | Assumption |
|---|---|
| 1 | Investors can buy/sell any security at competitive market prices (no taxes/fees) and borrow/lend at the risk-free rate. |
| 2 | Investors hold only efficient portfolios of risky assets — those that give max E[R] for a given SD. |
| 3 | Investors have homogeneous expectations: they agree on the volatilities, correlations, and expected returns. |
From Assumptions to the Market Portfolio
If everyone has the same beliefs and everyone holds the tangent portfolio, then the aggregate of all investor demand equals the market portfolio (every risky asset, value-weighted). So Tangent Portfolio = Market Portfolio.
The CAPM Formula
CAPM
E[Ri] = rf + βi × (E[RMkt] − rf)
where βi = Cov(Ri, RMkt) / Var(RMkt) — the market beta, since the market is the efficient portfolio.
The Capital Market Line (CML)
CML: the straight line from rf through the market portfolio in (SD, E[R]) space. Optimal portfolios for any risk tolerance lie on this line. Slope = Sharpe ratio of the market.
✏️ Quick Check — Theory
What's the role of the "homogeneous expectations" assumption?
It ensures all investors compute the same tangent portfolio. In equilibrium, that common tangent must equal the market portfolio (aggregate of all holdings).
✏️ Quick Check — Theory
What's the difference between the Capital Market Line and the Security Market Line?
CML: E[R] vs SD, only for efficient portfolios. SML: E[R] vs β, applies to every asset.
11.8 Determining the Risk Premium & the SML
Key Insight
The Security Market Line (SML) plots required return vs beta. Every asset lies on it — that's CAPM.
The Security Market Line (SML)
Plot β on the x-axis and E[R] on the y-axis. The SML is a straight line:
• Intercept = rf
• Slope = Market Risk Premium = E[RMkt] − rf
Under CAPM, every fairly-priced asset plots on the SML. Above → underpriced (high return for its risk). Below → overpriced.
• Intercept = rf
• Slope = Market Risk Premium = E[RMkt] − rf
Under CAPM, every fairly-priced asset plots on the SML. Above → underpriced (high return for its risk). Below → overpriced.
Beta of a Portfolio (Recap)
Portfolio Beta
βP = Σ xi βi
Portfolio Expected Return via CAPM
E[RP] = rf + βP × (E[RMkt] − rf)
Summary of the CAPM
The three big takeaways:
1. Only systematic risk (β) earns a premium.
2. The efficient portfolio = the market portfolio (under CAPM assumptions).
3. Every asset's required return lies on the SML.
1. Only systematic risk (β) earns a premium.
2. The efficient portfolio = the market portfolio (under CAPM assumptions).
3. Every asset's required return lies on the SML.
✏️ Quick Check — Math
rf=3%, E[RMkt]=9%. A stock has β=1.4. Its required return?
3% + 1.4 × (9% − 3%) = 3% + 8.4% = 11.4%
✏️ Quick Check — Theory
A stock plots above the SML. What does that mean?
It's offering a higher return than CAPM requires for its β → underpriced. Investors will buy it, pushing the price up (and the return down) until it lies on the SML.
✏️ Quick Check — Math
Portfolio: 40% in stock A (β=0.6), 60% in stock B (β=1.5). rf=2%, MRP=6%. E[RP]?
βP = 0.4(0.6) + 0.6(1.5) = 0.24 + 0.90 = 1.14. E[RP] = 2% + 1.14 × 6% = 8.84%
📐 Formulas — Chapter 11
Portfolio Weights
xi = Valuei / Total Value | Σ xi = 1
Portfolio Return
RP = Σ xi Ri
Expected Portfolio Return
E[RP] = Σ xi E[Ri]
Covariance
Cov(Ri, Rj) = E[(Ri − E[Ri])(Rj − E[Rj])]
Correlation
ρ = Cov(Ri, Rj) / (σi σj) ∈ [−1, +1]
Two-Stock Portfolio Variance
Var(RP) = x₁²σ₁² + x₂²σ₂² + 2 x₁ x₂ ρ σ₁ σ₂
Large Portfolio (Equal-Weight)
Var(RP) → Avg Covariance as N → ∞
Sharpe Ratio
Sharpe = (E[RP] − rf) / SD(RP)
Beta of Asset i vs Portfolio P
βiP = Cov(Ri, RP) / Var(RP)
Portfolio Beta
βP = Σ xi βi
CAPM
E[Ri] = rf + βi × (E[RMkt] − rf)
🃏 Flashcards — Chapter 11
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Chapter 12 — Estimating the Cost of Capital
Core Question
Chapters 10–11 told us that the CAPM determines the cost of capital. Chapter 12 is the hands-on counterpart: how do you actually produce the three numbers — rf, the market risk premium, and β — and combine them into a usable discount rate?
The Three Inputs You Must Estimate
| Input | What it is | Where it comes from (Ch 12) |
|---|---|---|
| rf | Risk-free rate | §12.2 — a Treasury yield matched to the horizon |
| E[RMkt] − rf | Market risk premium | §12.2 — historical average or forward-looking DDM |
| βi | Stock's systematic-risk loading | §12.3 — linear regression of excess returns |
Get those three, plug into the CAPM, and you have the equity cost of capital — the rate used to discount a project's equity cash flows.
Navigate
📈 12.1 Equity Cost of Capital
CAPM as a recipe · the Disney worked example
🌐 12.2 The Market Portfolio
Value-weighting · indexes · rf · the MRP debate
β 12.3 Beta Estimation
Regression · slope = β · intercept = α · R²
12.1 The Equity Cost of Capital
Key Insight
The cost of capital of any investment is the expected return of the best available alternative with the same risk. The CAPM makes "same risk" precise: same beta ⇒ same expected return.
Why a Cost of Capital Exists at All
When a firm invests, its shareholders give up the chance to invest that money elsewhere. The return they sacrifice on an equally-risky alternative is the project's opportunity cost of capital. If the project can't beat that alternative, shareholders are worse off — so it becomes the hurdle rate (the discount rate) for the project's cash flows.
The CAPM's contribution is to answer the hard part — "what return do equally-risky alternatives earn?" — using a single, observable risk measure: beta.
The CAPM Equation (the Security Market Line)
CAPM / Security Market Line
ri = rf + βi × ( E[RMkt] − rf )
The three ingredientswhat you must estimate
rf — risk-free Treasury yield
E[RMkt] − rf — market risk premium
βi — sensitivity to the market
E[RMkt] − rf — market risk premium
βi — sensitivity to the market
The intuitionwhat each does
rf is the floor — even zero-risk money earns this
MRP is the price of one unit of market risk
β is how many units of that risk you bear
MRP is the price of one unit of market risk
β is how many units of that risk you bear
The Security Market Line, Drawn
Takeaway: The SML is a straight line — required return rises linearly with beta. Intercept = rf (β = 0); slope = the market risk premium. Every correctly-priced asset lies on this line.
Worked Example — Disney's Equity Cost of Capital
Inputs: rf = 3%, E[RMkt] = 8% ⇒ MRP = 5%, βDisney = 1.29.
Step 1 — risk premium for Disney: β × MRP = 1.29 × 5% = 6.45%
Step 2 — add the risk-free floor: 3% + 6.45% = 9.45%
So Disney's equity cost of capital is 9.45%. This is the discount rate for Disney's equity cash flows — and for any new Disney project with the same business risk as the firm overall.
Step 1 — risk premium for Disney: β × MRP = 1.29 × 5% = 6.45%
Step 2 — add the risk-free floor: 3% + 6.45% = 9.45%
So Disney's equity cost of capital is 9.45%. This is the discount rate for Disney's equity cash flows — and for any new Disney project with the same business risk as the firm overall.
Why Beta — and Not Total Volatility?
From Chapters 10–11: investors hold diversified portfolios, so they're only compensated for risk they cannot diversify away — systematic risk, captured by β. A stock with huge total volatility but low β is mostly firm-specific noise; the market pays no premium for that noise. Cost of capital depends on β, never on total SD.
✏️ Quick Check — Math
rf = 3%, market expected return = 8%, β = 1.29. Equity cost of capital?
rE = 3% + 1.29 × (8% − 3%) = 3% + 6.45% = 9.45%
✏️ Quick Check — Theory
Stock A: β = 1.5, total SD = 25%. Stock B: β = 0.7, total SD = 40%. Which has the higher CAPM cost of capital?
Stock A. CAPM depends only on β. A's higher β gives it a higher required return, despite B's larger total volatility (which is mostly diversifiable firm-specific risk).
✏️ Quick Check — Math
A defensive utility has β = 0.5. With rf = 2% and MRP = 6%, what is its cost of capital? Compare to a cyclical with β = 1.6.
Utility = 2% + 0.5×6% = 5.0%. Cyclical = 2% + 1.6×6% = 11.6%. The cyclical demands 6.6 pp more — the price of 3× the market-risk exposure.
🧮 CAPM Calculator
12.2 The Market Portfolio
Key Insight
To use the CAPM you need a concrete "market" and a number for the market risk premium. This section builds both — and shows why the choices matter.
Constructing the Market Portfolio
In theory the market portfolio holds every risky asset. Each security appears in proportion to its share of total market value:
Market Capitalization of stock i
MVi = (Shares Outstandingi) × (Price per Sharei)
Portfolio Weight (value-weighted)
xi = MVi / Σj MVj
Two equivalent definitions of value-weighting: (1) hold each stock in proportion to its market cap, or (2) hold the same fraction of every company's shares. Both give the identical portfolio, and both make it passive — when prices move, the weights re-adjust by themselves, so no trading is required.
Worked Example — A 3-Stock Value-Weighted Portfolio
| Stock | Price | Shares | Market Cap | Weight |
|---|---|---|---|---|
| General Electric | $30 | 2 M | $60 M | 50% |
| Home Depot | $20 | 1.5 M | $30 M | 25% |
| Cisco | $10 | 3 M | $30 M | 25% |
| Total | $120 M | 100% |
If GE's price doubles to $60, its market cap → $120M and its weight → 120/180 = 67% automatically. You never traded. That is what "passive" means.
Three Weighting Schemes — Don't Confuse Them
Value-weighted
Weight ∝ market cap. Passive. Used for CAPM. (S&P 500)
Price-weighted
Hold ONE share of each. High-price stocks dominate. (DJIA)
Equal-weighted
Same $ in each. Needs constant rebalancing — not passive.
Market Indexes & Proxies
Holding every risky asset on Earth is impossible, so we use a market proxy:
| Index | Covers | Weighting | Good proxy? |
|---|---|---|---|
| S&P 500 | 500 largest US stocks | Value-weighted | ✅ Standard for CAPM |
| Wilshire 5000 | ~entire US market | Value-weighted | ✅ Most complete |
| Dow Jones (DJIA) | Only 30 stocks | Price-weighted | ❌ Poor proxy |
| ETFs (SPY, VTI) | Track an index 1:1 | Inherit the index | ✅ Cheap "buy the market" |
Common mistake: the DJIA is the index quoted on the news, but it's price-weighted — a $300 stock gets 30× the weight of a $10 stock regardless of company size. Don't use it as your CAPM market proxy.
The Market Risk Premium (MRP)
Market Risk Premium
MRP = E[RMkt] − rf
This is the reward for bearing one unit of market risk — the slope of the SML. Estimating it well matters as much as estimating β.
Choosing the Risk-Free Rate
Use a Treasury yield whose maturity matches the project horizon:
• short projects (≈1 yr) → T-bill yields
• long projects (5–30 yr) → Treasury bond yields (most common in capital budgeting)
• some firms average the two.
• short projects (≈1 yr) → T-bill yields
• long projects (5–30 yr) → Treasury bond yields (most common in capital budgeting)
• some firms average the two.
Approach 1 — Historical Average
Average the market's realized excess return over the risk-free rate. The horizon you pick changes the number:
| Period | Excess vs 1-yr T-bills | Excess vs 10-yr T-bonds |
|---|---|---|
| 1926–2015 | 7.7% | 5.9% |
| 1965–2015 | 5.0% | 3.9% |
The catch: even with 90 years of data the standard error is ≈ 2% (a 95% CI of ±4%). And it's backward-looking. Many practitioners now use 5–6% as a forward-looking estimate.
Approach 2 — Fundamental (Dividend-Discount) Method
Forward-Looking E[RMkt] (constant-growth DDM)
E[RMkt] = (Div1 / P0) + g = Dividend Yield + Expected Growth
Plug the current market dividend yield and a long-run dividend-growth estimate into the constant-growth model, then subtract rf for the MRP. It's forward-looking but very sensitive to the assumed g — best used as a cross-check on the historical figure.
✏️ Quick Check — Theory
Why is the S&P 500 a better market proxy than the DJIA?
Two reasons: (1) 500 stocks vs only 30 — far broader; (2) value-weighted (mimics the true market portfolio) vs price-weighted (an artifact of the DJIA's 1896 design that over-weights high-priced shares).
✏️ Quick Check — Math
Three stocks have market caps $80M, $120M, $200M. What weight does each get in the value-weighted portfolio?
Total $400M. A = 80/400 = 20%, B = 120/400 = 30%, C = 200/400 = 50%.
✏️ Quick Check — Math
Market dividend yield = 2%, expected long-run growth = 5%, 10-yr Treasury = 3%. Implied MRP via the fundamental approach?
E[RMkt] ≈ 2% + 5% = 7%. MRP = 7% − 3% = 4%.
✏️ Quick Check — Theory
You're valuing a 25-year project. Which Treasury yield should you use for rf, and why?
A long-term (20–30 yr) Treasury bond yield — the rf horizon should match the project's horizon. Using a 3-month T-bill would understate rf when the yield curve slopes upward.
12.3 Beta Estimation
Key Insight
Beta is the slope of the best-fitting line through a scatter of the stock's excess returns against the market's excess returns. We estimate it from historical data, assuming β is reasonably stable over time.
Step 1 — Do the Returns Move Together?
Before computing anything, look at the data. Plotting Callaway Golf's weekly returns against the S&P 500 (2017–2021) shows they move in the same direction — but Callaway swings far more (its weekly SD ≈ 51% vs the S&P's ≈ 19%). That extra amplitude is exactly what β captures.
Takeaway: same direction, bigger amplitude. β quantifies "how much bigger".
Step 2 — Fit the Best Line (Linear Regression)
Plot each week's excess returns (return − rf) of the stock against the market's, and fit the line that minimises the squared distances to the points:
Regression Equation
(Ri − rf) = αi + βi(RMkt − rf) + εi
β = slope
% change in stock excess return per 1% change in market excess return
α = intercept
Average return above the CAPM prediction; should be ≈ 0
ε = residual
Firm-specific noise; averages to 0; uncompensated
Takeaway: the line's slope is beta — how much the stock moves per 1% market move. Slope > 1 amplifies the market, < 1 dampens it. The intercept is alpha.
Interpreting β
| β | Reading | Examples |
|---|---|---|
| 0 | No systematic risk | T-bills (~0) |
| 0 < β < 1 | Defensive — dampens market swings | Utilities, food, healthcare |
| 1 | Moves one-for-one with market | The market portfolio |
| > 1 | Aggressive / cyclical | Tech, autos, banks, small-caps |
| < 0 | Moves opposite the market (rare) | Some gold miners; hedges |
How Much Does β Explain? — R²
R² = the fraction of the stock's variance explained by the market. For Callaway, R² ≈ 0.36 → only ~36% of its weekly variance is market risk; the other ~64% is firm-specific noise. For a broad index, R² ≈ 1. A low R² doesn't make β wrong — it just means more uncertainty around the estimate.
Practical Caveats
β is estimated, not knownsample size matters
Typical practice: 2 years of weekly or 5 years of monthly returns. The 95% CI on β is often ±0.3 or wider — use the point estimate but remember the noise.
α is usually just noisepast ≠ future
A positive α one period rarely persists. CAPM treats α as zero in expectation — no risk-adjusted free lunch.
✏️ Quick Check — Math
A stock's excess return rises on average 2.4% for every 1% rise in the market's excess return. What is its beta?
β = 2.4 — the regression slope. A very aggressive/cyclical stock.
✏️ Quick Check — Theory
A stock's historical α is +2%. What does it mean — and should you bake it into the cost of capital?
It returned 2% more than CAPM predicted for its risk in the past. Don't add it to the cost of capital — α is rarely persistent and is usually within the standard error of zero. CAPM assumes α = 0 going forward.
✏️ Quick Check — Theory
Stock X has SD = 50% but R² = 0.10 against the market. What does that imply?
~90% of X's volatility is firm-specific (diversifiable) — the market won't compensate it. Its β is likely small; only the systematic 10% drives required return.
✏️ Quick Check — Math
Regression gives β = 0.8, α = +0.05%/week. 10-yr Treasury = 3%, MRP = 5%. CAPM cost of capital? Trust the α?
rE = 3% + 0.8 × 5% = 7%. The weekly α (~2.6%/yr) is almost certainly within the standard error of zero — treat it as noise, don't add it.
📐 Formulas — Chapter 12
CAPM / Security Market Line
ri = rf + βi × (E[RMkt] − rf)
Market Risk Premium
MRP = E[RMkt] − rf
Market Capitalization
MVi = Shares Outstanding × Price per Share
Value-Weight
xi = MVi / Σ MVj
Fundamental MRP (DDM)
E[RMkt] = (Div1/P0) + g
Beta Regression
(Ri − rf) = αi + βi(RMkt − rf) + εi
Beta (interpretation)
βi = ΔRiexcess / ΔRMktexcess (regression slope)
🃏 Flashcards — Chapter 12
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Chapter 13 — Investor Behavior & Capital Market Efficiency
Core Question
CAPM predicts every investor holds the market portfolio. Real investors over-trade, under-diversify, and chase trends. When does that actually move prices? Only when the biases are systematic, not random.
The Logical Chain of This Chapter
| § | Question | Answer |
|---|---|---|
| 13.1 | What does an efficient market imply? | Every stock's α = 0; competition enforces it |
| 13.2 | Can ordinary investors beat it? | Only with unique info + low costs — so be passive |
| 13.3 | What do real investors actually do? | Under-diversify & over-trade — hurting returns |
| 13.4 | Do their mistakes break the CAPM? | Only if the biases are systematic |
Navigate
⚖️ 13.1 Competition & Capital Markets
Alpha · SML deviations · competition kills α
📰 13.2 Information & Rational Expectations
Informed vs uninformed · why be passive
🙋 13.3 Behavior of Individual Investors
Under-diversification · overtrading · the data
🌀 13.4 Systematic Trading Biases
Disposition · attention · mood · herding
13.1 Competition and Capital Markets
Key Insight
A stock's alpha measures how far its expected return sits above the Security Market Line. Competition among investors drives every alpha to zero — and that is exactly the statement "the market portfolio is efficient".
Where This Chapter Sits
In Chapter 11 we derived the CAPM from the assumption that all investors hold efficient portfolios with homogeneous (identical) beliefs. That gave us a clean conclusion: the market portfolio is efficient and every stock plots on the Security Market Line. But that derivation rested on strong assumptions. Chapter 13 asks the realistic follow-up: what if investors don't behave that way? Do real investors actually drive prices to the CAPM ideal — and if not, can anyone profit from the gap? This first section sets up the analytical tool we need for the whole chapter: the alpha of a stock.
The Market Portfolio May Not Be Efficient
If the average investor does not hold the true market portfolio, the market portfolio need not lie on the efficient frontier — there could be another portfolio with the same volatility but a higher expected return. This is the situation drawn in the textbook's Figure 13.1 ("An Inefficient Market Portfolio"): the market sits inside the frontier, not on it.
Whenever the market portfolio is not efficient, some stocks must offer more expected return than the CAPM predicts (given their risk) and others less. Those gaps are exactly what alpha measures — and they are the opportunity that competition then attacks.
Defining Alpha
Recall the CAPM "required return" for stock i: the return justified by its systematic risk, ri = rf + βi(E[RMkt] − rf). The alpha is the difference between the return investors actually expect the stock to earn and this required return — i.e. its vertical distance from the Security Market Line:
Alpha = Expected Return − CAPM Required Return
αi = E[Ri] − ( rf + βi × (E[RMkt] − rf) )
α > 0
ABOVE the SML — return beats the risk-justified level — under-priced — buy
α = 0
ON the SML — fairly priced
α < 0
BELOW the SML — over-priced — sell / short
Takeaway: dots above the SML have positive α (buy); below, negative α (sell). In equilibrium every dot must sit exactly on the line.
Profiting from Non-Zero Alpha Stocks
Suppose you currently hold the market portfolio and you spot a stock with positive alpha. You can improve your portfolio's risk-return trade-off: buy more of the positive-alpha stock (and finance it by selling a bit of negative-alpha stocks, or by reducing your market holding). Because the positive-alpha stock offers more return than its beta requires, this tilt raises your expected return without a proportionate rise in risk. Every investor who notices the same opportunity does the same thing.
The price-adjustment mechanism: the collective buying of the positive-alpha stock pushes its price up. A higher price today means a lower expected return going forward, so the alpha shrinks. Symmetrically, collective selling of negative-alpha stocks pushes their prices down, raising their future expected return until that alpha rises back to zero. The process stops only when every alpha has been competed away.
Key equivalence: "no positive-alpha opportunities remain" is the same statement as "the market portfolio is efficient" and the same statement as "every stock lies on the SML (α = 0)". The three are logically identical — proving one proves all three. This is precisely how the competitive CAPM equilibrium of Chapter 11 arises from individual investors chasing alpha.
Trading on News or Recommendations
A natural question: "if a stock has a positive alpha, why can't I just buy it and profit?" The catch is timing. By the time public news or an analyst recommendation reaches you, thousands of other investors have already received it and traded — pushing the price up and competing the alpha away before you can act. To profit you would need information that is not already reflected in the price, which leads directly into §13.2.
Worked Example
rf = 4%, MRP = 5%. Two stocks:
• Green Leaf: E[R] = 12%, β = 1.50 → CAPM = 4 + 1.50×5 = 11.5% → α = 12 − 11.5 = +0.5% (buy)
• Hand-Mover: E[R] = 7%, β = 0.85 → CAPM = 4 + 0.85×5 = 8.25% → α = 7 − 8.25 = −1.25% (sell)
• Green Leaf: E[R] = 12%, β = 1.50 → CAPM = 4 + 1.50×5 = 11.5% → α = 12 − 11.5 = +0.5% (buy)
• Hand-Mover: E[R] = 7%, β = 0.85 → CAPM = 4 + 0.85×5 = 8.25% → α = 7 − 8.25 = −1.25% (sell)
✏️ Quick Check — Math
Stock: E[R] = 14%, β = 1.6, rf = 3%, MRP = 6%. Alpha?
CAPM = 3 + 1.6×6 = 12.6%. α = 14 − 12.6 = +1.4% → above the SML → buy.
✏️ Quick Check — Theory
In a perfectly competitive efficient market, what is the alpha of every stock?
Zero. Competition forces every stock onto the SML. "Market efficient" ⇔ "all α = 0".
✏️ Quick Check — Theory
A stock has α < 0. What does an investor do, and what happens to the price?
Sell / short. Selling pressure pushes the price down, which raises its expected return until α returns to 0.
🧮 Alpha Calculator
13.2 Information and Rational Expectations
Key Insight
§13.1 showed competition drives alpha to zero. §13.2 asks the deeper question: under what circumstances can a specific investor actually profit from a non-zero-alpha stock before it disappears? The answer turns on who has information.
The Setup: Who Trades Against Whom?
The CAPM was derived assuming homogeneous expectations — every investor agrees on every stock's expected return, volatility, and correlations. Reality is different: investors hold different information and interpret it differently. To analyse this we split the market into two groups.
Informed investorshave information not yet in prices
They can identify which stocks have non-zero alpha and trade on it. Their buying/selling is what moves prices toward fair value — they are the engine of price discovery.
Uninformed investorsonly know what's already public
They have no edge over the price. Any stock they think is "cheap" looks identically cheap to everyone — so it is not actually cheap.
The Crucial Asymmetry
Trading is a zero-sum game in gross terms: for every investor who buys a stock that subsequently outperforms, there is a counterparty on the other side who sold it and underperformed. So whenever an informed investor earns a positive alpha by exploiting their information, someone must be earning a negative alpha as the loser of that trade. The losers are, on average, the uninformed.
Implication: if you trade without superior information, you are systematically taking the other side of the informed investors' winning trades. You don't just "earn the average" — you earn the average minus the informed investors' gains minus your trading costs.
Rational Expectations
The textbook resolves this with the assumption of rational expectations: all investors correctly interpret and use the information contained in market prices, and they understand their own informational position. A rational uninformed investor reasons: "I have no edge. If I trade actively I will lose to the informed and pay costs. The one portfolio whose alpha I can be sure is zero is the market portfolio itself."
The self-reinforcing conclusion: rational uninformed investors choose to hold the market portfolio (or a passive index). Because they do, they are not mis-pricing anything — so the informed investors' profit opportunities come only from each other, and competition among the informed keeps prices efficient. Passive investing is the rational equilibrium response to not having information.
Example 13.1 — Avoiding Being Outsmarted
Suppose you have no information advantage but you decide to actively pick stocks anyway. You learn that other investors in the market possess a great deal of information you are missing. Result: on any trade you make, the better-informed counterparty has the edge. Over many trades you will, on average, capture the average market return minus what the informed extract minus your costs — i.e. you will underperform a simple buy-and-hold of the market portfolio. The lesson: without superior information, the best strategy is to not try to outsmart the market.
When Can the Uninformed Beat the Market?
Two conditions must BOTH hold:
1. You possess information that other investors do not have (a genuine, non-public edge).
2. That information is valuable enough to overcome transaction costs — commissions, bid-ask spreads, price impact, and taxes.
For the overwhelming majority of individual investors, neither condition is met — so the rational choice is passive, low-cost indexing.
1. You possess information that other investors do not have (a genuine, non-public edge).
2. That information is valuable enough to overcome transaction costs — commissions, bid-ask spreads, price impact, and taxes.
For the overwhelming majority of individual investors, neither condition is met — so the rational choice is passive, low-cost indexing.
Active Trading Is Negative-Sum After Costs
Aggregate over all investors: the gross alphas must sum to zero (every trade has two sides). Therefore the average actively-managed dollar earns the market return before costs and the market return minus costs after them. Active management as a whole is a negative-sum game; passive index funds avoid the drag entirely. This is the core practical takeaway of the section.
✏️ Quick Check — Theory
Why is holding the market portfolio the right strategy for an uninformed investor?
Without an information edge you can't reliably find positive-α stocks. The market portfolio is the one portfolio guaranteed to have α = 0 (it is the market). Active picking, after costs, is expected to lose to it.
✏️ Quick Check — Theory
A friend says he beat the market by 4% last year. What three questions should you ask before crediting him with skill?
(1) Risk-adjusted? 4% could just be β > 1 in a rising market. (2) After costs & taxes? Gross alpha often vanishes net. (3) How many years? One year is luck; skill needs a long track record.
✏️ Quick Check — Theory
You randomly buy 5 and sell 5 stocks daily. Versus buy-and-hold of the market, expected result?
Worse. Before costs, random trading averages the market (α = 0). After commissions, spreads, and taxes you systematically underperform.
13.3 The Behavior of Individual Investors
Key Insight
§13.2 concluded the rational uninformed investor should hold the market and trade rarely. §13.3 documents what individual investors actually do — and it's the opposite. The question for market efficiency: do these mistakes cancel out, or do they push prices?
1 — Under-Diversification
One of the most important results of Chapters 10–11 is that diversification reduces risk at no cost to expected return. Yet the evidence is striking: the typical individual investor is severely under-diversified. Studies of brokerage accounts find the median household holds only 3–4 individual stocks, and a large fraction hold just one or two. Three distinct behavioral explanations are offered.
Familiarity Bias
Investors over-weight investments they feel they know: their employer's stock, locally-headquartered companies, and well-known consumer brands. Familiarity feels like information, but it is not — recognising a company tells you nothing about whether its stock is mis-priced. The result is a concentrated, under-diversified portfolio. The extreme case is employer stock: an employee whose 401(k) is loaded with company shares has their human capital (salary, job security) and their financial capital exposed to the same single firm — a catastrophic correlation if the firm fails (e.g. Enron employees).
Relative Wealth Concerns
Investors care about their performance relative to their peers, not just in absolute terms. To track the same outcomes as their neighbours/colleagues, they tend to hold the same popular stocks the people around them hold. This herding into a common subset further concentrates portfolios and amplifies under-diversification across a whole community.
The cost of all three: firm-specific (idiosyncratic) risk is diversifiable, so the market pays no premium for bearing it. Under-diversified investors take on extra volatility and receive zero extra expected return in exchange — they are strictly worse off than a diversified investor with the same expected return.
2 — Excessive Trading & Overconfidence
Even setting diversification aside, individuals trade far too much. The overconfidence hypothesis explains why: people systematically overestimate their own ability to pick winners and the precision of their information. Believing they have an edge they don't actually have, they trade aggressively — and every trade incurs commissions, bid-ask spreads, and (for winners) taxes.
The textbook's Figure 13.3 shows U.S. stock-market annual share turnover climbing over 1970–2021 — investors collectively churn their portfolios many times over. Barber & Odean's landmark study of tens of thousands of brokerage accounts produced two now-classic findings:
- The more an investor traded, the worse they did. Sorting investors into turnover groups, the lowest-turnover group roughly matched the market (~17–18%/yr) while the highest-turnover group earned only ~7%/yr — after costs (Figure 13.4 below).
- Men trade ~45% more than women and, consistent with overconfidence, underperform them — single men are the worst of all.
Sensation Seeking
Beyond overconfidence, some investors trade for the thrill of it — "sensation seeking". Researchers find that individuals who engage in other sensation-seeking behaviors (e.g. more speeding tickets) also trade their portfolios more actively. Trading is partly entertainment, and entertainment has a cost: the underperformance documented above.
Takeaway: the highest-turnover group earns ~10 pp/year less than the lowest, and the market index beats every active group. Trading costs + overconfidence destroy returns.
3 — From Individual Behavior to Market Prices
Here is the key analytical point of the section. We have established that individuals make real mistakes (under-diversifying, over-trading). Does this mean the CAPM is wrong and prices are inefficient? Not necessarily — it depends entirely on whether the mistakes are correlated across investors.
The crucial distinction:
• If individual departures from the CAPM are random and uncorrelated — some investors over-buy a stock while others over-sell it — the mistakes cancel out in aggregate. The average investor still effectively holds the market portfolio, so prices remain efficient and the CAPM still describes them. The mistaken individuals hurt only themselves.
• If the departures are systematic — many investors making the same error in the same direction at the same time — the mistakes do not cancel. Aggregate demand is distorted, prices can be pushed away from fundamental value, and the CAPM can fail.
• If individual departures from the CAPM are random and uncorrelated — some investors over-buy a stock while others over-sell it — the mistakes cancel out in aggregate. The average investor still effectively holds the market portfolio, so prices remain efficient and the CAPM still describes them. The mistaken individuals hurt only themselves.
• If the departures are systematic — many investors making the same error in the same direction at the same time — the mistakes do not cancel. Aggregate demand is distorted, prices can be pushed away from fundamental value, and the CAPM can fail.
So the existence of behavioral mistakes is not automatically a problem for market efficiency. The real question is whether the biases are shared and correlated. That is precisely what §13.4 examines: the specific biases that are systematic enough to potentially move prices.
✏️ Quick Check — Theory
CAPM says hold the market portfolio. What does that imply about trading frequency?
Trade very rarely. A value-weighted market portfolio re-balances itself as prices move. The only necessary trades are when you add or withdraw savings — never for "stock picking".
✏️ Quick Check — Theory
Your aunt holds only her employer's stock "because she knows it well". Which principle is she violating, and what's the cost?
Diversification. She bears huge firm-specific risk — and it's correlated with her job income. The market pays no premium for this; it could be diversified away for free.
✏️ Quick Check — Theory
Why are RANDOM individual mistakes not a problem for the CAPM?
Independent random errors cancel in aggregate; the average investor still holds ≈ the market portfolio, so prices stay efficient. Only correlated (systematic) biases can move prices.
13.4 Systematic Trading Biases
Key Insight
§13.3 showed only systematic (correlated) biases can move prices. §13.4 catalogues the specific biases that are shared widely enough to matter — and then asks whether they actually break the CAPM in practice.
1 — The Disposition Effect (Hanging On to Losers)
The single most documented systematic bias. Investors have a strong tendency to hold on to stocks that have lost value and sell stocks that have risen — the opposite of what a rational, tax-aware investor should do. Shefrin and Statman named this the disposition effect.
The evidence (Odean): tracking thousands of accounts, Odean measured the proportion of gains realised versus the proportion of losses realised. Investors realised their gains at a ~50% higher rate than their losses — they sold winners eagerly and clung to losers. The psychological driver: selling a loser forces you to admit a mistake and "realise" the loss; holding lets you hope it recovers.
Why it is doubly costly:
• Tax: in most tax systems you should defer gains (defer the tax) and realise losses early (harvest the deduction). The disposition effect does the exact opposite — it accelerates the tax bill and forgoes the loss deduction.
• Capital allocation: it traps money in chronically under-performing stocks.
This bias is systematic — it pushes most investors the same way — so it is a candidate for actually affecting prices (it contributes to short-term return "momentum").
• Tax: in most tax systems you should defer gains (defer the tax) and realise losses early (harvest the deduction). The disposition effect does the exact opposite — it accelerates the tax bill and forgoes the loss deduction.
• Capital allocation: it traps money in chronically under-performing stocks.
This bias is systematic — it pushes most investors the same way — so it is a candidate for actually affecting prices (it contributes to short-term return "momentum").
2 — Investor Attention, Mood & Experience
Attention
Individuals are far more likely to buy stocks that grab attention — those in the news, with extreme one-day returns, or abnormally high volume. They can only buy from the few stocks they're watching, but can sell anything they own, so attention drives net buying.
Mood
Mood measurably shifts trading and prices: the "sunshine effect" — sunnier weather at a city's exchange correlates with higher returns; major losing-team sporting results depress the next day's local returns.
Experience
Investors over-weight their own personal experience: those who lived through strong markets invest more aggressively for life. Past personal returns shape risk-taking more than the full historical record should.
Each of these can be systematic: weather and famous sports outcomes hit many investors in a region simultaneously; the same attention-grabbing stocks are salient to everyone at once. That correlation is what gives them the potential to move prices.
3 — Herd Behavior
Investors make similar trading mistakes because they are actively imitating each other rather than acting on their own analysis. Two distinct mechanisms:
Information cascadesinferring info from others' actions
Each investor reasons "so many people are buying — they must know something I don't" and copies them, ignoring their own information. The crowd's behaviour snowballs even when it started from a tiny, possibly wrong, signal.
Relative-performance concernsprofessional fund managers
Managers are judged against peers/benchmarks. Being wrong alone is career-ending; being wrong with everyone else is forgivable. So they cluster their trades around the consensus — institutionalised herding.
Herding is the textbook example of a correlated bias: by construction every herder's trade moves the same way at the same time. That is exactly the condition (from §13.3) under which a bias can push prices far from fundamental value — and is a leading explanation for asset-price bubbles and crashes.
Implications of Behavioral Biases
Putting it together: behavioral biases can move prices, but only under specific conditions, and even then exploiting them is hard.
When prices ARE affected
A behavioral bias moves prices only if a large number of investors share it (it is systematic) and it is not offset by informed arbitrageurs.
Limits to arbitrage
To profit from others' mistakes you must take the other side — but arbitrage is risky and capital-constrained, so mis-pricings can persist longer than a trader can stay solvent.
Bottom line for the practitioner: behavioral biases mean the CAPM is not perfect and prices can drift from fundamental value, sometimes for extended periods. But for estimating the cost of capital and doing capital budgeting, the CAPM remains the best tractable model: it is simple, transparent, hard to manipulate, and errors in forecasting a project's cash flows almost always swamp any modest CAPM mis-pricing.
✏️ Quick Check — Theory
What is the disposition effect and why is it harmful?
Selling winners early and holding losers too long. It hurts after-tax returns (triggers gains tax, forgoes loss harvesting) and ties up capital in poor performers.
✏️ Quick Check — Theory
If investors are biased but RANDOMLY (some too optimistic, some too pessimistic), does CAPM still hold?
Yes. Random uncorrelated biases cancel in aggregate. CAPM only breaks when biases are systematic — many investors making the same error simultaneously.
✏️ Quick Check — Theory
Why is herd behavior especially dangerous for market efficiency?
Each investor's trade is correlated with everyone else's — they move together. That's the definition of a systematic bias: it doesn't cancel, so it can push prices far from fundamentals.
✏️ Quick Check — Theory
A small-cap rallies and dominates the news; retail investors pile in. Which bias is most at work?
Investor attention (reinforced by herding). Attention-grabbing stocks attract disproportionate retail buying, often briefly pushing price above fair value.
📐 Formulas — Chapter 13
Alpha of a Stock
αi = E[Ri] − ( rf + βi(E[RMkt] − rf) )
CAPM Required Return (reference)
riCAPM = rf + βi(E[RMkt] − rf)
Decision rules: α > 0 → buy · α < 0 → sell · α = 0 → fairly priced.
Market portfolio efficient ⇔ αi = 0 for every stock i.
Market portfolio efficient ⇔ αi = 0 for every stock i.
🃏 Flashcards — Chapter 13
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