第三章 风险与资本成本 — 知识点讲解

1. Core Idea

In the previous chapter we computed average returns and variances using historical data. Here we instead work with future possible returns and their probabilities. Two concepts do the work:

• Expected return, E(R): the probability-weighted average of all possible returns — what you earn "on average";
• Variance σ² / standard deviation σ: the volatility of returns — how far the actual outcome may deviate from the expectation.

Why would anyone hold Stock U (expected return 20%) when Stock L promises 25%? The answer must be risk — L's actual outcome could be much better or much worse.

2. Expected Return

Suppose there are only two states of the economy, boom and recession, each with probability 50%:

$$E(R_U) = .50 \times 30\% + .50 \times 10\% = 20\%$$

$$E(R_L) = .50 \times (−20\%) + .50 \times 70\% = 25\%$$

In general, with n possible returns:

$$E(R) = \sum_{i=1}^{n} p_i \times R_i$$

3. Risk Premium

Projected (expected) risk premium = expected return on the risky asset − risk-free rate.

With $R_f = 8\%$:

• Stock U: $20\% − 8\% = 12\%$
• Stock L: $25\% − 8\% = 17\%$

Unlike the historical (ex post) premium of Chapter 12, this is a forward-looking premium based on projected returns.

4. Variance and Standard Deviation

Three steps:

1. Compute the deviation in each state: $R_i − E(R)$
2. Square each deviation and multiply by the state's probability
3. Sum the products to get the variance; standard deviation = √variance

For Stock U (E(R) = 20%, two states at 50% each):

$$\sigma^2 = .50 \times (.30 − .20)^2 + .50 \times (.10 − .20)^2 = .50 \times .01 + .50 \times .01 = .01$$

$$\sigma = \sqrt{.01} = .10 = 10\%$$

Stock L's variance is far larger (deviations of ±.45):

$$\sigma_L = \sqrt{.2025} = .45 = 45\% \qquad \sigma_U = \sqrt{.01} = .10 = 10\%$$

5. Unequal Probabilities (Examples 13.1 & 13.2)

Suppose a boom occurs only 20% of the time (so recession = 80%), and $R_f = 10\%$. Probabilities must still sum to 1:

$$E(R_U) = .80 \times 30\% + .20 \times 10\% = 26\%$$

$$E(R_L) = .80 \times (−20\%) + .20 \times 70\% = −2\%$$

Risk premiums: U = $26\% − 10\% = 16\%$; L = $−2\% − 10\% = −12\%$. Expected returns and risk premiums can be negative — unusual, but possible.

Variances (Example 13.2):

$$\sigma^2_L = .80 \times (−.18)^2 + .20 \times (.72)^2 = .1296 \quad \Rightarrow \quad \sigma_L = 36\%$$

$$\sigma^2_U = .80 \times (.04)^2 + .20 \times (−.16)^2 = .0064 \quad \Rightarrow \quad \sigma_U = 8\%$$

6. Data Card: L vs U (Equal Probabilities)

L offers a higher expected return but much more risk; U always pays at least 10%. Which to buy depends on personal risk preferences — there is no single right answer.

7. Historical vs. Projected Approach

The formulas look alike, but here the probabilities are given inputs, closer to the real information set of an investor making a decision.

8. Three Key Takeaways

1. Expected return is not a guaranteed return — it is only a probability-weighted average; a single security's actual outcome can deviate dramatically (L ranges from −20% to +70%).
2. Return and risk must be viewed together — a 25% expectation is meaningless without the 45% standard deviation; the risk premium is the reward for bearing that risk.
3. Probabilities must sum to 1 — states are exhaustive and mutually exclusive; changing the probability assumptions changes both E(R) and σ², and E(R) can be negative.

1. Core Idea

Capital market history is the starting point for the risk–return trade-off: what returns did different assets earn historically, and how much variability (risk) did they carry? The data come from the classic Ibbotson–Sinquefield studies covering 94 years, 1926–2019, on five portfolios:

All returns are nominal, pretax (not adjusted for inflation or taxes); the year-to-year change in the CPI is recorded as the inflation measure, so real returns can be computed.

2. The Historical Record: What $1 Grew To

$1 invested at the end of 1925 grew to (Figure 12.4):

Small caps earned the most but were the most erratic (worst for about the first 10 years); T-bills were the steadiest but grew the slowest. Extreme single-year returns: small stocks +142.87% in 1933 (large-company +53.99% that year); T-bills 14.71% in 1981; long-term government bonds 40.36% in 1982; worst year −43.34% for large-company stocks in 1931 (2008's −37.00% second).

3. Average Returns and Risk Premiums: The First Lesson

Historical average return = sum of yearly returns ÷ 94. For large-company stocks the 94 returns sum to about 11.37, so 11.37/94 = .121, or 12.1% — the best guess for a randomly picked year.

Real returns ≈ nominal − inflation: T-bills ≈ 0.5%; small stocks ≈ 16.3% − 2.9% = 13.4% (Rule of 72: purchasing power doubles about every 5 years); large-company stocks ≈ 9.2%. Risk premium = excess return = average return minus the risk-free (T-bill) return: large-company = 12.1% − 3.4% = 8.7%; nominal and real premiums differ only slightly, so only nominal premiums are reported.

First lesson: Risky assets, on average, earn a risk premium — there is a reward for bearing risk.

4. Measuring Variability: Historical Variance and Standard Deviation

Historical variance = average squared deviation of actual returns from the average (divisor T − 1; deviations always sum to zero — a built-in check):

$$\text{Var}(R) = \sigma^2 = \frac{1}{T-1}\left[(R_1-\bar{R})^2 + \cdots + (R_T-\bar{R})^2\right], \qquad \text{SD}(R) = \sigma = \sqrt{\text{Var}(R)}$$

Worked example: returns 10%, 12%, 3%, −9% → average 4%; deviations .06, .08, −.01, −.13; squared sum = .0036+.0064+.0001+.0169 = .0270 → σ² = .027/3 = .009, σ = √.009 = 9.487%.

Example 12.2: Supertech average 17.5%, σ = 29.86%; Hyperdrive average 5.5%, σ = 13.28% — Supertech is more than twice as volatile.

For the full record of σ across assets, see the table in Section 6: small stocks σ = 31.5% vs. T-bills 3.1% — more than 10 times — return and variability go hand in hand.

5. The Normal Distribution and the Second Lesson

Return distributions are roughly mound-shaped and symmetric, so the normal distribution (fully described by average and σ) is a good approximation: within ±1σ ≈ 2/3; within ±2σ ≈ 95%; beyond 3σ < 1%.

Large-company stocks (12.1% ± 19.8%): about 2/3 probability of a return in −7.7% to 31.9%; about 5% chance outside −27.5% to 51.7% (±2σ) — one year in three falls outside the normal range. Example 12.3: small-company stocks (16.3% ± 31.5%), about 1/3 probability outside −15.2% to 47.8%; by symmetry a 1/6 chance of losing more than about 15% (once every six years) — not for investors who cannot afford the risk.

Using history to price risk: T-bills at about 0.2% in early 2020 imply a minimum required return of 0.2% + 8.7% = 8.9% for large-company risk, and 0.2% + 12.9% = 13.1% for small-company risk. 2008 in the record: the S&P 500 plunged 37% (485 of 500 stocks down), worse only than 1931 (−43%), Nov 2007–Mar 2009 cumulatively −50%; long-term Treasuries +40% that year — evidence that diversification matters. International check: 17 countries, 1900–2005, U.S. premium 7.4% ranked 8th (world average 7.1%).

Second lesson: The greater the potential reward, the greater is the risk.

6. Arithmetic versus Geometric Averages

Buy a stock at $100; it falls to $50 (−50%) then rises back to $100 (+100%): arithmetic average = (−50% + 100%)/2 = 25%; geometric average = 0%. Both are correct — the arithmetic average answers "what did you earn in an average year?" (single-year forecasts); the geometric answers "what was your average compound return per year?" (true wealth growth). Geometric ≤ arithmetic always.

$$\text{Geometric average} = \left[(1+R_1)(1+R_2)\cdots(1+R_T)\right]^{1/T} - 1$$

Four steps: ① add 1 → ② multiply → ③ raise to 1/T → ④ subtract 1. Example (10%, 12%, 3%, −9%): (1.10 × 1.12 × 1.03 × .91)^(1/4) − 1 = 3.66% < arithmetic 4%.

Example 12.4: S&P 500, 1926–1930 (11.62%, 37.49%, 43.61%, −8.42%, −24.90%): product = 1.5158 → 1.5158^(1/5) − 1 = 8.67% (equivalently PV = $1, FV = $1.5158, N = 5). Example 12.5: $1 → $9,243.90 over 94 years ⇒ 9243.90^(1/94) − 1 = 10.2%.

Approximation (in decimals): Geometric ≈ Arithmetic − ½ variance. Large-company stocks: .121 − .039/2 = .101 ≈ 10.2% (σ = .198, variance .198² ≈ .039).

The more volatile the investment, the bigger the gap between the geometric and arithmetic averages.

7. Which Average for Forecasting? — Blume's Formula

If you know the true arithmetic average, use it to forecast future wealth (arithmetic 10% ⇒ $1,000 for 10 years → $2,593.74); with only estimates, the arithmetic average is too high for long horizons (optimistic) and the geometric too low for short horizons (pessimistic).

Blume's formula (N years of data, forecasting a T-year average return, T < N):

$$R(T) = \frac{T-1}{N-1}\times\text{Geometric} + \frac{N-T}{N-1}\times\text{Arithmetic}$$

Example: N = 25, arithmetic 12%, geometric 9% → R(1) = 12%, R(5) = (4/24)×9% + (20/24)×12% = 11.5%, R(10) = (9/24)×9% + (15/24)×12% = 10.875%. Practical rule: up to about a decade → arithmetic; a few decades (retirement planning) → split the difference; many decades → geometric. Unless stated otherwise, "average return" means the arithmetic average.

8. Three Key Takeaways

1. Risk is rewarded, and greater reward means greater risk — Lesson 1: risky assets earn a risk premium on average (8.7% large-company, 12.9% small-company); Lesson 2: potential reward rises with risk.
2. Risk is measured by the standard deviation — historical variance = sum of squared deviations ÷ (T − 1), σ its square root; under a normal approximation ±1σ covers about 2/3 of outcomes; small stocks σ = 31.5% vs. T-bills 3.1%.
3. Arithmetic for single-year forecasts, geometric for long-run compounding — geometric ≤ arithmetic (gap grows with volatility); forecast T-year returns with Blume's formula as a weighted blend.

Tags: 考点

1. Core Idea

Most investors hold a portfolio of assets rather than a single stock. §13.2 builds a two-asset portfolio from a high-risk stock (L, expected return 25%, std. dev. 45%) and a low-risk stock (U, expected return 20%, std. dev. 10%), establishing the two central conclusions of this section:

• The portfolio expected return is a weighted average of the assets' expected returns (simple, always true)
• The portfolio variance is NOT a weighted average of the assets' variances (not simple — the key exam point)

Portfolio variance must be computed state by state: first find the portfolio return in each state of the economy, then compute the expected value and variance of those state returns.

2. Portfolio Weights

The most convenient way to describe a portfolio is to list the percentage of total portfolio value invested in each asset — the portfolio weight x. With $50 in one asset and $150 in another, total value $200:

• $x_1 = \$50/\$200 = .25$, $x_2 = \$150/\$200 = .75$
• The weights must sum to 1.00, because all of your money is invested somewhere

Here the weights are .25 and .75; an equally weighted portfolio (half in each) uses weights .50 and .50.

3. Portfolio Return in Each State

With half your money in L and half in U (weights .50/.50) and only two states of the economy — recession and boom, each with probability .50:

The portfolio return in a state is the weighted average of the assets' returns in that state, using the portfolio weights.

4. Portfolio Expected Return

Two routes lead to the same answer, 22.5%:

Route 1 (state method): E(R_P) = .50 × 5% + .50 × 40% = 22.5%

Route 2 (weighted average): E(R_P) = .50 × E(R_L) + .50 × E(R_U) = .50 × 25% + .50 × 20% = 22.5%

Generalizing to n assets: E(R_P) = x₁E(R₁) + x₂E(R₂) + … + xₙE(Rₙ). The portfolio expected return is a linear combination of the assets' expected returns — this holds no matter what.

5. Portfolio Variance: The State Decomposition Method (Key Point)

Simple intuition suggests σ_P = .50 × 45% + .50 × 10% = 27.5% — the textbook states plainly: this approach is completely incorrect! The correct method is state decomposition: first compute the portfolio return in each state (5% and 40%), then apply the variance definition:

$$\sigma_P^2 = .030625, \qquad \sigma_P = \sqrt{.030625} = .175, \text{ or 17.5%}$$

Note 17.5% < 27.5%: the portfolio standard deviation is below the weighted average of the individual standard deviations — the portfolio dilutes risk. The variance on a portfolio is not generally a simple combination of the variances of the assets in the portfolio.

6. Risk Offsetting and a Zero-Variance Portfolio

Try different weights: 2/11 (about 18%) in L and 9/11 (about 82%) in U:

• Recession: R_P = (2/11) × (−20%) + (9/11) × 30% = 20.91%
• Boom: R_P = (2/11) × 70% + (9/11) × 10% = 20.91%

The return is the same no matter what happens — this portfolio has a zero variance! Combining assets into portfolios can substantially alter the risks faced by the investor. This is the crucial observation that motivates the discussion of diversification in the following sections — and the reason the "portfolio standard deviation" question cannot be answered by intuition.

7. Case Data Card

L offers a higher expected return but more risk (70% gain or −20% loss possible); U is safer (at least 10% in every state). Which stock to buy depends on personal preferences — but the portfolio's expected return lies between the two, while its risk can fall well below either, even to zero.

8. Key Takeaways

1. Returns add linearly; variances do not — the portfolio expected return is a weighted average and always holds; portfolio variance must be computed state by state.
2. Portfolios reshape risk — with the right weights, portfolio risk can be far below the weighted average of asset risks; with weights 2/11 and 9/11 the variance is exactly zero.
3. Exam pattern in three steps — ① compute the portfolio return in each state from the weights → ② find the expected return E(R_P) → ③ compute variance as Σ prob × (deviation)² and take the square root for the standard deviation.

1. Core Idea

Combining assets into a portfolio generally produces risk below the weighted average of the individual risks — this is the diversification effect. Its size is governed by the correlation coefficient between asset returns: the lower the correlation, the stronger the risk reduction.

Key insight up front: a portfolio's expected return is simply the weighted average of asset expected returns (linear), but portfolio risk is not a weighted average of asset risks (nonlinear). This asymmetry is the mathematical source of the diversification effect.

2. Portfolio Expected Return: Just a Weighted Average

Portfolio weight xᵢ = dollars invested in asset i ÷ total portfolio value; all weights sum to 1 (all money is invested somewhere):

$$E(R_P) = x_1 \times E(R_1) + x_2 \times E(R_2) + \cdots + x_n \times E(R_n)$$

Example: Stock L, E(R) = 25%, σ = 45%; Stock U, E(R) = 20%, σ = 10%. Equally weighted (50/50) portfolio:

$$E(R_P) = .50 \times 25\% + .50 \times 20\% = 22.5\%$$

No surprise at the return level — a portfolio cannot create expected return out of thin air; it is always the weighted average.

3. Portfolio Variance: Intuition Fails Here

Intuition suggests the portfolio standard deviation is the weighted average of the two standard deviations:

$$\sigma_P = .50 \times 45\% + .50 \times 10\% = 27.5\%$$

This is completely incorrect! Portfolio variance is not a simple combination of asset variances. The correct approach: compute portfolio returns state by state, then the variance. Two equally likely states:

• Recession: $R_P = .50 \times (−20\%) + .50 \times 30\% = 5\%$
• Boom: $R_P = .50 \times 70\% + .50 \times 10\% = 40\%$

$$\sigma_P^2 = .50 \times (.05 − .225)^2 + .50 \times (.40 − .225)^2 = .030625$$

$$\sigma_P = \sqrt{.030625} = .175 = 17.5\%$$

The portfolio σ is 17.5% — far below the "intuitive" 27.5%. Why did risk disappear? Because L and U move in opposite directions: in a recession L loses 20% while U gains 30%; in a boom L gains 70% while U gains only 10%. One asset's loss is offset by the other's gain.

4. The Role of the Correlation Coefficient

L and U move in exactly opposite directions; their returns have a correlation coefficient of ρ = −1 (perfect negative correlation, the theoretical extreme). The correlation coefficient sets the "ceiling" for diversification:

Dramatic demonstration — put 2/11 (≈18%) in L and 9/11 (≈82%) in U:

• Recession: $R_P = (2/11) \times (−20\%) + (9/11) \times 30\% = 20.91\%$
• Boom: $R_P = (2/11) \times 70\% + (9/11) \times 10\% = 20.91\%$

Identical returns in both states — zero variance, zero risk. This is the limit of diversification at ρ = −1. Assets with ρ = −1 barely exist in practice, but the principle holds: the lower the correlation, the more risk a portfolio can eliminate.

5. The Evidence: Another Lesson from Market History

Table 13.7 (equally weighted portfolios of randomly selected NYSE stocks, average annual standard deviations):

A single randomly selected NYSE stock has a standard deviation of about 49% per year; with 100 stocks it falls to about 20% — a reduction of roughly 60%. But the marginal benefit diminishes: most of the effect is realized with 10 securities; after about 30 there is little remaining benefit; by 500 the curve is flat (19.27%) and cannot go lower — the remaining risk cannot be diversified away.

6. Diversification and the Two Types of Risk

Why can't all risk be eliminated? The answer lies in the two types of risk:

• Unsystematic risk (= diversifiable risk = unique / asset-specific risk) affects a single asset or a small group (lawsuits, strikes, the success or failure of new products). The unsystematic parts of different assets are unrelated to each other, so in a large portfolio they wash out — a portfolio with many assets has almost no unsystematic risk.
• Systematic risk (= nondiversifiable risk = market risk) affects almost all assets (GDP, interest rates, inflation). It is a common factor that no number of additional stocks can remove.

$$\text{Total risk} = \text{Systematic risk} + \text{Unsystematic risk}$$

Diversification eliminates the unsystematic part; systematic risk is an unremovable floor (the flat part of Figure 13.1). This sets up the systematic risk principle of §13.6: the market rewards only systematic risk, since unsystematic risk can be eliminated at almost no cost.

7. A Real-World Illustration: the S&P 500 in 2019

The S&P 500 rose about 22% in 2019. Individual stocks told a very different story: AMD up about 130%, Lam Research up 120%, Target up 101% — while Abiomed fell 49%, Macy's fell 39%, and Occidental Petroleum fell 29%. The lesson is clear: diversification cannot guarantee you catch every winner, but it sharply reduces your exposure to extreme outcomes, both good and bad.

8. Three Key Takeaways

1. Portfolio risk ≠ weighted average of risks — diversification works because asset returns are not perfectly positively correlated; the lower ρ, the stronger the effect; at ρ = +1 diversification is useless, and at ρ = −1 risk can theoretically fall to zero
2. Unsystematic risk is diversifiable, systematic risk is not — the former approaches zero as the number of stocks grows; the latter is an unremovable floor
3. Diversification has diminishing returns — roughly 10 stocks capture most of the benefit; beyond 30 there is almost nothing left; at 100 stocks risk has already fallen from about 49% to about 20%

1. Core Idea

Where does risk come from? Not from what is expected — from surprises, unanticipated events. If an investment always delivers exactly what you expect, it is perfectly predictable and, by definition, risk-free. The total return on any asset therefore has two parts:

$$R = E(R) + U$$

• $E(R)$ — expected return: the part the market anticipates, already discounted into the price;
• $U$ — unexpected return: the part from new information revealed during the year; positive or negative, averaging zero over time.

Key corollary: an announcement is not news. Any announcement splits into two parts:

$$\text{Announcement} = \text{Expected part} + \text{Surprise (innovation)}$$

The part the market already anticipated has no effect on price — only the surprise matters. This is why "good news" can sometimes be bad news.

2. Decomposing the Surprise

The unexpected part U can itself be split into two sources:

$$R = E(R) + m + \epsilon$$

• $m$ — the systematic portion (letter m, since systematic risks are often called market risks);
• $\epsilon$ (epsilon) — the unsystematic portion, unique to the individual firm.

What makes $\epsilon$ important is that it is unrelated to the unsystematic portions of other assets — the precondition that allows diversification to eliminate it.

3. Systematic vs. Unsystematic Risk

Two remarks: the line between the two is never perfectly exact — every enterprise is part of the economy, so even the most narrow news ripples outward (the tale of the kingdom lost for want of a horseshoe nail). Still, some risks are clearly far more general than others. And the true source of risk is the unanticipated event, the surprise itself.

A classifying exercise (using the book's Flyers example): announcements about interest rates and GDP are important for nearly all companies — systematic; news about Flyers' president, its research, or its sales is of specific interest to Flyers alone — unsystematic.

4. Evidence from Market History

Table 13.7: average annual standard deviations of equally weighted portfolios of randomly selected NYSE stocks

Key facts:

• A single randomly chosen NYSE stock carries a standard deviation of about 49% per year;
• With 100 stocks, the standard deviation falls by about 60% — from 49% to roughly 20%;
• With 500 securities the standard deviation is 19.27%, close to the historical figure of about 20% for a portfolio of 500 large common stocks;
• 10 securities capture most of the diversification benefit; by 30 or so, little benefit remains (30 → 40 stocks: only 20.87% → 20.46%).

5. Why Unsystematic Risk Is Eliminated

The chain of logic:

1. Unsystematic events are company-specific (product successes, lawsuits, strikes);
2. In a large portfolio, positive firm-specific events lift some stocks while negative ones depress others;
3. The positive and negative effects wash out at the portfolio level, leaving a small net effect on total value.

The principle of diversification: spreading an investment across many assets eliminates some of the risk. "Diversifiable risk" and "unsystematic risk" are used interchangeably; systematic risk affects almost all assets, so it survives no matter how many stocks are held — "systematic risk" and "nondiversifiable risk" are likewise interchangeable.

The key sentence worth memorizing: unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk — for such a portfolio, essentially all of the risk is systematic.

6. The Limit of Diversification: Total Risk Decomposition

$$\text{Total risk} = \text{Systematic risk} + \text{Unsystematic risk}$$

• Diversifiable risk (the blue area in Figure 13.1): the part that disappears quickly as stocks are added;
• Nondiversifiable risk: the minimum level of risk that cannot be eliminated, no matter how diversified the portfolio.

The 2019 S&P 500 story: the index rose about 22% for the year, yet individual results varied wildly — biggest winners included AMD (up about 130%), Lam Research (up 120%), and Target (up 101%); losers included Abiomed (down 49%), Macy's (down 39%), and Occidental Petroleum (down 29%). Diversification reduces exposure to extreme outcomes, both good and bad — a portfolio always contains winners and losers; the index return reflects the average after firm-specific events cancel out.

This is another lesson from capital market history: diversification reduces risk, but only up to a point — some risk is diversifiable and some is not.

7. Case Revisit and Reflection (课程思政)

The case (early 2020): Boeing announced sales about 2% below analysts' estimates and EPS about 6% below; Norwegian Cruise Lines reported EPS of −$.99 versus the −$.28 estimate (prior year: +$.83); Chinese internet giant Tencent reported first-quarter revenue and profit above estimates. The market reaction defied intuition: Tencent's stock fell about 5%, Boeing jumped about 6%, and NCL rose about 1.6%. Why? The market prices surprises, not announcements: amid the COVID-19 lockdown, Boeing's results were "not as bad as they could have been"; NCL announced strong demand for cruise bookings for Q4 2020 through 2021 with pricing within historical ranges; and on Tencent's announcement day, the market as a whole fell.

Reflection:

• "Preventing systemic financial risks" is the bottom-line consensus of China's financial regulation. The first task of macro policy and regulation is to prevent the accumulation and spread of systematic risks (excessive leverage, shadow banking, financialization of real estate) — because once systemic risk erupts, it hits the whole economy and no one can escape. This gives the concept of systematic risk real-world weight.
• For individual investors: unsystematic risk can be eliminated through diversification, while systematic risk can only be borne and managed through asset allocation (stocks plus bonds, long-term regular investing) — "don't put all your eggs in one basket" rests on solid finance theory. Also stay alert to the dangers of going all-in on a single asset, illegal fundraising and scams — improve financial literacy and invest rationally.

8. Three Key Takeaways

1. Risk comes from surprises — anticipated information is already priced in; only the surprise (innovation) is the true source of risk and of price changes.
2. Diversification eliminates only unsystematic risk — unsystematic events wash out across a large portfolio, but systematic risk cannot be diversified away; total risk = systematic + unsystematic.
3. Diversification has its limits — about 10 stocks capture most of the benefit, and beyond 30 little remains; for a well-diversified investor, essentially all remaining risk is systematic.

1. Core Idea: The Systematic Risk Principle

The total risk of an investment can be decomposed into two parts:

$$\text{Total risk} = \text{Systematic risk} + \text{Unsystematic risk}$$

• Systematic risk: economy-wide risk (GDP, interest rates, inflation) affecting many assets — also called market risk or nondiversifiable risk; it cannot be eliminated by diversification.
• Unsystematic risk: risk unique to a single asset or small group — also called unique, asset-specific, or diversifiable risk; firm-specific events "wash out" in a large portfolio.

The systematic risk principle: the reward for bearing risk depends only on an investment's systematic risk — because unsystematic risk can be eliminated at virtually no cost (by diversifying), there is no reward for bearing it. Implication: the expected return on an asset depends only on its systematic risk, not its total risk.

2. Defining the Beta Coefficient

The beta coefficient (β) measures how much systematic risk a particular asset has relative to an average asset:

• An average asset has a beta of 1.0
• β = .50 → half as much systematic risk as an average asset
• β = 2.0 → twice as much
• A risk-free asset has a beta of zero (no systematic or unsystematic risk)

Caveat: not all betas are created equal — different providers use somewhat different estimation methods, and significant differences sometimes occur. It is wise to look at several sources.

3. Table 13.8: Betas of Selected Companies

(Source: Yahoo! Finance, 6/11/2020)

Reading: assets with larger betas have greater expected returns — Johnson & Johnson (β = .70) investors earn less, on average, than Sherwin-Williams (β = 1.23) investors. For example, Boeing's reported beta is 1.45, about one and one-half times the systematic risk of a typical stock.

4. Example 13.5: Total Risk versus Beta

• A has greater total risk but substantially less systematic risk;
• Since total risk = systematic + unsystematic, A must have greater unsystematic risk;
• By the systematic risk principle, B earns a higher risk premium and expected return despite less total risk.

Key exam point: "more total risk" does not mean "higher risk premium" — the market prices only systematic risk.

5. Portfolio Betas: A Weighted Average

A portfolio beta is calculated exactly like a portfolio expected return — a weighted average of the individual betas:

$$\beta_P = x_1\beta_1 + x_2\beta_2 + \cdots + x_n\beta_n$$

Example: half in Shopify (β = 1.59) and half in Coca-Cola (β = .55):

$$\beta_P = .50 \times 1.59 + .50 \times .55 = 1.07$$

Example 13.6: invest $1,000 (A), $2,000 (B), $3,000 (C), $4,000 (D) — total $10,000, weights .10/.20/.30/.40:

$$\beta_P = .10 \times .80 + .20 \times .95 + .30 \times 1.10 + .40 \times 1.40 = 1.16$$

Portfolio expected return: $E(R_P) = .10 \times 8\% + .20 \times 12\% + .30 \times 15\% + .40 \times 18\% = 14.9\%$; β_P = 1.16 > 1 → more systematic risk than an average asset.

6. The Market Portfolio and the Market Risk Premium

Market portfolio: a portfolio made up of all the assets in the market, with expected return denoted E(R_M).

• Since all assets must plot on the security market line (SML), so must the market portfolio;
• The market portfolio is representative of all assets in the market → average systematic risk → β_M = 1;
• The slope of the SML is the market portfolio's reward-to-risk ratio:

$$\text{SML slope} = \frac{E(R_M) - R_f}{\beta_M} = E(R_M) - R_f$$

The term E(R_M) − R_f is the market risk premium — the reward the market offers for bearing an average amount of systematic risk.

7. From Beta to the CAPM

In a well-functioning, active market, every asset must offer the same reward-to-risk ratio (risk premium ÷ beta), so all assets plot on the same line — the security market line (SML). Rearranging gives the capital asset pricing model (CAPM):

$$E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i$$

The expected return has three components: ① the pure time value of money, R_f; ② the reward for bearing systematic risk, [E(R_M) − R_f]; ③ the amount of systematic risk in the asset, β_i.

Example 13.8: R_f = 4%, market risk premium 8.6%, β = 1.3:

$$E(R) = 4\% + 1.3 \times 8.6\% = 4\% + 11.18\% = 15.18\%$$

If beta doubles to 2.6, the risk premium doubles to 22.36% and the expected return rises to 26.36% — the risk premium is proportional to beta.

Example 13.7 (valuation check): with R_f = 6%, SWMS (β = 1.3, E(R) = 14%) offers a reward-to-risk ratio of ≈ 6.15%, while Insec (β = .8, E(R) = 10%) offers only 5% — Insec's expected return is too low for its risk → its price is too high → overvalued.

8. Key Takeaways

1. The market rewards only systematic risk — total risk = systematic + unsystematic; unsystematic risk can be diversified away for free, so expected returns and risk premiums are determined by beta alone.
2. Three beta anchors — the average asset has β = 1 and the risk-free asset has β = 0; β < 1 means low systematic risk, β > 1 means high; a portfolio's beta is the weighted average of its assets' betas.
3. The market portfolio has β = 1 and the market risk premium is E(R_M) − R_f — CAPM: E(R_i) = R_f + [E(R_M) − R_f] × β_i, with the risk premium proportional to beta; beta is the bridge from risk measurement to the required return (cost of capital).

1. Core Idea: The Systematic Risk Principle

Total risk decomposes into systematic risk and unsystematic risk:

• Systematic (market) risk: affects nearly all assets (GDP, interest rates, inflation); cannot be eliminated by diversification
• Unsystematic (unique / asset-specific) risk: affects a single asset or a small group; can be virtually eliminated by diversification at almost no cost

The systematic risk principle: The reward for bearing risk depends only on the systematic risk of an investment. Because unsystematic risk can be eliminated by diversifying, the market does not reward risks that are borne unnecessarily.

Corollary: An asset's expected return (and risk premium) depends only on its systematic risk — not on its total risk.

2. Beta: Measuring Systematic Risk

The beta coefficient (β) measures how much systematic risk an asset has relative to an average asset:

• An average asset has β = 1.0; β = 0.50 means half as much systematic risk; β = 2.0 means twice as much
• A risk-free asset has β = 0 (no systematic or unsystematic risk)

Selected betas (Yahoo! Finance, 6/11/2020): Coca-Cola .55, McDonald's .66, Johnson & Johnson .70, Visa .93, Mastercard 1.08, Walt Disney 1.11, Sherwin-Williams 1.23, Shopify 1.59.

Caveat: different providers use somewhat different estimation methods, so compare several sources.

3. The Reward-to-Risk Ratio

Text data: Asset A has E(R_A) = 20%, β_A = 1.6; risk-free rate R_f = 8%.

Portfolio combining A with the risk-free asset (25% in A, 75% risk-free):

$$E(R_P) = .25 \times 20\% + .75 \times 8\% = 11\%$$ $$\beta_P = .25 \times 1.6 = .40$$

Borrowing at 8% to invest 150% of wealth in A (a levered portfolio — weights can exceed 100%):

$$E(R_P) = 1.50 \times 20\% - .50 \times 8\% = 26\%$$ $$\beta_P = 1.50 \times 1.6 = 2.4$$

All combinations (E(R), β) plot on one straight line; its slope is the reward-to-risk ratio:

$$\text{Slope} = \frac{E(R_A) - R_f}{\beta_A} = \frac{.20 - .08}{1.6} = .075, \text{ or } 7.5\%$$

Asset A offers a risk premium of 7.5% per "unit" of systematic risk.

Compare Asset B (β_B = 1.2, E(R_B) = 16%):

$$\frac{E(R_B) - R_f}{\beta_B} = \frac{.16 - .08}{1.2} = .0667, \text{ or } 6.67\% < 7.5\%$$

Arbitrage logic: B offers inadequate compensation for its systematic risk. Investors buy A and sell B — A's price rises and its expected return falls, B's does the reverse; trading continues until both plot on the same line.

4. Market Equilibrium and the Security Market Line (SML)

Fundamental result: In an active, competitive, well-functioning market, the reward-to-risk ratio must be the same for all assets:

$$\frac{E(R_A) - R_f}{\beta_A} = \frac{E(R_B) - R_f}{\beta_B}$$

Market portfolio: made up of all assets in the market, so β_M = 1 (average systematic risk). The SML slope is:

$$\text{SML slope} = \frac{E(R_M) - R_f}{\beta_M} = E(R_M) - R_f$$

where E(R_M) − R_f is the market risk premium.

Plotting expected return against beta, all assets must lie on a single line: the security market line (SML) — the relationship between systematic risk and expected return in financial markets. An asset above the line (C) rises in price and falls in expected return until it lands on the line; one below the line (D) adjusts in the opposite direction.

5. The CAPM Formula

For any asset i with expected return E(R_i) and beta β_i, it must plot on the SML, which yields the capital asset pricing model (CAPM):

$$\boxed{E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i}$$

Expected return consists of three parts:

1. Pure time value of money: R_f — the reward for waiting without taking any risk
2. Reward for bearing systematic risk: E(R_M) − R_f — the market premium for bearing an average amount of systematic risk
3. Amount of systematic risk: β_i — the asset's risk relative to the average asset

CAPM works for portfolios too: compute the portfolio beta as a weighted average (β_P = x₁β₁ + x₂β₂ + …) and plug it in. E.g., 50% Shopify (1.59) + 50% Coca-Cola (.55) → β_P = .50×1.59 + .50×.55 = 1.07.

6. Worked Examples

Example 13.8: R_f = 4%, market risk premium = 8.6%, stock beta = 1.3:

$$\text{Risk premium} = 1.3 \times 8.6\% = 11.18\%$$ $$E(R_i) = 4\% + 11.18\% = 15.18\%$$

If beta doubles to 2.6: risk premium doubles to 22.36%, so E(R_i) = 4% + 22.36% = 26.36%.

Self-test 13.4: R_f = 8%, E(R_M) = 16% (market risk premium 8%), beta .7:

$$E(R_i) = 8\% + .7 \times 8\% = 13.6\%$$

A stock with E(R) = 24% has a risk premium of 16% = twice the market risk premium → β = 2.0.

7. Applications: Alpha, Valuation, and Cost of Capital

Alpha: the distance between an asset's actual return and the SML. On the SML, an asset earns exactly what it should for its beta; a positive alpha is a return in excess of the CAPM benchmark (plots above the line); a negative alpha is undesirable.

Over/undervaluation (Example 13.7): R_f = 6%.

Insec offers insufficient expected return for its risk → its price is too high → overvalued relative to SWMS (SWMS undervalued relative to Insec); expect Insec's price to fall.

Preview of the cost of capital: shareholders can always invest in financial markets themselves, so a new project must offer an expected return no worse than what the market offers for the same beta — otherwise it has a negative NPV. The SML gives the "going rate" for bearing risk in the economy, the basis for the discount rate (cost of capital) on a project.

8. Key Takeaways

1. Only systematic risk is rewarded — unsystematic risk is diversified away for free, so expected return depends on beta alone
2. Market equilibrium pricing — in active markets every asset offers the same reward-to-risk ratio and plots on the SML
3. CAPM is the cornerstone of valuation — E(R_i) = R_f + β_i × (market risk premium), the basis for cost of capital and NPV decisions

Course ideology: The SML embodies the market law of "risk and return in proportion" — the market does not reward unnecessary risk, guiding investors to assess risk rationally and diversify rather than speculate blindly; equilibrium pricing keeps markets fair, as overvaluation is eventually corrected by the market.

1. Core Idea: What Is an Efficient Market?

Stock prices fluctuate widely from year to year, at least partly because new information arrives and investors reassess asset values. A market is "efficient" if prices adjust quickly and correctly when new information arrives. More precisely:

In an efficient capital market, current market prices fully reflect available information — based on available information, there is no reason to believe that the current price is too low or too high.

2. Price Behavior in an Efficient Market: The FCC Example

F-Stop Camera Corporation (FCC) has secretly developed an autofocusing system twice as fast as existing ones; capital budgeting shows a large positive NPV. The key assumption: the system's existence is inside information only.

Suppose the announcement is made in a Wednesday-morning press release. In an efficient market, Wednesday afternoon's price already reflects that information — buying Wednesday afternoon and selling Thursday would yield no profit (if it did, the market would have taken a full day to digest one press release).

Figure 12.14 shows three possible adjustment paths (Day 0 = announcement day):

In an inefficient market, buying right after the announcement and selling days later is a positive NPV activity — the price is too low for several days.

3. The EMH and the Zero-NPV Implication

The efficient markets hypothesis (EMH) asserts that well-organized capital markets, such as the NYSE, are efficient at least as a practical matter — inefficiencies may exist, but they are relatively small and uncommon.

The key implication for participants:

$$\text{NPV} = \text{Market value} - \text{Cost} = 0$$

If prices are neither too low nor too high, the difference between an investment's market value and its cost is zero — all investments in an efficient market are zero NPV investments. Investors get exactly what they pay for, and firms receive exactly what their stocks and bonds are worth when they sell them.

4. What Makes a Market Efficient: Investor Competition

Efficiency comes from competition among investors. Many people spend their entire lives searching for mispriced stocks — studying past prices and dividends, earnings, debt, taxes, industries, planned investments, sensitivity to the economy, and more. The profit motive is a powerful incentive: know more, profit from good news (buy) and avoid bad news (sell).

The logical consequence: mispriced stocks become fewer and fewer, and the market becomes increasingly efficient. An equilibrium emerges in which just enough mispricing remains for those best at identifying it to make a living — for most other investors, information gathering and analysis does not pay.

5. Common Misconceptions about the EMH

Misconception 1: "Throwing darts at the financial page" matches professional analysts. Efficiency does NOT mean it doesn't matter how you invest — a random dart thrower might land every dart in one or two high-risk genetic engineering stocks. Would you really put all your money there?

Misconception 2 (The Wall Street Journal dartboard contest): in 147 five-and-one-half-month contests from July 1990 to September 2002, four professional money managers beat the dartboard 90 times (90 to 57). Does this prove markets are inefficient? No: - Darts naturally select average-risk stocks, while pros pick riskier stocks to win — so on average, pros should win; - The pros' picks are announced publicly at the start, and this publicity may boost prices — a partially self-fulfilling prophecy.

Misconception 3: "Prices move daily, so the market can't be efficient." Price fluctuation is perfectly consistent with efficiency: investors are bombarded with information every day, and price movements are a reflection of that information flow. In a rapidly changing world, the absence of price movements would suggest inefficiency. Efficiency means the price a firm obtains for its stock is "fair" — the relevant information has been "priced out."

6. The Three Forms of Market Efficiency

The forms differ in what information is reflected in prices:

The forms are nested: weak ⊂ semistrong ⊂ strong. As for the strong form: casual observation (e.g., prior knowledge of a takeover attempt can be very valuable) suggests inside information does exist — the FCC example explicitly assumed the market was not strong form efficient; whether using it is lawful or ethical is a separate issue. The semistrong form is the most controversial.

7. Historical Evidence and a Reflection: Efficiency Is Built, Not Given

Capital market history suggests three things: 1. Prices respond rapidly to new information, and the response is at least not grossly different from what an efficient market would produce; 2. The future of prices, especially in the short run, is difficult to predict from publicly available information; 3. If mispriced stocks exist, there is no obvious way to identify them — simpleminded schemes based on public information will probably not succeed.

Course reflection (课程思政): Market efficiency is not born but built through institutions. China's registration-based IPO reform centers on information disclosure so prices reflect information more fully; strict enforcement against insider trading safeguards the fairness of "one information set for all"; investor protection reminds us that efficiency has an ethical twin — fairness and order. The more efficient the market, the more the fantasy of speculation should yield to rational, value investing.

8. Key Takeaways

1. Price is the mirror of information — in an efficient market, prices adjust quickly and correctly; investors get exactly what they pay for, and firms sell at exactly what their securities are worth.
2. Zero NPV is the signature of efficiency — prices neither too low nor too high → market value − cost = 0 → no free lunch; competition that erases excess returns is the mechanism that makes markets efficient.
3. Efficiency comes in layers and rests on evidence — weak (past prices) → semistrong (public information) → strong (inside information), with higher layers harder to sustain; when information is "priced out," there is no guaranteed shortcut to buying low and selling high.

1. Core Idea: Why the Cost of Equity Is the Hardest to Estimate

Of all the components of the cost of capital, the cost of equity is the most difficult to determine: unlike debt, there is no directly observable market quote for the return that equity investors require — it must be estimated.

The key bridge (§14.1): the return an investor receives on a security is the cost of that security to the company that issued it. "Required return," "appropriate discount rate," and "cost of capital" are used interchangeably — and the cost of capital depends primarily on the use of the funds, not the source.

The textbook offers two approaches:

1. The dividend growth model: $R_E = D_1/P_0 + g$ (from Chapter 8; only for firms that pay dividends with steady growth)
2. The SML (CAPM) approach — the focus of this note, built directly on the security market line from Chapter 13

2. The CAPM Formula: Three Inputs

Chapter 13's core conclusion: the expected return on a risky investment depends on three things —

1. The risk-free rate, $R_f$
2. The market risk premium, $E(R_M) - R_f$
3. The asset's systematic risk relative to the average, its beta, β

$$\boxed{R_E = R_f + \beta_E \times (R_M - R_f)}$$

The second term is the equity risk premium: the market "price" per unit of systematic risk ($R_M - R_f$) times the amount of systematic risk the stock carries ($\beta_E$). By tying the expected return directly to systematic risk, this is exactly where the CAPM beats the dividend growth model.

3. Estimating the Three Inputs in Practice

Note: different providers estimate beta differently (e.g., in the Eastman example Yahoo! reported 1.39 while Value Line reported 1.25) — check multiple sources and use judgment.

4. Worked Example: Walt Disney (Main-Text Demonstration)

$R_f$ = 1.53% (T-bills), market risk premium = 7%, β = 1.11:

$$R_E = .0153 + 1.11 \times .07 = .0930, or 9.30\%$$

Walt Disney's cost of equity is about 9.30%.

5. Worked Example: Alpha Air Freight (Example 14.1)

β = 1.2, market risk premium = 7%, risk-free rate = 6%:

$$R_E = .06 + 1.2 \times .07 = .144, or 14.4\%$$

Cross-check with the dividend growth model: $D_0$ = $2, g = 8%, $P_0$ = $30, so $D_1 = \$2 \times 1.08 = \$2.16$:

$$R_E = \$2.16/\$30 + .08 = .152, or 15.2\%$$

The two estimates are reasonably close, so we average them: ≈ 14.8% as Alpha's cost of equity.

6. Comparison with the Dividend Growth Model

Dividend growth model ($R_E = D_1/P_0 + g$):

• Advantage: simple to understand and easy to use
• Drawbacks: applicable only to firms that pay dividends; assumes constant dividend growth; very sensitive to g (a one-percentage-point upward revision of g raises the estimated cost of equity by at least a full percentage point); no explicit adjustment for risk, so it is hard to tell whether the estimated return is commensurate with the risk

The SML approach fixes both defects: it explicitly adjusts for risk and applies to companies other than those with steady dividend growth, making it useful in a wider variety of circumstances.

7. Limitations and Practical Pointers

1. Two inputs must be estimated: the market risk premium and beta. The 7% premium rests on about 100 years of returns on particular portfolios; different time periods, stocks, or markets yield very different estimates — and poor estimates mean an inaccurate cost of equity
2. The past is used to predict the future: economic conditions can change quickly, so history may not be a good guide
3. Cross-validation: ideally both approaches apply and produce similar answers, boosting confidence; also compare with results for similar companies as a reality check
4. Example: Eastman Chemical (§14.4) — CAPM gives $R_E = .0153 + 1.25(.07) = .1028$, or 10.28% (using Value Line's beta of 1.25); the dividend discount model gives 7.58%; the two are averaged to 8.93%
5. Industry practice: in the interview with Hershey's CFO Samuel Weaver — a survey of 10 food processing competitors showed that all 10 used the CAPM to compute the cost of equity; Hershey itself prefers the dividend growth model because its dividends grow steadily

8. Core Takeaways

1. The cost of equity cannot be observed — it must be estimated — the return shareholders require is the firm's cost of equity, the hardest input in the WACC
2. Three CAPM inputs — $R_E = R_f + \beta \times$ (market risk premium); estimation errors in β and the risk premium translate directly into errors in the cost of equity
3. Cross-check the two methods — the SML approach adjusts for risk explicitly and applies broadly; combining it with the dividend growth model and comparing against similar firms is how estimation error is kept in check in practice

1. Core Idea: Why the Cost of Equity Must Be Estimated

The return an investor receives on a security is the cost of that security to the company that issued it. But the cost of equity is the most difficult piece of the cost of capital puzzle: the capital market does not directly reveal the return equity investors require. Unlike the cost of debt (observable via the yield to maturity), the cost of equity must be estimated. The textbook offers two approaches:

1. The dividend growth model (DDM) — this section, based on the Chapter 8 dividend growth model;
2. The SML/CAPM approach — based on the security market line.

Following Section 14.1, the terms required return, appropriate discount rate, and cost of capital are used more or less interchangeably — an investment has a positive NPV only if its return exceeds its cost of capital. And one of the most important lessons bears repeating: the cost of capital depends primarily on the use of the funds, not the source.

2. The Dividend Growth Model: Formula and Derivation

From Chapter 8: if dividends grow forever at a constant rate g, the current price per share is:

$$P_0 = \frac{D_1}{R_E - g} = \frac{D_0 \times (1+g)}{R_E - g}$$

where $D_0$ is the dividend just paid, $D_1 = D_0 \times (1+g)$ is next period's projected dividend, and $R_E$ is the required return on equity (E for equity).

Rearranging to solve for $R_E$ gives the DDM cost of equity (Eq. 14.1):

$$R_E = \frac{D_1}{P_0} + g$$

Interpretation: cost of equity = expected dividend yield ($D_1/P_0$) + dividend growth rate (g, the capital gains component).

3. Implementing the Approach: Three Inputs

4. Worked Example: Greater States Public Service

A large public utility paid a dividend of $4 per share last year ($D_0 = \$4$). The stock sells for $60 per share ($P_0 = \$60$), and dividends are expected to grow at 6% per year forever. What is the cost of equity?

Step 1: Project next year's dividend

$$D_1 = D_0 \times (1+g) = \$4 \times 1.06 = \$4.24$$

Step 2: Apply the DDM formula

$$R_E = \frac{D_1}{P_0} + g = \frac{\$4.24}{\$60} + .06 = .1307$$

Result: The cost of equity is 13.07%.

5. Estimating g: Two Ways

Method 1 — Historical growth rates. Observe dividends over the past five years, compute year-to-year growth rates, and average them:

Method 2 — Analysts' forecasts. Forecasts are available from a variety of sources (e.g., growth estimates can be found at zacks.com); since different sources give different estimates, one approach is to obtain several and average them.

6. Advantages and Disadvantages

Advantage: Simplicity. It is both easy to understand and easy to use.

Disadvantages (four):

1. Applicable only to companies that pay dividends — useless in many cases;
2. Constant-growth assumption is unrealistic — dividends never grow at exactly a constant rate; the model truly fits only reasonably steady growers;
3. Very sensitive to the estimated growth rate — for a given price, revising g upward by just one percentage point raises the estimated cost of equity by at least a full percentage point (and somewhat more, since $D_1$ is usually revised upward as well);
4. No explicit risk adjustment — unlike the SML approach, there is no direct allowance for the riskiness of the investment, so we cannot judge whether the estimated return is commensurate with risk.

Evidence from practice (The Hershey Company): Samuel Weaver, formerly director of financial planning and analysis at Hershey, reports that a survey of 10 food-processing industry peers showed nearly all of them used the CAPM for the cost of equity — yet Hershey uses the dividend growth model, because the company pays dividends and enjoys steady, stable dividend growth projected within its strategic plan. When the model's conditions hold, the DDM is the appealing choice.

7. Cross-Checking with the SML Approach: Two Examples

The two approaches rely on different assumptions, so estimates differ; when they are close, we gain confidence in the estimate.

Example 14.1 — Alpha Air Freight: $\beta = 1.2$, market risk premium 7%, risk-free rate 6%, $D_0 = \$2$, g = 8%, $P_0 = \$30$.

• SML: $R_E = .06 + 1.2 \times .07 = .144$, or 14.4%;
• DDM: $D_1 = \$2 \times 1.08 = \$2.16$; $R_E = \$2.16/\$30 + .08 = .152$, or 15.2%;
• The estimates are reasonably close, so average them: $R_E \approx$ 14.8%.

Eastman Chemical (worked illustration): risk-free rate 1.53%, market risk premium 7%, Value Line beta 1.25.

• SML: $R_E = .0153 + 1.25 \times .07 = .1028$, or 10.28%;
• DDM: Eastman has paid dividends for only a few years, so its historical dividend growth is unreliable — the book substitutes analysts' EPS growth forecast of 4.26% for g: $R_E = \$2.48(1+.0426)/\$77.83 + .0426 = .0758$, or 7.58%;
• Both estimates are plausible, so average them: $R_E \approx$ 8.93% (later used in Eastman's WACC).

Key point: Different estimates are the norm, not a surprise — each method rests on different assumptions. Two remedies: discard the estimate that seems unreasonable, or average the two. If a firm pays few dividends (like Eastman), DDM credibility falls and earnings growth can proxy for dividend growth. In the best of all worlds, both approaches are applicable and give similar answers — then we have some confidence in the estimate, and we might also compare results with those for other similar companies as a reality check.

8. Three Key Takeaways

1. The cost of equity cannot be observed; it must be estimated — the DDM is one of the two standard approaches, and its formula is a single line: $R_E = D_1/P_0 + g$ (dividend yield + growth).
2. Estimating g is the Achilles' heel of the DDM — g is the only estimated input (historical averages, arithmetic vs. geometric, or analysts' forecasts), and $R_E$ is highly sensitive to it.
3. Know the DDM's boundary conditions — it applies only to dividend-paying firms with steady dividend growth and contains no explicit risk adjustment; in practice it is cross-checked against the SML approach and the two estimates are averaged.

1. Core Idea: Why These Two Costs Are "Observable"

Chapter 13 looked at securities from the investor's point of view; Chapter 14 flips to the issuing firm's point of view: the return an investor receives is the cost of that security to the company that issued it. Hence required return, appropriate discount rate, and cost of capital are used more or less interchangeably.

One principle bears repeating throughout the chapter: the cost of capital depends primarily on the use of the funds, not the source.

Intuition: for a risk-free project, the cost of capital is simply the risk-free rate; the riskier the project, the higher its cost of capital (i.e., its required return). The project's own risk determines its cost of capital — not where the money comes from.

The hard part is equity: in §14.2 the cost of equity (R_E) cannot be observed directly and must be estimated either with the dividend growth model (R_E = D₁/P₀ + g) or the SML (R_E = R_f + β_E × (R_M − R_f)). By contrast, the cost of debt and the cost of preferred stock can be observed or computed directly — that is what makes §14.3 comparatively easy.

2. The Cost of Debt: Definition and Two Ways to Observe It

The cost of debt (R_D) is the return the firm's creditors demand on new borrowing — note the emphasis on "new": it is a marginal cost, not a historical one.

Unlike equity, R_D can normally be observed directly or indirectly, and there is no need to estimate a beta for the debt:

Either way, the rate we want is quoted in the financial markets — no SML, no beta estimation.

3. The Trap: Coupon Rate ≠ Today's Cost

The coupon rate is irrelevant here. It tells us roughly what the firm's cost of debt was when the bonds were issued — a historical cost, not today's cost. So we must look at the yield on the debt in today's marketplace (YTM).

The exam distinction to remember: - Book cost of debt (coupon rate) ≠ market cost of debt (YTM) - Capital cost calculations always use the market value (YTM); the coupon rate is only used to compute cash flows

This is exactly the answer to Concept Question 14.3a: the coupon rate is a bad estimate because it is a historical rate.

4. Example: General Tool's Cost of Debt

General Tool issued a 30-year, 7% bond 8 years ago, so 22 years remain to maturity; the bond currently sells at 96% of face value, or $960.

Because the bond sells at a discount, its YTM is somewhat above 7% — but not much above, because the discount is fairly small. With annual coupons, the yield to maturity is about 7.37% (technique from Chapter 7):

$$R_D = 7.37\%$$

Check: at r = 7.37%, $70 × [1 − 1/(1.0737)²²] / .0737 + $1000/(1.0737)²² ≈ $960, matching the market price. The 7% coupon rate is useless here — it reflects the market level of 8 years ago.

5. Aftertax Cost of Debt: The Interest Tax Shield

The discount rate must be expressed on an aftertax basis to match aftertax cash flows. Interest paid by a corporation is tax deductible, while payments to stockholders are not — the government effectively pays part of the interest.

Example: a firm borrows $1 million at 9%, corporate tax rate T_C = 21%:

• Annual interest = $90,000
• Tax saving = .21 × $90,000 = $18,900
• Aftertax interest bill = $90,000 − $18,900 = $71,100
• Aftertax interest rate = $71,100 ÷ $1 million = 7.11%

$$R_D \times (1 - T_C) = 9\% \times (1 - .21) = 7.11\%$$

So the WACC debt term is written (D/V) × R_D × (1 − T_C). Note: preferred dividends, like common dividends, are not tax deductible — no tax adjustment applies to them.

Extension (preview of §14.4): B.B. Lean Co. has equity worth $28M (85.76% of capital) and debt worth $4.65M (14.24%), with its debt priced to yield 11% and a 21% tax rate:

$$WACC = .8576 \times .1318 + .1424 \times .11 \times (1 - .21) = .1254, or 12.54\%$$

The debt term uses the aftertax .11 × (1 − .21) = 8.69% — this is exactly where the interest tax shield enters the WACC.

6. The Cost of Preferred Stock: A Perpetuity Formula

Preferred stock pays a fixed dividend every period forever, so a share is essentially a perpetuity (Chapters 6 and 8). The cost of preferred stock is therefore straightforward (Formula 14.3):

$$R_P = \frac{D}{P_0}$$

where D is the fixed dividend and P_0 is the current price per share. The cost of preferred stock equals its dividend yield.

Alternative: preferred stock is rated much like bonds, so R_P can be estimated from the required returns on other, similarly rated preferred shares. If preferred stock is part of the capital structure, the WACC extends to (Formula 14.7):

$$WACC = (E/V) \times R_E + (P/V) \times R_P + (D/V) \times R_D \times (1 - T_C)$$

(In the Eastman Chemical illustration, the firm had no preferred stock, so that term simply dropped out.)

7. Example: Alabama Power's Cost of Preferred Stock

On December 6, 2019, Alabama Power Co. had two issues of $100-par ordinary preferred stock trading on the NYSE:

The two estimates are close, so Alabama Power's cost of preferred stock appears to be about 4.55%. Note that both prices ($102.50, $104.70) exceed the $100 par value, so the dividend yield runs below the stated preferred dividend rate.

8. Three-Way Comparison and Key Takeaways

Three Key Takeaways:

1. Use YTM, not the coupon rate, for debt — the coupon is a historical cost; the YTM is today's market cost; book value ≠ market value
2. Interest deductibility is a hidden subsidy for debt — aftertax cost = R_D × (1 − T_C); the higher the tax rate, the lower the aftertax cost; preferred and common dividends get no such subsidy
3. Preferred stock is a perpetuity — R_P = D/P₀, its dividend yield; of the three sources, only equity requires model-based estimation, while debt and preferred costs are observable

Self-Test (Textbook Concept Questions)

1. Why is the coupon rate a bad estimate of a firm's cost of debt? — It reflects the historical cost when the bonds were issued, not today's market cost
2. How can the cost of debt be calculated? — Use the YTM on the firm's outstanding bonds, or the interest rate on similarly rated newly issued bonds
3. How can the cost of preferred stock be calculated? — R_P = D/P₀ (the dividend yield); or observe required returns on similarly rated preferred shares

1. Core Idea

The weighted average cost of capital (WACC) is the cost of capital for the firm as a whole — the required return on the overall firm. Because a firm raises capital in a variety of forms (common equity, debt, and sometimes preferred stock) with different costs, the WACC averages these costs using market-value weights.

The WACC has two interpretations: stock-price maintenance — the overall return the firm must earn on its existing assets to keep its stock price intact; and discount rate — the required return on any new investment with essentially the same risk as existing operations (the same "risk class").

Two fundamental principles: the cost of capital depends primarily on the use of the funds, not the source; the firm's target capital structure (a fixed debt-equity ratio) is taken as given.

Opening example: BASF uses value-based management and expects to earn its WACC plus a premium — it pegged its cost of capital at 9% in 2020, down from the 10% used in 2018–2019.

2. Capital Structure Weights

Weights must use market values, not book values:

Example: equity $200M + debt $50M → V = $250M, so E/V = 80% and D/V = 20%.

Practical notes: short-term liabilities are often ignored in favor of total capitalization (long-term debt + equity); non-traded debt is valued by discounting at yields on similar traded debt. Book values are a last resort for privately held firms — treat the answer "with a grain of salt."

3. The Aftertax Cost of Debt

Interest is tax deductible while dividends are not, so the government pays part of the interest. Since we discount aftertax cash flows, the discount rate must also be aftertax:

$$\text{Aftertax cost of debt} = R_D \times (1 - T_C)$$

Example: borrow $1M at 9%, T_C = 21% → annual interest $90,000; tax saving 0.21 × $90,000 = $18,900; aftertax interest $90,000 − $18,900 = $71,100 → aftertax rate 7.11% = 9% × (1 − 0.21).

The coupon-rate trap: use the current market rate on new borrowing — for a firm with public debt, the yield to maturity on outstanding bonds. The coupon rate reflects the cost of debt when the bonds were issued, not today's cost (e.g., General Tool's 30-year 7% bond selling at 96% of par, $960, has a YTM of about 7.37% — that is the cost of debt).

4. The WACC Formula

$$\text{WACC} = (E/V) \times R_E + (D/V) \times R_D \times (1 - T_C)$$

• R_E (cost of equity): dividend growth model $R_E = D_1/P_0 + g$ (e.g., Greater States: D₁ = $4 × 1.06 = $4.24, R_E = $4.24/$60 + 6% = 13.07%); or SML/CAPM $R_E = R_f + \beta_E (R_M - R_f)$ (e.g., Disney: 1.53% + 1.11 × 7% = 9.30%)
• R_D (pretax cost of debt): the observable current market rate, i.e., the YTM
• T_C (corporate tax rate): the (1 − T_C) term captures the interest tax shield

With preferred stock (cost $R_P = D/P_0$, its dividend yield, e.g., Alabama Power: $4.52/$102.50 = 4.41%):

$$\text{WACC} = (E/V) \times R_E + (P/V) \times R_P + (D/V) \times R_D \times (1 - T_C)$$

5. Worked Example: B.B. Lean Co. (Example 14.4)

Given: 1.4M shares × $20 = $28M equity; debt with $5M face value quoted at 93% → $4.65M market value; R_f = 8%, market risk premium 7%, β = 0.74; debt YTM = 11%; tax rate 21%.

$$\text{WACC} = 0.8576 \times 0.1318 + 0.1424 \times 0.11 \times (1 - 0.21) = 0.1254 = \textbf{12.54%}$$

6. Book vs. Market Weights: Eastman Chemical

Eastman (EMN): cost of equity via SML (β = 1.25, R_f = 1.53%, premium 7%) = 10.28% and via dividend growth model = 7.58%; average = 8.93%. Cost of debt: weighted average over 10 long-term bond issues → 2.96% (book) and 2.98% (market).

Key lesson: the market-to-book ratio is usually well above 1 (about 1.76 for Eastman). Book equity values overstate the debt weight and systematically underestimate the WACC — the fundamental reason for using market-value weights.

7. Uses of the WACC

（1）Capital budgeting — the warehouse problem: project costs $50M and saves $12M aftertax per year for 6 years. Target D/E = 1/3 → E/V = 0.75, D/V = 0.25; R_E = 20%, R_D = 10%, T_C = 21%:

$$\text{WACC} = 0.75 \times 0.20 + 0.25 \times 0.10 \times (1-0.21) = 0.1698 = 16.98\%$$

$$\text{NPV} = -\$50 + \$12 \times \frac{1 - 1/(1.1698)^6}{0.1698} = -\$50 + 12 \times 3.5915 = -\$6.90\text{M} \rightarrow \textbf{Reject}$$

The WACC is the correct discount rate only for projects in the same risk class as the firm (Example 14.5: D/E = 0.5 → E/V = 2/3, D/V = 1/3; WACC = 22.10%; growing-perpetuity PV = $5M/(0.2210 − 0.05) = $29.2M).

（2）Performance evaluation — EVA: $100M of capital × 12% WACC = $12M; if cash flow from assets falls short of $12M, value is being destroyed. BASF's value-based management and Stern Stewart's EVA both benchmark value creation against the WACC.

8. Three Key Takeaways

1. Every piece matters — market-value weights (E/V, D/V), the aftertax cost of debt R_D(1−T_C), and a cost of equity cross-checked via DGM and SML;
2. Coupon rate ≠ cost of debt — use the current YTM; book values are a fallback that understates the WACC;
3. The WACC is a firm-wide hurdle — valid only for projects with risk similar to existing operations; otherwise switch to the pure play or subjective approach.

1. Core Idea: Cost Depends on the Use of Funds, Not the Source

The firm's WACC is the overall required return on the firm as a whole. Strictly speaking, the WACC is the appropriate discount rate only when the proposed investment is a replica of the firm's existing operations — i.e., when the project and the firm belong to the same risk class.

Rule of thumb: a pizzeria opening a new location, a retailer opening a new store, a manufacturer expanding production — use the WACC. For projects with clearly different risk — the WACC fails.

2. The SML vs. the WACC: Two Errors from a Single Cutoff (Figure 14.1)

Setup: risk-free rate $R_f = 7\%$, market risk premium $R_M - R_f = 8\%$; an all-equity firm with $\beta = 1$, so WACC = cost of equity = 15%.

$$\text{SML:} \quad R = R_f + \beta \times (R_M - R_f)$$

Error 1: profitable low-risk projects get rejected — Project A earns 14%, above its 11.8% required return (it plots above the SML and is a good investment), yet fails the 15% cutoff. Error 2: unprofitable high-risk projects get accepted — Project B earns 16%, below its 16.6% required return (it plots below the SML), yet passes the 15% cutoff.

3. Long-Run Consequences and Divisional Cost of Capital

• A firm that evaluates everything with its WACC will simultaneously accept unprofitable investments and become increasingly risky (low-risk projects are rejected, high-risk ones are attracted, so the asset mix grows riskier over time).
• The same problem hits multi-division firms: imagine a corporation with a regulated electric company (low risk) and an electronics manufacturing operation (high risk). The firm's overall cost of capital is really a mixture of two divisional costs.
• If the divisions compete for funds under one WACC cutoff, the riskier division wins (its returns are higher, with risk ignored), while the less glamorous division's great projects are starved. Large U.S. corporations are well aware of this problem, and many work to develop separate divisional costs of capital.

4. The Pure Play Approach

Since we cannot observe betas of untraded investments directly, we look outside the firm for other investments in the same risk class and use the market-required return on them as the discount rate.

• Example (electricity division): identify publicly traded electric companies; suppose a typical one has $\beta = 0.80$, AA-rated debt, and a capital structure of roughly 50% debt / 50% equity — build a WACC for that typical electric company and use it for the division.
• Example (new line of business): if McDonald's launched McPhones, it would look at companies already in consumer electronics (Apple, Samsung). Neither is ideal: Apple also does music, TV, and content production; Samsung also sells TVs and appliances — a true pure play is hard to find.

Limitation: no suitable companies may exist at all, making an objective discount rate very difficult to determine; still, being aware of the issue at least reduces the mistakes of a blanket WACC cutoff.

5. The Subjective Approach

Given the difficulty of objectively setting discount rates for individual projects, firms often adjust the overall WACC subjectively. Suppose the firm's WACC is 14%:

• Effect (Figure 14.2): compared with a single WACC cutoff, the magnitude of potential error shrinks — Project A would be accepted under the WACC cutoff but is rejected once classified as high risk.
• Takeaway: some risk adjustment, even a subjective one, is probably better than none.
• In principle, each project's required return should be determined objectively; in practice, unavailable information or excessive cost often limits us to subjective adjustments.

6. Positioning the Two Approaches

7. Three Key Takeaways

1. The WACC is a benchmark, not a master key — it belongs to the risk class of the firm's existing assets; projects of different risk need different discount rates.
2. Risk must be priced — a blanket WACC cutoff simultaneously rejects good low-risk projects and accepts bad high-risk ones, making the firm riskier over time.
3. Some adjustment beats none — when pure plays are unavailable, the subjective approach is a practical second best.

1. Core Idea

The CAPM states the required return as $E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i$. Of its three inputs, R_f is observable and the market risk premium can be estimated from history; the only firm-specific unknown is beta. In practice beta is estimated by an OLS regression of excess returns — the market model:

$$R_{it} - R_{ft} = \alpha_i + \beta_i [R_{Mt} - R_{ft}] + \varepsilon_t$$

The slope is the estimated beta (systematic risk); the intercept is Jensen's alpha. This is one modeling chain running from raw prices to a discount rate:

Prices → monthly returns → excess-return regression → β → CAPM cost of equity → WACC → project discount rate

2. Data Preparation

Three steps: 1. Monthly returns: R_t = P_t/P_{t−1} − 1, computed on adjusted prices 2. Rate conversion: T-bill quotes are annualized; convert to monthly: R_f,m = (1 + R_f,annual)^(1/12) − 1 (a rough ÷12 is acceptable) 3. Excess returns: stock return − R_f and market return − R_f, aligned into two columns for the regression

3. Estimating Beta by Market-Model Regression

$$\beta_i = \frac{\text{Cov}(R_i - R_f,\ R_M - R_f)}{\text{Var}(R_M - R_f)} = \text{regression slope}$$

Reading the output: - α (intercept) — Jensen's alpha, the "excess" return. If CAPM held exactly, α = 0; α > 0 means the asset plots above the SML, α < 0 below it - Residuals ε — the part of the monthly return unexplained by the market model (firm-specific news); they feed the appraisal ratio = α ÷ standard deviation of residuals, the standard measure for evaluating mutual fund performance - R² — the fraction of the stock's variance explained by the market (systematic risk)

Caution: not all betas are created equal. Providers use different methods, and estimates can differ materially. Eastman (EMN): Yahoo! reports β = 1.39, while Value Line (which shrinks extreme betas toward 1) reports 1.25 — the textbook uses 1.25. Boeing's Yahoo! beta is 1.45, about one and a half times the systematic risk of a typical stock. The average asset has β = 1.0. Sample betas (Table 13.8): Coca-Cola 0.55, McDonald's 0.66, Johnson & Johnson 0.70, Visa 0.93, Mastercard 1.08, Disney 1.11, Sherwin-Williams 1.23, Shopify 1.59.

Window debate: 36 vs. 60 months — shorter windows are more current but noisier; longer windows are more stable but may be stale. Daily, weekly, quarterly, or annual data are also options, each with trade-offs.

4. From Beta to the Cost of Equity (CAPM)

$$R_E = R_f + \beta (R_M - R_f)$$

• Example 13.8: R_f = 4%, market risk premium 8.6%, β = 1.3 → R_E = 4% + 1.3 × 8.6% = 15.18%; if beta doubles to 2.6, the risk premium doubles to 22.36% → R_E = 26.36%
• Eastman: R_f = 1.53% (T-bill), β = 1.25 (Value Line), premium 7% → R_E = 1.53% + 1.25 × 7% = 10.28%; cross-checked with the dividend growth model (7.58%); both plausible → average 8.93%

Portfolio beta is a weighted average like portfolio expected return: β_P = Σx_i·β_i. Half Shopify (1.59) + half Coca-Cola (0.55) → β_P = 0.50×1.59 + 0.50×0.55 = 1.07; Example 13.6 with weights 10/20/30/40% and betas 0.80/0.95/1.10/1.40 → β_P = 1.16 (> 1, above-average systematic risk).

5. WACC: Connecting Beta to the Firm Level

$$WACC = (E/V) \times R_E + (D/V) \times R_D \times (1 - T_C)$$

Example 14.4 (B.B. Lean Co.):

$$\text{WACC} = 0.8576 \times 0.1318 + 0.1424 \times 0.11 \times (1-0.21) = \textbf{12.54%}$$

Details: - The cost of debt is the current market rate (YTM), not the coupon rate (a historical cost) - Weights must be market values: Eastman's book-value WACC is 6.35% (E/V = 0.61), but the market-value WACC is 7.04% (E/V = 0.71) — book equity overstates the debt weight and systematically understates the WACC

6. Excel Implementation

1. Build three monthly-return columns (stock, S&P 500, T-bill), then two excess-return columns
2. Data → Data Analysis → Regression: Y range = stock excess returns, X range = market excess returns; check the "Residuals" box (needed later for the appraisal ratio)
3. Read the output: the slope (X Variable 1 coefficient) = β; the intercept = α; compare R² across alternative estimates
4. Appraisal ratio: compute the standard deviation of the saved residuals, then α ÷ STDEV(residuals)
5. WACC worksheet: enter β, R_f, and the market risk premium → R_E = R_f + β × premium; then enter E, D, R_D, T_C → WACC; the same sheet can compare 36-month vs. 60-month betas

7. Python Implementation

import statsmodels.api as sm

# Inputs: stock_excess / mkt_excess are monthly excess-return columns (T-bill adjusted)
X = sm.add_constant(mkt_excess)            # constant = Jensen's alpha
model = sm.OLS(stock_excess, X).fit()
alpha, beta = model.params                 # intercept = alpha, slope = beta
r2 = model.rsquared
appraisal = alpha / model.resid.std()      # appraisal ratio = alpha / residual std

def wacc(E, D, rE, rD, T):                 # Example 14.4: B.B. Lean
    V = E + D
    return (E / V) * rE + (D / V) * rD * (1 - T)

print(f"beta={beta:.3f}, alpha={alpha:.4f}, R2={r2:.3f}")
print(f"WACC={wacc(28e6, 4.65e6, .1318, .11, .21):.2%}")   # → 12.54%

8. Three Key Takeaways

1. Beta is a regression slope, not a look-up number — the window (36 vs. 60 months), return frequency, market index, and provider's method all change the estimate (Eastman: 1.39 vs. 1.25); cross-check several sources before use
2. Alpha and the residuals have independent value — α > 0 means the asset plots above the SML; the appraisal ratio (α ÷ residual standard deviation) is the standard metric for fund-manager performance
3. One chain, end to end — prices → regression β → CAPM cost of equity → market weights + aftertax debt cost → WACC (B.B. Lean 12.54%, Eastman 7.04%); the WACC is valid only for projects in the same risk class as the firm's existing operations