Lesson 11 ended with a promise hanging in the air: if diversification eliminates unsystematic risk for free, the market only pays a premium for systematic risk. Fine—but how much does it pay? How do we turn that statement into a number?Today, we make good on that promise. We will explore Markowitz’s efficient frontier, introduce beta—the number that measures each stock’s exposure to systematic risk—and arrive at the CAPM (Capital Asset Pricing Model), the most widely used (and most debated) formula in modern finance. By the end, that k rate that appeared so mysteriously in the Gordon model in Lesson 10 will finally have a father, a mother and a birth certificate.1. The efficient frontier: a menu of the best possible worldsTake every possible combination of risky assets—infinitely many portfolios, each with its own risk (σ) × expected return pair. Plot them on a chart and they form a cloud. Markowitz realized that, within this cloud, most portfolios are simply poor choices: for the same level of risk, another portfolio offers a higher return; for the same return, another carries less risk. We say these portfolios are dominated.Once the dominated portfolios are discarded, what remains is the cloud’s upper edge: the efficient frontier—the set of portfolios that deliver the maximum expected return for each level of risk. A rational investor will always choose from the frontier; which point they choose depends solely on their appetite for risk.And what happens when we add the risk-free asset to the menu (our old acquaintance, the government bond—Lessons 3 to 5)? The final piece of magic: combining the risk-free asset with a single special portfolio on the frontier (the tangency portfolio, which theory identifies with the market portfolio—in practice, a broad-based index) outperforms every other combination. The powerful conclusion is that investors do not need to pick stocks one by one—they need to choose the allocation: how much to put in the diversified market, and how much in the risk-free rate.Under Markowitz’s theory, a portfolio is considered efficient when...Answer: It offers the highest possible expected return for its level of risk—(equivalently: the lowest risk for its return). Portfolios outside the frontier are dominated: someone else is offering more for less. Efficiency does not mean an absence of risk—it means an absence of waste.2. Beta: the speedometer of systematic riskIf a diversified investor is exposed only to systematic risk, the relevant question about any stock is no longer "how much does it swing?" (σ, its total risk), but "how much does it swing WITH the market?". That is beta (β): the sensitivity of a stock’s return to movements in the market portfolio (technically, the stock’s covariance with the market divided by the market’s variance—in practice, the slope of the line relating the stock’s returns to those of the index).Here is the reading guide:β = 1: the stock tends to move with the market. If the Ibovespa rises 10%, the stock tends to rise ~10%.β > 1: amplifier (aggressive). A β of 1.6 tends to turn a +10% market move into +16%—and a −10% move into −16%. Typical of cyclical stocks: discretionary retail, construction companies and airlines (hello, Captain Túlio from Lesson 11!).β < 1: defensive. A β of 0.5 cushions the blow: the stock tends to rise and fall by half as much as the market. Typical of utilities (power, water and sanitation) and companies with stable demand.β ≈ 0: detached from the market—the risk-free asset itself is an example. (A negative β—an asset that rises when everything else falls—is a valuable rarity; certain hedges come close to this.)Notice the conceptual split: a junior mining company may have extremely high σ (enormous total risk), but if much of that risk is SPECIFIC—risk that a diversified investor can eliminate for free—its β may be modest. Volatility measures the risk of someone who concentrates; beta measures the risk of someone who diversifies.A stock has a beta of 1.5. On a trading day when the market rises 2%, the expected move in the stock, based on its beta, is approximately...Answer: 3.0%—beta is the multiplier of the market’s move: 1.5 × 2% = 3%. The reverse is also true: if the market falls 2%, the expected move is −3%. Beta magnifies moves in both directions—there is no amplifier that works only for good news.3. CAPM: the formula that prices riskPut the pieces together: (1) only systematic risk is rewarded; (2) beta measures the amount of systematic risk. The conclusion follows naturally—the required return on an asset must be the risk-free rate plus a premium proportional to its beta:E[R] = Rf + β × (E[Rm] − Rf)Here, Rf is the risk-free rate, E[Rm] is the expected return on the market portfolio, and the difference (E[Rm] − Rf) is the market risk premium—how much the market pays above the safe rate to anyone willing to bear the full amount of systematic risk (β = 1).3.1 Worked example—and the return to GordonSuppose Rf = 10% per year (the long end of our yield curve) and the market risk premium is 6% per year:Aggressive stock, β = 1.2: E[R] = 10% + 1.2 × 6% = 17.2%Defensive stock, β = 0.6: E[R] = 10% + 0.6 × 6% = 13.6%And look who shows up: in Lesson 10, we discounted Elétrica Luz do Sul at k = 14% "because the business risk justified it." Now the math ties together from the inside: a defensive utility with β ≈ 0.67 gives k = 10% + 0.67 × 6% ≈ 14%. The Gordon model’s k is CAPM’s E[R]—valuation and portfolio theory are the same building viewed from different doors. This is also how companies estimate their cost of equity: the minimum hurdle that projects and acquisitions must clear to create value.3.2 The SML and alpha: the yardstick for cheap and expensiveCAPM draws a line—the SML (Security Market Line)—on a chart of expected return × beta: it starts at Rf (β = 0) and passes through the market portfolio (β = 1). It is the fair-return yardstick: for each beta, the return required by systematic risk.What about assets that fall off the line? If your analysis (a well-built Gordon model, for example) projects a return for a stock above what the SML requires for its beta, that difference is alpha (α): excess return not explained by risk—the signature of an undervalued stock (or a talented manager). Below the line, alpha is negative: the return is insufficient for the risk, making the stock overpriced. In CAPM equilibrium, alphas should be zero; hunting for them is the business of active management.Consider a risk-free rate of 10% per year and a market risk premium of 6% per year. Under CAPM, the required return on a stock with a beta of 1.5 is...Answer: 19.0%—E[R] = 10% + 1.5 × 6% = 19%. The 24% option multiplies beta by the market’s entire return (1.5 × 16%) instead of multiplying it only by the PREMIUM—the most common CAPM calculation error.Under CAPM, the required return premium on a stock is proportional to its...Answer: Systematic risk, measured by beta—the market does not pay for risk that investors can eliminate for free through diversification (specific risk). This is the central thesis linking Lessons 11 and 12: σ measures total risk; β measures the risk that is rewarded.4. CAPM on trial: uses and limitationsA moment of intellectual honesty to close this section. CAPM is ubiquitous—used for corporate cost of capital, fund evaluation, valuation reports and regulatory arbitrations—because it is simple, logical and disciplined. At the same time, it has faced decades of empirical criticism: betas change over time, the market risk premium is a difficult estimate, and documented anomalies (size, value and momentum) suggest that a single factor does not tell the whole story—hence the multifactor models that populate modern research.The mature professional approach is to use CAPM as a disciplined starting point—a common language for discussing risk and return—and never as an oracle. Just as with Gordon, the model’s value lies less in the final number than in the assumptions it forces us to make explicit.“All models are wrong, but some are useful.”- George Box, statistician5. Tying it all together—and wrapping up equitiesThe essentials of the lesson:The efficient frontier brings together the non-dominated portfolios: maximum return for each level of risk. With the risk-free asset, the decision becomes an allocation between the safe rate and the diversified market.[ m]{Beta} measures systematic risk: β = 1 moves with the market; β > 1 amplifies (cyclicals); β < 1 cushions (defensives). σ is the risk of someone who concentrates; β, the risk of someone who diversifies.CAPM: E[R] = Rf + β × market risk premium—the price of systematic risk, the source of the Gordon model’s k and companies’ cost of equity.The SML is the yardstick for fair return by beta; alpha is the deviation from it—the trophy of active management and the signal that an asset is cheap or expensive.CAPM is a compass, not an oracle: start with it, make your assumptions explicit and understand its criticisms.With that, we wrap up equities: we know what a stock is (Lesson 9), how much it is worth (Lesson 10) and how risk and return balance each other (Lessons 11–12). In the next lesson, the course’s final continent: derivatives and futures markets—where you do not buy the asset itself, but the COMMITMENT tied to the asset, and where exporters, speculators and treasuries play the market’s fastest game of chess.Exercise for class discussionThe three candidates. Let’s think it through together.Your investment manager uses Rf = 10% and a market risk premium of 5% per year. The research team projected returns for three stocks (using the models from Lesson 10) and calculated their betas:Defensora S.A. (water and sanitation): β = 0.5; projected return = 14% p.a.Mediana S.A. (consumer goods): β = 1.0; projected return = 15% p.a.Turbina S.A. (construction): β = 1.6; projected return = 21% p.a.Discuss with your group:(a) Using the SML, calculate the REQUIRED return for each one. Then calculate alpha (projected − required). Which ones are undervalued according to the CAPM yardstick?(b) Turbina has both the highest alpha AND the highest projected return. Does that settle the argument in its favor? What does a beta of 1.6 imply for a year in which the market falls 10%? (Estimate each stock’s return in that scenario using Rf and the betas.)(c) The committee has a conservative mandate: losing more than 10% in a year is unacceptable for the client. How does this change the choice—and what does it teach us about alpha versus suitability?(d) A junior analyst argues: "Defensora has the lowest projected return, so eliminate it." Another counters: "Per unit of systematic risk, it is the best buy." Who is right, and why?(e) What weaknesses in the ASSUMPTIONS (the Lesson 10 projections, estimated betas and 5% premium) could undermine the conclusions? How would you protect the decision against them?Instructor guidance: (a) Required returns: Defender 10 + 0.5×5 = 12.5%; Median 10 + 1×5 = 15%; Turbine 10 + 1.6×5 = 18%. Alphas: +1.5 percentage points, 0 and +3 percentage points — Defender and Turbine are above the SML (undervalued); Median is right on the line, fairly priced. (b) It does not end there: with the market falling 10% (market return −10%, realized premium of −20 percentage points), the CAPM expectation is 10 + β×(−20): Defender ≈ 0%; Median ≈ −10%; Turbine ≈ −22% — the higher alpha comes bundled with twice the systematic exposure. (c) With a −10% floor, Turbine violates the mandate in plausible scenarios: the choice shifts to Defender (positive alpha AND a compatible profile) — alpha is desirable, suitability is mandatory; without suitability, the client sells at the bottom and turns risk into a realized loss (echoing Lessons 4 and 8). (d) The second point: absolute return ignores the price of risk; Defender's alpha per beta (1.5/0.5 = 3 alpha per unit of beta) is twice Turbine's (3/1.6 ≈ 1.9) — rejecting an investment based on gross return is like comparing runners without looking at the race distance. (e) Alphas of +1.5 to +3 percentage points fall within the margin of error of Gordon projections (Lesson 10 showed that a 1 percentage-point change in g moves value by double digits), historical betas change, and the 5% premium is a debatable estimate — safeguards: sensitivity analysis, a minimum margin of safety before acting, and diversification across the investment theses instead of going all-in on the largest one. Bottom line: CAPM organized the conversation — it did not settle it; that is precisely its role.Exercises1) (New - CESGRANRIO style) Assume a risk-free rate of 8% per year and an expected return on the market portfolio of 14% per year. Under the CAPM, the required return on a stock with a beta of 1.25 is:A) 15.5%.B) 17.5%.C) 14.0%.D) 10.0%.E) 22.0%.2) (New - CESGRANRIO style) A stock has a required return of 18% per year in a market where the risk-free rate is 10% per year and the market risk premium is 5% per year. Under the CAPM, the stock's beta is:A) 1.6.B) 0.8.C) 1.8.D) 2.25.E) 1.0.3) (New - CESGRANRIO style) Under the CAPM, an asset with a beta of zero should offer an expected return equal to:A) the risk-free rate.B) zero.C) the return on the market portfolio.D) the market risk premium.E) twice the risk-free rate.4) (New - CESGRANRIO style) The beta of the market portfolio itself is, by definition, equal to:A) 1.B) 0.C) −1.D) its standard deviation.E) the risk-free rate.5) (New - FGV style) In Markowitz portfolio theory, the efficient frontier consists of portfolios that:A) maximize expected return for each level of risk and are not dominated by any other combination.B) completely eliminate systematic risk through diversification.C) have the lowest expected return in the opportunity set.D) contain only the risk-free asset.E) have a correlation of +1 among all their assets.6) (New - FGV style) Shares of electric utility and sanitation companies typically have a beta below 1. This indicates that such shares tend to:A) fluctuate less than the market, dampening both gains and declines in the index.B) fluctuate more than the market, amplifying its movements.C) always earn more than the market portfolio.D) be immune to recessions and systemic crises.E) have zero idiosyncratic risk.7) (New - CEBRASPE style, judge the statement) Judge the statement: "According to the CAPM, the required return premium on a stock is proportional to its total risk, measured by the standard deviation of returns, since investors demand compensation for all the volatility they bear."( ) Correct ( ) Incorrect8) (New - CEBRASPE style, judge the statement) Judge the statement: "Under the CAPM, an asset whose projected return lies above the Security Market Line has positive alpha, suggesting a return higher than that required for its level of systematic risk."( ) Correct ( ) IncorrectAnswer Key1) A — Premium = 14% − 8% = 6%; E[R] = 8% + 1.25 × 6% = 15.5%. Option B (17.5%) multiplies beta by the market return (1.25 × 14%) instead of by the premium — the classic CAPM error.2) A — β = (18% − 10%) / 5% = 8/5 = 1.6. This is the CAPM read backward: the asset's required premium (8 percentage points) divided by the market premium (5 percentage points).3) A — With β = 0, the asset carries no systematic risk — and idiosyncratic risk is not compensated. What remains is exactly Rf: this is the case of the risk-free asset itself.4) A — The market compared with itself has a sensitivity of 1: this is the definition that anchors the entire beta scale.5) A — Efficiency means non-dominance: maximum return for each level of risk (or minimum risk for each return). The frontier does not eliminate systematic risk (option B) — it simply avoids wasting it.6) A — β < 1 means dampening in both directions: it rises less in upswings and falls less in downturns. Defensive does not mean immune (option D): in a systemic crisis, it falls too — just by less.7) Incorrect — The CAPM compensates only for SYSTEMATIC risk, measured by beta. Total risk (σ) includes idiosyncratic risk, which a diversified investor eliminates for free — and no one pays a premium for something that can be eliminated at no cost.8) Correct — Above the SML = projected return higher than the return required for that beta = positive alpha, an indication that the stock is undervalued (or that the projection is overly optimistic — both interpretations should be considered together).
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