In the previous lesson, you learned how to buy a piece of a company. One essential question remained: how much should you pay for it?B3 trading gives you a price every second. But price is what you pay; value is what you get — and the art of estimating the value of a stock, known as valuation, is what separates investing from betting. In this lesson, you will learn the two approaches the market has used for a century: the Gordon model, which calculates value from future dividends (and is, at heart, our old fixed-income pricing with an elegant twist), and multiples, the everyday comparative shortcut used on trading desks.A note on intellectual honesty before we begin: no model produces THE right value. They produce disciplined estimates — and, as you will see, discipline matters more than precision.1. The principle behind it all: future cash flows brought back to todayRemember the LTN? Its value was the present value of a future R$ 1.000. A stock follows the same principle: the value of any asset is the present value of the cash flows it will generate. The difference comes down to three twists:The stock’s cash flows (dividends) are uncertain — no one guarantees them.There is no maturity date: the cash flows are, in theory, perpetual.The discount rate k does not appear on a screen: it must reflect the business’s risk — the riskier the company, the higher k and the lower the present value.Where does k come from? For now, keep the conceptual formula in mind: risk-free rate + business risk premium. Turning that phrase into a number is the job of CAPM — the star of Lesson 12.2. Perpetuity: the most elegant formula in financeIf a stock paid a constant dividend D forever, its value would be the sum of all the present values — which mathematics compresses into a fraction of rare elegance:V = D / kA stock that will pay R$ 6,00 a year forever, discounted at 15% per year, is worth 6/0,15 = R$ 40,00. It is that simple: the sum of infinitely many terms fits into a division.3. The Gordon model: a growing perpetuityConstant dividends forever are unrealistic — companies grow (at least in line with inflation). Myron Gordon incorporated perpetual growth g into the perpetuity:V = D1 / (k − g)Here, D1 is the dividend expected in the next year (note: if the figure is the dividend that has just been paid, D0, project it forward one step: D1 = D0 × (1+g)), k is the discount rate, and g is the perpetual growth rate of dividends.3.1 Worked exampleElétrica Luz do Sul is expected to pay R$ 6,00 per share in dividends next year. Its dividends are growing steadily at 4% per year, and the business risk justifies a required return of 14% per year.V = 6 / (0,14 − 0,04) = 6 / 0,10 = R$ 60,00If the stock trades at R$ 48, the model suggests it is undervalued; at R$ 75, it is overvalued. This comparison between estimated value and the market price is at the heart of fundamental analysis.3.2 The existence condition and the sensitivity testThe formula requires k > g — perpetual growth greater than the discount rate would produce infinite value, an economic absurdity (no company can grow faster than the economy forever). In practice, perpetual g stays close to the economy’s long-term nominal growth rate.Now for a lesson in humility. Rework the Luz do Sul calculation by changing ONE parameter at a time:g from 4% → 5%: V = 6/0,09 = R$ 66,67 (+11%)k from 14% → 15%: V = 6/0,11 = R$ 54,55 (−9%)A one-percentage-point change in uncertain assumptions moved “fair value” by double digits. The lesson: the Gordon model does not produce a single number — it produces a range, and forces the analyst to make explicit the assumptions behind the price being paid. That transparency is precisely what makes it valuable.A stock is expected to pay a dividend of R$ 6,00 next year, with perpetual growth of 5% per year. If the required return is 15% per year, the stock’s value under the Gordon model is...Answer: R$ 60,00 — V = 6/(0,15 − 0,05) = 6/0,10 = 60. The R$ 40,00 option is the perpetuity WITHOUT growth (6/0,15): forgetting g in the denominator is the classic mistake.The Gordon model only produces valid results when...Answer: The discount rate (k) is greater than perpetual growth (g) — with g ≥ k, the denominator reaches zero or becomes negative and “value” explodes to infinity or turns negative, an absurd result. No company can grow faster than its discount rate forever.3.3 When the Gordon model does NOT workThe model assumes a mature company with stable payouts and predictable growth — the profile of utilities, mature banks and concession businesses. It breaks down for companies that do not pay dividends (they reinvest everything — the value exists, but lies in retained earnings rather than D1); high-growth companies (current g > k, requiring multi-stage models); and companies with highly volatile earnings (what perpetual dividend would you project for a cyclical company?). In these cases, the market discounts complete cash flows (DCF) or turns to the lesson’s second approach.4. Multiples: the comparative shortcutIf the Gordon model is a precision scale, multiples are a quick ruler: instead of projecting cash flows, you compare the stock price with some fundamental measure — and with its peers in the same sector.4.1 P/E — Price/EarningsThe most famous multiple: P/E = stock price / earnings per share (EPS). The intuitive interpretation: how many years of current earnings you are paying for the company. P/E 8 = eight years; P/E 30 = thirty.A high P/E does not shout “overpriced!” — it shouts “the market expects growth”: investors are paying a lot for current earnings because they believe future earnings will be higher. A low P/E may be a bargain — or a declining company that is deservedly discounted (the “value trap”). The multiple raises the question; it does not answer it. Technical caveats: P/E does not work with negative earnings and can mislead when earnings are cyclical or inflated by nonrecurring events.4.2 P/BV — Price/Book ValueIt compares the price with book value per share. P/BV < 1: the stock trades below book value — the market is paying less than R$ 1 for every R$ 1 of equity. That may be an opportunity... or the market may be signaling that the company’s equity generates inadequate returns (or is overstated on the books). It is widely used for banks, whose business is essentially their balance sheet.4.3 Dividend yieldDY = dividends paid over the last 12 months / price. The stock’s implicit “interest rate”: a 6% DY means that, assuming the payout is maintained, you receive 6% a year in dividends on the price you paid. Notice the elegant connection with the Gordon model: rearranging the formula gives k = D1/V + g — the stock’s expected return is yield PLUS growth. The two ends of the lesson meet.4.4 Relative valuation in practiceThe standard approach: the company is worth the peers’ multiple × its fundamental metric. If the power sector trades at a P/E of 8 and Luz do Sul earns R$ 7,00 per share, relative valuation suggests 8 × 7 = R$ 56,00. The method’s strength: it is quick and anchored to real prices. Its weakness: if the entire sector is expensive, you will “validate” the excess — multiples compare; they do not establish fundamental value.A company reports earnings per share of R$ 4,00, while its sector peers trade at an average P/E of 12. Under relative valuation, the stock’s reference value is...Answer: R$ 48,00 — value = peers’ multiple × fundamental metric = 12 × 4 = 48. The R$ 3,00 option divides instead of multiplying (4/12 × 9?) — pay attention to the direction of the calculation: P/E × E = P.A stock trades at a P/BV of 0,7. The correct interpretation of this multiple is that...Answer: The market prices the stock below its accounting book value, which requires investigating why — a P/BV of 0,7 means R$ 0,70 for every R$ 1 of equity. It may be a bargain, or the equity may generate poor returns (or be worth less than the books say). A multiple raises the question; it does not deliver the verdict.5. Price × value: the analyst’s synthesisThe two approaches complement each other: the Gordon model (absolute valuation) says how much the company is worth on its own; multiples (relative valuation) say how much it is worth compared with its neighbors. When both point in the same direction — and the market price is well below both estimates — the analyst has found what Benjamin Graham called a margin of safety: the gap between estimated value and the price paid, which provides a cushion against the inevitable errors in the assumptions.And what if the methods disagree? Excellent: the divergence reveals which assumption is driving the disagreement — and that is where real analysis begins.“Price is what you pay; value is what you get.”- Warren Buffett6. Bringing it all togetherThe essentials of the lesson:Stock value = PV of future dividends — the fixed-income principle applied to uncertain, perpetual cash flows with a discount rate that incorporates risk.Perpetuity: V = D/k. Gordon: V = D1/(k − g), requiring k > g, a mature company and stable payouts — and always accompanied by sensitivity analysis, since small changes in g and k can significantly move value.Multiples: P/E (years of earnings; high = growth expectations), P/BV (price vs. book value; standard for banks), DY (the stock’s yield — and k = DY + g ties everything together). Relative valuation = peers’ multiple × fundamental metric.Absolute + relative + margin of safety = the three pillars of fundamental decision-making.In the next lesson, we tackle the variable that has been hanging over everything: risk. What it is, how it is measured (volatility, correlation), and why diversification is the only free lunch in finance — the springboard to the CAPM of Lesson 12, which will finally tell us where k comes from.Exercise for classroom discussionThe analysts’ duel. Let’s think it through together.Elétrica Luz do Sul currently trades at R$ 48,00. Two analysts presented their valuations to the committee:Absolute Analyst (Gordon): D1 = R$ 6,00; g = 4%; k = 14% → V = R$ 60,00. “It is 20% below value. Buy.”Relative Analyst (multiples): EPS = R$ 7,00; average P/E for the power sector = 8 → V = R$ 56,00. “Confirmed: it is also cheap relative to its peers. Buy.”At the meeting, the portfolio manager plays devil’s advocate. Discuss with your group:(a) “What if perpetual growth is 3%, not 4%? What if regulatory risk justifies a k of 15%?” Recalculate the Gordon model under both scenarios. Does the buy thesis survive?(b) “The entire power sector may be cheap or expensive. What does the peers’ P/E of 8 NOT tell us?”(c) The two methods produced similar values (60 and 56). Is that strong evidence that the “true” value is somewhere in that range — or could the two errors be correlated? Think about what happens to k, g and the sector’s P/E if interest rates in the economy rise.(e) What margin of safety would you require before buying—and what would make you sell after you had bought?Guidance for the instructor: (a) g = 3%: V = 6/0.11 = 54.55; k = 15%: V = 6/0.11 = 54.55 (the same calculation by coincidence of the parameters)—the thesis survives, but the margin falls from 20% to 12%; combining both deviations (k = 15% and g = 3%): V = 6/0.12 = 50, a margin of 4%, effectively zero: the investment depends on optimistic assumptions. (b) Relative P/E only positions the company WITHIN the sector: if the entire sector is depressed or euphoric, the multiple validates the collective mistake—comparison is not a fundamental. (c) The errors ARE correlated: rising interest rates increase k (bringing down the Gordon valuation) and compress P/E ratios across the sector (bringing down the relative valuation) at the same time—two methods pointing to the same conclusion are not two independent pieces of evidence; connect this to the yield curve from Lesson 3 and k = Rf + premium. (d) DY = 6/48 = 12.5%; implied return k = 12.5% + 4% = 16.5% nominal. Against IPCA + 7% (≈ 11–12% nominal with 4–5% inflation), the ~4–5 percentage-point premium exists, but it is compensation for the uncertainty surrounding g and the business—the right question is whether that premium is sufficient, and it sets up the risk-premium discussion in Lesson 12. (e) There is no answer key: the goal is to get the class to articulate its required margin (e.g., buy only below 45–50) and its exit discipline (the price reaching fair value, assumptions deteriorating)—valuation becomes a process, not a number.Exercises1) (Original—in the style of CESGRANRIO) A stock is expected to pay a dividend of R$ 8.00 next year. Assuming perpetual growth of 4% per year and a required return of 14% per year, the stock's value under the Gordon model is:A) R$ 80.00.B) R$ 57.14.C) R$ 200.00.D) R$ 44.44.E) R$ 100.00.2) (Original—in the style of CESGRANRIO) A company has just paid a dividend of R$ 3.00 per share (D0). Its dividends grow by 6% per year, and the return required by investors is 12% per year. Under the Gordon model, the stock's value is:A) R$ 53.00.B) R$ 50.00.C) R$ 25.00.D) R$ 26.50.E) R$ 47.17.3) (Original—in the style of CESGRANRIO) A stock is trading at R$ 50.00 and is expected to pay a dividend of R$ 5.00 next year, with estimated perpetual growth of 3% per year. Under the Gordon model, the annual return implied by this price is:A) 13%.B) 10%.C) 3%.D) 8%.E) 16%.4) (Original—in the style of CESGRANRIO) A company reports earnings per share of R$ 4.00, while comparable companies in its sector trade at an average P/E ratio of 12. Under relative valuation, the stock's reference value is:A) R$ 48.00.B) R$ 16.00.C) R$ 3.00.D) R$ 30.00.E) R$ 12.00.5) (Original—in the style of CESGRANRIO) A stock trading at R$ 60.00 paid R$ 3.00 in dividends per share over the past 12 months. Its dividend yield is:A) 5%.B) 20%.C) 3%.D) 8%.E) 12%.6) (Original—in the style of FGV) Regarding the price-to-earnings ratio (P/E), which statement is correct?A) A high P/E usually reflects expectations of earnings growth and does not, by itself, indicate that the stock is overvalued or undervalued.B) The higher the P/E, the cheaper the stock is relative to its earnings.C) P/E is the most appropriate metric for valuing companies with recurring losses.D) P/E measures the proportion of earnings distributed as dividends.E) Companies in the same sector necessarily have the same P/E.7) (Original—in the style of CEBRASPE, judge the statement) Judge the following statement: "The Gordon model assumes constant perpetual dividend growth and requires the discount rate to be higher than the growth rate, making it more suitable for mature, stable dividend-paying companies than for fast-growing companies or companies that do not pay dividends."( ) True ( ) False8) (Original—in the style of CEBRASPE, judge the statement) Judge the following statement: "A price-to-book ratio below 1 necessarily indicates that a stock is undervalued, constituting an objective buy recommendation."( ) True ( ) FalseAnswer Key1) A — V = 8/(0.14 − 0.04) = 8/0.10 = 80. Option B (57.14) is the no-growth perpetuity (8/0.14): the trap is forgetting g in the denominator.2) A — First project the dividend: D1 = 3.00 × 1.06 = 3.18. Then: V = 3.18/(0.12 − 0.06) = 3.18/0.06 = 53.00. Using D0 directly in the formula leads to option B (50.00)—the model's most frequently tested trick.3) A — Rearranging the Gordon model: k = D1/P + g = 5/50 + 0.03 = 0.10 + 0.03 = 13%. Expected return is yield plus growth—the formula is being used "backward."4) A — Relative valuation: value = peers' P/E × EPS = 12 × 4 = 48.5) A — DY = 3/60 = 5% per year. Yield is the implicit "interest rate" represented by the income received on the price paid.6) A — A high P/E incorporates growth expectations; a low one may indicate a bargain or deserved decline. The multiple opens the investigation; it does not close it. And when a company has losses (option C), P/E simply does not apply.7) True — This is an accurate portrait of the model: a perpetuity with constant growth, the condition k > g, and its natural habitat in utilities and mature dividend payers; outside that setting, use multistage models or DCF.8) False — P/B < 1 only means that the price is below book value. It may be an opportunity, or it may reflect a low-return asset base (or assets overstated on the books)—the "value trap." No multiple, on its own, is an objective recommendation.
Reazioni
0
0 Commenti
Diventa il primo a commentare