In the previous lesson, you learned HOW futures work: daily settlement, margin, long and short positions. What remained was the billion-dollar-per-trading-day question: how MUCH SHOULD a futures contract be worth? The dollar future at R$ 5,26 when spot is R$ 5,20 — are those 6 cents the market's guess about the exchange rate? (Spoiler: no. And the answer will surprise you.)This is the course's final content lesson — and it was designed as a reunion. The interest-rate curve from Lesson 3, present value from Lesson 4, beta from Lesson 12 and the mechanics from Lesson 13 come together here to close the loop: pricing futures and setting up the three classic hedges in the Brazilian market — dollar, index and DI.1. The principle behind it all: cost of carryThink of two ways to "have the Ibovespa three months from now":Route A: buy the index portfolio TODAY, in the spot market, and hold it for 3 months. Cost: the spot price PLUS whatever your money would have earned at the interest rate (the opportunity cost of capital tied up in the asset).Route B: buy the FUTURES contract expiring in 3 months and leave the money earning interest while you wait.Both routes end in exactly the same place: you hold the index three months from now. Therefore, they must cost the same — if they do not, the arbitrageur from Lesson 13 steps in, buys the cheaper route, sells the more expensive one and pockets the risk-free difference until prices realign. This reasoning gives us the master formula for futures pricing:Futures price = spot price + cost of carrying the asset until expirationFor financial assets, the cost of carry is essentially the interest rate for the period (less any income paid by the asset along the way, such as dividends). The implication that overturns conventional wisdom: the futures price is NOT the market's forecast of the spot price — it is the spot price compounded by the cost of carry. A futures price above spot does not mean "it will rise"; it means "interest rates exist."Under normal conditions, the price of an equity index futures contract exceeds the index's spot price because...Answer: It reflects the cost of carry, essentially the interest for the period until expiration — it is arbitrage, not expectations, that ties futures and spot prices together. Dividends, in fact, work in the OPPOSITE direction (whoever holds the portfolio receives them, reducing the cost of carry).2. Index futures: the formula in actionIgnoring dividends for simplicity, the index future is worth the spot price compounded at the interest rate through expiration:F = S × (1 + i)^(du/252)Example: the Ibovespa spot index at 130.000 points, an interest rate of 15% per year, and expiration in 63 business days (3 months):F = 130.000 × (1,15)^(63/252) = 130.000 × 1,0356 ≈ 134.620 pointsThe roughly 4,600-point difference is not optimism: it is three months of CDI priced in. And notice the dynamic: as expiration approaches, the exponent shrinks and the future converges toward spot — on the expiration date, F = S. The difference between the two, known as the basis, disappears at expiration by construction. (Between initiation and expiration, however, the basis fluctuates — what is known as basis risk, which is why real-world hedges are highly effective but rarely perfect.)3. Dollar futures: interest-rate parityFor foreign exchange, carry involves two currencies — and the cost is the interest-rate differential between them:F = S × (1 + i Brazil)^(t) / (1 + i US)^(t)Example with a 1-year term: spot dollar at R$ 5,20, interest rates of 15% in Brazil and 4% in the US:F = 5,20 × 1,15 / 1,04 = R$ 5,75The arbitrage intuition (the "covered interest-rate parity"): anyone holding reais can either (a) invest them at 15% at home, or (b) buy dollars today, invest them at 4% abroad and sell the proceeds in the future. The two routes must produce the same result in reais — and the futures exchange rate is precisely the rate that makes them equal. Since Brazilian interest rates are chronically higher, the dollar future generally trades above spot — and those 6 cents from Lesson 13 (5,26 versus 5,20) were simply the 90-day interest-rate differential, not a forecast of a rise.The spot dollar is at R$ 5,00, the Brazilian interest rate is 12% per year and the US rate is 4% per year. Under interest-rate parity, the 1-year dollar future should be worth approximately...Answer: R$ 5,38 — F = 5,00 × 1,12/1,04 = 5,3846. Option B (5,60) compounds only at the Brazilian interest rate and forgets to subtract the return earned by the dollar abroad — the classic parity mistake.4. DI futures: trading the interest-rate curveNow we come to the most heavily traded contract on B3 — and to the promised reunion with Lesson 3. DI futures trade, for each maturity, the average interest rate expected through that date. The interest-rate curve is read from them: each maturity (Jan/27, Jan/28...) is a point on the ETTJ.The mechanics have an elegant symmetry. The contract has a fixed final value of 100.000 points at expiration, and its PU (unit price) today is that amount discounted at the rate being traded — the SAME calculation used for the LTN in Lesson 4:PU = 100.000 / (1 + rate)^(du/252)Example: expiration in 252 business days, with a traded rate of 13% → PU = 100.000/1,13 = 88.495,58 points.And here is the inversion that confuses every beginner (and appears on every trading-desk exam): rates and PU move in opposite directions — the old seesaw from Lesson 4. That is why the jargon goes: anyone who is long rates is short PU (profiting if interest rates RISE), while anyone who is short rates is long PU (profiting if interest rates FALL). DI futures turn the interest-rate curve — the chart we learned to read in Lesson 3 — into something that can be bought and sold.A market participant believes Copom will raise interest rates far beyond what the curve currently prices in and wants to profit from that scenario in DI futures. They should be...Answer: Long rates (short PU), profiting if rates rise — the rate in the denominator pushes PU down when it rises: the Lesson 4 seesaw applied to the contract. The dual terminology (rate × PU) exists precisely because the two move in opposite directions.5. The three classic hedges, from start to finishWith all three pricing frameworks in hand, here is the Brazilian portfolio manager's complete toolbox:5.1 Currency hedge (dollar futures)The hedge from Lesson 13, now with the pricing logic behind it: the exporter sells futures (locking in revenue); the importer buys them (locking in costs). Both now understand that the locked-in price includes the interest-rate differential — locking in R$ 5,26 with spot at R$ 5,20 is not "making 6 cents": it means receiving the carry determined by parity.5.2 Equity portfolio hedge (index futures)A portfolio manager with an equity portfolio fears a broad market decline but does not want to (or cannot) sell everything — because of costs, taxes or long-term conviction. The solution: sell index futures, temporarily neutralizing systematic risk. How many contracts? This is where the beta from Lesson 12 comes in:N = portfolio beta × (portfolio value / financial value of one contract)Example: a portfolio worth R$ 13 million with beta 1,2; the Ibovespa future at 130.000 points with a point value of R$ 1,00 (a contract worth R$ 130.000):N = 1,2 × (13.000.000 / 130.000) = 120 short contractsIf the market falls 10%, the portfolio (beta 1,2) is expected to lose roughly 12% (R$ 1,56 million) — and the 120-contract short position should generate approximately the same amount through daily settlements. Systematic risk has been transferred; what remains is idiosyncratic risk (small in a diversified portfolio) and basis risk. Notice the elegance: beta has moved from CAPM into an operational instruction.5.3 Interest-rate hedge (DI futures)A treasury desk holding a large position in long-term fixed-rate bonds (high duration — Lesson 7) fears a sell-off in the curve. Instead of selling the portfolio, it goes long rates (short PU) in DI futures: if interest rates rise, the losses on the bonds are offset by positive DI settlements. It is the immunization from Lesson 7 implemented with derivatives — without unwinding the portfolio. The reverse is also true: anyone planning to raise funds in the future and worried about higher interest rates can lock in today's debt cost on the curve.A portfolio manager oversees an equity portfolio worth R$ 13 million with a beta of 1,5 and wants to temporarily neutralize market risk by selling index futures, whose contract is worth R$ 130.000. The number of contracts to sell is...Answer: 150 — N = 1,5 × (13.000.000/130.000) = 1,5 × 100 = 150 contracts. Option B (100) forgets the beta: a portfolio that amplifies the market by 1,5× needs 1,5× more protection. The beta from Lesson 12 is the hedge ratio.“The futures market does not predict the future: it prices the present carried through time. Once you understand that, you stop looking for prophecies in prices and start looking for arbitrage opportunities.”- Derivatives desk summary6. Tying it all together — and tying up the courseThe essentials of the lesson:Cost of carry: futures = spot + cost of carry (interest, less income from the asset). Arbitrage — not expectations — ties the two prices together; the basis converges to zero at expiration.Dollar futures: interest-rate parity, F = S × (1+i BR)/(1+i US) — the premium over spot is the interest-rate differential, not a forecast of a rise.DI futures: PU = 100.000 discounted by the rate (the LTN calculation!); long rates = short PU, profiting when interest rates rise — the curve from Lesson 3 has become a tradable asset.The three hedges: currency (exporter sells, importer buys), portfolio (sell the index, sized according to beta) and interest-rate (DI offsetting the portfolio's duration).And this is where the entire course comes full circle. Look at the path we have taken: we began with a map of the financial system (Lesson 1), learned the language of interest rates and the curve (Lessons 2–3), priced bonds and mastered their risks (Lessons 4–8), assessed companies and the price of risk (Lessons 9–12) and ended by managing time and uncertainty with derivatives (Lessons 13–14). If there is one idea that ties everything together, it is this: every financial asset is a future cash flow viewed through an interest rate — and the difference between instruments lies in who promises the cash flow, how certain it is and what price is paid for the risk. Good luck on the exam, good work — and good investing!Exercise for classroom discussionTrading-desk day. Let's think it through together — this is the course's capstone exercise.You are the senior trader at an asset manager. On the same morning, three requests come in. Today's data: spot dollar R$ 5,20; Brazilian interest rates 15% p.a.; US interest rates 4% p.a.; Ibovespa 130.000 points (futures contract = index × R$ 1,00); DI Jan/28 trading at 13% p.a.I. The firm’s equity fund (R$ 26 million, beta 1.2) wants full protection against market risk throughout the election period without selling a single share.II. An importer client is about to purchase US$ 1 million worth of equipment, payable in 1 year, and wants to lock in the cost in reais today.III. The firm’s fixed-income fund is heavily invested in long-term fixed-rate securities, and the committee is concerned that the yield curve may steepen in the coming weeks; selling the securities would incur costs and crystallize losses.For each case, answer: (a) which futures contract and which SIDE (buy/sell; for DI, rate or PU)? (b) the calculation: how many contracts, or what price does the transaction lock in? (c) which risk does the transaction REMOVE, which does it NOT remove, and what cash-flow impact (daily settlement!) does the client need to understand before signing?And the course’s closing question: (d) the three transactions use three different contracts — but notice that the SAME present-value logic solves all three calculations. State that logic in one sentence.Guidance for the instructor: I — Sell Ibovespa futures: N = 1.2 × 26,000,000/130,000 = 240 contracts; removes systematic risk (temporary beta ≈ 0), does NOT remove idiosyncratic risk or basis risk, and the fund needs cash for daily settlements if the market RISES during the hedge (hedging means giving up part of the upside). II — Buy dollar futures (US$ 1,000,000 = 100 US$ 10,000 minis or 20 US$ 50,000 full-size contracts); the locked-in price is determined by interest-rate parity: F = 5.20 × 1.15/1.04 = R$ 5.75 → locked-in cost of R$ 5.75 million; removes foreign-exchange risk but does not remove the commercial risk of the purchase, and the client must understand that if the dollar FALLS, she will owe daily settlements (and "lose" the decline) — locking in a price means giving up both sides of the lottery. III — Take a long position in the rate (short PU) on DI in a size compatible with the portfolio’s duration: when interest rates rise, positive DI settlements offset the decline in the fixed-rate securities; removes (approximately) interest-rate market risk but does not remove the credit or liquidity risk of the securities, and cash will be required for DI settlements if the curve FLATTENS. (d) Expected summary sentence, in some variation: "pricing any futures contract means discounting (or carrying forward) the spot price at the interest rate for the period — present value and future value are the same calculation viewed from opposite sides"; this is the thesis that opens Lesson 4 and closes Lesson 14 — the entire course in one line.Exercises1) (Original - CESGRANRIO style) The spot index stands at 100,000 points and the interest rate is 10% per year. Ignoring dividends, the fair price of the futures contract on this index, maturing in 1 year, is:A) 110,000 points.B) 100,000 points.C) 90,909 points.D) 121,000 points.E) 105,000 points.2) (Original - CESGRANRIO style) The spot dollar is quoted at R$ 5.00. One-year interest rates are 15% per year in Brazil and 5% per year in the United States. Under covered interest-rate parity, the one-year dollar futures price should be approximately:A) R$ 5.48.B) R$ 5.75.C) R$ 5.00.D) R$ 4.57.E) R$ 6.00.3) (Original - CESGRANRIO style) A DI futures contract maturing in 252 business days is trading at a rate of 12% per year. Given that the contract’s settlement value is 100,000 points, its PU (unit price) is approximately:A) 89,285.71 points.B) 88,000.00 points.C) 112,000.00 points.D) 89,000.00 points.E) 88,495.58 points.4) (Original - FGV style) In the DI futures market, a participant seeking to hedge against (or profit from) a RISE in interest rates should take which position?A) Long rate, which is equivalent to being short PU.B) Long PU, which is equivalent to being long rate.C) Short rate, profiting from higher interest rates.D) Neutral, since DI futures are not sensitive to interest rates.E) Long PU and long rate simultaneously.5) (Original - FGV style) A fund heavily invested in long-term fixed-rate securities wants to hedge against a steepening of the yield curve through derivatives, without selling the securities. The appropriate DI futures transaction is to:A) Take a long-rate position (short PU), so that the settlements offset the securities’ decline if interest rates rise.B) Take a long-PU position, increasing the gain if interest rates rise.C) Sell the least liquid contracts available.D) Buy Ibovespa index futures.E) Sell dollar futures.6) (Original - CESGRANRIO style) A portfolio manager wants to neutralize the market risk of an equity portfolio worth R$ 13,000,000.00, with a beta of 1.2, by selling index futures contracts valued at R$ 130,000.00 each. The number of contracts to sell is:A) 120.B) 100.C) 156.D) 84.E) 130.7) (Original - CEBRASPE style, judge the item) Judge the following statement: "If the futures contract price significantly exceeds the spot price plus the cost of carry, arbitrageurs will tend to sell the futures contract and buy the asset in the spot market, earning a profit without directional risk and forcing the prices to converge again."( ) Correct ( ) Incorrect8) (Original - CEBRASPE style, judge the item) Judge the following statement: "The fact that dollar futures trade above the spot dollar indicates that the market, unanimously, expects the US currency to appreciate by the time the contract expires."( ) Correct ( ) IncorrectAnswer Key1) A — F = 100,000 × 1.10 = 110,000 points: the spot price carried forward at the period’s interest rate. Option C discounts instead of compounding — the calculation is going in the wrong direction.2) A — F = 5.00 × 1.15/1.05 = 5.476 ≈ R$ 5.48. Option B (5.75) uses the differential from another example; option E compounds only at the Brazilian interest rate, overlooking the return on the dollar abroad.3) A — PU = 100,000/(1.12)^(252/252) = 100,000/1.12 = 89,285.71 points. Option E (88,495.58) is the PU at a rate of 13% — exactly the example worked through in class, planted as a distractor for anyone memorizing numbers instead of doing the calculation.4) A — Rate rises → PU (the present value of the 100,000) falls: anyone seeking to profit from higher rates goes long rate, which is the same as going short PU. The two terms exist because the prices move as mirror images.5) A — This is a duration hedge using DI: being long rate means that positive settlements when interest rates rise offset the decline in the fixed-rate securities. Being long PU (option B) would DOUBLE the exposure instead of neutralizing it.6) A — N = beta × (portfolio value / contract value) = 1.2 × (13,000,000/130,000) = 1.2 × 100 = 120 contracts. Without the beta, the hedge would be undersized for a portfolio that amplifies market movements.7) Correct — This is cost-of-carry arbitrage in action: sell what is expensive (the futures contract), buy what is cheap (the spot asset plus carry), and lock in the difference. That is what keeps the formula working in practice.8) Incorrect — The futures premium over the spot price reflects the INTEREST-RATE DIFFERENTIAL between the currencies (covered parity), not a forecast of appreciation. With Brazilian interest rates above US rates, dollar futures will trade above the spot dollar even if the market expects the exchange rate to remain stable — the central lesson of the class.
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