In Lesson 4, you discovered the seesaw: rates rise, prices fall. You also saw that maturity amplifies the effect - a 5-year LTN fell 8.5% under the same shock that caused a 1-year LTN to fall 1.8%. That left the million-dollar question: can you predict, with a single number, how much a bond will move?You can. That number is called duration, and it is probably the most widely used concept among fixed-income managers worldwide. In this lesson, you will learn how to calculate it, use it to estimate gains and losses in seconds - and understand its built-in correction, convexity. This is the most mathematical lesson in the course so far. It is also the one that most clearly separates those who trade fixed income from those who merely invest in it.1. The right question: what is a bond's TRUE maturity?Start with a puzzle. Two bonds mature in exactly 3 years:Bond A is an LTN: it pays R$ 1,000 in a single payment at maturity.Bond B pays R$ 100 coupons at the end of each year and returns R$ 1,100 in year 3 (principal + final coupon).Both "mature in 3 years." But are they equally sensitive to interest rates? No! Bond B returns part of your money earlier - and money returned sooner spends less time exposed to rates. Maturity alone can be misleading.Macaulay duration corrects for this: it is the average time to receive the cash flows, weighted by the present value of each one. Rather than asking, "When does the bond mature?", it asks, "How long, on average, until my money comes back?"The immediate implication, which answers half the questions on an exam: for zero-coupon bonds (such as LTNs), duration equals maturity - all the money comes back at once, at the end. For coupon-paying bonds, duration is always shorter than maturity.An LTN (a zero-coupon bond) matures in exactly 4 years. Its Macaulay duration is...Answer: 4 years, equal to its maturity, because there is a single cash flow at maturity - with no interim payments, the weighted average time is the maturity itself. This equality applies only to zero-coupon bonds.2. Calculating Macaulay durationLet's calculate the duration of Bond B: annual R$ 100 coupons, R$ 1,100 returned in year 3, and a market rate of 12% per year.Step 1 - present value of each cash flow:Year 1: 100 / 1.12 = R$ 89.29Year 2: 100 / (1.12)² = R$ 79.72Year 3: 1,100 / (1.12)³ = R$ 782.96Bond price = sum of present values = R$ 951.96Step 2 - each cash flow's weight in the price:Year 1: 89.29 / 951.96 = 9.4%. Year 2: 79.72 / 951.96 = 8.4%. Year 3: 782.96 / 951.96 = 82.2%.Step 3 - average timing, weighted by those weights:D = 1 × 0.094 + 2 × 0.084 + 3 × 0.822 = 2.73 yearsIn other words: although the bond matures in 3 years, the money's "center of gravity" is at 2.73 years. For interest-rate sensitivity purposes, it is as if the bond were a 2.73-year LTN.3. Modified duration: the damage multiplierMacaulay duration measures time. To turn it into price sensitivity, we divide it by one plus the yield and obtain modified duration:Dmod = D / (1 + i)In our example: Dmod = 2.73 / 1.12 = 2.44And now the formula managers use all day:% price change ≈ − Dmod × change in yield (in percentage points)Try it: if the rate rises from 12% to 13% (+1 percentage point), Bond B's price falls by approximately 2.44 × 1 = −2.44%.Checking the long way: repricing the three cash flows at 13%, the price falls from R$ 951.96 to R$ 929.17 - a decline of −2.39%. The duration estimate missed by 0.05 percentage points. Not bad for mental multiplication - and we will soon see where that small difference comes from.Returning to Lesson 4: a 2-year LTN at 12% has D = 2 and Dmod = 2/1.12 = 1.79. Under that lesson's +2 percentage point shock, the estimate is −3.57%; the exact figure was −3.48%. The math checks out.A bond has a Macaulay duration of 5 years and trades at a yield of 10% per year. If the market rate rises by 1 percentage point, the approximate price decline estimated by modified duration is...Answer: 4.55% - Dmod = 5/1.10 = 4.545; change ≈ −4.545 × 1 percentage point ≈ −4.55%. The 5% option forgets to divide by (1+i): it uses Macaulay duration instead of modified duration, the most common mistake.4. What makes duration rise or fallThree factors determine any bond's duration:Maturity: the longer the maturity, the greater the duration. (The dominant factor.)Coupon: the HIGHER the coupons, the LOWER the duration - more money comes back early, bringing the center of gravity closer. That is why an NTN-B Principal (zero-coupon) is more sensitive than an NTN-B with semiannual interest payments and the same maturity.Market rate: the higher the rate, the shorter the duration - heavier discounting reduces the present value of distant cash flows, pulling the center of gravity closer.You now have the full vocabulary for the Lesson 5 paradox: the NTN-B 2060 moves so sharply because it combines an extremely long maturity with small coupons (or none, in the Principal) - the recipe for maximum duration in Brazil's bond market.Two bonds have the same issuer, the same yield and the same 10-year maturity. Bond X pays generous semiannual coupons; Bond Y is zero-coupon. Regarding interest-rate sensitivity, it is correct to say that...Answer: Y is more sensitive because its duration is greater than X's - for a zero-coupon bond, duration = 10 years; with generous coupons, money comes back sooner and duration falls well below 10. The same maturity does NOT mean the same risk: that is the lesson's central message.5. Convexity: the curvature that works in your favorWhere did the difference between the estimated −2.44% and the actual −2.39% come from? Duration draws a straight line tangent to the price-yield relationship. But the true relationship is a curve - convex, bowing downward.The practical consequence is elegant: for large yield changes, the duration line overstates losses and understates gains. The actual price falls LESS than duration predicts when rates rise, and rises MORE than it predicts when rates fall.In other words: convexity is the investor's friend. Between two bonds with the same duration, the one with greater convexity performs better in both scenarios - which is why the market is willing to pay a little more for it. For small shocks (up to about 1 percentage point), duration alone does the job; for large shocks, managers add the convexity adjustment to the estimate.A manager used modified duration to estimate an 8% decline in a bond's price following a sharp rise in interest rates. Considering the effect of convexity, the actual decline will tend to be...Answer: Less than 8%, because the price curve is convex - the duration line overestimates declines and underestimates gains. In large shocks, convexity always "gives back" part of the estimated loss.6. What it is used for in practice: immunization and managementDuration is not just a thermometer - it is the steering wheel. Two uses dominate the market:Immunization: anyone with an obligation due on a specific date (paying R$ 10 million in 5 years, for example) builds a portfolio with duration equal to the liability horizon. That way, the two opposing risks offset each other: if rates rise, the portfolio's price falls, but reinvesting cash flows earns more; if rates fall, the price rises and offsets weaker reinvestment returns. The perfect case is a matched zero-coupon bond: a 5-year LTN for a 5-year liability eliminates the problem by design. That is the logic of pension funds - and of Beatriz from Lesson 5.Active management: those who expect interest rates to fall lengthen the portfolio's duration (more long-dated NTN-Bs, more long-dated fixed-rate bonds) to ride the price appreciation; those expecting rates to rise shorten it (more LFTs, more short-term securities). A portfolio's duration is simply the average duration of its bonds, weighted by the value of each position - a single number that summarizes the risk of millions invested across different bonds.“Duration is fixed income's steering wheel: it does not change the road rates take, but it determines how much each turn shakes the passenger.”- Portfolio management desk summary7. Tying it all togetherThe lesson's essentials:Macaulay duration is the average timing of cash flows weighted by their present values - the money's "center of gravity." For zero-coupon bonds, duration = maturity; with coupons, it is always shorter.Modified duration (D/(1+i)) converts time into sensitivity: ΔP% ≈ −Dmod × Δyield.Duration increases with maturity and decreases with higher coupons and higher rates.Convexity corrects the duration line: actual losses are smaller and actual gains are greater than the linear estimate - which is why it is desirable.Immunization matches a portfolio's duration to the liability horizon; active management lengthens duration when rates are expected to fall and shortens it when they are expected to rise.In the next lesson, we will wrap up the fixed-income section by bringing all the pieces together - mark-to-market, duration and the yield curve - in strategy and risk management: how professionals build, protect and stress-test a real fixed-income portfolio.Classroom discussion exerciseThe time-locked vault. Let's think it through together.A company's treasury department needs to pay R$ 10 million in exactly 5 years (when one of its own debentures matures). The money already exists and is invested. Three directors support three different strategies:Director A: "LFT. No surprises, it tracks the Selic rate, and we can sleep easy."Director B: "An LTN maturing in 5 years, matched to the payment. We lock in the rate today and forget about it."Director C: "An NTN-B 2045, with a duration of around 12 years. Rates will plunge in the coming years; we will ride the appreciation and pay the debt with money left over."Discuss with your group:(a) What is the approximate duration of each strategy? Which one is immunized against the 5-year liability?(b) Director A's strategy looks safest. What hidden risk does it carry, considering the accumulated value in 5 years if Copom cuts the Selic rate in half?(c) If Director C is right and rates fall 2 percentage points, estimate the gain on his position using duration. And if he is wrong and rates rise 2 percentage points, what happens to the ability to pay the debt?(d) Why is Strategy B called immunization "by design"? What makes it immune to both rising and falling rates?(e) Is there any scenario in which C's choice would be defensible for a corporate treasury department? What separates risk management from a bet?Guidance for the instructor: (a) A: duration ≈ zero (daily floating-rate instrument, with no meaningful price sensitivity); B: duration = 5 (zero-coupon) = horizon → immunized; C: duration ≈ 12, mismatched by +7 years. (b) Reinvestment risk: the LFT locks in no rate whatsoever — if the Selic rate falls by half, the amount accumulated over 5 years may fall short of the target; price safety is not target safety. (c) Gain ≈ +2 × 12 = +24% (less after fine-tuning, more with favorable convexity); in the opposite scenario, ≈ −24%: the company may reach year 5 without the R$ 10 million — a solvency risk, not a volatility risk. (d) The 5-year zero-coupon bond delivers exactly R$ X on the exact date of the liability: there is no reinvestment or early sale — both risk channels disappear at once. (e) Only if the company deliberately sets aside risk capital for the bet, with limits and a stop — and then it is no longer the liability's savings. Conclusion: treasury matches duration to the liability (hedge); anyone who mismatches it is engaging in active management with other people's money — the ethical and technical boundary between the two is the size of the mandate.Exercises1) (Original - CESGRANRIO style) An LTN (zero-coupon bond) maturing in 3 years has a Macaulay duration of:A) 3 years.B) less than 3 years, due to the discounting of cash flows.C) more than 3 years, due to compound interest.D) 1.5 years, half the term.E) impossible to determine without the market rate.2) (Original - CESGRANRIO style) A bond has a Macaulay duration of 5 years and is traded at a rate of 10% per year. Its modified duration is approximately:A) 4.55.B) 5.00.C) 5.50.D) 4.00.E) 0.50.3) (Original - CESGRANRIO style) A bond with a modified duration of 6 experiences a 0.5-percentage-point increase in the market interest rate. The approximate change in its price, as estimated by duration, is:A) −3.0%.B) −6.0%.C) +3.0%.D) −0.5%.E) −12.0%.4) (Original - FGV style) A bond pays an annual coupon of R$ 100,00 and returns R$ 1.100,00 (principal plus final coupon) at the end of 2 years. At a market rate of 10% per year, its price is R$ 1.000,00, with the present values of its cash flows equal to R$ 90,91 (year 1) and R$ 909,09 (year 2). The Macaulay duration of this bond is approximately:A) 1.91 years.B) 2.00 years.C) 1.50 years.D) 1.10 years.E) 2.10 years.5) (Original - FGV style) All other factors remaining constant, the Macaulay duration of a fixed-income bond:A) decreases as the coupon rate increases.B) increases as the coupon rate increases.C) is independent of the coupon rate.D) decreases as the maturity increases.E) is always equal to the maturity.6) (Original - CESGRANRIO style) A pension fund must meet a concentrated stream of benefit payments in 8 years and wants to immunize its fixed-income portfolio against interest-rate fluctuations. To do so, it should structure the portfolio so that:A) its duration is approximately 8 years, equal to the liability horizon.B) its duration is as low as possible, concentrated in LFTs.C) its duration is as high as possible, maximizing convexity.D) all bonds mature in no more than 1 year, with positions rolled over.E) the portfolio contains only fixed-rate bonds maturing in more than 20 years.7) (Original - CEBRASPE style, judge the statement) Judge the statement: "Because of the convexity of the price-yield relationship, the actual loss on a fixed-rate bond following a sharp increase in interest rates tends to be greater than the loss estimated by modified duration."( ) Correct ( ) Incorrect8) (Original - CEBRASPE style, judge the statement) Judge the statement: "The duration of a fixed-income portfolio can be obtained by taking the weighted average of the durations of its constituent bonds, weighted by each bond's share of the portfolio's value."( ) Correct ( ) IncorrectAnswer Key1) A — Zero-coupon: a single cash flow at maturity, so the weighted average term is the term itself. Duration = 3 years, regardless of the rate.2) A — Dmod = D/(1+i) = 5/1,10 = 4,545. Forgetting to divide by (1+i) leads to answer B — the classic mistake.3) A — ΔP% ≈ −Dmod × Δi = −6 × 0,5 = −3,0%. Negative sign: the rate rose, the price fell (the seesaw from Lesson 4, now quantified).4) A — Weights: 90,91/1.000 = 9,09% in year 1 and 909,09/1.000 = 90,91% in year 2. D = 1 × 0,0909 + 2 × 0,9091 = 1,909 years. Less than the 2-year term, as with every coupon bond.5) A — Higher coupons return more money earlier and pull the center of gravity closer: duration falls. A longer term increases duration (answer D reverses this), and duration = term applies only to zero-coupon bonds (answer E).6) A — Immunization = matching the portfolio's duration to the liability horizon (8 years): price risk and reinvestment risk then offset each other. Minimizing duration (answer B) protects the price, but leaves reinvestment risk completely exposed.7) Incorrect — It is the opposite: convexity causes the actual price to fall LESS than the duration line predicts when rates rise (and to rise more when they fall). The curvature works in the investor's favor.8) Correct — Portfolio duration is the weighted average (by the market value of positions) of individual durations — this is exactly how trading desks summarize the interest-rate risk of an entire portfolio in a single number.
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