Over the course of five lessons, we assembled the fixed-income puzzle piece by piece: rates and the yield curve (Lesson 3), pricing fixed-rate bonds (Lesson 4), floating-rate and indexed securities (Lesson 5), private credit (Lesson 6), and duration (Lesson 7). Today we fit the pieces together: we will organize all the risks that have appeared along the way into a single map — and learn the strategies professionals use to build portfolios that can withstand them.This is the lesson that turns knowledge into judgment. By the end, when looking at any fixed-income portfolio, you will know to ask the two questions that matter: what risks is it exposed to? and were those risks chosen or simply overlooked?1. The five-risk map“Fixed income has no risk” is the most expensive sentence in Brazil’s financial market. It does — at least five kinds, and every security carries its own combination of them.1.1 Market risk (or interest-rate risk)The old familiar risk from Lessons 4 and 7: a bond’s price fluctuates when the yield curve moves. Measured by duration. Who suffers most: long-dated fixed-rate bonds and NTN-Bs. Who barely suffers: the LFT. It only hurts those who sell before maturity — but it really hurts.1.2 Credit riskDefault, the subject of Lesson 6: the issuer may fail to pay. Mitigated by guarantees, ratings, diversification and, in the case of bank deposits, the FGC — always within its limits. Federal government bonds denominated in local currency are the benchmark for minimum risk.1.3 Liquidity riskThe risk of not being able to sell quickly at a fair price. It shows up in the size of the bid-ask spread: a government bond can leave the portfolio within seconds at a minimal spread; a debenture issued by a small company may take weeks — or require a painful discount to sell today. Liquidity is like oxygen: nobody notices it until it runs out, and it runs out precisely in crises, when everyone wants to sell at the same time.1.4 Reinvestment riskThe overlooked risk — and the most treacherous one. Anyone receiving cash flows over time (semiannual coupons, short-term CDBs renewed at maturity, LFTs accumulating the Selic rate) will need to reinvest at future rates, which nobody can guarantee. An investor who “lives off income” from short-term securities yielding 15% a year will discover this risk on the day the renewal rate comes in at 9%. Notice the elegant symmetry: market risk punishes those who need to sell when interest rates RISE; reinvestment risk punishes those who need to reinvest when interest rates FALL. The immunization strategy from Lesson 7 works precisely because it sets one risk against the other.1.5 Inflation riskThe risk of earning a nominal return while losing purchasing power. A fixed-rate bond yielding 12% in a year when inflation reaches 13% produced — under the Fisher equation from Lesson 5 — a negative real return. Structural protection: IPCA-linked securities.An investor holds LFTs and daily-liquidity CDBs, renewing them continuously to “live off income.” The MOST relevant risk in this strategy is...Answer: Reinvestment, because future reinvestment rates may fall — the portfolio has almost no duration (no significant market risk), FGC coverage and daily liquidity. Its exposed flank is income melting away along with the Selic rate. There is no risk-free portfolio: there is only a portfolio with hidden risk.A debenture issued by a small company only finds a buyer at a substantial discount to its reference price when the investor needs to sell it urgently. This is... riskAnswer: Liquidity — the problem is neither the interest rate nor default, but the lack of buyers willing to pay a fair price when the sale is needed. The gauge is the spread between bid and ask prices.2. Marking to market × marking along the curveWith the risk map in hand, we revisit an accounting decision that regularly makes headlines: how should the value of a security be recorded in a portfolio?Marking to market (MtM): every day, the security is worth the price at which it could be sold today — repriced according to the yield curve. It is the honest snapshot. Mandatory for investment funds (so that unitholders who leave and those who remain are treated fairly) and for Tesouro Direto.Marking along the curve (accrual): the security is recorded at its purchase price, “accreting” at the contracted rate day after day in a smooth line toward the R$ 1,000 it will be worth at maturity. It ignores the market along the way. Permitted in specific situations — typically for those who declare the intention and ability to hold until maturity (as in the case of banks and insurers for certain portfolios, and an approach individual investors may adopt for securities they do not intend to sell).The golden rule for avoiding confusion: marking does not change the security — it changes the lens. The same fixed-rate bond is an electrocardiogram through the MtM lens and a gentle ramp through the curve lens. If you ARE going to hold it to the end, the ramp is your economic reality. If you MAY need to sell, the only truth is the electrocardiogram — and hiding behind the ramp is self-deception.Investment funds are required to mark their assets to market primarily to...Answer: Ensure fair treatment for unitholders who enter and leave on different dates — without MtM, those redeeming would receive an unrealistic value, transferring gains or losses to those who stayed. Marking protects unitholders, even if it makes statements nerve-racking.3. The three portfolio architecturesOnce we understand the risks and have duration as our steering wheel, how should we spread money across maturities? Three classic designs dominate practice:3.1 Bullet: everything concentrated on the targetAll maturities concentrated around a single point — typically the time horizon of the goal. This is the Lesson 7 immunization strategy by construction: a liability due in 5 years, with securities maturing in 5 years. Maximum precision for ONE goal with a fixed date; no flexibility for whatever happens along the way.3.2 Barbell: the two ends of the dumbbellHalf the portfolio in very short-term securities (liquidity, opportunity), half in very long-term securities (yield, appreciation when rates fall) — and nothing in between, like the plates on a dumbbell. A certification-exam favorite: for the SAME average duration, the barbell has greater convexity than the bullet — it performs better in large shocks, in both directions. The price: greater reinvestment risk at the short end and greater market risk at the long end, coexisting in the same portfolio.3.3 Ladder: one rung per yearMoney divided among regular, successive maturities — 1, 2, 3, 4 and 5 years, for example. Each year, one rung matures and is reinvested at the end of the ladder at the prevailing rate. The effect: reinvestment risk is spread over time (you never renew everything at a bad rate), there is always a nearby maturity providing natural liquidity, and the portfolio is never fully exposed to a single point on the curve. It is the default strategy for investors without a single fixed date — and the most recommended approach for beginners.For the same average portfolio duration, how does the barbell strategy (very short-term + very long-term securities) differ from the bullet strategy (concentrated maturities)?Answer: Greater convexity, performing better during large interest-rate swings — spreading cash flows across the extremes increases the curvature of the price-rate relationship. The other alternatives reverse the barbell’s trade-offs: it HAS more reinvestment risk at the short end and more market risk at the long end.An investor distributes resources equally among securities maturing in 1, 2, 3, 4 and 5 years, reinvesting each maturity at the longest term. The main benefit of this “ladder” is...Answer: Diluting reinvestment risk and generating periodic liquidity — each year only a fraction is reinvested (never everything at a bad rate), and a maturity is always approaching. The ladder does not affect credit or taxation, and its duration is intermediate by construction.4. The stress test: the professionals’ questionProfessional trading desks do not ask “what does it yield?” before asking “how much does it lose if everything goes wrong?”. The tool is the stress test: apply a standardized adverse scenario to the portfolio and measure the damage.The simplified process, which you already know how to perform with the course tools:1. Calculate the portfolio’s duration (weighted average — Lesson 7).2. Choose the shock: for example, +3 p.p. across the entire curve (a magnitude seen in recent Brazilian crises).3. Estimate: loss ≈ modified duration × 3%. A portfolio with Dmod 6 loses 18%; one with Dmod 1, 3%.4. Ask: can the owner of this money see that number on the statement without selling at the bottom? If not, the risk is wrong for the client — no matter how attractive the yield.Complete the analysis with the layers duration does not capture: what if the issuer of the largest holding delays payment (credit)? What if 20% of the portfolio needs to be raised within a week (liquidity)? What if inflation overshoots its target (purchasing power)? A portfolio is understood only when all five questions on the map have an answer.“Amateurs ask how much it yields. Professionals ask how much it loses, when it loses, and who can withstand the loss.”- Risk-management rule of thumb5. Tying it all together — and closing the fixed-income sectionThe essentials of the lesson:The five risks: market (duration), credit (default), liquidity (exit spread), reinvestment (future rates), and inflation (purchasing power). Every security carries a combination; there is no zero-risk security, only hidden risk.Market and reinvestment are mirror images: one punishes you when rates rise, the other when they fall — and immunization sets them against each other.MtM × curve: same economics, different lenses; funds mark to market to protect unitholders; the curve lens is honest only for those who truly hold until maturity.The three architectures: bullet (precision for a fixed date), barbell (greater convexity at the cost of both risks at the ends), ladder (diluted reinvestment risk + periodic liquidity).The stress test captures the discipline: Dmod × shock, plus the questions about credit, liquidity and inflation.And here our journey through fixed income comes to an end. In the next lesson, we change worlds: we enter equities — where there is no promise of cash flow, “maturity” does not exist, and a company’s value must be discovered, not calculated. Get ready: the tools are changing, but the present-value reasoning you have mastered will remain the soul of everything.Exercise for classroom discussionThe foundation committee. Let’s think it through together.You have joined the investment committee of a family foundation with R$ 12 million, which has three obligations:I. R$ 2 million in 1 year (already-contracted renovation of its headquarters).II. R$ 4 million in 5 years (a promised scholarship fund).III. R$ 6 million with no set date (a perpetual endowment that must preserve purchasing power and generate income for operating expenses).The portfolio inherited from the previous management team: 100% in NTN-B 2045 (duration ≈ 12), purchased because “it was the one paying the most.”Discuss with your group:(a) Apply the map: which of the five risks does the inherited portfolio impose on each of the three obligations? Which obligation is in the most dangerous position?(b) Run the stress test: with Dmod ≈ 12, what happens to the portfolio under a +2 p.p. shock? What does this mean for next year’s renovation?(c) Design the new policy: what strategy (bullet, barbell, ladder — or a combination) and which securities would you assign to each liability? Justify your choices using the concepts from Lessons 4 to 8.(d) For the perpetual endowment, is a long-dated NTN-B the villain or the hero? What changes compared with the one-year liability?(e) The previous treasurer argues: "By maturity, in 2045, no one will have lost anything — mark-to-market is just noise." In what sense is he right, and what is the fatal flaw in his argument for THIS foundation?Guidance for the instructor: (a) Liability I is the critical one: money needed in one year is exposed to a duration of 12, creating extreme market risk plus liquidity risk if the asset has to be sold; II has a smaller but real mismatch; III is the only liability for which a long-dated NTN-B makes sense (inflation protection, perpetual horizon). (b) Loss ≈ 12 × 2 = ~24% (slightly less because of convexity): the renovation might have had to be paid for by selling at the bottom of the market — the stress test reveals that the risk is not "volatility" but failing to meet the liability. (c) A defensible answer: I → a very short bullet (LFT/one-year fixed-rate bond matched to the liability); II → a five-year bullet (matched short-term LTN/NTN-B, immunization from Lesson 7); III → a core allocation to long-dated NTN-Bs (real protection) + a laddered segment to provide funding income, reducing reinvestment risk. (d) The hero in a perpetual portfolio: there, the "flaw" (high duration) becomes an advantage (locking in a long-term real interest rate), and mark-to-market truly becomes noise — the same security plays a different role depending on the liabilities, which is the lesson's central thesis. (e) He is right ONLY for money that can wait until 2045; the flaw is that two of the three liabilities cannot — solvency has a deadline, and mark-to-market is the price of exiting at the wrong time. Takeaway: investment policy starts with the liabilities, never with the rate displayed in the shop window.Exercises1) (New question - CESGRANRIO style) An investor had to sell, before maturity, a fixed-rate bond purchased months earlier, realizing a loss because interest rates had risen during that period. The risk that materialized in this transaction was:A) liquidity risk.B) credit risk.C) market risk.D) reinvestment risk.E) inflation risk.2) (New question - CESGRANRIO style) Reinvestment risk in a fixed-income portfolio is MOST relevant when the portfolio:A) is concentrated in short-term floating-rate securities and securities with frequent coupon payments, in a falling-rate environment.B) is concentrated in long-term fixed-rate, zero-coupon securities held to maturity.C) consists exclusively of long-term inflation-linked securities.D) has a duration identical to the liability horizon.E) is marked to model rather than marked to market.3) (New question - FGV style) A significant difference between the purchase price and the sale price (bid-ask spread) of a thinly traded debenture in the secondary market is a typical manifestation of:A) market risk.B) liquidity risk.C) credit risk.D) reinvestment risk.E) foreign-exchange risk.4) (New question - FGV style) Regarding mark-to-market valuation in investment funds, it is correct to state that it:A) is optional, allowing the manager to choose between market and book valuation according to the desired return.B) eliminates share-price volatility by smoothing the value of the assets until maturity.C) ensures equitable treatment among investors who buy and redeem at different times by reflecting the daily realizable value of the assets.D) applies only to the equity securities in the portfolio.E) records securities at their acquisition cost adjusted by the contracted rate.5) (New question - FGV style) Two portfolios have the same average duration of five years. Portfolio Alfa concentrates maturities around five years (bullet); Portfolio Beta combines one-year and nine-year securities (barbell). In the face of large interest-rate swings, Portfolio Beta is expected to show:A) perfect immunization against market risk.B) lower convexity, with larger losses in any scenario.C) behavior identical to Portfolio Alfa because their durations are equal.D) greater convexity, with relatively better performance both when rates rise sharply and when they fall sharply.E) elimination of reinvestment risk at the short end.6) (New question - CESGRANRIO style) The ladder strategy, with maturities distributed at regular intervals and maturing securities successively reinvested, has as its main benefits:A) spreading reinvestment risk over time and generating periodic liquidity.B) maximizing the portfolio's duration and convexity.C) eliminating the credit risk of private issuers.D) guaranteeing returns above CDI in any scenario.E) concentrating maturities exactly at the liability horizon.7) (New question - CEBRASPE style, judge the statement) Judge the following statement: "A portfolio composed exclusively of LFTs and one-day repurchase agreements is, in practice, free of significant market risk but remains exposed to reinvestment risk in scenarios where the Selic rate falls."( ) True ( ) False8) (New question - CEBRASPE style, judge the statement) Judge the following statement: "In a simplified stress test, the estimated loss on a fixed-income portfolio in response to a parallel shock to the yield curve can be approximated by multiplying the portfolio's modified duration by the size of the shock, with the actual loss tending to be somewhat smaller because of convexity."( ) True ( ) False9) (New question - FGV style) A foundation's fixed-income portfolio, valued at R$ 20 million, has the following composition by market value: 40% in LFTs (modified duration practically zero), 35% in LTNs with a modified duration of 2.6, and 25% in NTN-Bs with a modified duration of 9.0. In the lesson's simplified stress test, with a parallel shock of +3 percentage points to the curve, the portfolio's estimated loss is approximately:A) R$ 632,000.00.B) R$ 1,350,000.00.C) R$ 2,320,000.00.D) R$ 1,896,000.00.E) R$ 3,160,000.00.10) (New question - CESGRANRIO style) A manager with R$ 5 million wants to build a barbell portfolio with a modified duration of 4, combining only LFTs (modified duration considered zero) and a long-term NTN-B with a modified duration of 10. The manager compares this portfolio with a bullet alternative with the same modified duration. The amount to allocate to the NTN-B and the correct assessment for a +2-percentage-point shock to the curve are:A) R$ 2.0 million; an estimated loss of about R$ 400 thousand, greater than the estimate for the bullet, because of the long end.B) R$ 2.0 million; an estimated loss of about R$ 400 thousand, equal to the estimate for the bullet, with the barbell's actual loss tending to be smaller because of its greater convexity.C) R$ 3.0 million; an estimated loss of about R$ 600 thousand, with the short end eliminating reinvestment risk.D) R$ 2.0 million; an estimated loss of about R$ 1.0 million, because the relevant duration is that of the long-term security.E) R$ 2.5 million; an estimated loss of about R$ 500 thousand, because the barbell divides the capital equally between the two ends.11) (New question - CEBRASPE style, judge the statement) Judge the following statement: "Replacing mark-to-market valuation with book (accrual) valuation does not change the market risk of a fixed-rate security, only the way its fluctuation shows up in the records; if the security has to be sold before maturity, the amount obtained will be the market price, not the value recorded on the accrual curve."( ) True ( ) False12) (New question - FCC style) An investor built a ladder with five equal rungs of fixed-rate securities, maturing in 1, 2, 3, 4, and 5 years, all purchased at a rate of 14% per year. Right after the purchase, rates across the entire curve fall to 10% per year and stay there. At the end of the first year, the maturing rung is reinvested in a 5-year security. Approximating the portfolio's rate by the weighted average of the rungs' rates (before income tax) and assuming IPCA inflation of 4.5% per year, the portfolio's real rate in the second year is approximately:A) 13.20%.B) 8.70%.C) 5.26%.D) 9.09%.E) 8.33%.Answer Key1) C — Selling early when interest rates are higher means market risk (interest-rate risk), the risk measured by duration. Credit risk would be default; liquidity risk would be the difficulty of finding a buyer at a fair price.2) A — Reinvestment risk hurts when substantial amounts of money come due early (short-term securities and coupons) and must be reinvested at falling rates. A long-term zero-coupon security held to maturity (choice B) is precisely the case with NO reinvestment risk.3) B — A wide bid-ask spread is the classic indicator of illiquidity: the market charges a high price for providing an immediate exit. It has no necessary connection with default or with the level of interest rates.4) C — Mark-to-market valuation exists to protect investors: those who enter and those who exit trade at the real value of the assets on that day. Choices B and E describe book valuation — which is not permitted as a general rule for funds.5) D — The classic result: with equal durations, the barbell portfolio has greater convexity (cash flows spread across the extremes) and outperforms the bullet portfolio in large shocks, in either direction. The cost appears in the risks at the two ends — which choice E incorrectly denies.6) A — A ladder never has to roll over the entire portfolio at an unfavorable rate (it spreads out reinvestment risk), and there is always a rung maturing (providing natural liquidity). Concentrating maturities at the liability horizon (choice E) describes a bullet portfolio.7) True — LFTs and repurchase agreements have virtually zero duration (making market risk irrelevant), but future returns follow the Selic rate: if it falls, income falls with it. This is a portrait of the hidden risk in a "safe portfolio."8) True — Loss ≈ modified duration × shock is exactly the calculation used in the simplified stress test (Lesson 7), and convexity makes the actual loss slightly smaller than the linear estimate — the curve works in the investor's favor.9) D — Portfolio modified duration = 0.40 × 0 + 0.35 × 2.6 + 0.25 × 9.0 = 0.91 + 2.25 = 3.16. Loss ≈ 3.16 × 3% × R$ 20 million = R$ 1,896,000 (the actual loss tends to be somewhat smaller because of convexity). Choice A uses a 1 p.p. shock; B considers only the NTN-B; C takes the simple average of the three durations (3.87), without weighting by value; E removes the LFTs from the weighting (3.16 / 0.60 ≈ 5.27), as if the 40% in LFTs were not part of the portfolio.10) B — Weight in the NTN-B: w × 10 = 4 → w = 40% → R$ 2.0 million (and R$ 3.0 million in LFTs). Estimated loss ≈ 4 × 2% × R$ 5 million = R$ 400 thousand, the same as the bullet, because the duration is equal. The difference lies in convexity: with cash flows spread across the extremes, the barbell tends to lose less than the straight line predicts. Choice C swaps the weights (60% at the long end would give a duration of 6) and also denies the reinvestment risk of the short end; D applies the long-term security's duration to the entire portfolio; E assumes a barbell is always split half and half (which would give a duration of 5); A reverses the effect of convexity.11) True — "Valuation does not change the security; it changes the lens." Book valuation smooths the records but does not eliminate the exposure: whoever sells before maturity receives the price the market pays on that day. The trap lies in thinking that changing the accounting criterion reduces the risk. It does not reduce it; it only hides it.12) E — In the second year, 4 rungs (80%) remain at 14% and the reinvested rung (20%) yields 10%: nominal rate ≈ 0.8 × 14% + 0.2 × 10% = 13.2%. Real rate by Fisher: 1.132 / 1.045 − 1 ≈ 8.33%. Choice A forgets inflation; B subtracts the rates instead of dividing the factors; C assumes the entire portfolio is renewed at 10%, as with a rolled-over short-term CDB (precisely the risk the ladder dilutes); D ignores the reinvestment of the maturing rung.
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