In the previous lesson, we mastered fixed-rate bonds: the rate locked in, the price moving with the yield curve, and mark-to-market valuation causing nasty surprises. Today we meet the other two members of the government bond family — and they could hardly be more different from each other.On one side, the LFT (Tesouro Selic): the tamest bond in the market, which barely fluctuates and has therefore become synonymous with an emergency fund. On the other, the Tesouro IPCA+ (NTN-B): the only one that guarantees returns above inflation — and, paradoxically, the most volatile of all when marked to market. Understanding why the “safest against inflation” is also the one that swings the most in the short term is the central objective of this lesson.1. The concept that ties everything together: the VNALTN has a fixed nominal value of R$ 1.000. LFT and NTN-B do not: their redemption value grows over time, tracking an index. This growing value has a name: VNA — Updated Nominal Value.Here is how it works: both started out worth R$ 1.000 on a base date in 2000. Since then:The VNA of LFT is adjusted, business day after business day, by the Selic rate.The VNA of NTN-B is adjusted, month by month, by the change in IPCA.After more than two decades of adjustments, these VNAs are now worth several thousand reais each. The bond’s market price is therefore a quotation applied to the VNA:Price = VNA × QuotationIf the quotation is 100%, the bond trades “at par” — it is worth exactly the VNA. A quotation below 100% is a discount; above 100%, a premium. And the rate traded by the market is embedded in the quotation.The Updated Nominal Value (VNA) of LFT and NTN-B is adjusted, respectively, by which indexes?Answer: Selic rate and IPCA — LFT tracks the daily Selic rate (which is why it is called “Tesouro Selic”), while NTN-B tracks official inflation as measured by IPCA (which is why it is called “Tesouro IPCA+”).2. LFT (Tesouro Selic): the bond that rides an escalatorLFT is a pure floating-rate bond: its VNA rises every business day by exactly what the Selic rate earns. If Selic is at 15% per year, the VNA rises by the 0,0555% per business day we calculated in Lesson 3 — day after day, without surprises, like an escalator.2.1 So does LFT have no market price?It does, and here is the subtle point that often appears on exams. LFT trades at a small premium or discount to the VNA — generally a few hundredths of a percentage point, something like Selic + 0,05% or Selic − 0,02%. This rate reflects supply and demand for the bond and the market’s perception of risk over the term.The practical consequence is that mark-to-market valuation exists for LFT, but it is tiny. If the discount fluctuates from 0,02% to 0,10%, the price moves by fractions of a percent — nothing like the 8,5% plunge in the long-term LTN we saw in Lesson 4. During rare episodes of market stress (such as in 2002 and at points during 2020-2021), the discount on LFTs widened enough to produce negative days for DI funds, surprising everyone — the exception that proves the rule.2.2 What it is used forBecause it tracks the policy rate with almost no volatility, LFT is the natural instrument for emergency funds and short-term cash: daily liquidity, with no meaningful risk of having to sell at a loss because of mark-to-market valuation. The price of that peace of mind? No premium: you earn the Selic rate, no more and no less — and if Copom cuts the rate, your return falls along with it.An investor needs to build an emergency fund with daily liquidity and minimal price volatility. Among government bonds, the most suitable option is...Answer: LFT (Tesouro Selic), with a floating rate tied to the policy rate — the VNA grows daily with Selic and mark-to-market valuation is minimal. Fixed-rate bonds and long-term NTN-Bs may be sharply down precisely on the day the money is needed.3. Tesouro IPCA+ (NTN-B): the contract against inflationNTN-B promises what no other bond can: a guaranteed real interest rate. Anyone who buys “IPCA + 6% per year” and holds it to maturity will receive the inflation accumulated over the period, whatever it may be, PLUS 6% per year in real purchasing-power gains.The mechanism is as follows: the VNA takes care of inflation (rising with IPCA every month), while the rate agreed at purchase — the “+ 6%” — comes from the discount in the quotation, just as the discount on LTN embedded its fixed rate.3.1 Two versions of the same bondTesouro IPCA+ (NTN-B Principal): zero coupon — pays everything at once at maturity. Its structure is identical to LTN’s, except that it is based on a VNA that grows with IPCA.Tesouro IPCA+ com Juros Semestrais (NTN-B): pays semiannual interest coupons throughout its life and the principal at the end. It is preferred by those seeking periodic income — but each coupon received is taxed and must be reinvested.3.2 Real interest × nominal interest: the Fisher equationThis is the most important moment of the lesson. The relationship between nominal interest (what appears on the statement), real interest (what your purchasing power gains) and inflation is NOT a simple subtraction — it is a division:(1 + nominal i) = (1 + real i) × (1 + inflation)Example 1 — from nominal to real: a fixed-rate bond pays 12% per year and inflation was 5%. The real gain was:real i = 1,12 / 1,05 − 1 = 6,67% per year (not the 7% produced by the naive subtraction!)Example 2 — from real to nominal: an NTN-B contracted at IPCA + 6%, in a year when IPCA ends at 4%, produces a nominal return of:nominal i = 1,06 × 1,04 − 1 = 10,24% per year (not 10%!)The difference may seem small, but over periods of 10, 20, 30 years — the typical maturities of NTN-Bs — the compounding effect is enormous. And exam writers LOVE offering simple subtraction as a trap answer.An investment returned 15% nominally in a year when inflation was 10%. According to the Fisher equation, the real gain was approximately...Answer: 4,55% — real i = 1,15/1,10 − 1 = 0,04545. The 5% option is the simple subtraction (15% − 10%), the classic trap: the higher inflation is, the more inaccurate the subtraction becomes.4. The NTN-B paradox: protection that swingsNow for the apparent contradiction. If NTN-B protects against inflation, why does it plunge (or soar) when marked to market?Because the “+ 6%” is, in practice, a fixed-rate component — the REAL interest rate. And fixed-rate bonds, as you learned in Lesson 4, have prices that move like a seesaw: if the market real interest rate rises from 6% to 7%, the price of your 6% NTN-B falls. And because NTN-Bs have the longest maturities in the market (2035, 2045, 2060...), the maturity effect we saw in Lesson 4 appears here in its most extreme form: a 2060 NTN-B can fluctuate by more than 10% in a single quarter.In short, NTN-B offers two guarantees at maturity (inflation + the contracted real rate) and no guarantee along the way. It is the perfect bond for long-term goals with a fixed date — retirement is the classic example — and the worst possible choice for money that may need to be withdrawn at any time.An NTN-B was purchased at IPCA + 6% per year. Months later, the real interest rate traded for the same maturity rose to IPCA + 7%. The market price of this bond...Answer: Fell, because the real rate acts as a fixed-rate component — the same seesaw from Lesson 4 applies here, now to real interest. The IPCA adjustment protects the VNA, not the quotation.A subtle exam trap: “NTN-B guarantees a real return” is conditionally true — it guarantees a real return of IPCA + the contracted rate FOR THOSE WHO HOLD IT TO MATURITY. Anyone who sells earlier may even suffer a nominal loss if the market real interest rate has risen enough. The same logic as LTN, now in a “real” version.5. The family portrait: the three bonds side by sideTo wrap up the government bond section, here is the comparison that brings everything together:LTN (Tesouro Prefixado) — index: none (fixed nominal rate). Mark-to-market risk: high, increasing with maturity. Typical use: locking in a rate when a decline in interest rates is expected; goals with a date matching the maturity.LFT (Tesouro Selic) — index: daily Selic rate. Mark-to-market risk: minimal. Typical use: emergency fund, cash, waiting for opportunities.NTN-B (Tesouro IPCA+) — index: IPCA + fixed real rate. Mark-to-market risk: the highest in the market at long maturities. Typical use: retirement and long-term goals with purchasing-power protection.The guiding question for any allocation is: what is the time horizon for the goal, and what happens if I need to exit early? Once that question is answered, the right bond almost chooses itself.“A fixed-income investor does not choose between earning more or less; they choose which risk they are willing to carry: interest-rate risk, inflation risk or opportunity risk.”- Fixed-income desk synthesis6. Bringing it all togetherThe essentials of the lesson:VNA is the nominal value that grows with the index: daily Selic for LFT, monthly IPCA for NTN-B. Price = VNA × quotation, with the traded rate embedded in the quotation’s premium or discount.LFT is the pure floating-rate bond: it rides an escalator, has minimal mark-to-market volatility and is ideal for an emergency fund — it earns the Selic rate and nothing more.NTN-B guarantees real interest (IPCA + rate) at maturity; the relationship between nominal interest, real interest and inflation follows the Fisher equation — division, never subtraction.The real-rate component of NTN-B behaves like a fixed-rate bond: when real interest rises, the price falls — and extremely long maturities amplify everything.In the next lesson, we move beyond government bonds and into private credit: CDBs and debentures — where, in addition to market risk, a new character enters the scene: credit risk. The spread we studied in Lesson 1 will finally be priced.Exercise for classroom discussionThe case of the three sisters. Let’s think it through together.Three sisters each inherited R$ 100.000 and asked for your help as a consultant:Ana, 28, wants to leave the money invested while she gathers the courage to start a business — she may need to withdraw it all in three months or in three years; she does not know.Beatriz, 35, will use the money exactly 4 years from now as a down payment on an apartment whose contract she has already signed. She believes Selic will fall significantly during that period.Carolina, 35, will not need the money for 25 years: it is her retirement fund. Her biggest fear is “turning 60 and finding that the money cannot buy what it buys today.”Discuss with your group:(a) Which government bond is suitable for each sister, and why?(b) What happens to each of them if Selic plunges over the next two years? And what if the market real interest rate rises sharply?(c) Should Beatriz choose a fixed-rate bond maturing in 4 years or in 10 years? What would the mismatch in maturities mean for her?(d) Carolina saw that NTN-B 2050 “fell 12% last year” and became afraid. What argument would you use — and what question would you ask to test whether she can stay the course?(e) Is there a single, “correct” answer for all three? What besides the time horizon should be part of the conversation?Teacher guidance: (a) Ana → LFT (liquidity with minimal mark-to-market exposure); Beatriz → fixed-rate bond (or short-term NTN-B) maturing in four years: it locks in the high rate before the rate cuts she expects; Carolina → long-term NTN-B, the only instrument that guarantees purchasing power plus a real return over a 25-year horizon. (b) As the Selic rate falls, Ana earns less (the cost of a floating-rate investment), Beatriz celebrates the appreciation of her fixed-rate bond, and Carolina sees the NTN-B rise if the real interest rate moves in line; if the real interest rate rises, Carolina sees her account balance plummet — and that is where the distinction between a potential loss and a crystallized loss from Lesson 4 comes in. (c) Four-year maturity: matching the bond's term to the goal's time frame eliminates the risk of selling at the worst possible moment; the 10-year bond would leave her at the mercy of mark-to-market pricing when she buys the property. (d) The 12% drop is an exit price, not a loss for someone who holds the bond until 2050; the test question is: "can you NOT look at the app for a few years?". (e) There is no single answer: tolerance for fluctuations, the need for partial liquidity, diversification across maturities (a ladder) and taxation all factor into the discussion. Bottom line: a bond is neither good nor bad — it is suitable or unsuitable for the goal.Exercises1) (Original - CESGRANRIO style) The Treasury Financial Note (LFT) is a federal government bond whose return is linked to:A) the daily change in the Selic rate.B) the monthly change in the IPCA, plus semiannual interest payments.C) a fixed rate determined at the issuance auction.D) the exchange-rate variation of the US dollar.E) the Reference Rate (TR), plus 6% per year.2) (Original - CESGRANRIO style) For indexed government bonds, the trading price is obtained by applying a quote to the Updated Nominal Value (VNA). When the quote is below 100%, the bond is said to be traded at:A) a discount, and the transaction's effective rate exceeds the index's variation.B) a premium, and the transaction's effective rate exceeds the index's variation.C) a discount, and the transaction's effective rate is lower than the index's variation.D) a premium, and the effective rate equals the index's variation.E) par value, with no effect on the transaction's rate.3) (Original - CESGRANRIO style) An investment posted a nominal return of 12% in a year when inflation measured by the IPCA was 5%. According to the Fisher equation, the investment's real return was approximately:A) 6.67% per year.B) 7.00% per year.C) 5.83% per year.D) 17.60% per year.E) 2.33% per year.4) (Original - FGV style) An NTN-B Principal was purchased at a rate of IPCA + 6% per year. Over a 12-month period in which the IPCA accumulated to 4%, the bond's nominal return, for an investor who held it between equivalent dates, was approximately:A) 10.24% per year.B) 10.00% per year.C) 6.00% per year.D) 24.00% per year.E) 9.80% per year.5) (Original - FGV style) Regarding the behavior of government bonds under mark-to-market accounting, select the correct statement.A) The long-term NTN-B tends to have the greatest price volatility among Treasury bonds because it combines a fixed real rate with extended maturities.B) The LFT has high price volatility because it follows the Copom's decisions.C) The LTN is not subject to mark-to-market pricing because it has a fixed nominal value of R$ 1,000.00.D) Adjusting the VNA for the IPCA prevents any fluctuation in the NTN-B's price.E) Floating-rate and fixed-rate bonds have the same sensitivity to changes in interest rates.6) (Original - CESGRANRIO style) The fundamental difference between the NTN-B (Treasury IPCA+ with Semiannual Interest) and the NTN-B Principal is that the former:A) pays semiannual interest coupons over the life of the bond, while the latter concentrates the entire payment at maturity.B) is adjusted by the IGP-M, while the latter is adjusted by the IPCA.C) has a floating Selic rate, while the latter has a fixed rate.D) is not subject to mark-to-market pricing, unlike the latter.E) has a fixed nominal value of R$ 1,000.00, while the latter has a VNA.7) (Original - CEBRASPE style, judge the item) Judge the following statement: "By guaranteeing a return linked to the IPCA plus a contracted real rate, the NTN-B ensures that the investor cannot suffer a nominal loss, even if the bond is sold before maturity."( ) True ( ) False8) (Original - CEBRASPE style, judge the item) Judge the following statement: "Because its Updated Nominal Value is adjusted daily by the Selic rate, the LFT has low price volatility, limited to variations in the premium or discount at which the bond is traded."( ) True ( ) FalseAnswer Key1) A — The LFT is a pure floating-rate bond tied to the daily Selic rate. Option B describes the NTN-B; option C describes the LTN. Building this mental "family portrait" resolves half the questions on the subject.2) A — A quote below 100% means a discount: the investor pays less than the VNA and therefore, in addition to the index's variation, pockets the difference — the effective rate exceeds the index. It is the same principle as the LTN's discount, applied to a nominal value that grows.3) A — i real = 1.12/1.05 − 1 = 6.67%. Option B (7%) is the simple subtraction, the standard trap; option D incorrectly multiplies the effects.4) A — i nominal = 1.06 × 1.04 − 1 = 10.24%. Option B (10%) is the simple sum — the Fisher equation compounds; it does not add.5) A — A fixed real rate plus extremely long maturities means the greatest sensitivity in the market. The LFT (option B) is precisely the opposite; the LTN (option C) has a fixed nominal value, but its PRICE fluctuates; the IPCA adjusts the VNA, it does not lock the quote (option D).6) A — Semiannual coupon versus zero coupon: that is the only structural difference. Both are IPCA + real rate, both have a VNA, and both are subject to mark-to-market pricing.7) False — The guarantee of IPCA + real rate applies at maturity. In an early sale, the market price prevails: if the real interest rate has risen enough, the loss may even be nominal.8) True — This is an accurate description of the LFT: the VNA tracks the Selic rate day by day, and price fluctuations are limited to small changes in the premium/discount — minimal, though not nonexistent (as seen during episodes of market stress).
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