Inflation, Rates & the Yield Curve

Phase 7 · Week 59 · ~20 hours

Where it sits. A 300 basis point rise in the discount rate cuts a growth company's value by 45%, with nothing whatever changing inside the business. That arithmetic arrives at the end of this module, and everything before it builds the machinery that sets the rate.

This is the third module of the macro phase, and the one that connects the machine you have built to the number that sits under every valuation you will ever do. M7.01 built the economy from transactions and taught you that credit is the volatile part of spending. M7.02 put the central bank on top of that machine and taught you how the policy rate (the price of the shortest, safest money) is made, and it ended on a promise: that the policy rate is the "gravity" of every asset price. This module keeps that promise, in two movements.

The first movement is inflation, because inflation is what the central bank is chasing, the thing its whole apparatus exists to control, and you cannot read a central bank, a bond, or a valuation without being able to read an inflation print cold: what inflation actually is, the four different engines that produce it, the rival ways we measure it (and why the Fed and the RBI measure it differently and even watch different numbers), the three pathologies where it goes wrong (deflation, hyperinflation, stagflation), and the single most famous and most-abused relationship in macro, the Phillips curve.

The second movement is the yield curve, the term structure of interest rates, which is where the single policy rate M7.02 produced gets stretched out across time into the entire spectrum of rates that actually discount cash flows: what the curve is, what drives its shape, why an inverted curve has predicted almost every US recession, how to read the four ways it moves (bull and bear, steepener and flattener), and finally duration and the gravity of valuation, the precise, arithmetic reason a higher discount rate compresses every asset price and crushes long-duration assets (growth stocks, long bonds) the hardest, tied explicitly back to the DCF and terminal-value work you did in Phase 3. By the end you can look at an inflation release and a yield curve and know what each is telling you about the gravity under every price. M7.04 takes the rate you now understand and runs the government's debt arithmetic through it; M7.05 walks the whole apparatus through its historical breakdowns. But this is the module where "rates" stop being a headline and become a tool you operate.

Suggested hour budget. Core content: 9h · worked examples (decompose an inflation print; read and classify a curve; duration and the price move; same company at two discount rates): 3h · practice set (inflation-type, measure-choice, base-effect, curve-reading, steepener/flattener, duration, gravity): 4.5h · applied mini-project (build a live inflation-and-curve read for India and the US): 2.5h · flashcards + mastery check: 1h.

App link. This module feeds the Macro & Cycles branch and seeds four drills that recur through the program. An inflation-print decomposition drill (an index series and component weights in; the YoY, the sequential-momentum annualized rate, the base-effect call, and the headline-vs-core split out). A curve-reading drill (a snapshot of yields in; the shape, the 2s10s and 10y–3m spreads, and the recession-signal read out, wired to a bank of real US Treasury and India G-sec curves). A steepener/flattener classification drill (two curve snapshots in; the correct one of the four labels, bull/bear × steepener/flattener, and its likely driver out, timed). And a duration/valuation-sensitivity calculator that hands the discount rate straight to the app's valuation sandbox: this is the module where the macro branch and the valuation branch physically join, because the discount rate this module explains is the same input the sandbox uses to price a company. It closes the loop the whole program is built on. Macro sets the gravity; gravity sets the price.

Learning objectives

By the end you can:

1. Define inflation from first principles as a fall in the purchasing power of money (a rise in the general price level), and distinguish it cleanly from the three things beginners confuse it with: disinflation (a falling inflation rate), deflation (a falling price level), and a relative price change in one good.
2. Classify inflation by its engine: demand-pull, cost-push, monetary, and expectations-driven (built-in). State the mechanism of each, give a real example, and explain why the type dictates whether a central bank should fight it or look through it.
3. Explain how inflation is measured and why the choice matters: CPI construction and its four biases; headline versus core and why central banks watch core; the US PCE versus CPI difference (formula, weights, scope) and why the Fed prefers PCE; India's CPI versus WPI and why India moved its anchor from WPI to CPI around 2015. Then read any inflation print as a rate of change, decomposing base effects from genuine momentum.
4. Diagnose the three pathologies: deflation and Irving Fisher's debt-deflation spiral; hyperinflation and its fiscal-monetary mechanism (Weimar, Zimbabwe, Venezuela); and stagflation, why it is the hardest regime to fight, and what the 1970s taught.
5. Argue the Phillips curve evenhandedly: the original inflation–unemployment tradeoff, why it broke down in the 1970s, the expectations-augmented (Friedman–Phelps) version and the vertical long-run curve at the natural rate, and the live modern debate over whether the curve is "flat," "dead," or merely non-linear.
6. Read a yield curve: define the term structure, decompose a long yield into expected future short rates plus a term premium, classify the shape (normal, flat, inverted, humped), and compute and interpret the 2s10s and 10y–3m spreads for both the US Treasury and India G-sec curves.
7. Use the inversion signal and the four curve moves: explain why an inverted curve has led almost every US recession, state the record and the honest caveats (lead time, term-premium distortion, the 2022–24 episode), and classify any curve move as a bull/bear steepener or flattener with its likely driver.
8. Compute duration and cash out "rates as gravity": estimate a bond's price change for a rate move via modified duration, and demonstrate with a worked same-company-two-discount-rates DCF why a higher rate compresses all valuations and hits long-duration assets (growth equities, long bonds, unprofitable tech) hardest. That closes the loop to M3.05's terminal value.

Prerequisites & connections

Builds on. M7.02, where this module begins exactly where the last ended: the policy rate is made there and stretched across time here; the mandate differences (the Fed targets 2% PCE, the RBI targets 4% headline CPI) are stated there and dissected here (§4.5–4.6); and the closing "rate is gravity" idea is proved here (§4.14). M7.01, for the money-and-credit machine and, in particular, the quantity-theory intuition behind monetary inflation (§4.2) and the endogenous-money reason "printing" need not be inflationary. M3.01, time value of money: every claim in the second half of this module is present-value arithmetic; the growing-perpetuity (Gordon) formula and discounting are assumed cold. M3.03, the cost of capital: the risk-free rate that anchors CAPM and WACC is a point on the yield curve, so this module explains where that first input comes from and what moves it. M3.05, terminal value, sensitivity, and reverse DCF: you already computed how sensitive a valuation is to the discount rate; §4.14 supplies the macro cause of that sensitivity and reunites it with duration. M0.03, real versus nominal (Fisher): inflation is the wedge between nominal and real, and the Fisher equation (nominal ≈ real + expected inflation) is the hinge of §4.1 and §4.10.

Feeds forward. M7.04, fiscal policy and sovereign debt: the r versus g debt-dynamics arithmetic runs on the very rate and inflation numbers this module teaches, and hyperinflation (§4.8) is the violent end of the fiscal-dominance story M7.04 formalizes. M7.05, cycles, bubbles, and crises: the yield-curve inversion signal (§4.11) is a leading indicator you will apply to 2008 and 2020, and stagflation (§4.8) is the regime behind the 1970s case study. M7.06, FX and the dollar: real-rate differentials (nominal rate minus expected inflation) are a prime driver of currencies, and the term premium and curve shape reappear in the global bond market. M7.07, the dashboard capstone: the CPI/core-PCE prints, the policy-rate path, the 10y yield, and the 10y–2y and 10y–3m spreads are core dashboard indicators, read exactly as this module teaches: rate-of-change, versus expectations, mapped onto the cycle. And every valuation module downstream inherits §4.14: the discount rate is the gravity, and you now know what moves it.

The one-sentence version of this module. Inflation is the rate at which money loses its purchasing power (measured several rival ways for good reasons, produced by four different engines, and pathological at both extremes), and the yield curve is that same inflation-and-rate story stretched across time into the term structure of interest rates, whose level is the gravity on every asset price and whose slope is one of the few honest recession forecasts the market gives away for free.

Core teaching content

4.8 The three pathologies: deflation, hyperinflation, stagflation

A little inflation, stable and expected, is the healthy state: it greases the labour market (firms can cut real wages by holding nominal wages flat, avoiding demoralizing nominal cuts), it keeps a safe distance from the deflation trap, and it is why every major central bank targets a small positive number (2% in the US and euro area, 4% in India) rather than zero. Inflation becomes pathological at both extremes and in one nasty combination. All three pathologies are worth understanding mechanically, because each is a regime where the ordinary rules break and where fortunes (and companies) are made and destroyed.

Common mistakes & how experts think differently

1. Confusing disinflation with deflation. The single most common error. "Inflation fell to 3%" means prices are still rising, just more slowly (disinflation). It does not mean prices are falling (deflation). A beginner hears "inflation is coming down" and pictures cheaper groceries; prices are still going up. Experts keep the ladder straight (inflation, up fast; disinflation, up slowly; deflation, down; reflation, pushed back up) because the policy and portfolio implications of each are completely different.

2. Treating a single dramatic price as "inflation." Petrol jumps, onions spike, and the headline screams inflation. But a relative price change in one good is not a change in the general price level. Experts ask: is this one price moving, or is the whole basket moving? A supply shock in one item is a cost-push relative move the central bank usually looks through; broad-based increases across the basket are the real thing. The composition, not the drama, is the signal.

3. Reading only the year-over-year headline and missing the momentum turn. YoY inflation is a twelve-month average, sluggish and backward-looking. It can keep printing "high" for months after prices have actually stopped rising (and vice versa). Experts always compute sequential momentum (MoM, 3m/3m annualized) to see what inflation is doing right now, and they explicitly strip base effects before concluding a trend has changed. Amateurs get whipsawed by base effects and mistake an optical move for a real one.

4. Quoting CPI when the Fed targets PCE (and forgetting core). Endless confusion comes from comparing "CPI at 3.5%" to the Fed's "2% target". But the Fed's target is on PCE, which runs a few tenths lower, and the Fed operationally watches core PCE. Experts know which gauge each central bank actually steers by (Fed: core PCE; RBI: headline CPI), quote the right one, and read an ensemble (headline, core, trimmed-mean) rather than a single number.

5. Thinking "money printing" must mean inflation. Carried over from M7.02 and worth repeating because it is so persistent: QE creates reserves (base money trapped in the banking system), not spendable broad money, which is why a decade of QE produced below-target inflation. Inflation needs new lending and spending, not parked reserves. The 2021 inflation came from QE plus fiscal transfers into spending accounts plus supply shocks: money reaching the real economy, not the balance-sheet expansion per se. Experts track broad money and spending (and the fiscal stance), not the central-bank balance sheet alone.

6. Believing the Phillips curve is either a reliable law or completely dead. Both extremes are wrong. The naive 1960s "stable exploitable menu" was refuted in the 1970s; but the mechanism (slack and expectations push on inflation, supply shocks matter) is sound. Experts use it as a framework for thinking (is the labour market tight, are expectations anchored, is there a supply shock?), never as a forecasting formula, and they treat NAIRU as an unobservable, drifting estimate rather than a known number.

7. Getting "bull" and "bear" backwards on the curve. "Bull" means bond prices up = yields DOWN; "bear" means yields up. Beginners intuit the opposite (associating "bull" with "up = higher yields") and misclassify every curve move. Experts anchor it to price: a bull market in bonds is a rally, bonds rally when yields fall, so a "bull steepener" is falling yields with the front end falling most.

8. Reading an inverted curve as "sell now." Inversion is a recession forecast with a 6–24-month lead, not a timing signal, and equities often rise for a year or more after inversion. Experts weight it heavily as a warning and a reason to check their risk posture, but they neither panic-sell on the inversion nor dismiss it because "the recession hasn't come yet". They hold the term-premium and 2022–24 caveats honestly rather than either trusting or mocking the signal.

9. Applying US macro frameworks unchanged to India. The inversion signal, the Phillips curve, even the headline-vs-core logic all work differently in India: food-dominated CPI, supply-shock-driven inflation, an informal labour market, an SLR-distorted G-sec curve, headline (not core) targeting. Experts localize every framework to India's structure rather than importing the American textbook whole; a mechanical transplant is a classic and costly error.

10. Discounting a stock at "whatever rate" without seeing that the rate is the whole game. Beginners treat the discount rate as a technical input to be set once and forgotten. Experts know it is the gravity: that a growth stock is mostly terminal value, that terminal value is acutely sensitive to the rate through the (WACC − g) denominator, and that therefore a modest change in the risk-free rate can swing a long-duration valuation by tens of percent. They stress-test every valuation across a range of rates and treat any price that only works at permanently-low rates as a bet on gravity staying weak.

Worked examples

Worked Example 1: Decompose an inflation print (India CPI)

You are handed the monthly India CPI release. Read it like an analyst: separate momentum from the year-over-year headline, strip the base effect, and split headline from core. (Index levels and weights are illustrative, modeled on India CPI's ~2012=100 scale and its published weights. Pull the actual MOSPI series; the method is the point.)

The data. CPI (Combined) index: June last year (the base, twelve months ago) = 185.0; May this year = 192.0; June this year = 192.5. Component YoY inflation this month: food & beverages +6.0% (weight 46%), fuel & light +5.0% (weight 7%), core (ex food & fuel) +3.0% (weight 47%).

Step 1: the year-over-year headline. YoY = 192.5 / 185.0 − 1 = 0.0405 = 4.05%. Sits comfortably inside the RBI's 2–6% band, below the 4% target's upper reaches. At first glance, unremarkable.

Step 2: the sequential momentum. Month-over-month: 192.5 / 192.0 − 1 = 0.260%. Annualize it (compound): (1.00260)¹² − 1 = 3.17% (or the quick approximation 0.260% × 12 = 3.13%). The read: current momentum (~3.2% annualized) is running below the 4.05% YoY. Inflation right now is cooler than the backward-looking headline suggests: the YoY is being held up by price rises from earlier in the twelve-month window. A YoY-only reader would miss this; the momentum reader sees disinflation in train.

Step 3: the base effect. Project next month. Suppose last July's index was unusually high at 187.5 (a monsoon-driven food spike that has since passed), and this month-to-next momentum is normal, say +0.3%, taking the index to 192.5 × 1.003 ≈ 193.1. Then next month's YoY = 193.1 / 187.5 − 1 = 2.97%. The headline will drop from 4.05% to ~3.0%, but not because inflation slowed (momentum was a normal +0.3%); it fell because the year-ago base was high. This is a favourable base effect, and an analyst flags it explicitly so the RBI's expected "sharp fall in July CPI" is read as arithmetic, not as fresh disinflation.

Step 4: headline versus core. Reconstruct the headline from components: 0.46 × 6.0% + 0.07 × 5.0% + 0.47 × 3.0% = 2.76% + 0.35% + 1.41% = 4.52%. (The small gap from the 4.05% index-based figure reflects rounded weights and levels; the decomposition is the lesson.) The read: the headline (~4.5%) sits well above core (3.0%) because a food spike (46% of the basket at 6%) is doing almost all the work: food alone contributes 2.76 points of the 4.5. Core, the persistent, demand-driven signal, is a benign 3.0%. The verdict an analyst writes: "Headline elevated but food-driven; core contained at 3%; momentum ~3% annualized; a favourable base effect will pull the July headline toward 3%. Underlying inflation is not a problem — this is a supply-side food story the RBI can look through unless it broadens into core or un-anchors expectations." That one sentence (engine, momentum, base effect, core) is the professional read, and it points to an RBI on hold or easing rather than hiking.

Worked Example 2: Read and classify a yield curve (US Treasury, with an India contrast)

Two snapshots of the US Treasury curve, six months apart. Classify each shape, compute the two key spreads, and name the move between them. (Yields illustrative, in %; pull live values from the US Treasury par-yield curve or FRED: DGS3MO, DGS2, DGS10, etc.)

Step 1: the spreads and shape at Date 1. 2s10s = 10y − 2y = 4.20 − 4.90 = −0.70% (−70 bp). 10y−3m = 4.20 − 5.40 = −1.20% (−120 bp). Both spreads are negative → the curve is inverted (short rates well above long). The read: the market expects the Fed to have to cut, a classic restrictive-policy, recession-warning configuration. (Note the 30-year at 4.40 above the 10-year: the very long end is often less inverted, giving a "U" shape common in deep inversions.)

Step 2: the spreads and shape at Date 2. 2s10s = 4.05 − 4.10 = −0.05% (−5 bp), essentially flat, all but dis-inverted. 10y−3m = 4.05 − 4.40 = −0.35% (−35 bp), still mildly inverted but far less so. The curve has moved from deeply inverted toward flat.

Step 3: name the move. Compute the changes: 3m −1.00, 2y −0.80, 5y −0.40, 10y −0.15, 30y +0.05. Yields fell across the front and belly → a bull move (bond prices up). The short end fell far more than the long end (3m −100 bp vs 10y −15 bp) → a steepener. Therefore this is a bull steepener. The driver: the market has begun pricing imminent rate cuts, and the front end (which tracks policy) rallied hardest as it discounted the coming easing, dragging the curve out of its inversion. The analyst's flag: a bull steepener dis-inverting a long-standing inversion is historically the "here it comes" configuration, because the cuts get priced when the downturn (or its clear approach) has arrived. This is not an all-clear; it is often the opposite.

Step 4: the India contrast. An India G-sec snapshot might read: 91-day T-bill 6.30, 2y 6.35, 5y 6.55, 10y 6.60, 30y 6.85, a normal, upward-sloping curve (10y − 91d = +30 bp). Two lessons. First, the level is structurally higher than the US curve: India's higher inflation demands higher nominal yields for the same real return (the Fisher point, §4.1). Second, India's curve is normally upward-sloping and rarely inverts, and for the structural reasons in §4.10, even when it flattens you should not read it as a US-style recession signal. Use the 10-year G-sec (6.60%) as the rupee risk-free rate for discounting Indian cash flows; do not trade the India slope as a growth forecast.

Worked Example 3: Duration and the price of a rate move (US Treasury and India G-sec)

Compute how much two bonds move when yields rise 100 bp, and see duration and convexity as numbers. (Synthetic bonds with clean coupons; the arithmetic is exact.)

The bonds. Both pay an annual coupon on ₹1,000 (or $1,000) face and are priced at par (so their yield equals their coupon):

• Bond A: 10-year, 4% coupon, yield 4%, price 1,000.
• Bond B: 2-year, 4% coupon, yield 4%, price 1,000.

Step 1: reprice Bond A after a +100 bp move (yield 4% → 5%). Price = 40 × [1 − 1.05⁻¹⁰]/0.05 + 1000 × 1.05⁻¹⁰. With 1.05⁻¹⁰ = 0.61391: annuity factor = (1 − 0.61391)/0.05 = 7.7217; coupons = 40 × 7.7217 = 308.87; face = 1000 × 0.61391 = 613.91; price = 922.78. Price change = (922.78 − 1000)/1000 = −7.72%.

Step 2: reprice Bond B after the same move. Price = 40/1.05 + 1040/1.05² = 38.10 + 943.31 = 981.41. Price change = −1.86%.

Step 3: the duration reading. Bond A fell 7.72%, Bond B fell 1.86% for the identical 100 bp move: the 10-year moved ~4.1× as much as the 2-year. That ratio is the duration story: computing Macaulay duration gives 8.44 years for Bond A (modified = 8.44/1.04 = 8.11) versus 1.96 years for Bond B (modified 1.89). The modified duration predicts the move directly: %ΔP ≈ −ModDur × Δy = −8.11 × 1% = −8.11% for Bond A, −1.89 × 1% = −1.89% for Bond B, close to the exact repricings.

Step 4: see the convexity. Notice Bond A's predicted −8.11% is slightly worse than the exact −7.72%. That 0.4% gap is convexity: the price–yield relationship is curved, so straight-line duration overstates the loss on a rate rise (and understates the gain on a fall), a small, favourable error for the bondholder. The lesson: longer bonds have higher duration and move much more (the quantified "long bonds move more"); duration is an excellent linear estimate; convexity is the second-order polish. An India 10-year G-sec at a ~6.6% yield has a modified duration around 7 and would fall on the order of 7% for a 100 bp yield rise: identical mechanics, higher yield level. Duration does not care about the country; it cares about when the cash arrives.

Worked Example 4: Same company, two discount rates: rates as gravity (the DCF bridge)

This is the quantitative heart of §4.14. Value ONE synthetic growth company at two discount rates, an 8% WACC and an 11% WACC (a 300 bp rise, the kind a hiking cycle delivers), and decompose the hit to see the duration effect inside a single valuation. (Synthetic company "Helios," labeled synthetic; all figures in $m.)

The company. Helios generates free cash flow of $100m next year (Year 1), growing 10% per year for 10 years (the explicit high-growth phase), then settling to a terminal growth of 4% forever. This is a classic long-duration, growth profile. Most of the value will sit far in the future.

Step 1: value the explicit 10-year FCF stream (the "near" cash flows). Using the growing-annuity present value, PV = C₁/(r − g_s) × [1 − ((1+g_s)/(1+r))ⁿ] with C₁ = 100, g_s = 10%, n = 10:

• At r = 8%: PV = 100/(0.08−0.10) × [1 − (1.10/1.08)¹⁰] = (−5000) × (1 − 1.2014) = (−5000)(−0.2014) = $1,007m.
• At r = 11%: PV = 100/(0.11−0.10) × [1 − (1.10/1.11)¹⁰] = 10,000 × (1 − 0.9135) = 10,000 × 0.0865 = $865m.
• Change in the near cash flows: (865 − 1,007)/1,007 = −14.1%.

Step 2: value the terminal value (the "far" cash flows). Year-10 FCF = 100 × 1.10⁹ = $235.8m; Year-11 FCF = 235.8 × 1.04 = $245.2m. Terminal value at Year 10 = 245.2/(r − 0.04), discounted back 10 years:

• At r = 8%: TV₁₀ = 245.2/0.04 = 6,131; PV = 6,131/1.08¹⁰ = 6,131/2.1589 = $2,840m.
• At r = 11%: TV₁₀ = 245.2/0.07 = 3,503; PV = 3,503/1.11¹⁰ = 3,503/2.8394 = $1,234m.
• Change in the terminal value: (1,234 − 2,840)/2,840 = −56.6%.

Step 3: assemble, and read the gravity.

The lesson, in numbers. A 300 bp rise in the discount rate, with nothing about the business changed and only the gravity moving, cut Helios's value by 45%. And look where the damage fell: the near cash flows lost only 14% (they are short-duration), but the terminal value lost 57% (it is a long-duration object, and the (WACC − g) denominator widened from 4% to 7%, a 75% widening that alone nearly halves it). Because Helios is mostly terminal value (74% of its worth at the 8% rate), the whole company fell almost as hard as its terminal value did. That is duration inside an equity, and that is the gravity of rates made arithmetic.

Now the contrast that completes the point. A mature cash-cow whose value sits mostly in near cash flows would behave like the top row, falling ~14% rather than ~45%, because its cash is short-duration. (Be honest about the nuance: a stable Gordon perpetuity is also long-duration, with 100/(0.08−0.04) = 2,500 falling to 100/(0.11−0.04) = 1,429, a 43% drop, so "value" protects you only insofar as the cash actually arrives soon; a low-growth stock that is still all-future-cash is long-duration too. Duration is about timing, not the "growth/value" label.) The takeaway that powers the rest of your investing: raise the risk-free rate and the terminal-value-heavy, long-duration assets (high-growth software, unprofitable tech, long bonds) fall the hardest, exactly as they did in 2022. The yield curve sets the rate; the rate is the gravity; the terminal value is what it pulls on hardest. Stress-test every valuation you build across a range of discount rates, and never own a price that only makes sense if gravity stays weak forever.

Worked Example 5: Breakeven inflation and the real rate (US TIPS)

The bond market prices expected inflation for you, for free: subtract the inflation-protected yield from the nominal yield and you have the market's forecast. Read it, then see why it is called a "breakeven." (Yields illustrative, in %; pull live values from FRED: DGS10 for the nominal 10-year and DFII10 for the 10-year TIPS real yield.)

The data. The 10-year nominal Treasury yields 4.30%; the 10-year TIPS (Treasury Inflation-Protected Security, whose principal is indexed to CPI so it pays a real yield) yields 1.85% real.

Step 1: the quick breakeven (the market convention). breakeven ≈ nominal − real = 4.30 − 1.85 = 2.45%. This is the breakeven inflation rate, the average annual CPI inflation over the next ten years that the market is implicitly pricing. Every trading desk quotes it as exactly this simple subtraction.

Step 2: the exact breakeven (Fisher). The subtraction drops a cross-term. Fisher exactly: (1 + nominal) = (1 + real)(1 + breakeven), so breakeven = 1.0430/1.0185 − 1 = 2.41%. The ~4 bp gap between 2.45% and 2.41% is the real × inflation cross-product the shortcut ignores: trivial at US rate levels, larger where rates are high (which is why the shortcut is rougher for India). The read: the market expects ~2.4% CPI inflation a year for a decade. Since the Fed targets 2% PCE and PCE runs ~0.3–0.4 pp below CPI (§4.5), a 2.4% CPI breakeven is ≈ 2.0% PCE, so expectations sit essentially on the Fed's target: anchored. (Quoting the 2.4% CPI figure against the 2% PCE target would falsely scream "expectations un-anchored", the §4.5 gauge-mismatch trap.)

Step 3: why "breakeven"? The real return, three ways. Buy the nominal bond and your real return depends on what inflation actually turns out to be (realized real = (1 + nominal)/(1 + realized inflation) − 1):

• Inflation lands at 3.0% (above breakeven): 1.0430/1.030 − 1 = 1.26% real, below the 1.85% you could have locked risk-free in the TIPS. You would have been better off in the TIPS.
• Inflation lands at 2.0% (below breakeven): 1.0430/1.020 − 1 = 2.25% real, above the TIPS. The nominal bond wins.
• Inflation lands exactly at the breakeven, 2.41%: 1.0430/1.0241 − 1 = 1.85%, identical to the TIPS.

The lesson. That third line is the whole point: 2.41% is the inflation rate at which the nominal bond and the inflation-linked bond deliver the same real return; they break even. Above it, inflation eats the nominal bond's fixed coupon and TIPS win; below it, nominals win. So the breakeven is simultaneously (a) the market's inflation forecast, (b) the hurdle your own inflation view must beat to prefer one bond over the other, and (c), honestly, a touch higher than pure expected inflation, because it also embeds a small inflation-risk premium (compensation for inflation uncertainty), so breakevens read expectations slightly high. This is the Fisher hinge of §4.1 turned into a live, tradeable number, and the cleanest daily gauge of where inflation expectations, the subject of this module's whole first half, actually sit.

Worked Example 6: A curve steepener and its P&L (India G-sec)

A rates desk does not merely read the curve; it positions for the curve's next move. Build the classic "2s10s steepener," weight it so only the slope matters, and compute the profit when the curve steepens. (Synthetic G-secs with clean modified durations; ₹ in crore; carry and financing ignored, since the trade and the arithmetic are the point.)

The setup. You expect the G-sec curve to steepen (the 10y−2y spread to widen). The position that profits from a steepening is long the front, short the back: buy the 2-year, short-sell the 10-year. But a naked pair would also swing with the level of rates (a parallel move), on which you have no view, so you duration-weight it to be DV01-neutral (immune to a parallel shift), leaving P&L driven only by the change in slope.

• 2-year G-sec: modified duration 1.9. You go long ₹100 cr.
• 10-year G-sec: modified duration 8.0. You go short the amount that matches the 2-year leg's rate sensitivity.

Step 1: size the hedge (DV01-neutral). Match "rupee duration" (modified duration × notional) across the legs: short notional = 100 × 1.9/8.0 = ₹23.75 cr. Check a +1 bp parallel move: the long 2y loses 1.9 × 0.01% × 100 = ₹0.019 cr; the short 10y gains 8.0 × 0.01% × 23.75 = ₹0.019 cr; net ≈ ₹0. The level is hedged; only the slope can move the P&L.

Step 2: the curve steepens 15 bp, case A (a bull steepener). The 2-year falls 20 bp and the 10-year falls 5 bp (spread widens −5 − (−20) = +15 bp):

• Long 2y: %ΔP = −1.9 × (−0.20%) = +0.38% on ₹100 cr → +₹38.0 lakh.
• Short 10y: the bond gains −8.0 × (−0.05%) = +0.40% on ₹23.75 cr = +₹9.5 lakh, but you are short, so P&L = −₹9.5 lakh.
• Net P&L = 38.0 − 9.5 = +₹28.5 lakh.

Step 3: the same 15 bp steepening, case B (a bear steepener). Now the 2-year rises 5 bp and the 10-year rises 20 bp (spread again widens 15 bp):

• Long 2y: −1.9 × (+0.05%) × ₹100 cr = −₹9.5 lakh.
• Short 10y: the bond loses −8.0 × (+0.20%) = −1.60% on ₹23.75 cr; short, so P&L = +₹38.0 lakh.
• Net P&L = −9.5 + 38.0 = +₹28.5 lakh, identical.

The lesson. Both cases pay exactly ₹28.5 lakh, because a DV01-neutral steepener's P&L collapses to one clean formula: P&L ≈ (rupee duration per leg) × Δ(spread) = ₹190 cr × 0.0015 = ₹28.5 lakh (₹190 cr of rupee duration, since 1.9 × 100 = 8.0 × 23.75, times a 15 bp widening). The level of rates dropped out by construction; whether the steepening arrived as a rally (bull) or a sell-off (bear) is irrelevant to first order. The trade expresses a pure view on the slope, which is exactly what "positioning for a bull/bear steepener" (§4.12) means in P&L terms. (Two honest footnotes, both from §4.13: convexity makes the 10-year leg's true move a shade different from the linear estimate, and carry, the net yield earned or paid to hold the legs, is a separate P&L the desk tracks. For the modest moves here, the duration arithmetic is the story.)

Practice set

Problems 1–4 are guided (framework applied with scaffolding), 5–9 independent, 10–12 exam-condition (12 minutes each, closed book). Levels, rates, and weights are illustrative "as of mid-2026 — verify"; use the numbers given. Write your full answer before reading the solution.

P1 (guided: name the engine). For each scenario, name the inflation engine (demand-pull, cost-push, monetary, or expectations-driven) and give a one-line reason: (a) a government at war has its central bank print money to cover its deficit, and prices rise 30% a month; (b) OPEC cuts output, oil doubles, and prices across the economy rise while growth slows; (c) unemployment is at a 50-year low, wages are climbing fast, and consumer demand outruns what firms can produce; (d) after three years of 8% inflation, unions now demand 8% raises automatically and firms raise list prices 8% every January regardless of demand; (e) a failed monsoon spikes Indian vegetable prices and lifts headline CPI, though core holds at 4%.

Solution. (a) Monetary: deficit monetization exploding the money supply; this is the hyperinflation template (§4.8), the monetary engine run to destruction. (b) Cost-push: an energy supply shock raising input costs economy-wide while depressing output (the stagflation recipe). (c) Demand-pull: aggregate demand outrunning capacity at a positive output gap; the "good problem" kind the rate tool is designed to cool. (d) Expectations-driven (built-in): a wage-price spiral running on belief, no fresh shock needed; the self-fulfilling engine central banks fear most. (e) Cost-push (a food supply shock). Note that it is a relative price move showing up in headline while core stays anchored, exactly the food-driven print the RBI looks through unless it broadens into core or un-anchors expectations.

P2 (guided: CPI biases). Name the CPI bias in each, then answer the summary questions: (a) the index keeps pricing beef at its old weight after shoppers switched to chicken when beef got dear; (b) this year's phone has a far better camera, but the index treats the whole price rise as inflation; (c) shoppers moved to warehouse clubs and online discounters and pay less, but the index samples traditional stores; (d) a blockbuster drug enters the basket years after launch, missing its early price falls. Summary: (e) in which direction do these biases push measured CPI, and what did the Boskin Commission estimate?

Solution. (a) Substitution bias: a fixed basket can't capture switching away from what got relatively dear; the largest bias, and the main reason the Fed prefers PCE. (b) Quality-change bias: part of the price buys more product; fought with hedonic adjustment. (c) Outlet-substitution bias: the saving from cheaper outlets is missed. (d) New-product bias: late basket entry misses the early price declines and consumer-surplus gains. (e) All four tend to make CPI overstate true inflation; the Boskin Commission (1996) put the overstatement at roughly 1.1 percentage points a year (since reduced by BLS fixes, but a few tenths likely remain).

P3 (guided: which gauge, and why). (a) Which inflation gauge does the Fed target and operationally watch, and give two reasons it prefers that gauge to CPI? (b) Which gauge does the RBI target, and why headline rather than core? (c) If US CPI prints 3.6% and the CPI–PCE wedge is about 0.4 points, roughly what is PCE, and is the Fed nearer its target than the CPI headline suggests?

Solution. (a) The Fed targets 2% headline PCE and operationally watches core PCE. Two reasons it prefers PCE: (i) the formula, since PCE is chain-weighted (Fisher-ideal) and captures substitution, which CPI's fixed Laspeyres basket cannot, so PCE is a better cost-of-living measure and runs a few tenths lower; (ii) the scope, since PCE is broader, counting consumption paid on households' behalf (employer- and government-funded healthcare) that out-of-pocket CPI misses (a different weight structure follows, e.g. much less housing, much more healthcare than CPI). (b) The RBI targets 4% headline CPI ±2%. It is headline, not core, because food is ~46% of the Indian basket and household inflation expectations are set by the food and fuel prices people see daily, so an anchor that excluded food would disconnect from the inflation Indians actually experience and act on. (c) PCE ≈ 3.6% − 0.4% = 3.2%; yes, the Fed is closer to its 2% goal than the 3.6% CPI headline implies, which is the classic reason to quote the right gauge.

P4 (guided: CPI vs WPI in India). (a) Name two structural flaws that led India to drop WPI as its policy anchor. (b) Which committee recommended CPI, and roughly when did India adopt CPI-based flexible inflation targeting? (c) WPI is running +6% while CPI is +4%: what does that gap tell an equity analyst?

Solution. (a) Any two: WPI contains no services (which are more than half of India's GDP, so WPI literally can't see much of the price level); WPI is a wholesale/producer measure, not what households pay (wrong concept for a central bank whose job is household purchasing power); WPI is dominated by globally-traded commodities and fuel, so it swings with world prices and the exchange rate rather than domestic demand. (b) The Urjit Patel Committee (2014); India adopted CPI-based flexible inflation targeting around 2015–16 (RBI Act amended, MPC constituted 2016). (c) WPI above CPI means producer/input costs are rising faster than retail prices: companies are being squeezed and cannot fully pass costs through, a margin-compression warning for goods manufacturers (FMCG, autos, consumer durables, the Phase 5 sectors). Reading the two indices against each other locates where in the supply chain the price pressure sits and who is absorbing it.

P5 (independent: read the print). India CPI index: twelve months ago = 150.0; last month = 156.5; this month = 157.0. (a) Compute the YoY inflation rate. (b) Compute this month's MoM change and annualize it. (c) Do YoY and momentum agree? (d) If the index next month rises a normal +0.3% and the year-ago base for that month was an unusually low 150.2, compute next month's YoY and explain the base effect.

Solution. (a) YoY = 157.0/150.0 − 1 = 4.67%. (b) MoM = 157.0/156.5 − 1 = 0.319%; annualized = (1.00319)¹² − 1 = 3.90% (quick approx 0.319 × 12 = 3.83%). (c) No: momentum (~3.9%) is running below the YoY (4.67%), so inflation right now is cooler than the backward-looking headline; the YoY is held up by price rises from earlier in the twelve-month window (disinflation in train). (d) Next month's index = 157.0 × 1.003 = 157.47; YoY = 157.47/150.2 − 1 = 4.84%. The YoY rises to 4.84% despite normal +0.3% momentum, an unfavourable base effect: the low year-ago base (150.2) mechanically lifts the YoY. An analyst flags this as arithmetic, not fresh inflation, and points to the ~3.9% momentum as the true underlying pace.

P6 (independent: the three pathologies). (a) State Irving Fisher's debt-deflation spiral in three steps and name the reinforcing "zero-lower-bound" trap. (b) Give the fiscal-monetary mechanism of hyperinflation in two sentences and name Cagan's threshold. (c) Why is stagflation the hardest regime for a central bank, and what single 1970s→1980s episode is the canonical lesson?

Solution. (a) Spiral: falling prices raise the real burden of nominal debt → debtors sell assets and cut spending to service it (distress selling) → mass selling pushes prices down further → real debt rises again (loop). Trap: at the zero lower bound the nominal rate can't fall below ~0, so falling (negative) inflation forces the real rate up (real = nominal − inflation) exactly when the economy needs it lower, and monetary policy loses its lever. (b) A government that can't tax or borrow enough finances spending by printing money (monetizing the deficit), exploding the money supply and prices; as money visibly loses value, people spend it ever faster (velocity spikes) and the shrinking real value of money forces even faster printing, a self-reinforcing collapse. Cagan's threshold: inflation above 50% per month. (c) Stagflation pairs high inflation and high unemployment, which demand opposite medicine from the single interest-rate tool (hike to fight inflation → worsens unemployment; cut to fight unemployment → worsens inflation), so the central bank must choose which fire to fight. The canonical lesson: the 1970s oil-shock stagflation with un-anchored expectations, broken by Volcker's ~19–20% rates and the deep 1981–82 recession (unemployment 10.8%), proof that once expectations un-anchor, only a costly, credible, sustained tightening restores price stability.

P7 (independent: the Phillips curve). (a) What did the original Phillips curve claim, and what refuted it? (b) What did Friedman and Phelps add, and what shape is the long-run curve? (c) State the modern debate fairly in two or three sentences.

Solution. (a) The original curve (Phillips 1958; Samuelson–Solow 1960) claimed a stable, exploitable inverse tradeoff between inflation and unemployment, a "menu" a government could choose along. It was refuted by the 1970s stagflation, when high inflation and high unemployment occurred together, off any stable curve. (b) Friedman and Phelps added inflation expectations: inflation ≈ expected inflation − β(u − u*) + supply shock. The short-run tradeoff exists only via expectational error; once people expect the inflation, the short-run curve shifts up and unemployment returns to its natural rate (NAIRU). So the long-run Phillips curve is vertical: no permanent tradeoff, and pushing unemployment below NAIRU just produces accelerating inflation. (c) Fairly: the mechanism (slack and expectations push on inflation; supply shocks matter) is sound, but the stable, precisely-calibrated, exploitable tradeoff does not exist. The curve looked nearly flat/"dead" from the 1990s–2010s (unemployment fell to 3.5% with no inflation surge, on anchored expectations and globalization), then 2021–22 revived the debate, with revivalists arguing it is convex/non-linear (flat with slack, steep when very tight) and that supply shocks plus expectations did the work. Use it as a framework for thinking, never a forecasting formula; NAIRU is unobservable and drifts. And the whole logic is weaker still in India (informal labour, food-driven supply-side inflation).

P8 (independent: read the curve). A US Treasury curve reads: 3-month 5.30%, 2-year 4.60%, 10-year 4.10%, 30-year 4.30%. (a) Compute the 2s10s and 10y–3m spreads. (b) Classify the shape and state what it implies. (c) Give the mechanism by which this signal has predicted recessions, and two honest caveats.

Solution. (a) 2s10s = 4.10 − 4.60 = −0.50% (−50 bp); 10y−3m = 4.10 − 5.30 = −1.20% (−120 bp). (b) Both negative → the curve is inverted; short rates above long. It implies the market expects the central bank to cut rates, i.e. it is forecasting economic weakness/recession (with policy currently restrictive). (c) Mechanism (any coherent account): an inversion is the market pricing future cuts because it foresees weakness (a collective forecast that has been right); it is also causal via banks, whose borrow-short/lend-long margin gets squeezed by inversion, so they tighten credit and help cause the slowdown; and it signals policy has become restrictive, which is what ends expansions. Caveats (any two): long/variable lead time of 6–24 months (equities often keep rising after inversion, so it is not a "sell now" signal); term-premium distortion from QE/safe-asset demand may make a given inversion mean less; the 2022–24 episode inverted deeply yet the forecast recession did not arrive on schedule (a genuine misfire-or-delay, still debated); and the signal is unreliable on India's G-sec curve (SLR distortion, RBI operations, different cycles).

P9 (independent: name the move). Two curve-move cases. Case 1: the 2-year goes 3.0% → 4.2% and the 10-year 4.5% → 5.0%. Case 2: the 2-year goes 4.0% → 4.1% and the 10-year 4.2% → 4.9%. For each: (a) did yields rise or fall (bull or bear)? (b) did the curve steepen or flatten? (c) name the move and give its typical driver.

Solution. Case 1: (a) yields rose → bear. (b) the front end rose more (2y +1.2 vs 10y +0.5), so the spread narrowed from +1.50 to +0.80 → flatten. (c) Bear flattener, driver: the central bank hiking or expected to hike, selling off the front end, with the long end rising less because the market bets tightening will cool future growth/inflation (the classic mid-hiking-cycle move, e.g. 2022). Case 2: (a) yields rose → bear. (b) the long end rose more (10y +0.7 vs 2y +0.1), so the spread widened from +0.20 to +0.80 → steepen. (c) Bear steepener, driver: rising inflation expectations / term premium, heavy government bond supply (fiscal worry), or a growth/reflation surprise, all of which hit the long end. The rule that generates both answers: policy-driven moves hit the front end (bull steepener, bear flattener); long-run inflation/growth/supply-driven moves hit the long end (bear steepener, bull flattener).

P10 (exam, 12 min, duration). A 10-year zero-coupon G-sec yields 7.0%. (a) State its Macaulay duration and compute its modified duration. (b) Estimate its price change if the yield rises 60 bp. (c) Explain in one line why a 10-year zero moves more than a 10-year coupon bond. (d) Which asset carries more interest-rate risk, this 10-year zero or a 2-year bond, and why, in one line?

Solution. (a) A zero pays only at maturity, so Macaulay duration = maturity = 10 years; modified duration = 10/1.07 = 9.35. (b) %ΔP ≈ −ModDur × Δy = −9.35 × 0.60% = −5.6% (the exact repricing is about −5.4%; the ~0.2% difference is convexity, since duration slightly overstates the loss). (c) The zero puts all its cash at year 10 (duration = full maturity), while a coupon bond pays some cash earlier (duration < maturity), so the zero is longer-duration and more rate-sensitive. (d) The 10-year zero: its duration (9.35) is roughly five times the 2-year's (~1.9), so it moves ~5× as much for the same yield change; longer-dated, lower-coupon cash flows are more rate-sensitive.

P11 (exam, 12 min, gravity via terminal value). A company's terminal free cash flow (Year n+1) is $50m. (a) Compute the terminal value at a 9% WACC and again at a 12% WACC, both with 3% terminal growth, and the % change. (b) Recompute both at a higher 5% terminal growth and give the % change. (c) What do the two results together demonstrate about which assets rates hit hardest?

Solution. (a) TV = FCF/(WACC − g). At 9%: 50/(0.09 − 0.03) = 50/0.06 = $833m. At 12%: 50/(0.12 − 0.03) = 50/0.09 = $556m. Change = (556 − 833)/833 = −33.3% (a 300 bp rate rise cut the terminal value by a third, because the denominator widened from 6% to 9%, i.e. 50% wider). (b) At 9%: 50/(0.09 − 0.05) = 50/0.04 = $1,250m. At 12%: 50/(0.12 − 0.05) = 50/0.07 = $714m. Change = (714 − 1,250)/1,250 = −42.9%. (c) The higher-growth case (smaller WACC − g denominator) is more rate-sensitive (−42.9% vs −33.3%): the closer terminal growth sits to the discount rate, the more explosively the terminal value swings with rates. Since a growth company's value is mostly terminal value, rising rates compress long-duration/high-growth assets far more than short-duration cash-cows, the gravity of §4.14 made arithmetic. (Note the corollary: any valuation resting on g near WACC is dangerously fragile to a rate rise.)

P12 (exam, 12 min, synthesis). You are handed today's macro tableau for the US: core PCE momentum has slowed to ~2.5% annualized (YoY still 3.1% on base effects); the 3-month bill is 5.20%, the 2-year 4.30%, the 10-year 3.95%; and over the last quarter the 2-year fell 70 bp while the 10-year fell 20 bp. (a) What is the inflation read (level, momentum, base effect)? (b) Classify the curve shape and the quarter's move. (c) Put it together: what regime is this, what is the market expecting the Fed to do, and what does it imply for long-duration equities?

Solution. (a) Inflation is disinflating: core-PCE momentum (~2.5%) is running below the YoY (3.1%), and the gap is largely base effects (high year-ago readings dropping out), so underlying inflation is close to target and cooling, better than the headline YoY suggests. (b) Curve shape: 2s10s = 3.95 − 4.30 = −35 bp; 10y−3m = 3.95 − 5.20 = −125 bp → inverted (market pricing cuts / forecasting weakness). The move: yields fell (bull) and the front end fell far more (2y −70 vs 10y −20), so the curve steepened → bull steepener, the market pricing imminent cuts and dis-inverting the curve. (c) Regime: late-cycle disinflation with a restrictive, inverted curve beginning to price an easing pivot, the classic "the Fed is about to cut because growth is cooling" configuration (and historically, a bull steepener out of a long inversion often coincides with the downturn's onset). Market expectation: the Fed to cut over coming meetings. Implication for long-duration equities: falling rates reduce the gravity, so the discount rate drops, terminal values expand, and long-duration/growth assets (whose value is mostly terminal value) benefit most (§4.14), the mirror image of 2022. That holds provided the accompanying growth slowdown does not cut the cash flows themselves by more than the lower rate lifts their present value (the honest two-sided caveat, since a bull steepener out of inversion is often the recession's doorbell).

P13 (guided: the Fisher real rate). An Indian fixed deposit pays 7.5% nominal; CPI inflation is running 5.5%. (a) Give the quick approximate real rate. (b) Compute the exact Fisher real rate. (c) In one line, why must Indian nominal rates run above US nominal rates to deliver the same real return?

Solution. (a) Approximate: real ≈ nominal − inflation = 7.5 − 5.5 = 2.0%. (b) Exact: (1 + real) = (1 + nominal)/(1 + inflation), so real = 1.075/1.055 − 1 = 1.90%. The ~0.1 pp gap from the 2.0% shortcut is the real × inflation cross-term the approximation drops; it widens as inflation rises, which is why the shortcut is rougher in high-inflation economies. (c) By Fisher, nominal ≈ real + expected inflation, so India's higher structural inflation forces higher nominal rates just to leave the same real return: the rupee is expected to lose purchasing power faster, so lenders demand more nominal compensation (§4.1; this real-rate wedge is a prime FX driver in M7.06).

P14 (guided: breakeven inflation). The 10-year nominal Treasury yields 4.20%; the 10-year TIPS real yield is 1.70%. (a) Compute the approximate 10-year breakeven inflation. (b) Compute the exact breakeven (Fisher). (c) The Fed targets 2% PCE and CPI runs ~0.35 pp above PCE: is this breakeven consistent with roughly anchored expectations?

Solution. (a) breakeven ≈ nominal − real = 4.20 − 1.70 = 2.50%, the market's implied average CPI inflation over ten years. (b) Exact: breakeven = 1.0420/1.0170 − 1 = 2.46% (the ~4 bp gap from 2.50% is the Fisher cross-term). (c) A 2.46% CPI breakeven, minus the ~0.35 pp CPI–PCE wedge, is ≈ 2.1% expected PCE, essentially the Fed's 2% target, so expectations look roughly anchored. (Comparing the raw 2.46% CPI figure to the 2% PCE target would falsely suggest expectations are running hot, the §4.5 gauge-mismatch trap this comparison exists to avoid.)

P15 (independent: realized real return, and why it "breaks even"). You buy the 10-year nominal Treasury from P14 at 4.20% (the TIPS real yield is 1.70%, so the breakeven is ~2.46%). (a) If CPI actually averages 3.5% over the decade, compute your realized real return and say whether you beat or lagged the TIPS. (b) The same, if CPI averages 1.5%. (c) At what inflation rate do the nominal bond and the TIPS deliver the same real return, and what is that rate called?

Solution. (a) realized real = 1.0420/1.035 − 1 = 0.68%, far below the 1.70% you could have locked in the TIPS; you lagged, because realized inflation (3.5%) exceeded the 2.46% breakeven, so the fixed nominal coupon lost more purchasing power than the market had priced. (b) 1.0420/1.015 − 1 = 2.66%, above the TIPS; you beat it, because inflation (1.5%) came in below the breakeven and the nominal bond over-compensated. (c) At exactly the breakeven, ~2.46%: 1.0420/1.0246 − 1 = 1.70%, identical to the TIPS real yield, which is why it is the "breakeven." Above it, inflation-linked bonds win; below it, nominal bonds win; the breakeven is the hurdle your own inflation forecast must clear to choose between them.

P16 (independent: modified duration on a curve move). A 10-year G-sec has a modified duration of 7.2 and trades at ₹98.50 per ₹100 face. Its yield rises 45 bp. (a) Estimate the % price change and the new price. (b) Will the true repriced value be slightly above or below your estimate, and what is the effect called? (c) In the same sell-off a 2-year G-sec (modified duration 1.9) sees its yield rise 60 bp; compute its % price change, say which bond lost more, and state what you must compare to label the overall curve move a flattener or a steepener.

Solution. (a) %ΔP ≈ −modified duration × Δy = −7.2 × 0.45% = −3.24%; new price ≈ 98.50 × (1 − 0.0324) = ₹95.31. (b) Slightly above the estimate: the price–yield relationship is convex (§4.13), so modified duration overstates the loss on a yield rise, and the true price falls a little less, landing just above ₹95.31. (c) 2-year: %ΔP ≈ −1.9 × 0.60% = −1.14%. The 10-year lost far more (−3.24% vs −1.14%) despite a smaller yield move, because its duration is ~3.8× larger: "long bonds move more." But the flattener/steepener label is about the yield-spread change, not the price move: the 10y rose 45 bp and the 2y rose 60 bp, so the 10y−2y spread narrowed by 15 bp → yields up (bear) + spread narrowing (flattener) = a bear flattener. (Knowing only that the 10-year fell more in price is not enough to name the move, which is a classic trap.)

P17 (independent: a DV01-neutral steepener's P&L). A trader puts on a DV01-neutral 2s10s steepener: long $50m of the 2-year (modified duration 2.0) and short $12.5m of the 10-year (modified duration 8.0). (a) Verify the trade is DV01-neutral. (b) The curve steepens by 20 bp (the 10y−2y spread widens 20 bp); compute the P&L. (c) Would the P&L differ if that 20 bp steepening arrived as a bull steepener (yields down) versus a bear steepener (yields up)? Why?

Solution. (a) "Dollar duration" per leg = modified duration × notional: 2y = 2.0 × 50 = 100; 10y = 8.0 × 12.5 = 100. Equal → DV01-neutral (a +1 bp parallel move loses 100 × 0.0001 = $0.010m on the long 2y and gains the same on the short 10y, netting ≈ 0). (b) With dollar duration of $100m per unit yield per leg, P&L ≈ dollar duration × Δ(spread) = 100 × 0.0020 = $0.20m = $200,000. (c) No (to first order): because the trade is DV01-neutral the level (parallel) component nets out, so P&L depends only on the change in the spread. Any path with a +20 bp steepening pays ≈ +$200,000, bull or bear. (Convexity and carry create small second-order differences; §4.13.)

P18 (exam, 12 min, synthesis: inflation, curve, and gravity). Today's US tableau: the 10-year nominal Treasury is 4.50%, the 10-year TIPS real yield 2.00%; the curve reads 3-month 5.10%, 2-year 4.60%, 10-year 4.50%; over the past quarter the 2-year fell 15 bp and the 10-year fell 40 bp. A synthetic US growth firm, Aster (synthetic), has terminal free cash flow (Year n+1) of $80m and terminal growth 5%. (a) Compute the 10-year breakeven inflation and say whether ~2% PCE-target expectations look anchored (CPI runs ~0.35 pp above PCE). (b) Compute the 2s10s and 10y–3m spreads, classify the shape, and name the quarter's move with its driver. (c) Size Aster's rate sensitivity: compute its terminal value at an 8% WACC and again after a 300 bp rise to 11%, and the % change. (d) Tie it together in one sentence.

Solution. (a) Exact breakeven = 1.0450/1.0200 − 1 = 2.45%; minus the ~0.35 pp CPI–PCE wedge ≈ 2.1% expected PCE, essentially the 2% target → expectations look anchored. (b) 2s10s = 4.50 − 4.60 = −0.10% (−10 bp); 10y−3m = 4.50 − 5.10 = −0.60% (−60 bp) → the curve is inverted (mildly at 2s10s, more at the 10y–3m). The move: yields fell (bull), and the long end fell more than the front (10y −40 bp vs 2y −15 bp), so the spread narrowed by 25 bp → flatten. Hence a bull flattener, driver: a flight-to-quality / falling long-run growth-and-inflation move (a recession scare) while restrictive policy keeps the front end pinned (the 3-month still at 5.10%). (c) TV = FCF/(WACC − g): at 8%, 80/(0.08 − 0.05) = 80/0.03 = $2,667m; at 11%, 80/(0.11 − 0.05) = 80/0.06 = $1,333m; change = −50% (the WACC − g denominator doubled from 3% to 6%, halving the value, a vivid warning of how fragile a g of 5% against an 8% WACC is). (d) Inflation expectations are anchored, and the inverted, bull-flattening curve says the market smells a downturn and expects the Fed to ease, so the gravity (the ~4.5% risk-free rate) looks likelier to fall than rise, which would lift long-duration Aster most (its terminal value swings ~50% on a 300 bp move), the mirror image of 2022 — with the honest caveat that the very recession pulling rates down may cut Aster's cash flows by more than the lower rate lifts them.

P19 (guided: shift or movement, and which curve). For each event, say which curve moves and in which direction, whether output and the price level move together or in opposite directions, and which of the four inflation engines it belongs to. (a) A large, deficit-financed transfer payment lands in household bank accounts. (b) A cartel cuts oil output and the price of crude doubles for two years. (c) Wage and price setters raise next year's contracts by 8% because they now expect 8% inflation. (d) The price level falls and firms therefore sell more units along their existing cost structure. (e) A permanent productivity gain from a new general-purpose technology.

Solution. (a) AD shifts right. Output and the price level rise together; this is demand-pull (engine 1), and the fiscal source is what makes it a shift rather than a movement. (b) SRAS shifts left. The price level rises while output falls, the opposite-direction signature; this is cost-push (engine 2) and it is the stagflation case. (c) SRAS shifts left as well, because expected inflation is built into contracted costs at every output level; this is expectations (engine 4), and it is why an anchored expectation is a supply-side asset for a central bank. (d) A movement along AD, not a shift: the price level is the good's own price at the economy level, and only a price-level change produces a movement. Naming this a "demand increase" is the error EC1.01 flags for a single market and it inverts the predicted sign roughly half the time. (e) LRAS shifts right, and SRAS with it. Output rises and the price level falls, which is good deflation, the one case in §4.8 where a falling price level is not a warning. Engine 3 (monetary) is absent from every answer above because it works through AD over long horizons rather than as a distinct shock.

P20 (independent: size the two shocks). (a) Nominal aggregate demand grows 7.0% while potential output grows 2.5%. Compute the long-run inflation the arithmetic forces, and name the engine. (b) Nominal demand grows 4.0% while long-run supply falls 1.5%. Compute the change in the price level and in output, name the pathology, and say why no single policy lever fixes it. (c) In which of the two would a central bank raising rates be doing something a textbook would call straightforwardly correct, and why is the other case genuinely a judgment rather than a calculation?

Solution. (a) Real output cannot outrun potential for long, so the price level absorbs the remainder: 1.070/1.025 − 1 = 4.39%. That is demand-pull (engine 1) against a vertical LRAS. (b) 1.040/0.985 − 1 = 5.58% on the price level, with output −1.5%. That is stagflation: tightening to bring the 5.58% down deepens the 1.5% output loss, and easing to protect output ratifies the price rise, so the lever that helps one problem worsens the other. (c) In (a). The shock is pure demand, the two objectives point the same way, and slowing nominal demand fixes the inflation without any trade-off against the mandate's other half. In (b) the central bank has to decide how much output loss it will accept to stop a supply shock feeding into expectations, and that is a judgment about engine 4, not an arithmetic problem. It is also why §4.9's Phillips-curve debate is about supply shocks far more often than about demand ones.

Applied mini-project: a live inflation-and-curve read for India and the US

Task. Pull the latest inflation prints and government yield curves for both India and the US, and write a two-page "rates read" that turns four raw data pulls into an analyst's verdict on inflation, the curve, and the gravity on valuations in each economy. This is the exact skill §4.7 and §4.11 teach and the M7.07 dashboard capstone will assume you own; do it on live data so you are reading today's actual regime, not a frozen example.

Data to pull (all free, all primary):

• US inflation: the latest CPI release (BLS, bls.gov) and the latest PCE release (BEA, in the Personal Income and Outlays report). Note headline and core for each, and the CPI–PCE wedge. On FRED (fred.stlouisfed.org), plot CPIAUCSL, PCEPI, and PCEPILFE (core PCE), and compute the 3-month annualized momentum, not just the YoY.
• India inflation: the latest CPI (Combined) release (MOSPI/NSO, mospi.gov.in) and the latest WPI (Office of the Economic Adviser, eaindustry.nic.in). Note headline CPI, the food and core components, and the CPI–WPI gap. The RBI DBIE has the series.
• US yield curve: the US Treasury par-yield curve (treasury.gov) or FRED: DGS3MO, DGS2, DGS5, DGS10, DGS30. Compute the 2s10s (T10Y2Y on FRED) and 10y–3m (T10Y3M) spreads, and check the NY Fed recession-probability model (which uses 10y–3m). Optionally note the ACM term-premium estimate.
• India yield curve: the G-sec curve from CCIL (ccilindia.com) or RBI/worldgovernmentbonds.com, for the 91-day T-bill, 2y, 5y, 10y, 30y. Compute the term spread.

Required contents.

1. Two inflation reads (one US, one India), each following the §4.7 discipline: the surprise vs consensus (if available) → YoY vs 3-month momentum → base-effect flag → composition (which engine: food/energy/core-goods/core-services) → the one-sentence verdict and what it implies for the policy rate. For the US, state both CPI and core PCE and reconcile the wedge; for India, state CPI and the CPI–WPI gap and what it says about corporate margins.
2. Two curve reads, each following §4.11: the shape, the 2s10s and (US) 10y–3m spreads, the recession signal (with caveats for the US, and the "don't over-read the slope" caveat for India), and, comparing to a snapshot from ~3–6 months ago, the move named as one of the four (bull/bear × steepener/flattener) with its driver.
3. The gravity bridge (one paragraph each): state the current risk-free rate an analyst should use to discount cash flows in each economy (the 10-year yield), whether rates look likely to rise or fall from here, and therefore what the setup implies for long-duration vs short-duration equities in each market right now (§4.14).
4. The India–US contrast: are the two economies in the same inflation-and-rate phase or diverging? Explain why, using the structural differences (food-dominated CPI, headline targeting, higher structural rates, SLR-distorted curve).

Scoring rubric (self-score honestly; pass ≥11/14):

Reading & resources

• Aswath Damodaran, "The Fed and Interest Rates" and his risk-free-rate posts/classes (free on his NYU page and Musings on Markets blog): the definitive bridge from the policy rate and the curve to the discount rate in a valuation, from the valuation master himself. The perfect complement to §4.14. [Free] [Intermediate]
• FRED (fred.stlouisfed.org). The single most valuable habit this module can give you: plot the series yourself. CPIAUCSL/PCEPI/PCEPILFE (CPI, PCE, core PCE), T10Y2Y and T10Y3M (the two curve spreads, with recession shading), DGS10 (the 10-year), DFII10 (the 10-year real yield / TIPS), and the NY Fed's recession-probability series. Watching a spread cross zero into a shaded recession bar teaches more than any paragraph. [Free] [Beginner]
• The New York Fed, "The Yield Curve as a Leading Indicator" (newyorkfed.org): the primary source for the 10y–3m recession-probability model (Estrella–Mishkin lineage) and the ACM term-premium estimates; read the methodology page once. [Free] [Intermediate]
• RBI Monetary Policy Reports and the DBIE database; MOSPI/NSO CPI releases; the Office of the Economic Adviser WPI releases (rbi.org.in, mospi.gov.in, eaindustry.nic.in): your permanent primary sources for Indian inflation and the G-sec curve. Read one RBI MPR's inflation chapter to see a central bank do the §4.7 decomposition professionally. [Free] [Intermediate]
• The Urjit Patel Committee Report (2014, "Revision and Strengthening of the Monetary Policy Framework") (rbi.org.in): the document that moved India's anchor from WPI to CPI; read the executive summary to see the §4.6 argument made by the people who made the decision. [Free] [Intermediate]
• Milton Friedman, Money Mischief (1992): the clearest popular statement of "inflation is always and everywhere a monetary phenomenon," with the hyperinflation history; pair with his 1968 AEA address "The Role of Monetary Policy" for the expectations-augmented Phillips curve from the source. [Paid book / address is Free] [Intermediate]
• Irving Fisher, "The Debt-Deflation Theory of Great Depressions" (1933, Econometrica): nine pages, free (public domain), and still the clearest account of the deflation spiral (§4.8). [Free] [Intermediate]
• Marginal Revolution University, the inflation and business-fluctuations videos (mru.org): free, short, well-made coverage of inflation measurement, the Phillips curve, and the quantity theory at exactly this level. [Free] [Beginner]
• Zerodha Varsity, "Currency, Commodity, and Government Securities" and the macro modules (India): for the G-sec market, the yield curve, and Indian rate plumbing in the local context. [Free] [Beginner]
• Edward Chancellor (ed.), Capital Returns, and Marc Levinson, The Great Inflation and Its Aftermath (or Burton Malkiel's inflation chapters): for the 1970s stagflation as lived history and the capital-cycle lens on inflation regimes. [Paid] [Intermediate]
• Ben Bernanke, 21st Century Monetary Policy (2022): carried over from M7.02; its chapters on inflation dynamics, the flattening Phillips curve, and the 2020 framework are the best modern practitioner account of the debates in §4.4 and §4.9. [Paid] [Intermediate]
• Aswath Damodaran, The Little Book of Valuation, and McKinsey's Valuation chapter on cost of capital: to reconnect the rate to the DCF; re-read the terminal-value and cost-of-capital sections after this module and the "gravity" point will land with new force. [Paid] [Intermediate]
• Going deeper: the FRED breakeven and real-yield series (T10YIE, DFII10, T5YIFR) (fred.stlouisfed.org). Plot the 10-year breakeven inflation (T10YIE), the 10-year TIPS real yield (DFII10), and the 5-year-5-year-forward inflation expectation (T5YIFR, the Fed's favoured market gauge of long-run anchoring) against realized CPI. Watching breakevens jump on a CPI surprise is §4.7 and Worked Example 5 in one live chart, the fastest way to make "market-implied expected inflation" a thing you can see move. [Free] [Intermediate]
• Going deeper: Antti Ilmanen, Expected Returns (2011), the rates and inflation chapters. The rigorous practitioner treatment of the term premium, breakeven inflation, the inflation-risk premium, and duration as a priced risk factor; the bridge from this module's intuition to how a bond desk and a multi-asset investor actually decompose and price the curve. Demanding but definitive. [Paid] [Advanced]

The Modern Analyst's Addendum

Everything above teaches this skill from first principles, by hand — that is how you learn it, and the mastery check still tests it that way. This addendum shows how a working analyst amplifies the same skill today. It adds; it never replaces. (R1/R10)

AI-Augment this skill

{
  "skill": "Reading an inflation print for momentum and base effects, the gauge differences (CPI vs PCE, CPI vs WPI, headline vs core), the expectations-plus-term-premium decomposition of a long yield, curve shapes and the inversion signal, the four curve moves, duration and convexity, and rates as the gravity of valuation",
  "use": "This is the module with the most machine-readable primary data in the whole program, which changes what a tool is for: not to tell you numbers, but to help you build the pipeline that fetches them and to check the transformations you apply. Four uses earn their place. (1) PIPELINE DRAFTING — 'write a function that fetches the Treasury daily par yield curve and returns a tidy frame of tenor and yield', 'write a bootstrap that turns par yields into zero rates and asserts each par bond reprices to 100'. The assertion is the point: a bootstrap that does not reprice its own inputs is wrong, and the check is one line. (2) GAUGE DISCIPLINE — before any comparison, ask it to state which index each figure is on and what the wedge is: CPI against a PCE target, WPI against CPI, headline against core, a breakeven quoted on CPI against a target defined on PCE. §4.5 and Worked Example 5 both turn on this, and it is the error a tool can genuinely help you avoid because the answer is definitional rather than current. (3) DECOMPOSITION DRILLING — 'decompose this 10-year yield into expected average short rate plus term premium, and tell me which component I would need to source separately'; the honest answer is that the term premium is a MODEL output, and being told that is worth more than a number. (4) NAMING THE MOVE — feed it two curve snapshots and ask for the bull/bear and steepener/flattener classification with the reasoning; the rule is mechanical, so the answer is checkable. What it must never supply: today's 10-year yield, today's breakeven, today's CPI print, today's repo rate, or the current shape of either curve. All of those are one fetch away from a versioned source, and a stale one is undetectable inside correct-sounding prose.",
  "tools": ["Chat assistants — Claude, ChatGPT for gauge discipline, decomposition questioning and curve-move classification", "Coding assistants — for the par-to-zero bootstrap, the forward-rate engine, the duration/convexity functions and the fetch pipelines below", "Document/PDF AI — for the MOSPI CPI release's item tables, the RBI Monetary Policy Report's inflation chapter, and the New York Fed's ACM term-premium documentation"],
  "prompt": "Write a bootstrap that takes a par-yield curve at integer annual tenors and returns zero rates and discount factors, then assert that every input par bond reprices to exactly 100 under the recovered discount factors. Then add a forward-rate function f(a,b) from the zeros. Do not fetch or invent any yields — I will supply the curve. Separately: for each figure I paste, state which index it is measured on (CPI, core CPI, PCE, core PCE, WPI, HICP) and flag any comparison I have made between two different gauges.",
  "verify": "Four checks, and the first two are arithmetic identities that a wrong pipeline cannot pass. (1) The bootstrap must reprice its inputs: each par bond, discounted on the recovered factors, must come back to exactly 100. (2) The forwards must rebuild the spots: compounding the one-year forwards from year 1 to year n must reproduce the n-year zero to the last basis point, since forwards are a rearrangement of the zeros and not new information. (3) Duration must be checked against an exact repricing, never trusted alone — Worked Example 3's whole point is the 39 bp gap between the linear estimate and the truth, and the gap grows with the size of the move. (4) Every level carries its source and its release date: the Treasury par curve is published daily and dated, FRED series carry vintages, MOSPI's CPI has a release calendar. Then the verification that is about judgement rather than code: any claim you make about the inversion signal must be checked against how many episodes actually support it. §4.11 states the record honestly; the quantitative lens below shows what a record of that size can and cannot carry, and the answer constrains what you are allowed to say.",
  "diy": "The gate is unaided: you compute a YoY, a sequential annualised momentum and a base-effect projection from index levels; reconstruct a headline from component weights; classify a curve shape and compute both spreads; name a curve move from two snapshots; compute a breakeven by subtraction and by exact Fisher; apply modified duration to get a price change and say which way convexity biases it; and reprice a terminal value at two discount rates. Nothing in this callout is admissible in that room."
}

Modern Data Analysis

By hand first. You decomposed a CPI print into headline, core, momentum and base effect and wrote the one-sentence verdict; you classified two curve snapshots and named the move between them; you repriced a 10-year bond exactly and then again with duration, and saw the gap; you computed a breakeven by subtraction and by Fisher and worked out why it is called a breakeven; you valued Helios at two discount rates and watched the terminal value lose 57% while the near cash flows lost 14%. All of that stays by hand, and Worked Example 4 most of all: the terminal-value collapse is the single arithmetic fact that connects this phase to every valuation you will build.

Today's workflow. The quoted curve is an input, not the object. Three transformations turn it into something you can actually use, and each is a few lines. First, bootstrap. The Treasury publishes a par yield curve (the coupon that would price each maturity at 100), and par yields are not discount rates. Strip them into zero rates and discount factors by solving forward one tenor at a time, and now you have the thing that actually discounts a cash flow at year seven, which no point on the par curve gives you. Second, take forwards. The zeros contain, by arithmetic rearrangement, the market's own implied path of short rates: the one-year rate the curve implies for the year beginning five years from now. That path is what "the market expects cuts" means quantitatively, and it converts §4.10's expectations-hypothesis paragraph into a chart you can put a policy view against. Third, decompose the level. A nominal yield splits into a real yield plus a breakeven (Worked Example 5 does this), and separately into an expected-average-short-rate plus a term premium (§4.10), with the honest asymmetry that the first decomposition is a subtraction of two traded prices and the second requires somebody's model, which is why the New York Fed's ACM estimate is a published series with a method attached rather than an observation. Around all three sits the discipline the inflation half demands: every series is transformed the same way every month, same seasonal treatment, same annualisation convention, same base, so that a change in the number is a change in the world.

Tools & sources (IN + US). numpy for the bootstrap and the forwards, pandas for the panel, scipy.optimize if you fit a parametric curve, matplotlib for the curve and the forward path; US: the Treasury's Daily Treasury Par Yield Curve Rates (home.treasury.gov, published every business day as the official par curve, and the input the bootstrap consumes), FRED by id for DGS1MO, DGS3MO, DGS1, DGS2, DGS5, DGS10, DGS30 (the constant-maturity nominals), T10Y2Y and T10Y3M (the two spreads §4.11 names, published as series so you never subtract by hand), DFII5, DFII10 and DFII30 (TIPS real yields, the input to Worked Example 5), T5YIE, T10YIE and T5YIFR (breakevens and the five-year-forward five-year breakeven, the cleanest single gauge of long-run anchoring), CPIAUCSL and CPILFESL, PCEPI and PCEPILFE (the index the Fed's target is actually defined on), USREC (the NBER recession indicator, the series that makes the episodes countable rather than remembered), plus BLS CPI detailed tables for the item weights and BEA NIPA price tables for PCE, the Cleveland Fed's trimmed-mean CPI and inflation nowcasts and the Dallas Fed's trimmed-mean PCE for the ensemble §4.4 recommends, the New York Fed's ACM term-premium decomposition (free, daily, downloadable: the §4.10 dashboard item that lets you actually watch the component the inversion debate turns on), and the New York Fed's yield-curve recession-probability model driven by the 10y–3m spread; India: MOSPI for the CPI release, the item-level indices and the published weights that make Worked Example 1's reconstruction reproducible, the Office of the Economic Adviser (eaindustry.nic.in) for WPI, CCIL (ccilindia.com) for the traded G-sec market and its published zero-coupon yield curve, FBIL (fbil.org.in) for benchmark G-sec yields, the overnight MIBOR and the INR OIS curve, the RBI Handbook of Statistics for long G-sec yield history, and the RBI's Monetary Policy Report for the inflation projection path.

Now the asymmetry, and in this module it is sharp enough to change what an Indian analyst can do. The US breakeven is a subtraction of two traded prices, published daily, free, for a dozen tenors. India has no liquid inflation-linked bond market. The Inflation Indexed Bonds issued in 2013–14 never developed a traded curve, so there is no Indian breakeven at all, and market-implied inflation expectations simply do not exist as a number. What replaces them is survey-based: the RBI's Inflation Expectations Survey of Households and its Survey of Professional Forecasters, both free, both published on a bi-monthly or quarterly cadence with a lag. So the US analyst reads expectations off a screen every morning and the Indian analyst reads them off a survey every few months, and no amount of technique closes that gap. There is likewise no daily published Indian term-premium decomposition comparable to ACM. Long, cleaned G-sec and OIS histories are a paid product: CMIE, CEIC, Bloomberg or Refinitiv. Everything else named above is free.

# Par -> zero -> forward, and duration/convexity — recomputed in-session (R3)
import numpy as np
T, P = [0.25, 2, 5, 10, 30], [5.40, 4.90, 4.30, 4.20, 4.40]   # WE2 Date 1, read as PAR yields
par = np.interp(np.arange(1, 31), T, P) / 100
z, d = np.zeros(30), np.zeros(30)
for i, c in enumerate(par):                    # bootstrap: each par bond must price to 1
    d[i] = (1 - c * d[:i].sum()) / (1 + c)
    z[i] = d[i] ** (-1 / (i + 1)) - 1
f = lambda a, b: ((1 + z[b-1])**b / (1 + z[a-1])**a) ** (1 / (b - a)) - 1
print(f"{z[0]*100:.3f} {z[9]*100:.3f} {f(1,2)*100:.3f} {f(4,5)*100:.3f} {f(5,10)*100:.3f}")
#        5.186        4.169         4.601         3.420         4.075
def bond(c, y, n, F=1000):
    cf = [(t, F*c + (F if t == n else 0)) for t in range(1, n + 1)]
    Pv = sum(v / (1+y)**t for t, v in cf)
    D  = sum(t * v / (1+y)**t for t, v in cf) / Pv
    C  = sum(t * (t+1) * v / (1+y)**(t+2) for t, v in cf) / Pv
    return Pv, D / (1 + y), C
P0, MD, C = bond(0.04, 0.04, 10); P1, _, _ = bond(0.04, 0.05, 10)
print(f"{P1:.2f} {P1/P0-1:+.4%} {MD:.3f} {C:.3f} {-MD*.01:+.4%} {-MD*.01 + .5*C*.01**2:+.4%}")
#      922.78    -7.7217%      8.111   80.754      -8.1109%           -7.7071%

Verify. Prove every piece against the module's own worked figures before it goes near a live curve. Worked Example 1: YoY 192.5 / 185.0 − 1 = 4.0541%, MoM 0.2604%, compounded annualisation 3.1701%, projected next-month YoY 2.9747%, weighted reconstruction 0.46 × 6.0 + 0.07 × 5.0 + 0.47 × 3.0 = 4.52%. Problem 5: 4.6667%, 0.3195%, 3.9020%, 4.8409%. Worked Example 2: Date 1 spreads −70 bp and −120 bp, Date 2 −5 bp and −35 bp. Worked Example 3: Bond A prices 1,000.00 → 922.78, exact −7.7217%, Macaulay 8.435, modified 8.111; Bond B 981.41, −1.8594%, modified 1.886; the 10-year moved 4.15× the 2-year against a duration ratio of 4.30×. Problem 10's zero: modified duration 9.346, exact repricing on +60 bp −5.4383% against the linear −5.6075%. Worked Example 4: near cash flows 1,007.0 → 865.2 (−14.08%), year-10 FCF 235.8 and year-11 245.2, terminal PV 2,839.7 → 1,233.8 (−56.55%), total 3,847 → 2,099 (−45.43%), terminal share 73.8% → 58.8%. Worked Example 5: exact breakeven 1.0430 / 1.0185 − 1 = 2.4055%, realized real returns 1.26%, 2.25% and 1.85% at the breakeven. Worked Example 6: hedge notional ₹23.75 cr, rupee duration ₹190 cr, P&L ₹28.5 lakh in both cases. Problem 11: −33.3% and −42.9%. Problem 18: 2.45% breakeven, −10 bp and −60 bp spreads, terminal value −50%. All reproduced here to the stated precision. Then the two checks that catch a broken pipeline rather than a broken sum: the bootstrap must reprice each input par bond to exactly 100, and compounding the one-year forwards from year 1 to year n must return the n-year zero exactly. Forwards are a rearrangement of the zeros, so any discrepancy is a bug and not a discovery.

Quantitative lens

The module gives you the concepts; four of them become numbers only when you run the curve through code, and the fifth, the recession signal, becomes honest only when you count. Everything below is computed from the module's own illustrative curves and stated ranges; these are properties of those inputs, not measurements of any market or any historical record.

The curve you are quoted is not the curve you discount with. Take Worked Example 2's Date 1 snapshot as what it actually is, a par curve, the form the Treasury publishes, and bootstrap it. The par 10-year of 4.200% becomes a zero rate of 4.169%, 3 bp below; the par 30-year of 4.400% becomes 4.480%, 8 bp above. The sign flips along the curve because a par yield is a blend of all the zeros up to that maturity, so it is pulled toward wherever the curve has been rather than where it is. Three and eight basis points sound like rounding, and on a 30-year liability stream they are not: the discount factor, not the quoted yield, is what multiplies the cash flow, and the recovered d[30] = 0.2685 is the number that does the work. The reason to bootstrap is not precision for its own sake. It is that there is no quoted yield at all for year 7 or year 13, and a cash flow arriving then needs a rate.

Forwards turn "the market expects cuts" into a count. From those same zeros, the one-year rate implied for the year beginning one year out is 4.601% against a one-year spot of 5.186%, so the curve is pricing a 58 bp fall in the first year alone. Walk the whole implied path and it troughs at 3.420% in the year beginning at year 4, a total decline of 177 bp from the one-year spot, which is about seven cuts of 25 bp priced into that curve before it turns back up. That is what §4.11's "the market expects the central bank to have to ease" is, stated as the market states it. Now run the identical procedure on the module's India G-sec snapshot and the contrast is arithmetic rather than rhetorical: the implied one-year forward rises 6 bp in year 2, and f(5,10) at 6.672% sits essentially on the 10-year zero of 6.620%. The India curve prices no cuts at all, which is exactly §4.10's point that the two curves are not carrying the same kind of information, now as a number rather than a caution.

Convexity is not a footnote; it is 96% recoverable in one term. Worked Example 3 shows duration overstating Bond A's loss and calls the 0.4 pp gap convexity. Compute the convexity and the gap closes: modified duration 8.111 predicts −8.1109% against an exact −7.7217%, an error of +38.9 bp; the bond's convexity is 80.754, and adding ½ × C × (Δy)² gives −7.7071%, an error of −1.5 bp. The second-order term recovers 96.2% of the first-order error on a 100 bp move. Two things follow. The practical one: for anything larger than a small move, quoting a duration-only estimate is leaving a measurable, one-sided, always-in-the-same-direction error on the table, and it takes one extra term to remove. The conceptual one: convexity is worth something to the holder, because the error is favourable in both directions, which is why it is priced, and why a bond with negative convexity is a different animal.

Now the part the module states honestly in words and that deserves to be stated in numbers. §4.11 says inversion has preceded essentially every US recession of the last sixty-five years with a lead of 6 to 24 months. Both halves of that sentence are constrained by the same fact: there are only about eight such episodes. Take the module's own stated range as the range of observed leads and be as generous as the arithmetic allows. A range of 6 to 24 months implies a standard deviation of 5.196 months. With n = 8, the standard error of the mean lead is 1.84 months, so a 95% interval on the average lead is [10.7, 19.3] months, already wide. But the average lead is not the quantity anyone uses. The quantity anyone uses is the lead of the next episode, and the 95% prediction interval for that is [2.0, 28.0] months: a 26.1-month window, which is 1.45 times wider than the entire observed range. Read that again, because it is the whole finding. The interval within which the next lead should fall is wider than the set of leads ever observed. And pinning the mean lead down to ±1 month at 95% would require about 104 episodes: at roughly one per business cycle, several centuries of data that will never exist.

The hit rate is subject to the same arithmetic, and it is the more surprising half. Suppose the record is a perfect eight for eight. With zero failures in eight trials, the exact one-sided 95% lower bound on the true hit rate is 68.8%. That is, a signal that has never once failed in eight tries is statistically consistent with a true success rate of about seven in ten. Count the 2022–24 episode the module flags as a possible misfire, making it eight for nine, and the lower bound falls to 57.1%. Neither number says the curve is uninformative; both say that "has preceded essentially every recession" and "works about two-thirds of the time" are the same evidence. So the disciplined sentence is the module's own: a serious warning, weighted heavily, with a lead measured in quarters-to-years. The arithmetic explains why no sharper sentence is available, rather than leaving it as modesty.

One more thing, and it is the quiet one. The forward curve above looks like a precise reading of the market's mind. Part of it is a modelling choice you made without noticing. The quoted curve has points at 3 months, 2, 5, 10 and 30 years; everything between them is interpolation. Rebuild the same curve with piecewise-constant forwards between quoted tenors instead of linear-on-par (both entirely standard) and the one-year forwards move by up to 44 bp, with a mean absolute difference of 16 bp, and the implied trough shifts from 3.420% to 3.859%. Forty-four basis points is nearly two rate cuts of difference in "how low the market thinks rates go", produced by a choice that appears in no chart legend. The zeros at quoted tenors barely move, 0.5 bp at ten years, because those are pinned by the data; it is precisely the unquoted region, where the forwards are most eloquent, that is most nearly your own invention.

Honest limits. The curves used here are the module's illustrative snapshots, marked as such, and nothing above is a statement about any current market. The Treasury par curve is published daily and dated, and you should bootstrap today's, not this one. The bootstrap assumes annual coupons and annual compounding on integer tenors, where real Treasuries pay semi-annually and settle on real dates with real day counts; the mechanism is right and the third decimal is not. Reading a quoted curve as par yields is correct for the Treasury's published series and would be wrong for a set of secondary-market yields on off-the-run bonds. The forward path is the market's implied path under a pure expectations reading, and §4.10 is explicit that a term premium sits inside it, so "seven cuts priced" is shorthand for "seven cuts plus whatever the term premium is doing", and separating those needs the ACM model, whose output is an estimate with its own error. The small-sample arithmetic uses the module's own stated 6-to-24-month range under a uniform assumption, which is a deliberately generous stand-in for a distribution nobody has published in this addendum; a real study would use the actual dated episodes from USREC and the actual dated inversions, and the intervals would move. But not, given n = 8, by enough to change the conclusion, because the width is driven by the sample size and not by the distributional choice. Most importantly, none of this licenses the opposite error: a wide prediction interval is not a reason to ignore an inverted curve. It is a reason to say "the clock has probably started" and refuse to say when it stops.

Do it in code: bootstrap the published par curve rather than quoting it, assert that it reprices its own inputs, take forwards and count the cuts, decompose the nominal into real plus breakeven where a breakeven exists and say plainly that it does not exist for India, add the convexity term whenever the move is not small, and state the sample size next to any claim about the inversion record. See G1/FI1.03, G1/FI1.02 and G1/QM1.01.

Where this goes next: galaxy cross-links

• G1 · FI1.03 Term Structure & Bootstrapping does the bootstrap properly: semi-annual coupons, day counts, off-the-run adjustments, and parametric fits (Nelson-Siegel-Svensson) for the region between quoted tenors that the interpolation finding above is about.
• G1 · FI1.02 Duration & Convexity covers Macaulay, modified, effective and key-rate duration, the convexity term derived rather than bolted on, and why a portfolio's duration is not the average of its bonds' maturities.
• G1 · FI1.01 Bond Pricing & Yield Measures sets yield to maturity against current yield against horizon return, and shows why the yield you are quoted is not the return you will earn unless you reinvest at it.
• G1 · QM1.01 Probability, Distributions & Estimation covers confidence versus prediction intervals, the exact binomial bound behind the 68.8% figure, and what a sample of eight can and cannot be asked to support.
• G2 · DA1.05 Statistics in Code is about implementing those intervals rather than quoting them, and the habit of putting the sample size in the chart title.

Teach it back & journal

Feynman prompt. Write a one-page explainer for a smart 15-year-old titled "Why does the stock market fall when interest rates go up?" You must convey, in plain words and without jargon: that anything worth owning is worth the cash it will hand you in the future; that a higher interest rate means a rupee (or dollar) you will only get years from now is worth less to you today (because you could have earned that higher rate in the meantime); that this "shrinking of the future" hits things whose payoff is furthest away, such as a fast-growing company whose big profits are a decade off, or a 30-year bond, much harder than things paying you now; and the one image that ties it together, that an interest rate is like gravity on all prices, so when the central bank raises rates, everything gets a little heavier and prices fall, and the lightest, most far-off things fall the most. If your page needs the words "discount rate," "duration," or "terminal value," you don't own it yet: say it with time and money and gravity instead.

Journal prompt. Three entries. (1) Pull today's numbers and write the date beside each: the latest US CPI and core PCE, the latest India CPI, the US 10-year Treasury and 2-year yields (and the 2s10s and 10y–3m spreads), and the India 10-year G-sec yield. In one line each, note whether inflation is above or below target and whether each curve is normal, flat, or inverted. This is the first entry of a habit you will keep for years, and the "as of mid-2026" numbers you will verify against later. (2) Take one company you have studied (starter bank or your own list) and write two sentences: roughly what fraction of its value do you think sits in the terminal value (is it a long- or short-duration equity?), and therefore would a 200 bp rise in the risk-free rate hurt it more or less than the average stock? This is the §4.14 judgment made concrete on a real business. (3) Reflect honestly: before this module, if someone said "inflation came down," did you picture prices falling? Write, in your own words, the difference between disinflation and deflation, and one reason deflation is the more dangerous of the two. This is the misconception the module most wanted to fix, and writing the correction is how it sticks.

This module's flashcards and mastery quiz are wired into the app: see the node's Quiz and Reviews.