Week 32: Duration and Convexity — Bond Price Sensitivity Beyond Macaulay

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1. Why This Is Important

In Week 5 you learned that a bond is four numbers and a calendar, and that price is just discounted cash flow. In Week 18 you learned that the discount rate is the universal denominator of finance. In Week 31 you learned to read the curve. This week you learn the verb that connects those nouns: how much.

How much does my bond fund lose when the 10-year yield jumps 50 bps? How much does my pension liability move when the curve flattens? How much of a TLT drawdown is "normal" versus "regime change"? These are not abstract questions. They are the daily P&L of every fixed-income position on earth, and they are answered by two numbers — duration and convexity — calculated from the bond contract you already know.

You need to internalise four things from this lesson.

Duration is not "years until maturity." It is the

cash-flow-weighted average time until you get your money back, which equals the elasticity of price to yield. A 10-year coupon bond has a duration around 8, not 10. A 30-year zero-coupon bond has a duration of exactly 30. Confusing duration with maturity is the single most common mistake in fixed income, and it cost a generation of "10-year Treasury" buyers in 2022 the chance to measure their own risk before the move.

Convexity is not a footnote — it is the entire long end. For

small yield moves, the linear duration approximation is fine. For large moves (the 2022 yields-tripling type of move), convexity adds back the curvature you've been ignoring. On a 30-year Treasury, ignoring convexity over a 300 bps move misstates the P&L by several percentage points. Convexity is also positive for straight bonds — a free asymmetry in the holder's favour, which gets paid for in lower yields on the long end.

Key-rate duration is how professionals actually hedge. The

single-number "modified duration: 6.2 years" is a portfolio summary, not a hedging tool. Real fixed-income desks decompose duration along the curve — 2-year KRD, 5-year KRD, 10-year KRD, 30-year KRD — and hedge each bucket separately because the curve does not move in parallel. Twists, flatteners, and steepeners are the daily reality.

**2022 was duration risk made visible — the vol tail wagging

the dog.** TLT, the 20-year-plus Treasury ETF with a modified duration around 17, fell about 33% in 2022 — a worse calendar year than the S&P 500 in 2008. The bonds did exactly what their duration said they would do. The investors — many of whom thought "Treasuries are safe" — got blindsided because they never looked up their own duration. The 40-year tail of falling yields had wagged the entire risk-perception dog.

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2. What You Need to Know

2.1 Macaulay Duration — The Weighted Average Maturity

Frederick Macaulay defined duration in 1938 as the weighted average time until a bond's cash flows arrive, where each weight is the present value of that cash flow as a share of the bond's price:

$$ D_{\text{Mac}} \;=\; \frac{1}{P} \sum_{t=1}^{m N} \frac{t}{m} \cdot \frac{C_t}{(1+y/m)^{t}} $$

Three properties follow immediately from this formula.

• **A zero-coupon bond's Macaulay duration equals its maturity

exactly.** With only one cash flow at time $N$, the weighted average has only one entry: $D_{\text{Mac}} = N$. A 30-year STRIPS has duration 30 by construction.

• A coupon bond's duration is strictly less than its maturity.

The intermediate coupons pull the weighted average toward the present. A 30-year 5% coupon bond at par has Macaulay duration around 15.4 — about half the headline maturity number that names the bond.

• Higher coupon -> lower duration. More cash arrives early, so

the weighted average shifts left. The Week 32 image

week32_duration_curve.png shows three duration curves stacked

inside the same maturity axis: the 45-degree line is zero-coupon, the 5%-coupon curve sags below it, and the 10%-coupon curve sags further still.

Macaulay's units are years. That's the only number you can describe to a non-quant friend without losing them in calculus. But for hedging P&L, we want a different unit: percent price change per percent yield change. That is modified duration.

2.2 Modified Duration — Price Sensitivity in Percent

Modified duration is Macaulay duration divided by one period of yield compounding:

$$ D_{\text{mod}} \;=\; \frac{D_{\text{Mac}}}{1 + y/m} $$

Why this divisor? Because it converts "weighted average time" into the linear price-yield slope:

$$ \frac{\Delta P}{P} \;\approx\; -\,D_{\text{mod}} \cdot \Delta y $$

Read it out loud: *for a small yield move $\Delta y$, the percentage price change is about minus modified duration times the yield move.* A 10-year Treasury at 4% with $D_{\text{mod}} = 8.1$ years loses about 8.1% of its price for each 1% rise in yield, and gains 8.1% for each 1% fall. The minus sign is the inverse price-yield relationship from Week 5, now quantified.

Three working numbers worth memorising for April 2026:

• 2-year T-note, $D_{\text{mod}} \approx 1.9$. A 50 bps move moves

price by ~1%.

• 10-year T-note, $D_{\text{mod}} \approx 8.1$. A 50 bps move moves

price by ~4%.

• 30-year T-bond, $D_{\text{mod}} \approx 17.0$. A 50 bps move moves

price by ~8.5%.

Same yield curve, three very different sensitivities. This is why "I own Treasuries" is not a risk statement. The maturity matters.

2.3 Effective Duration — When Cash Flows Aren't Fixed

Modified duration assumes the cash-flow schedule is fixed. That is true for vanilla Treasuries and investment-grade corporates. It is false for callable bonds, putable bonds, and the entire mortgage-backed securities (MBS) universe, where the cash flows themselves move with rates — homeowners refinance when rates fall, extend when rates rise.

For these "rate-dependent cash flow" bonds, duration must be measured numerically by shocking the curve up and down and remeasuring price:

$$ D_{\text{eff}} \;=\; \frac{P_{-} \;-\; P_{+}}{2 \cdot P_0 \cdot \Delta y} $$

where $P_-$ is the price after a downward parallel shock of $\Delta y$, $P_+$ is the price after an upward shock, and $P_0$ is today's price. For a vanilla Treasury, effective duration matches modified duration to several decimals. For a 30-year MBS, effective duration is much shorter than modified duration when rates fall (the prepay option kicks in) and much longer when rates rise (extension risk). That asymmetric profile is negative convexity, and it is the reason MBS portfolios behaved so badly in 2022 — they extended into the rate hike instead of contracting away from it.

2.4 Convexity — The Curvature Term

Duration is a tangent line. The price-yield curve is a curve. The gap between them is convexity. Taylor-expand price as a function of yield around the current yield $y_0$, keep two terms instead of one, and divide by price:

$$ \frac{\Delta P}{P} \;\approx\; -\,D_{\text{mod}} \cdot \Delta y \;+\; \tfrac{1}{2} \cdot C \cdot (\Delta y)^{2} $$

with convexity defined as the second derivative scaled by price:

$$ C \;=\; \frac{1}{P} \cdot \frac{d^{2}P}{d y^{2}} $$

The image week32_convexity_payoff.png shows all three lines on top of each other for a 30-year, 5%-coupon bond at 5% yield: the true price-yield curve, the linear duration approximation, and the duration + convexity quadratic. For a 100 bps move the duration line is fine. For a 300 bps move (the 2022 type of move) the linear approximation undershoots the gain on the rally side and overshoots the loss on the sell-off side.

That asymmetry is the central feature: **for a straight bond, convexity is always positive, and it always works in the holder's favour**. A 100 bps rally produces a slightly bigger price gain than a 100 bps sell-off produces in price loss. The market knows this, which is why long-bond yields embed a small "convexity discount" relative to where pure expectations would put them.

2.5 Key-Rate Durations — The Curve Doesn't Move in Parallel

The single number $D_{\text{mod}} = 8$ assumes a parallel shift — every point on the curve moves by the same amount. In reality the curve twists. Short rates move on Fed policy. Long rates move on inflation expectations and term premium. The 2022 sell-off was a bear flattener (short rates up faster than long); the 2024 rally was a bull steepener (long rates down faster than short). Neither is parallel.

Key-rate duration (KRD) decomposes total duration into the contribution from each maturity bucket. Standard buckets are 3M, 2Y, 5Y, 10Y, 30Y. For a 10-year bullet Treasury, almost all of the duration sits in the 10Y bucket. For a 30-year MBS, duration is spread across 5Y, 10Y, and 30Y buckets because the prepay option makes the effective cash flow date sensitive to multiple parts of the curve. A bond fund manager hedging a portfolio doesn't short "Treasury duration" — they short specific futures contracts (TU at 2Y, FV at 5Y, TY at 10Y, US at 30Y) sized by KRD bucket. That is how professionals actually run a duration-neutral book.

For the retail investor: don't memorise KRDs. Just know that when your bond ETF prospectus reports a single duration number, it has already collapsed five-or-more dimensions into one, and that collapse is wrong by several percent of NAV in any non-parallel move. If you own a barbell (Week 31's barbell strategy), your KRD profile is fundamentally different from a bullet of the same total duration.

2.6 The 2022 TLT Drawdown — Duration Risk Made Visible

iShares 20+ Year Treasury Bond ETF (TLT) is the canonical long-bond proxy. Its modified duration sits around 17 years. In 2022 the 30-year Treasury yield rose from about 1.9% to about 4.0%, a move of 210 bps. Linear duration predicts:

$$ \frac{\Delta P}{P} \;\approx\; -\,17 \cdot 0.0210 \;=\; -35.7\% $$

Convexity gives some back. With long-bond convexity around 350, $\tfrac{1}{2}\cdot 350 \cdot (0.021)^2 = +7.7\%$. Net prediction: about $-28\%$. TLT actually delivered roughly $-31\%$ on price plus about $+2\%$ in coupons, for a total return near $-29\%$. The formula nailed it within a percent.

Two things to take from this. First, the math worked — there was no market dysfunction, no failure of the model. Second, the people holding TLT did not do this calculation in advance. Many bought "Treasuries" expecting safety in a Fed hiking cycle, never having asked the question "what is my duration." The vol tail wagged the dog: a regime that had run for 40 years made everyone forget that long Treasuries can lose a third of their value in a year. Duration is the antidote — but only if you actually compute it before you buy.

**Horace's view — the 2022 failure was a portfolio-shape problem, not a bond-math problem.** My own read on why long bonds did so much damage that year is that they were sitting inside what most people called the "diversified core" — the bond half of a 60/40, the long-duration sleeve of a target-date fund, the "safe" allocation in a balanced portfolio. The whole point of holding them there was that they were supposed to hedge the equity drawdown. Instead they correlated to it and added to it, because they were never genuine safety to begin with — they were a leveraged duration bet that worked for 40 years and then didn't. In the barbell shape I run now, the safety end is short-duration cash and bills plus gold, not long Treasuries. Long Treasuries, if I want them at all, sit on the asymmetric end as a specific rates trade with a thesis behind it, not as the portfolio's pillow. The diversified core had quietly become its own concentration risk — a concentrated bet on the 40-year disinflationary regime — and 2022 was the year that concentration printed.

2.7 Putting Duration to Work — Three Practical Rules

Match duration to your liability or your horizon. If you need

the money in 5 years, hold bonds with duration around 5. The reinvestment-rate risk and price risk cancel each other out roughly at the duration horizon — this is immunisation, the foundation of pension and insurance bond management.

**Use modified duration to size hedges, not to predict big

moves.** For yield changes within +/-50 bps, linear duration is accurate enough. Beyond that, use the convexity-corrected quadratic. Beyond +/-200 bps, reprice the bond directly from cash flows — the approximation breaks down. The interactive lab below lets you watch all three predictions side by side as you slide the rate shock.

**Treat "duration of my bond fund" as a pinned-up number on the

wall.** Open the prospectus, find effective duration, write it down. A 50 bps Fed surprise on Wednesday afternoon will cost you that number times one half percent. If that number scares you, shorten the fund. If not, you have the right ballpark for your fixed-income sleeve.

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3. Common Misconceptions

"Duration equals years to maturity." Only true for zero-coupon

bonds. Coupon bonds are always shorter-duration than their maturity.

"Modified duration is in percent." No — modified duration is

in years (it's the semi-elasticity), and the product of $D_{\text{mod}}$ with $\Delta y$ (in decimal) gives the percent price change.

"Convexity is a small correction; ignore it." True for +/-50

bps, dangerously false for +/-300 bps. The 2022 long-bond move would have been mispriced by 4-7 percentage points without convexity.

"Negative convexity is bad." Negative convexity earns the

holder more yield in normal regimes — that's why MBS yield more than Treasuries. The "bad" only shows up in extreme moves. You're being paid for the option you wrote.

"All Treasury bonds are equally safe." They are equally

credit-safe. Their *price-risk profiles are completely different* — a 2-year T-note and a 30-year T-bond are almost different asset classes from a duration standpoint.

"Duration changes slowly." It changes meaningfully with

yields. As yields rise, modified duration falls (the bond's effective horizon shortens). The number on your prospectus is a snapshot, not a constant.

**"My bond fund's duration is the average of holdings'

maturities."** No — it's the dollar-weighted average of holdings' modified durations, which is shorter than the maturity average for any coupon-paying portfolio.

"Convexity is always positive." True for vanilla bonds.

False for callable corporates, MBS, and any bond where the issuer or the underlying borrower holds a prepay option.

"I don't own bonds, so this doesn't apply." Equity valuation

is duration math too — a stock with cash flows growing forever has a much longer "equity duration" than a mature dividend payer, which is why high-multiple growth stocks crashed harder than value in 2022.

"The yield curve will move parallel." It almost never does.

Twists, steepeners, flatteners, and bull/bear variants of each are the normal regime. Single-number duration only describes parallel-shock P&L.

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4. Q&A Section

Q1: Why divide Macaulay by $(1+y/m)$ to get modified duration? Macaulay is in time units (years). Modified is the elasticity of price to yield — i.e., $-\frac{dP/dy}{P}$. A bit of calculus on the present-value formula shows the algebraic relationship is exactly $D_{\text{Mac}}/(1+y/m)$. You can also see it as a small discrete-compounding correction: at $y=0$, Macaulay equals modified; at higher yields they diverge.

Q2: How do I find the duration of my bond ETF? Check the fund company's website. iShares, Vanguard, and SPDR all publish "effective duration" prominently in fund characteristics. Do not confuse it with "average maturity." For TLT in April 2026, effective duration is around 16.7. For BND (total-bond market), about 6.3. For SHV (1-3 month T-bills), about 0.1.

Q3: Why do long bonds have positive convexity? The price-yield function $P(y)$ is a sum of $1/(1+y)^t$ terms. Each term is convex in $y$ (positive second derivative), and the longer the time $t$, the more convex. Long bonds amplify this curvature. Mathematically, convexity scales roughly with the square of duration, which is why convexity matters disproportionately at the long end.

Q4: What's a typical duration target for a retiree? There's no single answer, but the common pension-style rule is to match portfolio duration to your average liability date. A retiree with a 15-year horizon could justify intermediate (5-7 year) bond duration; younger savers with longer horizons can hold longer duration without it being inappropriate. Most "lifecycle" target- date funds default to 5-7 year duration in the bond sleeve and shorten gradually with age.

Q5: Did anyone make money on the 2022 bond move? Yes — anyone short long-duration Treasuries via TLT puts, ZROZ shorts, or paying-fixed in the swaps market. Several macro hedge funds (Brevan Howard, Element Capital) had banner years. The informational content was free (the Fed had told the market it would hike); the willingness to position against a 40-year trend was rare.

Q6: How does convexity get priced into long bond yields? The "convexity discount" — long bonds yield slightly less than pure expectations would say, because the holder is getting positive asymmetry for free. The discount is small (5-15 bps at the 30-year in normal regimes) but real. Inverse-floaters and other high-convexity instruments trade at noticeably wider discounts to fair-yield models.

Q7: What's "DV01" and how does it relate to duration? DV01 (or PV01) is the dollar price change for a 1 bps yield move: $\text{DV01} = D_{\text{mod}} \cdot P \cdot 0.0001$. It's how fixed-income desks size positions in dollar terms. A trader who "is long $1M DV01" makes/loses $1M per 1 bps move in their benchmark yield.

Q8: How do I hedge a bond portfolio's duration? Sell Treasury futures sized by DV01. The TY (10-year futures) contract has a DV01 around $80; selling enough TY contracts to match your portfolio's DV01 leaves you duration-neutral. For non-parallel shifts you'd ladder the hedge across TU/FV/TY/US contracts using key-rate DV01s.

Q9: Does duration apply to TIPS the same way? Yes, but to real yields, not nominal. TIPS modified duration is the sensitivity of price to real-rate moves. A 10-year TIPS has real duration around 8.5 — about the same number as a nominal Treasury — but it's measuring exposure to a different rate. In 2022 nominal yields exploded but real yields exploded more, which is why TIPS lost almost as much as nominals.

Q10: How does duration apply to equities? Stocks have effective duration too — measured by how price responds to discount-rate moves. High-multiple growth stocks (cash flows far in the future) have equity durations of 25+, while dividend-paying utilities are around 10-15. The 2022 Nasdaq drawdown wasn't a "tech bubble" so much as a duration repricing — the same Taylor expansion at work, just on equity cash flows.

**Q11: What's the difference between effective duration and option-adjusted duration?** Effective duration shocks the yield and remeasures price. Option-adjusted duration (OAD) shocks the yield curve in a model that accounts for embedded options (calls, prepay, etc.). For an MBS, OAD differs from modified duration because the prepay option shortens the effective cash flows. Same idea, more precise machinery.

Q12: How do I use the lab below? Slide maturity to see duration grow roughly linearly. Slide coupon to see duration shrink as more cash arrives early. Slide the rate shock and watch the three price predictions diverge — exact, linear- duration-only, and duration+convexity. The convexity-corrected prediction tracks exact almost perfectly out to +/-200 bps; beyond that, even quadratic Taylor breaks down and you should reprice from cash flows directly.