Duration and the Price of a Rate Move

What duration is, twice

Duration has two meanings and they are the same number. The first is mechanical: it is the weighted average time until the cash flows arrive, which is why a 10-year note has a duration of about eight years rather than ten — the coupons arrive earlier and pull the average down. The second is economic: it is the percentage change in price for a one percent change in yield, with a minus sign, which is why a duration of eight means a 8% loss for a 1% rise. The two are connected because the timing of the cash flows is what determines their sensitivity to the discount rate. Three consequences follow. Duration is additive across a portfolio, weighted by market value, which is what makes it a usable risk measure: you can state the interest-rate exposure of a mixed book in years. Duration is a first-order approximation that improves as the yield change gets smaller, so it is accurate for 10 and 20 basis points and progressively worse for 200. And duration rises with maturity and falls with coupon, because the coupon arrives sooner and shortens the average life of the cash flows — which is why a zero-coupon bond has a duration equal to its maturity and a high-coupon bond has one much shorter. The second-order term is convexity, and it is always helpful for a plain bond, because the price-yield relationship curves the right way: the price rises more for a fall in yield than it falls for a rise of the same size. Convexity is 0.5 × convexity × (Δyield)², added to the duration estimate. For a 10-year note with a convexity of 75 and a 100bp move that is 0.375% — small, and worth knowing because it scales with the square of the move, so it is 1.5% for a 200bp move and it is much larger on long bonds and on mortgage-backed securities, where prepayment makes the curvature work in the wrong direction. The move, priced three ways — Duration only: −8.1 × 1.00%: −8.1%, or $81,000 · Convexity: 0.5 × 75 × 0.01²: +0.375%, or $3,750 back ← · Price change: −7.73%: −$77,250 · Plus a year of 4.20% coupon: total return −3.53% The linear approximation breaks down in a severe move, and it breaks down in the direction that hurts on assets with negative convexity — mortgage-backed securities, where prepayments accelerate when yields fall and extend when they rise, so the price loses more than duration implies in both directions. Duration is a tool for normal moves and a starting point elsewhere.

Why this is the key to a whole portfolio

Duration is the reason a rate move reaches assets that have no coupon at all. An equity is a claim on cash flows arriving over decades, so its value is discounted over a much longer horizon than a 10-year bond, which makes it far more rate-sensitive in duration terms — the mechanism behind the de-rating of long-duration growth equities when yields rose in 2022, and behind the fact that long bonds and long-duration equities fell together that year. Translating an equity’s cash-flow profile into a duration is approximate, but it is the right frame: the same rise in the discount rate hurts both, and it hurts the one with the longest cash flows most. That gives a portfolio a single rate-exposure number, and it lets you see when two positions are the same bet in different clothing. A portfolio holding long government bonds and long-duration growth equities is not diversified against an inflation shock; it holds the same duration exposure twice, which is exactly what the regime lesson described in the language of quadrants. Conversely, a short-duration bond sleeve and a value-tilted equity book are both saying that the near-term cash flows matter more than the distant ones, which is a coherent position rather than a diversified one. The practical use is therefore threefold: state the book’s rate exposure in years, decide deliberately how much of it you want, and check that the equity and bond sleeves are not making the same bet twice. That last check is the one most often missed, and it is the one that cost conventional portfolios the most in the first tightening cycle after a long period of falling yields. • An equity is a long-duration claim; a rate rise compresses its value through the same mechanism as a bond. • Translate the book into a single rate exposure so two positions cannot hide the same bet. • Long bonds plus long-duration growth equities is one duration bet, not a diversification. • Short-duration bonds plus near-term cash-flow equities are a position, not a hedge. Real rates, not nominal, are the right discount for a claim on real cash flows. Inflation-linked bonds and equities are therefore compared on the real curve, which is why a rise in real yields hurts both and a rise driven by inflation expectations hurts the nominal bond most.

The same duration, a different exposure

Modified duration is a single number summarising sensitivity to a parallel shift in the curve, and two portfolios with the same number can behave completely differently to anything that is not parallel. The classic example is a **barbell** — very short and very long maturities — against a **bullet** — everything concentrated in the middle. At equal duration the barbell holds more long-end exposure and more short-end cash, and it therefore benefits more from a curve that steepens (short rates falling, long rates rising) and suffers more from a flattening. A single duration figure cannot distinguish them, which is why the number is a summary and not a risk report. The tool that does distinguish them is **key-rate duration**: the sensitivity of the portfolio to a move at one specific point on the curve, holding the rest fixed. A portfolio’s exposure is a small set of those numbers, and their shape tells you what happens when the curve twists rather than shifts. It is the standard way a rate portfolio is described internally, and it explains a fact that a single duration number cannot — that two funds with identical durations can disagree about whether a given meeting is good or bad news, because they are positioned at different points on the curve. Convexity complicates it further in a direction that matters for whole asset classes. Bonds generally have positive convexity, meaning their price rises more when yields fall than it falls when yields rise — a property that helps the holder. Assets with **negative convexity**, the clearest example being a mortgage-backed security, extend in duration as rates rise (fewer homeowners refinance, so the cash flows stretch out) and shorten as rates fall. That is why mortgages hedge badly: the exposure grows exactly when it is most expensive to hedge, and a portfolio built assuming constant duration underestimates the loss in a sustained rate rise. Two portfolios, one duration, different behaviour — Bullet: 5-year exposure only: Neutral to twists; exposed to the whole curve moving · Barbell: 1-year and 30-year: Higher convexity; gains more from steepening, loses more from flattening · Mortgage-backed paper: Negative convexity — duration extends as rates rise ← When someone quotes a single duration for a portfolio, the natural next question is where on the curve the exposure sits and whether the curve can twist. A parallel shift is the one move that makes the single number sufficient, and it is also the least common.

Convexity, and the curve in pieces

Duration is a first-order term, which is another way of saying it is accurate for small moves and increasingly wrong for large ones. The correction is **convexity**, the second derivative of price with respect to yield, and it is worth understanding because the error it corrects is systematic rather than random — and because some instruments deliberately have the sign of it reversed. For a plain bond the price-yield relationship curves in the holder’s favour. A yield fall produces a larger gain than a yield rise of the same size produces a loss, and a duration-only estimate therefore overstates the loss on a rise and understates the gain on a fall. The correction is roughly additive and always positive for a bond without options: price change ≈ (−duration × yield change) + (½ × convexity × yield change squared). For a small move the second term is negligible; for a two-hundred-basis-point move it is not, and in a stressed market the distinction is exactly what is being traded. **Negative convexity** is the case that matters most, and it is common rather than exotic. A callable bond and a mortgage-backed security have it for the same reason: the borrower holds an option to repay early, and they repay exactly when it is good for them — when yields fall. The result is that the holder has the upside truncated while the downside is not, because as yields fall the effective duration shortens and as yields rise it extends. In plain language, the position becomes a longer-duration asset precisely when duration is hurting. That is a risk sold for a yield, and it is worth checking on any fund whose description contains the word “income”. The second piece is the shape of the exposure rather than its size. A single duration number summarises a sensitivity to a parallel shift in the whole curve, and real moves are rarely parallel. **Key-rate duration** measures the sensitivity to a move at each point separately, which gives a vector instead of a scalar. Two portfolios can have identical duration and behave completely differently when the curve twists: one concentrated at the belly, one split between the front and the long end. A duration-matched hedge between them is matched only for a parallel move, and curve changes are the rule rather than the exception. That distinction underlies the bullet-versus-barbell decision. The same duration can be assembled from a single intermediate bond or from short plus long bonds, and the barbell carries more convexity and more exposure to the curve’s shape. Choosing between them is a view on the curve as much as on the level, which is why the choice should be written down as a view rather than made by accident. The practical version is short: price a large move both ways to see what convexity is worth on your instruments, check whether the convexity you hold is positive or negative before treating duration as the whole exposure, and replace the single duration number with a key-rate profile when the position is large enough for the curve’s shape to matter. • Convexity is the second-order term and it always favours a plain bond. • Callable bonds and mortgage-backed securities have negative convexity: the upside is truncated. • Key-rate duration replaces one number with a vector, which is what a twist tests. • Bullet and barbell can have the same duration and opposite curve exposure. The same asymmetry reaches further than bonds. A business whose value is a distant terminal value has convexity of its own in the discount rate, which is why the repricing of 2022 was larger for the longest-duration equities than a single sensitivity estimate would have implied.

Sources

• Duration, convexity and the price-yield relationshipStandard fixed-income analytics; Fabozzi, “Bond Markets, Analysis and Strategies”
• Duration as a common measure of rate exposure across asset classesStandard portfolio rate-risk practice
• Convexity in long bonds and mortgage-backed securitiesStandard literature on convexity and negative convexity in prepayable assets