1. What duration is supposed to do
An insurer matches the duration of its assets to that of promises due decades out, so a rate move damages both sides. If the assets’ duration is not what the fact sheet says, the match is not what the balance sheet says.
The study.
| Measure | Value |
|---|---|
| Funds | Nine |
| Window | January 2008 to August 2026, eighteen years |
| Trading days | 4,649 |
| A duration of seven means | a loss of seven percent per one percentage point rise in rates |
| Method | each fund’s daily return regressed on the daily change in Treasury yields; minus the slope is the duration |
| Source | daily prices, never a fact sheet |
2. The measurement works, on government bonds
The headline column is less innocent, since I chose each fund’s maturity from its mandate.
The Treasury ladder. “Declared in advance” is the estimate set before fitting; every measured value lands within four months, on t-statistics from 85 to 156, and rates explain 82 to 94% of daily variation. The last column comes from a joint regression on the two, five, ten and thirty year yield changes, told nothing about what each fund holds: four for four. Rerun from any start year between 2008 and 2013, the seven to ten year fund moves between 7.36 and 7.62.
| Treasury fund | Measured duration (years) | Declared in advance | Largest exposure, blind joint fit |
|---|---|---|---|
| 1 to 3 year | 1.68 | 1.8 | 2 year |
| 3 to 7 year | 4.19 | 4.5 | 5 year |
| 7 to 10 year | 7.36 | 7.5 | 10 year |
| 20 year and longer | 16.72 | 16.5 | 30 year, at 13.9 of a 16.6 total |
Only one raw extension figure reaches significance at 5% (high yield, t = 1.99); in the equity-controlled column none does, the largest being 1.1.
| Fund | Duration (years) | Rates explain | Extension | Extension, equities held constant |
|---|---|---|---|---|
| Treasury 1 to 3 year | 1.68 | 83.6% | −0.01 | 0.00 |
| Treasury 3 to 7 year | 4.19 | 93.7% | −0.07 | −0.04 |
| Treasury 7 to 10 year | 7.36 | 93.7% | −0.03 | +0.09 |
| Treasury 20 year and longer | 16.72 | 91.2% | −0.29 | +0.04 |
| Treasury inflation protected | 6.78 | 82.3% | +0.12 | +0.13 |
| Agency mortgage backed | 3.63 | 52.1% | +0.55 | +0.36 |
| US aggregate | 4.32 | 51.7% | +0.13 | +0.27 |
| Investment grade corporate | 5.17 | 28.7% | +1.05 | +0.40 |
| High yield corporate | −1.23 | 1.1% | +2.25 | +0.32 |
3. It degrades as credit is added
Yields rise when the economy strengthens, and strength narrows the spread demanded of risky borrowers, so the spread gain offsets the rate loss. Anyone hedging a liability on a published high yield duration is relying on a number that changes sign with the window.
Roughly a year of the investment grade fund’s apparent duration, and all of high yield’s, was cancelled by an exposure that is not a rate exposure.
| Measure | Investment grade | High yield |
|---|---|---|
| Duration, rates only (years) | 5.17 | −1.23 |
| Duration, equities held constant | 6.23 | +1.17 |
| Rates explain, rates only | 28.7% | 1.1% |
| Rates explain, with equities | not reported | 48.2% |
| Equity exposure | not reported | 0.397 (t = 17) |
High yield duration by sample window.
| Window | Duration (years) |
|---|---|
| From January 2008 (headline) | −1.23 |
| From 2013 | +0.97 |
| Before 2022 | −3.26 |
| From 2022 | +3.58 |
4. What I could not measure
Convexity is why mortgage portfolios lengthen exactly when rates rise. Two attempts to measure it failed, and that failure is the finding: daily fund prices cannot show it.
Two attempts at convexity.
| Test | What the record says |
|---|---|
| Direct: a squared yield change | Not one of the nine estimates is distinguishable from zero, the largest t-statistic in the set being 1.41 against the 1.96 a five percent test requires. The point estimates are not quoted, because reading their signs would be reading signs off noise, and one is negative for a portfolio theory says must be positively convex. An earlier version of this lab reported them without their t-statistics, which dressed nine coin flips as findings. |
| Indirect: duration on rate-up days less rate-down days | Agency mortgage +0.55 years and high yield +2.25, against roughly zero for the Treasury ladder. That is the textbook picture. |
| The confound | Rising-rate days are risk-on days: the daily correlation of the ten year yield change with the stock market is +0.28. Holding the stock market constant removes 86% of high yield’s gap, 62% of investment grade’s and 35% of the mortgage fund’s, after which none of the nine is distinguishable from zero. |
| The power | The mortgage fund’s equity-controlled gap is 0.36 years, on a 95% interval running from −0.29 to +1.00, and the smallest effect the test could reliably detect is 0.87 years. Real agency mortgage extension sits inside that interval, and so does zero. Investment grade would need an effect of 2.3 years before this data could see it. |
| The option-free control | Measured against the nominal ten year, the inflation-protected fund printed a gap of 0.66, larger than the mortgage fund’s own 0.55, which read as a falsification. That was the wrong curve for a bond whose coupons are indexed to inflation. Against the Treasury’s own real yield curve its gap falls to 0.12, its duration rises from 4.95 to 6.78 years and its explained variation from 52.7% to 82.3%. The ordering now runs the right way round, 0.12 for option-free government paper against 0.55 for mortgages, but the difference carries a t-statistic of −0.54, so it remains suggestive rather than established. |
| The mechanism itself | Not in question. Homeowners refinance when rates fall and stop when rates rise, so agency mortgages certainly do have negative convexity. What fails here is the measurement, and absence of evidence is not evidence of absence. |
5. The number that did move
A mortgage book sized on the pre-2022 number carried nearly three times the rate exposure it believed it had by the time rates finished rising.
Measured duration either side of 1 January 2022, a split date declared in advance. Sweeping all 49 monthly candidates from 2020 to 2024, the mortgage fund’s ratio ranges from 1.93 to 2.77, and the declared date is the most favourable of the 49.
| Fund | Before 2022 | From 2022 |
|---|---|---|
| Agency mortgage backed | 2.38 | 6.59 |
| US aggregate | 3.62 | 5.98 |
| Investment grade corporate | 4.12 | 7.66 |
| Treasury 7 to 10 year | 7.25 | 7.64 |
| Treasury 20 year and longer | 16.56 | 17.19 |
6. What this does not establish
The boundaries.
| Limit | What it costs |
|---|---|
| No option-adjusted spread | Stripping out the value of the borrower’s option, to reach the compensation actually paid for credit risk, requires security-level cashflows and an option model. Neither is free daily data. None is computed anywhere here. |
| Timing | The Treasury yield fix is struck before the equity market closes, so part of each day’s response lands in the next day’s return. Adding a lead and a lag raises the seven to ten year fund from 7.36 to 7.45, investment grade from 5.17 to 5.70 and high yield from −1.23 to −0.66, which halves the headline figure section 3 is built on. Every duration printed here is a lower bound, and the credit ones are the loosest. |
| Collinear maturities | The four maturities move together closely enough that individual key-rate durations are fragile, with variance inflation factors up to 29.6 and several negative loadings that no long-only portfolio can truly have. Only the largest loading per fund is used as evidence above. |
| Fund prices, not holdings | These are market prices, so they carry the premium or discount to holdings value, largest in stressed credit. A measured duration is a regression slope over a sample, not a property of any bond. |
| The curve | The yields are constant-maturity par yields, not a zero curve. |
| Survivorship | The nine funds all existed in 2007 and still exist, which is a survivor set, though a mild one for index vehicles this large. |
The further a portfolio sits from government bonds, the less its duration describes its behaviour, the more that behaviour shifts with the regime, and the more of its risk sits in something the duration number never mentions. The second-order refinement everyone quotes cannot be recovered from daily prices, which is worth knowing before someone reports it to three decimal places.
Provenance, bs-prov/1.0
- Method
- Nine bond funds, daily total returns in percent regressed on daily changes in Treasury constant-maturity yields in percentage points, 2 January 2008 to 13 August 2026, 4,649 trading days in the window (each fund regresses on 4,643 to 4,648 of them, depending on missing prices), Newey-West standard errors throughout. The two calendars are intersected before either is differenced, so a return and its yield change always span the same session gap. Key rate durations from a joint regression on the two, five, ten and thirty year changes. The rising-versus-falling duration gap comes from a single interaction regression, so the gap itself carries a standard error, and is repeated with the equity market fully interacted. Convexity is 200 times the quadratic coefficient; the lab proves that scale on every run against simulated zero-coupon bonds of known duration and convexity, and refuses to run if it fails. Regime split at 1 January 2022, declared in advance and then swept across all 49 monthly candidates from 2020 to 2024.
- Sources
- Yields from the Federal Reserve’s public database, series DGS2, DGS5, DGS10 and DGS30. Fund prices adjusted for distributions, fetched by pipeline/fetch_bond_etfs.py. Every statistic is computed by deterministic code in pipeline/rates_duration_lab.py and recorded in state/rates_duration.json; none is estimated by a language model.
- Corrections
- An adversarial review of the first version of this lab, run before publication, found six defects, all fixed here: convexity was scaled by two rather than two hundred; curvature estimates were reported without the t-statistics that show them to be indistinguishable from zero; the rising-versus-falling gap was computed from two separate regressions and so had no standard error of its own; that gap was presented as a convexity test without controlling for the equity channel that produces most of it; the regime split used a different maturity from the headline durations; and bond-market holidays matched two-session yield moves to one-session returns. The paper’s central claim reversed as a result: convexity is reported here as not measurable rather than as measured.
- Revision, 17 August 2026
- The inflation-protected fund was measured against the nominal ten year yield, which is the wrong curve for a bond whose coupons are indexed to inflation, and a reviewer caught it. It is now measured against the Treasury’s own real par yield curve, newly fetched by pipeline/fetch_treasury_curve.py. Its duration moves from 4.95 to 6.78 years, its explained variation from 52.7% to 82.3%, and its extension gap from 0.66 to 0.12, which removes the apparent falsification discussed in section 4 and replaces it with a control that behaves as theory predicts. No other fund is affected, and the paper’s conclusion is unchanged: the test still cannot settle the question.
- Honesty
- No option-adjusted spread. Durations are mild lower bounds because the yield fix precedes the equity close. Key-rate durations are collinear, so only the largest loading per fund is used as evidence. Fund market prices carry premium and discount to holdings value. The nine funds are survivors. The declared 2022 split is the most favourable of 49 candidates for the mortgage result, and the sweep range is reported above.
- Related
- Working Paper No. 31 (where high-yield credit risk went), No. 9 (the AI-CDO, tranched credit), No. 10 (backtesting three value-at-risk models), and No. 26 (a pre-registered falsification of a volatility model).