Guide · Bonds and debt funds · Updated 2026-09-23

Duration is a tangent, not a price: the estimate, its error, and the move at which convexity decides the loss

Duration is usually introduced as a definition and left there. This page computes it: for constructed bonds whose every payment is stated, for the four NSE G-Sec maturity-bucket indices as they stood in September 2026, and against eleven years of those indices' daily closes. It builds on the arithmetic of bond price and yield, including the clean and dirty price. Every figure is computed in the build of this page or read from a named file, and the closing note says how.

Two numbers called duration, answering two different questions

Take a bond that pays 3.50 rupees every six months for ten years and 100 at the end, priced at a 7 per cent yield compounded half-yearly, which is how Indian government bond yields are quoted. Its price is exactly 100, and the final payment alone carries 52 per cent of it in present value.

Macaulay duration asks when, on average, the money comes back. Weight each payment date by that payment's share of the price and the answer is 7.35 years. It is a time, shorter than maturity for any bond that pays coupons, because the coupons pull the balance point towards the present. A zero coupon bond has one payment, so its Macaulay duration is its maturity.

Modified duration asks how much the price changes for a small change in yield. Differentiate the price with respect to the yield and divide by the price: the result is Macaulay duration divided by one plus the yield per period, here 1.035, which gives 7.11. It is a percentage change in price per percentage point of yield; per basis point, 1,00,000 rupees of this bond loses about 71 rupees. The two figures sit close together only because one plus half of 7 per cent is close to one, which makes modified duration about 3.4 per cent smaller. They carry different units and answer different questions.

Duration rises with maturity, but ever more slowly: the 30-year bond has three times the maturity of the 10-year bond and 1.76 times its modified duration, and no par bond's Macaulay duration can exceed 14.79 years at this yield, the duration of a perpetuity. Convexity, the column that matters below, rises 3.9 times over the same step.

Macaulay duration is a horizon, and the arithmetic shows which one

Suppose the yield moves 100 basis points just after purchase and stays there, with every coupon reinvested at the new yield. A holder who sells within a year takes the price effect with little time to reinvest; a holder who waits to maturity has no price effect and keeps the reinvestment effect. Between the two, the effects cancel, and that horizon is the Macaulay duration.

At 7.35 years the value is 165.96 after a rise or a fall, against 165.87 if nothing moves: the holder ends fractionally ahead either way, which is convexity again. Shorter horizons lose on a rise; longer ones lose on a fall, through reinvestment. For a fund that holds its duration roughly constant the logic survives to first order: a rise costs about the modified duration times the move at once, and the higher yield earns back about the size of the move each year, so the break-even horizon is again close to the duration.

The first-order estimate is a tangent, and a tangent is only true where it touches

The duration estimate, minus modified duration times the change in yield, is the straight line touching the price curve at today's yield. The curve bends upward away from it on both sides, so whichever way the yield moves, the estimate prices the bond too low.

The error grows with the square of the move: 0.003 at 10 basis points, 0.31 at 100 and 2.63 at 300 for the 10-year bond, roughly a hundredfold for a tenfold move. It is larger on a fall than on a rise of the same size, because the curve steepens as yields fall. And it scales with convexity rather than duration: the 40-year bond has 1.88 times the modified duration of the 10-year bond, 4.8 times its convexity, and 4.1 times its error at a 300 basis point rise.

Adding half the convexity times the square of the move closes most of the gap, leaving 0.010 for the 10-year bond at 100 basis points. It does not close all of it for long bonds and large moves: at a 300 basis point rise the second-order estimate for the 40-year bond is -26.10 per cent against an exact -29.39, now too optimistic by 3.29, because the curvature itself changes along the way. At that size only the price rebuilt from the cash flows is reliable.

Convexity is the part of the curve the tangent leaves out, and it pays the holder both ways

Because the curve lies above its tangent, equal moves in opposite directions are not symmetric. For the 10-year bond a 100 basis point fall adds 7.44 per cent and a rise takes 6.80; at 300 basis points, 24.53 against 18.69. For the 40-year bond, 15.10 against 11.96, and 59.62 against 29.39. A plain fixed-coupon government bond, with no call feature to cap its price, gains more on a fall than an equal rise costs.

That is the precise sense in which convexity favours the holder, and it is not a hedge: the long bonds that carry the most of it also carry the largest losses, 29.39 per cent for the 40-year bond on a 300 basis point rise. Convexity makes the loss smaller than the straight line says, not small. A position that is short convexity receives the mirror image; the options version of that arithmetic is the gamma cost of short convexity.

The point where the error becomes the answer

The useful question is the size of move at which the straight line stops being good enough, and a tolerance in rupees per 100 of price turns it into a number.

For the 10-year bond the line stays within 10 paise per 100 rupees up to about 55 basis points and is a rupee out by about 171; for the 40-year bond those points arrive at 25 and 77.

Against the market's own record the two scales separate cleanly. The largest one-session fall in the 10-year benchmark index in the sample came on 8 February 2017, when the Monetary Policy Committee held the repo rate at 6.25 per cent and changed its stance from accommodative to neutral: 2.17 per cent, about 31 basis points of yield, at which the duration estimate for a 10-year bond is wrong by 3 paise. Day to day, duration is as good as exact. Over a cycle it is not. In the OECD's monthly series the average 10-year yield went from 7.40 per cent in June 2013 to 8.99 per cent in April 2014, 159 basis points, and rose 172 basis points between July 2020 and June 2022. Applied as a parallel rise of 159 basis points to the 15 years and above basket as it stood in September 2026, duration alone puts the loss at 18.93 per cent; the bonds reprice to a loss of 16.16. The 2.77 rupees per 100 between the two is the convexity term, and it is the difference between an estimate and a loss.

Measured: the long buckets move more on the day, and more again over the month

NSE publishes four G-Sec maturity-bucket indices: 4 to 8 years, 8 to 13, 11 to 15, and 15 years and above. Under the index provider's methodology document of August 2026, each holds up to three liquid government bonds in its band with more than 5,000 crore rupees outstanding, weighted 40:60 on turnover and amount outstanding, reviewed monthly and computed as a total return index with accrued interest on 30/360. The 10-year benchmark index follows the on-the-run 10-year bond, and its clean price variant leaves accrued interest out.

Durations first. Each constituent in the factsheets dated 31 August 2026, which set the baskets in force through September, was priced from its coupon, its maturity and the valuation yield FBIL published for 16 September 2026. The prices reproduce FBIL's published clean prices for all twelve bonds to within 0.13 paise per 100, which checks the convention before any duration is taken from it.

The long bucket is named for its floor of 15 years, but in September 2026 its shortest bond had 29 years to run and its longest 40. Its modified duration was still only 11.91, because at a yield near 7.7 per cent a long bond's duration is already close to the perpetuity ceiling.

The measured side uses 3,385 daily index close files. A pair of consecutive files counted as one session only when the clean price index's own change column agreed with the difference of its closes, which removed 3 pairs spanning archive gaps; the total return indices cannot be checked this way, because their change column leaves out accrual across weekends and holidays. A further 55 sessions on which the clean price index did not move, days the equity market traded and the government bond market did not, were set aside, leaving 2,628 sessions from 10 November 2015 to 18 September 2026.

Daily volatility rises with maturity, from 18.3 to 34.2 basis points over the full period and from 16.4 to 33.7 in the last twelve months. The horizon changes only the long bucket's answer. On the day it moved 1.00 times the 10-year index, less than the 11 to 15 year bucket's 1.05; over five sessions 1.24 times and over twenty-one 1.39, while the 4 to 8 year bucket stayed between 0.61 and 0.66. Either the long end responds over weeks or its daily valuations carry noise that washes out over a month, and these data cannot separate the two. Either way, a single session understates what a long portfolio does over the weeks that follow.

Single sessions are looser than the averages. Among the 100 largest sessions for the 10-year index, the long bucket moved further than the 4 to 8 year bucket, in the same direction, on 81; all four lined up strictly by maturity on only 24, one of them 4 May 2022, when an unscheduled policy decision raised the repo rate by 40 basis points to 4.40 per cent. In 2017 and 2018 the 11 to 15 year bucket was the more volatile of the two longest buckets; in every other year the long bucket was.

Measured against a yield: what could be obtained, and what it showed

An empirical duration needs a yield series matching the bonds. One exists: FBIL publishes a par yield curve and a valuation yield for every government bond on each Mumbai business day. Its Notification No. 10 of 28 December 2021, however, makes lagged data free to the public only for viewing on its website, so it was used here for a single date and not as a history, and the RBI's weekly yield statistics could not be retrieved as a history in this session. What could be obtained and cited is the OECD's monthly 10-year government bond yield for India, from its Main Economic Indicators as republished by the Federal Reserve Bank of St. Louis.

That series behaves as a monthly average: it explains 30 per cent of the variation in the benchmark index's month-end changes and 67 per cent of the variation in its monthly-average changes, so averages are used on both sides. It then has to pass a test. The 10-year benchmark clean price index should move by about its bond's modified duration, 6.77 in September 2026, for each percentage point of the 10-year yield. Across all 128 monthly changes it moved 4.67; by period, 4.95 from December 2015 to December 2019, 3.74 for 2020 to 2022, and 6.15 from January 2023 to July 2026, with 87 per cent of the variation explained. The worst misses are months in which the yield series moved while the index did not, such as May 2020, when the average fell 85 basis points and the index's average rose 1.6 per cent. That is the signature of a benchmark series switching from one bond to another, which a price index chain-links and a yield series does not. The switch dates could not be verified, so it is a reading, and the full-period figure is not used as an empirical duration.

In the recent period sensitivity rises with maturity, from 4.42 for the 4 to 8 year bucket to 9.38 for the long bucket, and the ratio of measured sensitivity to computed duration falls from 1.06 to 0.79. The benchmark index itself recovers only 0.91 of its bond's duration against this series, so the level of every ratio is somewhat understated; the fall from short to long is not. Yields around five years moved at least as much as the 10-year yield, and yields 30 to 40 years out moved about four-fifths as much. The comparison is indicative, because the baskets were rebalanced monthly through the period and the durations are September 2026's.

A stated duration is a sensitivity to the portfolio's own yield. Turned into a sensitivity to the 10-year yield that headlines report, it has to be multiplied by how far the portfolio's yield moves with the 10-year, and for a long portfolio that multiplier has recently been well below one.

The category names changed in February 2026, and the test did not

SEBI circular HO/24/13/15(2)2026-IMD-RAC4/I/5764/2026 of 26 February 2026 superseded the categories set by circular SEBI/HO/IMD/DF3/CIR/P/2017/114 of 6 October 2017, which had been consolidated as clause 2.6 of the Master Circular for Mutual Funds of 27 June 2024. It took effect on its date, gave existing schemes six months to comply, and requires a scheme's name to be the same as its category.

Every duration-named category was renamed to a term, from Long Duration Fund to Long Term Fund down to the Ultra Short Term Fund, and the test did not change. Each category is still defined by the Macaulay duration of the portfolio, the scheme information document must still explain Macaulay duration, and the circular states that Macaulay duration shall be mentioned at portfolio level. Its own clause 2.6.3.14, on investing the residual portion in InvITs, still names the Ultra-Short Duration Fund and the Low Duration Fund by their old titles. The Medium Term and Medium to Long Term categories keep the provision letting the manager cut portfolio duration to one year in an anticipated adverse situation, with written reasons placed before the trustees. The potential risk class each debt scheme carries grades interest rate risk on the same measure: under SEBI circular SEBI/HO/IMD/IMD-II DOF3/P/CIR/2021/573 of 7 June 2021, class I allows a Macaulay duration of up to one year, class II up to three years and class III any, a scheme's figure being its instruments' Macaulay durations averaged by their share of assets.

The sharpest case is the gilt category. The 2017 name, Gilt Fund with 10 year constant duration, said duration while its description said a constant maturity of 10 years. The 2026 name, 10-year Constant Maturity Gilt Fund, says maturity too, while the characteristic still requires the portfolio's Macaulay duration to be equal to 10 years. Those are two different portfolios. The 10-year benchmark bond had a Macaulay duration of 7.01 years on 16 September 2026; a coupon bond at par at 7 per cent needs 16.5 years to maturity to reach 10, and the other route is a 10-year zero coupon strip. A portfolio run to the name carries a modified duration near 6.8, one run to the characteristic near 9.7: about 43 per cent more loss for the same rise in its yield. A page still using the 2017 name was written before the circular or has not been revised since; one that equates the category with holding the 10-year bond has read the name, not the characteristic. Which of the two a scheme runs shows in its disclosed Macaulay duration.

A stated duration is a statement about loss per yield move

Reading a debt fund's disclosure takes three steps, each of them arithmetic.

One. Turn the weighted time into a sensitivity. Divide the disclosed Macaulay duration by one plus half the portfolio yield: at 7 per cent, a Macaulay duration of 7 years is a modified duration of 6.76.

Two. Multiply by the move. That is a first-order loss of about 6.76 per cent for a 100 basis point rise in the portfolio's own yield, or about 68 rupees per basis point on each 1,00,000 rupees.

Three. Correct for size and for which yield moved. Near 100 basis points, convexity gives back about 0.27 rupees per 100 on a portfolio like the 8 to 13 year basket; for a long portfolio and a move of 150 basis points or more it gives back whole rupees. And a 100 basis point rise in the headline 10-year yield has recently meant noticeably less than 100 in a long portfolio's own yield: the measured ratio was 0.79, or 0.87 after allowing for the yield series' own shortfall.

The same figures give the rise a year's income can absorb. To first order, a portfolio yielding y with modified duration D loses a year's yield when its yield rises by y divided by D within the year: 164 basis points for the 4 to 8 year basket in September 2026, 106 for 8 to 13, 83 for 11 to 15 and 64 for the long basket. It ignores the roll-down and the convexity that both help, and it is not a forecast; it is the size of the income cushion in the units that matter.

Where the arithmetic stops being the answer

The curve does not move in parallel. Every repricing above moves all yields together; the measured ratios, 1.06 at the short end and 0.79 at the long end, show they did not.

Duration drifts. It falls as a bond ages and rises as yields fall, 7.11 at 7 per cent and 7.24 at 6 per cent for the 10-year bond, so a disclosed figure is a snapshot.

Floating rate bonds answer to the reset. Their sensitivity is set by the time to the next coupon reset, not by maturity; the NSE maturity indices exclude them, along with inflation-linked and special securities.

Spreads and liquidity are outside it. Duration against a government yield sees nothing of a credit spread, and when an issuer's paper stops trading the loss is not a yield move at all; at the scale of a fund, that is what side-pocketing exists for.

Embedded options change the sign. A callable bond is capped near its call price, so its convexity turns negative as yields fall. The index bonds here each carry a fixed coupon and a single redemption date.

Duration is the fastest honest answer to one question: roughly how much a portfolio loses if its own yield rises by a modest amount. Convexity says how fast that answer degrades as the amount grows, and the thresholds above say where. Neither says whether yields will move. Rebuilding the figure from the cash flows, rather than taking a category name or one disclosed number on trust, is what makes the rest checkable.