Key-Rate Duration vs Modified Duration: One Number or Ten

Modified duration measures how far a bond's price moves when its yield moves, as a single figure, and it gets that figure by moving every rate at once. Key-rate duration drops the everything-at-once part. It moves one point on the rate curve, holds the rest still, and reports what happened. Do that at every point and the answers add back to the modified duration.

Both measures are running the identical experiment, and the only thing that separates them is how much of the curve is allowed to move while the experiment runs. A bond price is a pile of separately discounted payments. Freeze every rate except one, and the only payments that can react are the ones that particular rate was discounting. Unfreezing the rates one at a time and collecting the ten separate answers must land back on the answer that moving all ten together would have given. That is not a happy accident and it is not a rival measurement. It is one measurement, taken apart.

What is modified duration measuring, and what does it quietly assume while it measures?

Take the bond worked on throughout: the ten year bullet bond, Rs 1,000.00/- of face amount, an 8.50 per cent coupon paid once a year, ten payment dates, priced at par. The bond faces a simple question. If its yield rises by one percentage pointThe unit produced by subtracting one rate from another. A move from 8.50 to 9.50 is one percentage point. It is not one per cent of anything, and the two are not interchangeable., roughly how much of its price does it give up?

Modified duration is the answer to exactly that question, and on this bond it reads 6.5613. Modified duration is the per cent of its price the bond gives up for every percentage point the yield travels, and it is built in two steps that are covered separately. First the weighted average waiting time of the payments, the MACAULAY duration of 7.1191 years. Then a division by one plus the yield, and that division turns a waiting time in years into a price sensitivity. Seven point one one nine one over one point zero eight five gives 6.5613, and the moment that division happens the figure stops being a length of time and becomes a rate of price change.

Now look at what had to be true for that single figure to exist at all. One yield was applied to every one of the ten payments. The payment arriving next year and the payment arriving in ten years were discounted at the same 8.50 per cent. When the experiment ran, that one yield was nudged, and nudging it nudged all ten discountings together. A single yield is the only thing a MODIFIED duration was ever given to move, so a single MODIFIED duration can only describe a world in which every rate travels exactly the same distance.

The assumption is not a criticism of the measure. The assumption describes the measure's input. A measure handed one rate can only report on moves in one rate. Everything that follows comes from taking that assumption seriously rather than treating it as a technicality in a footnote.

What happens if one point on the curve moves and every other point is left alone?

Here is the everyday version, and it is worth holding on to before any arithmetic arrives. A shop pays six suppliers. The owner is told costs went up five per cent across the board, and from that one sentence she can work out what her month looks like. Now a different sentence: the cement supplier put prices up and nobody else did. The five per cent figure is suddenly useless to her, not because it was wrong, but because it was an average of six things and she now needs to know about one of them. To answer the second question she needs six numbers where she had one.

Key-rate duration is the six numbers. Pick one nodeA maturity at which a rate curve is actually pinned down by a quoted number, rather than filled in from the two on either side of it. on the rate curve. The rate at that node moves and nothing else does. The whole bond is then repriced. The percentage of price that changed, divided by the size of the move in percentage points, is the key-rate duration at that node, and a bond has one of them for every node worth bumping.

Key-rate duration is the same perturbation as before with a narrower thing moving: hold everything, move one point, measure what happened. The experiment did not change. The wording of the question did. And because the question got narrower, the answer got smaller and there are now several of them.

The curve used here is a dull one. The curve is flat at 8.50 per cent at every node, and that flatness is the only reason the bullet bond prices at exactly Rs 1,000.00/-: the coupon rate and every discount rate are the same number, so nothing is being pushed above or below par. A discount factorWhat one rupee arriving on a stated future date is worth today. Multiply a payment by it to get that payment's present value. at each node comes from that node's rate and no other. Every rate here compounds once a year, and every price below was worked on that clock.

What should two measures of the same thing actually be judged on?

A comparison whose tests are improvised as it proceeds is really two descriptions sharing a heading, so the criteria are set down before either measure is examined again. Five tests do the work here, and every one of them separates the two measures rather than flattering both.

A test that both sides pass identically has been wasted, so each of these five was chosen because the two measures answer it differently. The five are, in order: what moves during the experiment, how many figures come out of it, what unit those figures are quoted in, what questions each figure can actually answer, and what each one costs to produce and to read.

The third one looks like housekeeping and is not. Two measures quoted in different units cannot be added together or compared, so the unit test is what decides whether the relationship between these two is arithmetic or merely an analogy. Hold on to it until the grid arrives.

How do the two measures score on each of those five tests?

Run them side by side, in the same order, with no re-ordering to suit either side.

Read down the third row again. The third row is the one that turns this from a beauty contest into arithmetic. Both measures come out in the same unit. Not a similar unit, the same one: a percentage of price given up for each percentage point a rate moves. Quantities in the same unit can be added. The shared unit is why the ten key-rate figures can be summed at all, and it is why what follows is a containment rather than a contest.

Row four is the row that decides which measure is needed when. Modified duration answers one question well. Key-rate duration answers that same question, by adding its pieces back up, and then answers a second question modified duration cannot reach from any angle. Row five is the price of that reach: ten repricings instead of one, and a picture to interpret instead of a figure to quote.

Do the ten separate answers add back to anything?

The ten add back to the modified duration, exactly, and that is the whole relationship between the two measures in one sentence. Here is the ten year bullet bond's map, worked on its own flat 8.50 per cent curve, one node bumped at a time.

The last column adds as follows. Nought point nought seven two two, plus nought point one three three one, plus nought point one eight four zero, plus nought point two two six one, plus nought point two six nought five, plus nought point two eight eight one, plus nought point three nought nine eight, plus nought point three two six three, plus nought point three three eight three, plus four point four two two nine. The printed figures come to 6.5613 with nothing left over, and 6.5613 is the modified duration of the same bond.

The closing of that column is worth a moment. A column of rounded figures usually leaves a residue behind, and this one does not. Take the same column and stop at the ninth row, though, and the tidiness disappears. The nine printed figures add to 2.1384 and the nine unrounded ones add to 2.1385. The full column closes on the nose and a nine-tenths slice of the same column does not. A printed figure is a display of a number rather than the number itself. Where a total has to be relied on, work it from the unrounded parts.

Adding that expression over every node has a clear consequence. The division by one plus the yield is common to all ten, so it comes outside. Left inside is every waiting time weighted by its share of the price, and that is the definition of MACAULAY duration. So the sum is that same MACAULAY duration with the identical division applied afterwards, and that division is what defines a MODIFIED duration.

The identity makes the addition a check as well as a teaching point. If ten key-rate durations computed by hand or in a spreadsheet come to 6.4 against a modified duration of 6.5613, nothing interesting about the bond has been discovered. There is a mistake, and the addition has just found it.

How is each of those ten figures actually produced?

The expression above is the quick route. The slow route shows why the ten figures exist rather than asserting that they do, and it is worth watching once. Take the three year point. Add one basis point, a hundredth of one percentage point, to the rate at that node, and change nothing anywhere else on the curve.

Only one thing in the whole bond can react. The Rs 85.00/- arriving at three years was being discounted at the three year rate, so its present value moves: from Rs 66.5472/- down to Rs 66.5288/-, a difference of Rs 0.018397/-. The other nine present values are discounted at nodes nobody touched, so they sit exactly where they were. The price falls from Rs 1,000.00/- to Rs 999.9816/-, and the whole of that fall came from one payment.

Now divide. The price gave up 0.0018397 per cent of itself for a move of 0.01 percentage points, and 0.0018397 divided by 0.01 is 0.1840. 0.1840 is the three year key-rate duration, arrived at by moving a rate rather than by reading a formula. Every figure in the last column of that table can be reproduced this way, by bumping one node and repricing. Bumping and repricing is what makes each figure a measurement rather than an assertion.

Two more, to see the range. Bump the one year point by a basis point and the price settles at Rs 999.9928/-. Bump the ten year point by the same basis point and it settles at Rs 999.5579/-, a fall of Rs 0.4421/-. Same size of move, same bond, and the second one costs sixty one times what the first one did.

One honest wrinkle. The two routes agree very nearly rather than to the last digit. A bond's price does not respond to rates in a straight line, so a bump measured only as a rise lands slightly under the figure the expression gives. At the one year point the difference is invisible at four decimals. At the ten year point the rise-only measurement reads 4.4206 against the 4.4229 the expression produces. Measuring the bump both ways, as a rise and as a fall of the same size, and averaging the two brings all ten figures back to the last digit. The bend that causes the gap is convexity. Convexity is covered separately.

What does the shape of those ten figures show that the single figure cannot?

Look at the ten figures again, not as a column to add but as a shape. The ten figures are nowhere near equal. The one year point contributes 0.0722. The ten year point contributes 4.4229. The largest is sixty one times the smallest, and the run from one year to nine years climbs so gently that all nine of them together are still less than half of what the tenth date carries on its own.

Put a share on it, taking the bond's own modified duration as the base. 4.4229 divided by 6.5613 is 67.41 per cent. Two thirds of everything this bond feels about rates is attached to a single date, the one on which the Rs 1,000.00/- of face amount comes back. The nine interest dates share 32.59 per cent between them, and the two shares close on 100.00 with nothing left over. The ten year point on its own is 2.0682 times the other nine put together.

Here is a second angle on the same date that makes the concentration sharper rather than merely repeating it. Of the Rs 1,850.00/- this bond promises across its whole life, Rs 1,085.00/- lands on that final date. The final payment is 58.6486 per cent of the rupees. But that date carries 67.41 per cent of the sensitivity, a full 8.7591 points more than its share of the money. The gap is waiting time. The final payment is not only the biggest, it is also the one that has to wait longest, and both of those push a key-rate duration up.

Here is the everyday version of that shape. A household pays a little rent every month and put down a large deposit that comes back on the day the lease ends. Ask what a change in their landlord's terms costs them and the honest answer is almost entirely about one date. The monthly amounts matter, but not the way the deposit does, and an average across all the dates would hide precisely the thing worth knowing.

And now the consequence. Two bonds can report the identical MODIFIED duration of 6.5613 and have key-rate maps that look nothing whatever like each other, so a move concentrated at one part of the curve treats them completely differently while the single reported figure insists they are the same.

The zero coupon bond is the proof. The zero has one payment, landing at 7.1191 years, the ten year bullet bond's unrounded MACAULAY duration of 7.119062643353 years carried across on purpose. One payment means one node, so its entire map is a single entry of 6.5613 at that date and nothing anywhere else. Dividing a single waiting time by one plus the yield is all there is to a MODIFIED duration, so the zero reports 6.5613, the same figure the bullet reports. Two bonds. One reported figure. One map spread across ten dates, one map standing on a single date.

What does a move at one part of the curve do to two bonds reporting the same figure?

The last claim can be left as an assertion or it can be worked, and working it costs two paragraphs. Both moves below are assumed, and both are stated in full before they are used. Both are applied to the flat 8.50 per cent curve the two invented bonds sit on, and to nothing else.

Move one: a rise of 100 basis points at the ten year point, with every other point on the curve held exactly where it is. The ten year bullet bond has a payment landing on that date, so it feels the move: 4.4229 per cent of its price, on the first orderAn estimate built from the slope alone, which ignores the way a response curves. Close enough for a small move, and it drifts as the move gets bigger. reading its key-rate duration gives. The zero coupon bond has no payment at ten years at all. Its single payment lands at 7.1191 years, and the ten year rate discounts nothing it holds. Its answer is 0.0000 per cent. Not small. Nothing.

Move two: a rise of 100 basis points at the 7.1191 year point, again with the rest of the curve held still. Now it is the other way round. The whole of the zero's sensitivity was standing on that one date, so the zero gives up 6.5613 per cent of its price, all of it. The ten year bullet bond pays nothing at 7.1191 years, so its answer is 0.0000 per cent.

Two bonds. One reported MODIFIED duration of 6.5613 between them. Two declared moves, and under each one, one bond loses several per cent of its price while the other does not move at all. A single figure has no idea which part of the curve it came from, so no reading of one could have produced either of those answers.

Two caveats sit alongside those two answers. The first: both readings above are first order. Each one lays a straight line where the bond's real response bends. Reprice each bond in full. Move one costs the bullet 4.2067 per cent rather than 4.4229, and move two costs the zero 6.3226 per cent rather than 6.5613. Both first-order figures came out larger than the full repricing. Any duration measure carries the same known lean: on a rise in the yield the straight line overstates what is actually given up.

The second caveat concerns what the pair of bonds cannot be used to argue. Both bonds carry the identical 8.50 per cent yield here. An identical yield keeps the arithmetic legible. The underlying material carries no price for the bend in a bond's price response. Where people actually trade, a bond whose response bends more is a bond more people want, and wanting shows up in the price. So everything above describes what these two payment schedules do, and says nothing whatever about which of them anybody should hold. The figure that would settle that was never in the material to begin with.

So when is the single figure enough, and when does it stop being enough?

The honest answer is the one nobody expects after eight hundred words about maps: most of the time the single figure is enough, and reaching for ten of them is work with nothing at the end of it.

Consider a move that really is even across the curve. Every node rises by the same amount, each key-rate duration gets multiplied by that same amount, and the ten products add to exactly what the modified duration would have given on its own. Ten repricings, one chart, and an answer identical to the one already available. The map earned nothing.

The map earns its cost in one situation only: when the exposure in question is concentrated somewhere. A borrower whose refinancing all falls in the next two years cares about the near nodes and very little else. Somebody matching payouts twenty years out cares about the long endTrading shorthand for the far maturities on a curve, the ones ten years out and beyond. The near ones are the short end. and is close to indifferent about everything nearer than that. A hedgeA position taken on purpose to cancel part of an exposure somebody already carries, rather than to make money on its own account. placed at one maturity against an exposure sitting at another has a mismatch that only shows up node by node. A curve twistAny move in which the points on a rate curve do not all travel the same distance. One end can move while the other sits still, or the two ends can travel in opposite directions., by definition, moves different parts of the curve by different amounts, so a measure that assumed they all moved together has nothing to say about it.

And here is the part that is easy to skip. Whether the move in prospect is even across the curve is a separate question, and neither of these two measures answers it. Modified duration assumes evenness. Key-rate duration is silent on it and simply reports what each node would do. Whichever measure is in hand, the judgement about what kind of move to worry about comes from somewhere else entirely.

The reading rule, then: the single figure is quoted by default, the map is computed when the concern is attached to one part of the curve, and a map is never presented as more accurate than the figure it adds up to. A map is not more accurate. A map is more detailed, and detail is a different property from accuracy. The two get confused constantly.

What does neither of these two measures say?

Three things, and then one figure the underlying material cannot supply.

Neither measure says where any rate is going. Both are conditional arithmetic: given a move of this size at this point, here is the price response. The move is an input somebody supplies, and nothing in either measure has an opinion about what to supply. A key-rate map with ten precise figures on it can look like a forecast from across the room, and it is not one.

Neither measure carries the bend. Both are first order. Both lay a straight line against a response that curves, and both lean the same way while doing it: on a rise in the yield they overstate what is given up, and on a fall in the yield they understate what is gained. The lean was visible earlier when the full repricings came in at 4.2067 and 6.3226 against first-order readings of 4.4229 and 6.5613.

Neither measure says whether the borrower pays. Everything here is arithmetic on a payment schedule that was assumed to happen. Both bonds here were built with no credit element in them anywhere. Working out how a price answers a rate stays clean on instruments where repayment is simply assumed, and it stops two separate worries being blended into a single figure.

And one figure that cannot be produced, with the reason in its place

There is a Rs 5,000 crore fixed income holding in the underlying material, with a MODIFIED duration of 5.20 against a benchmarkA stated reference position that somebody's result gets measured against. It is chosen before the measuring starts rather than picked afterwards to flatter the result. at 4.80. The difference is 0.40, and that difference is a sensitivity rather than a length of time: MACAULAY duration is the one measured in years, and a gap between two MODIFIED durations inherits what a MODIFIED duration is. On an even rise of 100 basis points across the whole curve, that holding gives up about 5.20 per cent, or Rs 260 crore. Of that, Rs 240 crore came with the benchmark and Rs 20 crore is the part that was actually decided, and the two add back to Rs 260 crore. The two differ by thirteen times over, so which of them is meant should be named every time.

Now ask the obvious follow-up: what would an uneven move do to that holding? No such figure exists in the underlying material. The underlying material contains no uneven curve move for that holding at all. Producing a number would mean inventing the move as well as the answer to it. An invented move reads exactly like a measured one, and that is the whole problem. The two moves further up were declared out loud, applied to two invented bonds, and carried no further. Those two moves stop there.

Who actually reaches for the ten figures rather than the one?

Start with the household version, because it is the same shape and it costs nothing to picture. A couple carries two loans. One is a two year loan that gets repriced every twelve months. The other is a long home loan on a rate fixed for years yet. Somebody offers them a single average sensitivity for the pair. Under a general rise in borrowing costs that average serves them fine. Under a move where short borrowing costs jump and long ones sit still, the average is worse than useless. The average quietly attributes to the home loan a reaction the home loan is not going to have. Two numbers, one per loan, and the question answers itself.

The institutional versions are the same problem wearing a suit. A lender funding long lending with short deposits is exposed to a specific pair of points on the curve rather than to the curve in general, so a single sensitivity figure hides the very mismatch that matters. Matching one figure says nothing about matching a shape, so somebody running a bond fund against a benchmark can match the single figure exactly and still be positioned quite differently. And anybody who has placed a hedge at one maturity against an exposure at another has a residual that lives entirely in the difference between two key-rate maps.

The reading habit worth taking away is small. When a question names a part of the curve, a single figure is the wrong instrument, however precise it looks. When a question is about rates in general, the map is expensive detail. The measure should match the shape of the question rather than the sophistication the answer is meant to convey.

The ten year bullet bond, the zero coupon bond, the Rs 5,000 crore fixed income holding and the benchmark beside it are invented.
Educational material. Not advice on any investment, tax, budget or market position.