Barbell vs Bullet: Two Shapes, One Average Maturity

A barbell splits its market value between one short maturity and one long one, leaving the middle empty. A bullet shape puts everything at a single maturity. Hold the two at the same weighted average maturity and their MODIFIED durations land 0.0024 apart. The real separation is not sensitivity at all, but the number of dates the money comes back on.

Two arrangements can only be compared once something has been pinned. Whatever is pinned decides what a comparison can show, and pinning the average maturity turns this comparison into one about arrangement rather than about length. Pin nothing and the comparison is between a two year holding and a ten year one. Anybody could have predicted that answer. Pin the average and something less obvious happens: the two arrangements turn out to agree on almost every measure a report would carry, and to disagree on one that no report carries at all.

One discipline before the arithmetic starts. Every weight below is a share of market valueThe price a holding would change hands at today, rather than the amount its paperwork says is owed. Every weight here is a share of the first., and which base is meant is repeated in the same sentence as the weight. Two holdings with identical amounts owed, bought at different prices, do not carry identical weights. Build the average on amounts owed instead and the label above it stays put while the number underneath it moves. Nothing below would then reconcile.

What is a barbell, before anything gets compared to it?

A barbell puts its market value at two maturities, one near and one far, and holds nothing in the stretch between them. Two positions. Two repayment dates. A hole in the middle that is not an accident but the whole design.

The barbell used here is declaredSet out as a worked example, rather than lifted from any record of what anybody holds. here rather than recorded anywhere: half of its market value sits at 1.00 year and half at 3.00 years. Both legs are zero coupon government holdings priced on the invented SPOT curve this material carries, compounding once a year. The rate standing at the one year node is 5.90 per cent. The rate at the three year node is 6.55 per cent. Nothing at all sits at 2.00 years. Two bars of equal height at 1.00 year and 3.00 years, with the space between them left blank, is everything a barbell is.

The design does something particular to the money. Half of the market value comes back inside twelve months and is then in hand, unencumbered and early. The other half stays out for three full years. There is no middle position quietly averaging the two. The average exists only as arithmetic. No holding sits at 2.00 years for anybody to point at.

What is a bullet shape, taken entirely on its own?

In a bullet shape, a single maturity carries everything. One position. One repayment date. Nothing early and nothing late.

The bullet shape declared here holds all of its market value at 2.00 years, again as a zero coupon government holding on the same invented curve, again compounding once a year, at the 6.25 per cent standing at the two year node. Where the barbell is a pair of positions with a gap, the bullet shape is a single position with no gap to have.

And here is the naming point that saves an enormous amount of confusion later on. A bullet BOND is an instrument: one borrower, one date, the whole of the principal repaid then. A bullet SHAPE is an arrangement: a set of market value sitting at one maturity, whether that comes from one instrument or forty of them maturing together. A bullet shape is what is meant here, every single time. The word shape is welded on for that reason. A holding of one hundred separate bullet bonds all maturing in 2.00 years is a bullet shape. So is a single one.

What has to be pinned before the two can be compared at all?

Set a barbell at 1.00 year and 3.00 years against a bullet shape at 10.00 years and the comparison is not between two arrangements. The comparison is between a short holding and a long one, and the long one will look more sensitive to rates for the plainest reason there is: it is longer. The arrangement never entered the argument.

So the weighted average maturity is pinned at 2.00 years on both sides, and the pinning is done first, before any criterion is applied. Half of 1.00 year plus half of 3.00 years is 2.00 years. The whole of it at 2.00 years is 2.00 years. Same market value on both sides, same average, and now every remaining difference that turns up has only one place it could have come from. Pinning the average maturity is not a simplification made for convenience; it is the step that converts a comparison of lengths into a comparison of arrangements.

Everything below leans on this arithmetic, so the arithmetic is worth taking slowly. Weights are shares of market value. The barbell's shares are 0.5 and 0.5. Multiply each share by the maturity it sits at and add: 0.5 times 1.00 gives 0.50, and 0.5 times 3.00 gives 1.50, and those two make 2.00 years. The bullet shape has one share of 1.0 at 2.00 years, so its average is 1.0 times 2.00, or 2.00 years. Two routes, one destination, and the destination was chosen rather than discovered.

The choice of split is worth sitting with. A reader who meets these two arrangements for the first time often suspects the split was tuned to make the numbers agree. The split was tuned, in exactly one respect: the equal split is the only split of a 1.00 year and a 3.00 year holding that averages to 2.00 years. Move the split and the average moves with it immediately. The control below makes that movement visible. The equal split is not a convenient coincidence. The equal split is the single position at which the contest is fair.

Which six things are the two arrangements compared on?

A comparison with no stated list is a comparison in which whoever is writing gets to choose the ground after seeing the result. So the list is fixed first, applied to both sides in the same order, and every row gets answered even when the honest answer is that nothing can be reported.

The six are these. Where the market value sits. How many dates the money comes back on. Weighted average maturity. MACAULAY duration. MODIFIED duration, together with what a stated PARALLEL rise costs. And last, a movement that lifts one end of the curve further than it lifts the other. Four of those six rows come back nearly identical, one comes back plainly different, and one comes back empty, and the empty one is the row most treatments of this subject spend all their time on.

Here is the same grid with the figures written out. The last row decides how much weight the rest of them can carry, so the last row comes before any of the others.

Where do the two arrangements come back exactly the same?

Rows three and four both read 2.00 years on both sides, and a reader could easily count that as two separate agreements. The two rows are one agreement counted twice, and seeing why is worth more than either row on its own.

Row three is pinned. The weighted average maturity was set at 2.00 years before anything was compared, so its agreement carries no information whatsoever. Row four is the interesting one. Waiting time in years is what a MACAULAY duration reports, and each date gets its weight from where the present value of that payment sits. Every holding in both arrangements pays nothing at all before it matures. So for each of them, the only payment is the one at the end, all of the present value sits on that single date, and the average waiting time collapses to the maturity itself.

Averaging maturities and averaging MACAULAY durations are therefore the same operation run on the same column of numbers, so rows three and four cannot disagree here. Equality of the two rows is a property of these particular holdings and not a law about arrangements. Put a coupon anywhere in either side and the two rows separate at once: money starts arriving before maturity, the weighted waiting time drops below the maturity, and the two columns stop being copies of each other. The record's own ten year bullet bond does exactly that, and its MACAULAY duration is nowhere near ten.

How far apart do the two MODIFIED durations actually sit?

For a holding that pays nothing before it matures, on a curve compounding once a year, the MODIFIED duration is the maturity divided by one plus that holding's own SPOT rate. Not the average rate, not the two year rate for everything: each leg is divided by the rate standing at its own node.

Run it three times. At 1.00 year against the 5.90 per cent standing at that node, 1.00 divided by 1.0590 is 0.9443. At 3.00 years against 6.55 per cent, 3.00 divided by 1.0655 is 2.8156. At 2.00 years against 6.25 per cent, 2.00 divided by 1.0625 is 1.8824.

Now assemble. The barbell is half of 0.9443 and half of 2.8156, adding to 1.8799. The bullet shape is the single 1.8824. The gap between the two arrangements is 0.0024, two and a bit thousandths of one unit of MODIFIED duration. Carry that into a stated PARALLEL rise of 100 basis points and the barbell gives up about 1.8799 per cent of its market value while the bullet shape gives up about 1.8824 per cent, so the whole of the difference is 0.0024 percentage points.

Two honest notes on that pair of percentages. The first is that both percentages are a first order reading. One MODIFIED duration, one movement, one multiplication, and the arithmetic ends. A correction exists that turns the estimate into the exact answer, and a known direction exists in which the estimate leans without it. Both are covered separately. The second is that the small gap is not an accident of the split. Dividing by one plus the rate bends the relationship between maturity and sensitivity very slightly, so the value at the midpoint sits a shade above the average of the two ends. Across a stretch of curve only two years wide, that bend is worth 0.0024 and nothing more.

The default setting reproduces the worked example above exactly, and it was chosen for that reason. Then push the control one notch either way. The average maturity moves by 0.20 years for every notch, immediately and visibly, and the moment it moves the comparison has stopped being fair: the barbell is no longer being asked the same question as the bullet shape. The equal split is not one option among eleven; it is the only setting on the whole control at which the two arrangements are still comparable. Push all the way to either end and the barbell has quietly turned into a bullet shape of its own, sitting at 1.00 year or at 3.00 years, with a MODIFIED duration of 0.9443 or 2.8156 and an average maturity to match.

If the sensitivity barely differs, what does?

Row two, and it is not a sensitivity at all. The barbell hands money back on two dates, at 1.00 year and at 3.00 years. The bullet shape hands it back on one, at 2.00 years. Two against one.

Look at what that means for somebody standing at the start with a calendar. Twelve months from now, half of the barbell's market value has matured and is money in hand, free to be spent, parked or committed elsewhere. Twelve months from now, the bullet shape has returned nothing whatsoever, and will return nothing for another year after that. One arrangement pays at 1.00 year and the other does not, and no duration figure reports that fact.

The difference in repayment dates is structuralTrue because of the way a thing is put together, so it holds without anybody assuming what rates will do next.. It does not depend on a movement in rates, on a forecast, on a scenario or on any assumption at all. Set every rate to zero and the barbell still returns half its market value at 1.00 year. Independence from every assumption is what makes the repayment count the honest answer to what separates the two arrangements, and it is also what makes the count invisible: dispersionHow widely spread in time a set of repayment dates is. Two arrangements can share an average and still be spread very differently around it. is a spread around an average, and an average throws a spread away by construction.

Notice how thoroughly the summary measures hide it. Weighted average maturity: identical. MACAULAY duration: identical. MODIFIED duration: 0.0024 apart. Three measures, three near perfect agreements, and behind them one arrangement returns money in year one and the other returns nothing. A holder reading only those three lines has read three true statements and has learnt nothing about the question they most likely care about.

Why is one row of the grid left blank?

Row six is the row every treatment of this subject builds towards. The barbell holds positions at two ends of the curve. The bullet shape holds none, sitting entirely in the middle. Lift one end of the curve further than the other, therefore, and the two arrangements have to answer differently. A movement that lifts one end further than the other is the one comparison a barbell and a bullet shape exist to have, and most of the subject is written about it.

Nothing is reported in that row. Not a small figure, not an approximate one, not a range. The cell is drawn empty and the reason is printed inside it: no such scenarioA stated set of movements somebody has written down so that a result can be worked from it. Without one written down, no result exists to report. is recorded here, one end lifting further than the other is a subject covered elsewhere, and a figure produced in this cell would be a movement made up on the spot to fill it.

An invented reading is set in the same typeface as a measured one, sits in the same cell, and gets quoted onward by exactly the same people. The cell stays blank rather than being filled illustratively for that reason. Say the row had been filled with a plausible number. Nothing about the way it looked would tell a reader three steps downstream that it came from nowhere. A plausible number would be copied into a summary, then into a note, then into somebody's argument, and at no point would the chain carry a warning label. A blank cell cannot do that. A blank cell states what is not known, and a reader who wants the subject is pointed at where it is properly covered.

There were three ways to handle it and only one of them survives inspection.

What does this look like on an ordinary street?

Two households put Rs 4,00,000/- each into fixed deposits, and both of them describe what they have done as roughly two years. The first household halves it. One half matures next year. The other half runs on for three. The second puts the whole Rs 4,00,000/- into a single two year deposit.

Asked the average, both give the same answer. Two years, both of them, and neither is fudging. A different question separates them. How much comes back next year? The first household says Rs 2,00,000/-. The second says nothing at all. An average is built by throwing away exactly the information the second question asks for. The average was never the number that separated these two households, and it was never going to be.

Put a school fee due next July into the story and the two households stop being equivalent immediately. One of them has the money arriving. The other has a deposit that will not mature for another year and a choice about breaking it. Nothing in the phrase roughly two years says which household is which, and nothing in a MODIFIED duration of 1.8799 against 1.8824 would say it either.

Who actually reads these two rows, and for what?

Start with an analyst holding a fact sheetThe short published summary a holder of a pooled vehicle receives, which carries a handful of averages and rarely the dates behind them. for a debt scheme. A fact sheet typically prints a small set of averages: an average maturity, a rate sensitivity figure, perhaps a credit summary. Usually absent is the list of dates on which money is due back. So the analyst can see row three and row five of the grid above, cannot see row two, and has no way to reconstruct row two from what is printed. Two schemes reporting the same pair of averages can be arranged in genuinely different ways, and the fact sheet will not separate them. Disclosure duties, and how often they fall due, are set by the keeper named in the block below.

Next, a treasury desk at a lender that knows it must fund a large outflow in twelve months. For that desk row two is the only row in the grid that matters, and rows three, four and five are close to decoration. A barbell delivers half of its market value on the date the desk needs it. A bullet shape at 2.00 years delivers nothing then and must be sold instead, at whatever price the day happens to offer. Both arrangements will have reported the same average maturity in every internal summary the desk produces. The summary is not at fault. A summary is built to compress, and compression is exactly what it did.

Then a research analyst comparing two managers whose averages match. Concluding that the two managers are running the same thing under different badges is an easy step, and the pull towards it is strong. Two arrangements that agree on every published average can still be built from completely different dates, and the published averages are structurally incapable of showing that. The move is not to guess at the arrangement. The move is to notice that the question has not been answered and to ask for the repayment dates directly.

And a household, again. The household case never goes away. Anybody choosing between a laddered set of deposits and a single one is choosing between arrangements at the same average, and the deciding question is almost never sensitivity to rates. The deciding question is when the money is wanted.

Which of the two is the portfolio recorded here?

Neither. A reader who has spent this long on two arrangements will reasonably expect them to be pointed at something, so the answer needs saying plainly.

The recorded portfolio runs to Rs 5,000 crore and reads 5.20 in MODIFIED duration, against a benchmark reading 4.80. The record carries no list of what that portfolio holds: no maturities, no dates, no split. So the grid above cannot be run against it, and no sentence above should be read as hinting which of the two arrangements it resembles. A reading of 5.20 is as consistent with a barbell as with a bullet shape.

Two figures do survive that ignorance, and they are worth having precisely because they survive it. Take a stated PARALLEL rise of 100 basis points. The whole exposure gives up 5.20 per cent. Struck on a market value of Rs 5,000 crore, that is a loss of Rs 260 crore. The active part is the 0.40 by which 5.20 stands above 4.80, so it gives up 0.40 per cent, and that comes to Rs 20 crore. The whole exposure runs to thirteen times the active part and the pair gets swapped constantly, so which of the two is being quoted needs saying on every occasion. Neither figure needed any knowledge of arrangement to compute, and that is exactly why both can be stated while row six cannot.

The active figure looks like a verdict, and it is not one. Rs 20 crore is the size of what a stated movement would cost the part of the exposure sitting beyond the benchmark. It is a size, not a judgement. A portfolio carrying 0.40 more MODIFIED duration than its benchmark is carrying a positioning decisionA choice somebody already made about sitting longer or shorter than a comparison series. This material measures what such a choice is worth arithmetically and nothing beyond that. that somebody already made. The decision gets priced against one stated parallel movement, and then the arithmetic stops. Whether the gap should be there, whether anybody should shrink it or widen it, and what rates will do next are three separate questions, and a priced cost answers none of them.

The barbell, the bullet shape, the curve they are priced on, the Rs 5,000 crore portfolio and its benchmark are invented.
Educational material. Not advice on any investment, tax, budget or market position.