Ladder, Barbell and Bullet: The Three Portfolio Shapes

Where a bond portfolio's money sits along the maturities it holds is its shape. Equal amounts at a run of successive dates make a ladder. Money at one short date and one long date, with nothing in between, makes a barbell. The whole of it at a single date makes a bullet shape. All three can carry one weighted average maturity. The average never names the arrangement behind it.

Nothing a portfolio reports belongs to the portfolio itself. Each figure is borrowed from the holdings underneath it and then averaged, and each holding is counted in proportion to what it is worth today rather than what it promises to repay later. Averaging runs one way only. Two portfolios with not a single holding in common can produce the identical number, and that number is what a report carries. One method does all of it, applied three times over, and what the method quietly discards matters as much as what it keeps.

How is any portfolio level figure actually built?

Take the first holding, and any figure attached to that one holding: the date it repays, its waiting time in years, its sensitivity to a change in yield. Multiply that figure by the holding's market value weightA holding's share of what the whole is worth today at market prices, rather than its share of the amounts contracted to be repaid later.. The weight is that one holding's value today divided by the value of the whole today. Do the same for the second holding, and the third, and every one after that. Add the products. One procedure produces every portfolio level number below, and there is no second procedure hiding behind it.

Which base the weights divide by is not a detail, and getting it wrong produces a number that looks perfectly normal. Here are the two candidates side by side.

Take two holdings contracted to repay the same amount on the same day. Price one of them above the other and their weights part company immediately, even though the promise written into each is word for word identical. Built on face amounts instead, the same weighted average changes while the label on it does not, and a report will not always state which base sat underneath.

Now look at what the method throws away. Once the products are added, the individual holdings have gone out of the answer completely. The sum does not say how many there were. The sum does not say where any of them sat. The answer keeps the average and discards the arrangement, and no amount of staring at the average brings the arrangement back. Forgetting that is the commonest way a reported figure gets misread.

What is Weighted Average Maturity the average of?

Weighted average maturity averages dates, and only dates. Specifically, it averages the days on which the holdings are contracted to hand back their principal, with each date weighted by that holding's share of today's market value. Run the method from the block above, put a repayment date in the F slot, and the number that comes out is the weighted average maturity.

Weighted average maturity answers exactly one question: on average, how far away is the money contracted to come back. One distance in time is the entire content of the figure, and everything else a reader might want from it has to come from somewhere else. Notice in particular what is not in the definition. Nothing about coupons. Nothing about money arriving before the final date. The final date is the only date the definition looks at. A holding could pay out four fifths of its value in the first two years, and weighted average maturity would still count only its last date.

The narrowness is not a defect. A measure that averages repayment dates is a useful thing to have, especially when somebody wants to know when cash is contractually due back. The trouble starts when a second figure, built by the same method but on a different input, gets read as though it were the same measure. The second figure is portfolio duration.

Which Portfolio Duration does a report actually mean?

Two different figures travel under the name portfolio duration. A report silent on which of the two it carries cannot be used at all. Both are built by the method in the first block, and they differ only in what goes into the F slot.

Put each holding's MACAULAY durationThe average number of years spent waiting for a bond's money, with each date weighted by how much of the price arrives on it. It is built from a single bond's payments in the material on rate sensitivity. in the F slot and the sum is the portfolio's MACAULAY duration, a waiting time measured in years. Put each holding's MODIFIED durationA sensitivity rather than a stretch of time. It says roughly what percentage of its price a holding gives up when its yield rises by one percentage point. in the F slot instead and the sum is the portfolio's MODIFIED duration, a sensitivity carrying no unit of time whatsoever. How each of those two behaves on a single bond is settled under rate sensitivity, and both are used here rather than rebuilt.

Here is the trap sitting inside that last row, and it is worth slowing down for. Subtract one MODIFIED duration from another and the difference is quoted in the same units as a MODIFIED duration. The difference is a sensitivity too. A report that calls such a difference a number of years has borrowed the word from the other measure and is describing a sensitivity as though it were a length of time. Watch for it. The borrowing is common enough that it reads as normal English, and once it is in a sentence nobody downstream questions it.

Portfolio Duration vs Weighted Average Maturity: why do the two part company?

One holding already carried on this platform shows the whole reason in a single pair of numbers. The ten year bullet bond has a maturity of 10.00 years. Its MACAULAY duration is 7.1191 years. The two figures are not two readings of one quantity that happen to disagree. Each of them answers a question the other never asks, and the 2.8809 years standing between them is made entirely of coupons.

Think about where the bond's money actually arrives. A slice of it lands at the end of every year, and the last date carries the final slice together with the principal. Maturity ignores the early slices completely and reports the final date. The early slices are money that has already come back, so a waiting time cannot ignore them. Money landing ahead of the final date can only drag the average wait closer, and nothing in a schedule of payments can push it back out. The MACAULAY duration of a coupon paying bond is therefore always shorter than its maturity, and never the other way round.

Now build a portfolio out of that bond and one other holding, so the same gap shows up at the portfolio level. Put half the market value into the ten year bullet bond. Put the other half into a two year zero coupon holdingA holding that hands over nothing at all until its final date, so the whole of its money arrives on one day.. A zero coupon holding pays nothing before its final date, so its MACAULAY duration is exactly its own maturity, 2.00 years.

Weighted average maturity first. Half of 10.00 years and half of 2.00 years is 6.0000 years. Now the MACAULAY duration. Half of 7.1191 years and half of 2.00 years is 4.5595 years. Same portfolio, same day, two figures 1.4405 years apart, and both of them correct.

A reader checking that 4.5595 with a pen will land a ten thousandth away from it. Halving the printed 7.1191 gives 4.55955, and that prints as 4.5596. To more places than four, the MACAULAY duration carried by that bond reads 7.119062643353 years. Halving the unrounded figure gives 4.5595313, and that prints as 4.5595. The second route is the one taken, and 6.0000 less 4.5595313 is 1.4404687, printing as the 1.4405 above. A rounded display figure is never an input to the next step, and this is the smallest possible demonstration of why.

What do a ladder, a barbell and a bullet shape look like side by side?

Everything up to here has been method. Three arrangements make the method concrete, and all three are set out in full below. Each one holds government zero coupon holdings at three maturities, and each one is worth exactly the same amount of money as the other two.

The curve those maturities sit on is a teaching object, and it reckons its interest once a year. Three of its points are used below. Note that each rate is a SPOT rateThe single rate that discounts one payment landing on one future date, with nothing at all changing hands in between., the right kind of rate for a holding that makes exactly one payment.

Now the three arrangements themselves, declared as a grid so that nothing about them has to be inferred from a sentence. Each row read across shows where that arrangement's money sits.

Follow the three pictures left to right and notice that no two of them hold anything in common except the amount of money and, as the averages below show, the average. Every cell above divides by market value, and each of the three rows is worth the same total as the other two.

If three arrangements share one average, what is the average missing?

Work each one through with the method from the first block. The ladder averages 1.00 year, 2.00 years and 3.00 years at one third each, giving 2.00 years. The barbell averages 1.00 year and 3.00 years at one half each, giving 2.00 years. The bullet shape holds one maturity and nothing else, so its average is that maturity, 2.00 years.

The second figure does no better. A zero coupon holding pays nothing before its final date, so its MACAULAY duration is its own maturity, and averaging maturities and averaging MACAULAY durations become the identical calculation on these particular holdings. Three arrangements, one weighted average maturity of 2.00 years, one MACAULAY duration of 2.00 years, and neither figure can see any difference between them.

Do not read this as a fault in these particular numbers, chosen to be awkward. Collapsing is what an average is for. A single figure standing in for a whole distribution must lose the distribution, and the only surprise here is how completely it goes: an arrangement that hands money back in three instalments and an arrangement that hands it back in one have been reduced to the same two characters.

Does the third figure separate them?

There is one more portfolio level figure available, so it is fair to ask whether it does any better. For a holding of this kind the recipe is short: take the maturity, then divide it by the quantity one plus whatever SPOT rate belongs to that node. Because the three arrangements draw on three different points of the curve, the three holdings underneath them carry three different figures.

Weight those three holding figures the way the first block says and the arrangements come out as follows. The ladder, at one third each, reads 1.88074. The barbell, at one half of the first and one half of the third, reads 1.87993. The bullet shape holds only the two year holding, so it reads that holding's own figure, 1.88235. The widest gap between any two of the three is 0.00242, a little over two ten thousandths.

Two decimals fewer and the same three figures print as 1.8807, 1.8799 and 1.8824. Subtracting the outer two of those printed figures gives 0.0025, not the 0.0024 that the unrounded values give. Neither number is a mistake; they are the same quantity read at two different precisions, and five places are printed above so that the gap can be reproduced rather than taken on trust.

What can be claimed from a spread that narrow? Very little, and it is worth being exact about how little. On a parallel moveEvery point on the curve travelling the same distance together, in one step. It is an assumption, and it is labelled as one at every use., and to the first order, these three arrangements behave almost identically.

Two things are being left out there, and both get named rather than buried. The estimate is a first order one: a single multiplication, sensitivity against the size of the shift, and then it halts. A straight line laid against a curve does not land exactly, and it misses in the same direction every time. The lean, together with the correction that removes it, carries the name convexityThe curve that a straight line estimate misses, and the correction that puts the estimate right. It is worked out separately. and belongs to the material on rate sensitivity rather than here, where the rough estimate stands unsharpened.

The three arrangements above were set out holding by holding, so every figure attached to them can be rebuilt with a pen. The next object is a different one: a recorded portfolio, with a size and a MODIFIED duration but no list of what it holds. A figure with a holding list behind it and a figure without one cannot be read the same way, and the two are kept apart for that reason.

What does a portfolio level MODIFIED duration cost in rupees?

Rs 5,000 crore is what the recorded portfolio holds in fixed income, and 5.20 is the MODIFIED duration standing against that amount. The thing its performance is set against carries a MODIFIED duration of 4.80. Subtract one from the other and the portfolio sits 0.40 longer in MODIFIED duration. The 0.40 is a difference in sensitivity and not, whatever a report may call it, a number of years.

Suppose every point on the curve rises by 100 basis pointsThe unit rate changes are quoted in. One percentage point holds a hundred, so a move of 25 is a quarter of a percentage point and nothing has to be rounded away., one percentage point in all. A MODIFIED duration of 5.20 says the portfolio gives up about 5.20 per cent of its value, and on Rs 5,000 crore that is Rs 260 crore. So far this is one multiplication.

Now split the Rs 260 crore. Its two parts mean entirely different things. The build sits in three rows.

Look at the size of the two parts before anything else. One of them exists because the portfolio holds bonds at all, and it would have been there under any manager. The other exists because somebody chose to sit longer than the thing the portfolio is measured against. Most of the cost was never anybody's choice.

There is one cell that cannot be filled. A single sensitivity figure describes a single uniform shift and nothing else. The same Rs 5,000 crore under a move where the near end of the curve travels one way while the far end travels another therefore cannot be costed at all. No such movement has been described, so no rupee figure for it can be produced without inventing one. Every amount above answers one narrow question: the whole curve travels one distance in one step, and what is the bill. Change the question and the arithmetic above stops applying.

Which of the two rupee figures is meant?

Rs 260 crore and Rs 20 crore are both correct answers about the same portfolio on the same day under the same rise. The two figures answer different questions, and a sentence that does not say which question it is answering is not a shorter version of the truth. The shorter sentence is a different claim.

The two differ by thirteen times, and the consequence of muddling them is worth spelling out in both directions. To write that Rs 20 crore is what a 100 basis point rise everywhere costs the portfolio is to describe a Rs 5,000 crore book as though it held Rs 384.62 crore. Rs 384.62 crore is the only sum on which a MODIFIED duration of 5.20 produces Rs 20 crore. To write instead that the positioning decision cost Rs 260 crore is to charge one person for a market that would have moved with or without them. Neither sentence contains a wrong arithmetic step, and both are false.

One question the arithmetic settles, and three it does not. Somebody decided, some time ago and with no reference to anything written here, that the portfolio would sit longer than the thing it is measured against. The arithmetic settles exactly one question about that: the price tag on the decision, given a rise of a stated size applied everywhere at once. Whether it was a wise decision is not settled here. Whether the gap should be closed, widened or left exactly where it is, is not settled here either. Nobody knows what rates will do next, and no sensitivity figure claims to.

What does the same idea look like at household scale?

A household living on one salary has Rs 6,00,000/- set aside and three fixed deposit slips on the table. There are three sensible ways to fill them in, and the grid holds all three so that nobody has to keep four amounts in their head at once.

The final column is identical down the grid, exactly as it was for the three arrangements worked earlier. What the three feel like to the household is a different question, with three different answers. The first hands back money every single year, so a school fee falling due in year two is already covered. The second hands back nothing whatsoever in year two, so the same fee has to be met from somewhere else or the deposit has to be broken. The third hands back everything at once in year two. The fee is covered handsomely, and the household is left holding the entire amount on one day and deciding what to do with it. The average was never what mattered to this household, and it is not what matters to the portfolio either.

Why does the word bullet mean two different things here?

The double meaning catches careful readers. Both meanings are correct and neither is slang. A bullet bond is a single bond that repays all of its principal on one date, and the ten year bullet bond used earlier is exactly that. A bullet shape is a portfolio that holds its money at one maturity. One term describes an instrument, the other describes an arrangement, and they can come apart completely.

Take ten bullet bonds, one maturing in each of the next ten years, and put an equal amount of market value into each. Every single holding repays on one date, so every single holding is a bullet bond. The portfolio spreads its money evenly across ten successive maturities, and that makes it a ladder. A portfolio built entirely out of bullet bonds can be a ladder, a barbell or a bullet shape, and knowing that the instruments are bullets settles nothing about which. The term bullet shape is used here every time the arrangement is meant, and anything written from it should do the same.

Who reads these figures, and what do they actually do with them?

An analyst summarising a debt scheme from a single factsheet is the most common reader of a portfolio duration figure, and the most exposed. The factsheet gives one number. The analyst has to write a paragraph. The arithmetic above licenses a sentence about what a rise across the whole curve would cost, stated as a percentage of value with the assumption named. The figure was built by a method that discarded the arrangement, so it licenses no sentence at all about how the money is arranged.

A treasurer at a lender is reading the same kind of figure for a different reason. Money comes in on deposits with their own dates and goes out as loans with theirs, and the treasurer wants to know how far apart the two sides sit in sensitivity. For a lender the two rupee figures come apart in a useful way: one of them describes the exposure the balance sheet carries because it is a lender at all, and the other describes the part that traces to a choice somebody made about where to sit. How a regulated lender must value, disclose and stress that exposure is settled by rule rather than by arithmetic, and the rule sits with the regulators named above.

A household reads the same idea with no figures at all. Somebody with three deposit slips and a school fee due in year two has a shape problem, not an average problem. In all three cases the practical question is the same: does this figure answer what I am about to write down, or am I about to borrow it for a question it was never built for?

What can a Portfolio Duration figure never settle?

Which of the three arrangements the Rs 5,000 crore portfolio actually is cannot be settled. There is no list of what it holds. Not one bond, not one weight, not one date. Its MODIFIED duration of 5.20 arrives as a single fact with no arrangement standing behind it, and that is not a gap in the record so much as the ordinary condition of reading a portfolio from a report.

The honest position is neither of the easy ones, so two things have to be held at once. A MODIFIED duration figure fixes what a parallel move costs, completely and exactly, to the first order: that is a real thing to know, and the whole of the rupee arithmetic above is spent computing it. The same figure fixes nothing at all about how many holdings produced it, where along the dates they sit, or what follows if the short dates travel one distance while the long dates travel another. The figure is precise about one question and silent about another, and reading the silence as an answer is the error worth guarding against.

Where these numbers came from

Every rate above is assumed rather than read off a market. The curve is a teaching object, and it reckons its interest once a year. The convention is stated wherever the curve is used: the identical three rates compounded twice a year would price these holdings differently. The portfolio and the thing its rate sensitivity is set against carry no names. A name would hint at a list of holdings that could be looked up, and no such list exists. Every remaining figure here is arithmetic run on the maturities and the rates printed beside it, worked again at each point of use rather than carried across from somewhere earlier.