Fixed Income Portfolio Measures: Duration and Maturity

Feed the calculator one row per holding, with four figures in each row. Out come three portfolio level readings: weighted average maturity, a MACAULAY duration and a MODIFIED duration. Each is the same market value weighted sum run down a different column. Add a benchmark reading and a move size, and three rupee figures follow, every one of them labelled.

One arithmetic operation does the whole of the work here, and it is performed four times. Multiply each holding's figure by that holding's share of the total market value, then add. Nothing more elaborate happens anywhere in this calculator. The reason a calculator exists for something this simple is that the mistakes are made at the input stage and never at the arithmetic stage. A column gets read off the wrong heading, or a weight gets built on the wrong base, and the answer that comes back is plausible, checkable and wrong.

So the blocks that follow spend more time on where each number is found than on what is done with it. A MACAULAY duration and a MODIFIED duration each measure something already established elsewhere. What is rarely obvious is which of the two is sitting in the third column of a statement that has arrived by email.

What does this tool take in, and what does it hand back?

The calculator takes four boxes in every holding row, and two more that sit outside the grid altogether: one for the benchmark's MODIFIED duration, one for the size of the move being priced. Ten boxes carry the worked default, and every figure the calculator prints is built from those ten and from nothing else.

A reader who has seen a portfolio report will expect more boxes than ten. There is no box for the issuer, none for the sector, none for the coupon, none for the credit assessment, and none for what the holding cost when it was bought. None of those enters any of the seven outputs. If a figure is not in the ten boxes, it cannot be in the answer. Somebody asking why two reports disagree can be told exactly that.

Why does every weight here start from market value?

Two flats sit in the same building, and a household bought both for Rs 40,00,000/- each. One of them would now fetch Rs 46,00,000/-. The other would fetch Rs 34,00,000/-. Ask what share of the household's property wealth sits in the first flat and nobody in the room answers fifty per cent. The household divides Rs 46,00,000/- by the Rs 80,00,000/- the pair would fetch together, and calls it 57.5000 per cent. Nobody has to be taught this. Only when the same question arrives with a bond statement attached do people reach for the wrong denominator.

The denominator under every weight in this calculator is total market value, and face amount never appears in it. Two holdings that promise to repay the same amount can be worth very different sums today, and it is today's sum that decides how much each one pulls the portfolio's readings around. A weighted average built on face amounts is not a slightly rougher version of the right answer. An average built on face amounts is a separate quantity wearing a heading it has borrowed, and the printed result gives away nothing about which of the two it is.

The bond version of that picture is exactly as blunt. A holding that promises Rs 1,00,000/- and is carriedA holding is carried at the value a holder's own books put on it for a stated date, which need not be what was paid for it and need not be what it will eventually repay. at Rs 1,04,000/- and one that promises Rs 1,00,000/- and is carried at Rs 92,000/- do not weigh the same. The two holdings weigh the same only in a table built on the wrong column, and that table produces four outputs which all look completely ordinary.

Where is each of these six numbers actually found?

Where each number is found is the part of a calculator that usually gets left out, and it is the part where the errors live. Each row below says where the number is found. What each number means is covered separately, so not one row says it.

Two of those six rows deserve a second look. The norm a supervised holder carries a debt portfolio at decides what number is allowed to go in the market value box, so that row is the one an authority has a view on. The norm is rewritten periodically, so its current wording sits with the body that maintains it, and that body is named below. The move size row is the opposite case: nobody outside the room has any view on it at all. The move size is declared, and the calculator answers the question declared rather than a question about the future.

The two duration rows are where the day goes wrong, and the reason is entirely mechanical. On any holding statementThe periodic list a custodian or a fund administrator sends showing every position held, one line each, with a handful of measurement columns alongside. the MACAULAY column and the MODIFIED column sit side by side, both are quoted to four decimal places, and both look like the same species of number. One measures a stretch of waiting. The other measures a rate of change. Copy the pair in the wrong order and the portfolio figure comes out wrong by a factor nobody notices. The closing block below takes that error apart.

How does one operation turn a table into four portfolio figures?

Each row's share of the total market value is a weight for that holding, and the weights add to one. The weights run down the maturity column and are added: weighted average maturity. The identical weights run down the MACAULAY column and are added: the portfolio's MACAULAY duration. The same weights run down the MODIFIED column and are added: the portfolio's MODIFIED duration. Three different portfolio figures come out of one operation applied to three different columns, and the operation itself never varies by so much as a step.

The fourth output is not an average at all. The benchmark's MODIFIED duration subtracted from the portfolio's is the active difference. An active difference is a sensitivity and not a stretch of time. The active difference needs no weights, no grid and no holdings, and that turns out to matter a great deal below.

Notice what the third card is careful not to say. A MODIFIED duration is a rate of change and never a stretch of time, so the first two outputs carry the word years and the third one does not. Tidiness is not why the calculator prints those unit labels. A portfolio report that leaves them off is the exact document from which somebody later copies the wrong column.

What does the worked default return?

The grid above is loaded with two holdings and half the market value in each. The ten year bullet bond arrives with both of its duration figures already made, 7.119063 years and 6.561348. The second holding is a two year zero coupon claim priced off the recorded two year reading of 6.25 per cent, compounding once a year. Because it pays nothing at all before the day it repays, its MACAULAY duration is also 2.000000 years, and its MODIFIED duration is 2.000000 divided by 1.0625, or 1.882353.

The three readings are printed as plain text here as well as live in the panel. Weighted average maturity comes out at 6.0000 years, the portfolio's MACAULAY duration at 4.5595 years, and its MODIFIED duration at 4.2219. Hold the first two side by side for a moment: 1.4405 years separates two figures describing the same two holdings on the same day.

A word about the decimal places in that table, because a checking reader will otherwise land one ten thousandth away from the printed answer and assume the mistake is theirs. For the ten year bullet bond the record prints a MACAULAY duration of 7.1191 years, to four places. A four place 7.1191 averaged with 2.0000 gives 4.55955, and 4.55955 sits exactly on a half at the fourth decimal. Where it goes from there depends entirely on which rounding rule a given spreadsheet happens to use. So the grid is fed the underlying figure carried to six places, 7.119063, and the portfolio reading of 4.5595 follows from it without any coin toss. The same reasoning puts 6.561348 in the MODIFIED box rather than the recorded 6.5613, and 1.882353 rather than 1.8824.

Why is that first gap entirely the bullet bond's doing?

Take the two holdings apart and the answer falls out in one line. The ten year bullet bond repays in 10.000000 years, but its MACAULAY duration is 7.119063 years, so 2.880937 years of gap opens up between the two. Every rupee of coupon landing before the repayment date drags the average waiting time inward, and nine of them land. The two year zero coupon holding hands over nothing until the day it repays, so nothing drags anything inward, and the gap it opens measures 0.000000 years.

Half of 2.880937 is 1.4404685, and that is the whole of the portfolio's 1.4405 year gap, down to the last place. Nothing else contributes. There is nothing else to contribute. Once that is visible, the general rule needs no proving: a portfolio holding nothing but zero coupon claims shows no gap between the two averages at all, and a portfolio stuffed with high coupon holdings shows a wide one. The gap is a reading of how much of the promised money arrives early.

And the second step, from 4.5595 down to 4.2219?

That one happens row by row, before any averaging takes place. Each holding's MODIFIED duration is its own MACAULAY duration divided by one plus its own yield. The ten year bullet bond yields 8.50 per cent, so 7.119063 divided by 1.085 gives 6.561348. The two year zero coupon holding is priced off 6.25 per cent, so 2.000000 divided by 1.0625 gives 1.882353. Two different divisors, applied to two different rows, and only then does the weighting happen.

So there is no single yield by which the portfolio's MACAULAY duration can be divided to arrive at its MODIFIED one. 4.5595315 divided by 4.2218505 is 1.079984, and 1.079984 is neither 1.085 nor 1.0625. The ratio 1.079984 is a blend of the two divisors, weighted the same way as everything else in the calculator, and it belongs to this particular pair of holdings and to no other. A treatment that offers one yield and divides the portfolio figure by it has quietly assumed every holding sits at the same yield. The number 1.079984 is worth keeping to hand: what goes wrong in the closing block turns out to be about precisely that factor.

What is the word PARALLEL doing inside these figures?

PARALLEL is doing more work here than most readers expect, and it is doing it in two places rather than one. The obvious place is the rupee line: a MODIFIED duration multiplied by a move size prices a move in which the whole SPOT curve steps sideways, every point of it, by one common amount on one common day. The less obvious place is the average itself. Adding MODIFIED durations together with market value weights delivers the portfolio's true sensitivity on one condition only. Every holding's own yield must step by one common amount on one common day. The assumption has already done its work inside the fourth output long before any rupee figure appears.

Which is why the calculator prints the word beside the outputs instead of tucking it into a footnote. Suppose a rise in the two year reading were larger than a rise in the ten year one. The two holdings would then be reacting to two different moves, the weights would still add to one, the average would still print to four decimals, and it would no longer be measuring what its heading says. Nothing about the output would look different. An output that cannot be seen to be wrong is the whole reason the assumption sits on the face of the panel.

A move that is not parallel has no box in this calculator, and no figure comes back for one. A curve that twists asks something else, and what comes back is a different number entirely. Pricing a twist needs a stated shape for the move, part by part along the curve, and that work is covered separately.

Which of the three rupee figures is actually in view?

One move, three answers, and the whole point of the block is that each one arrives with its name attached. The portfolio's MODIFIED duration multiplied by the size of the move gives the WHOLE EXPOSURE, first as a percentage of total market value and then in rupees. The benchmark's MODIFIED duration multiplied by that same move gives the part that arrived with the benchmark, before anyone made a decision about anything. The active difference multiplied by the same move gives the ACTIVE PART.

The first and the third are the two figures most often confused, and the second is precisely what explains the distance between them, so the calculator returns all three and prints the label beside each. On the two holding grid above, a PARALLEL rise of 100 basis points prices out as follows.

The middle bar is longer than the top one, which surprises people the first time. The middle bar is only saying that this particular pair of holdings sits shorter in MODIFIED duration than the benchmark it is measured against, so the active part comes out on the minus side and pulls the whole exposure in. Retyping the benchmark box above as 3.50 rearranges all three figures, with the identity still closing.

What happens when there is no holding table at all?

Here is the awkward fact about this calculator, and it is worth stating plainly rather than designing around. The portfolio this material keeps coming back to holds Rs 5,000 crore of fixed income. Its MODIFIED duration reads 5.20. The benchmark it gets measured against reads 4.80. No holding table for it has ever been published. None exists. Written out in whole rupees that market value is Rs 50,00,00,00,000/-, and that is the form the box below takes. So the grid cannot be filled, and the honest thing to do is run the calculator with the grid empty and see what survives.

A vegetable seller counted the notes at the end of every day, so she knows exactly what she spent this month, down to the rupee. She never wrote the crates down, so what share went on tomatoes she cannot say. The total is real. The split was never written down, and no length of staring at one figure will conjure the other. The second panel below is that situation exactly: the rupee outputs still work, the two averages cannot, and which of the cells goes blank is the most useful thing in this walkthrough.

The empty cells carry the point. A portfolio level MODIFIED duration is what remains after the holding table has been thrown away, and it is enough to price a declared move and nothing more. How long the money is out for lives in the maturity column, and a portfolio level MODIFIED duration cannot say it. An average waiting time lives in the MACAULAY column, and it cannot supply that either. A blank cell with its reason written inside is a better thing to hand a reader than a figure invented to fill the space.

What will this calculator not return?

A tool that lists its refusals on its own face is more use than one that quietly returns something for every question. Five things get asked of this calculator that it cannot answer, and each one is turned down for a different reason.

The fourth row is the one worth dwelling on. A portfolio sitting longer in MODIFIED duration than its benchmark got there because somebody decided it should sit there. The calculator prices that decision under a declared move, and the pricing is arithmetic; whether the decision was a good one is not arithmetic, and nothing in the ten boxes could settle it. A verdict, a range around an output, a probability and a view on where a rate goes next all need material the ten boxes do not hold, so the calculator returns none of them.

Who actually sits down and uses this?

Three people, and they enter the calculator at different doors. An analyst opening a published fact sheet is handed a total and a portfolio level MODIFIED duration, and that is the end of the disclosure. No holdings, no maturities, nothing to average. The second panel is not a degraded version of the first for that reader; it is the whole of the toolkit, and knowing which two cells must stay blank is what stops a plausible average being invented to fill a report.

A treasury desk at a lender has the grid, so it reaches for the top panel instead. The desk is funding loans that run for years out of deposits that run for months, and before a committee meets somebody has to be able to say what a stated rise in rates costs the book. The whole exposure is the answer to that question. The book is not being measured against a benchmark at all, so nobody at that table is interested in the active part.

And somebody preparing a note on a manager wants only the third figure. The whole exposure describes the book, most of which arrived with the benchmark and reflects nobody's positioningWhere a holder has deliberately placed itself relative to the yardstick it is measured against, as opposed to the exposure it would carry by simply matching that yardstick. at all. The active part is the slice attributable to a decision. Handing that reader the whole exposure by mistake overstates what the manager did by thirteen times on the recorded portfolio. An error of that size survives a meeting.

Every figure in both panels answers to a box that can be retyped, so no output is hidden behind a control. The top panel adds two things on top of the boxes. One is a slider that walks the declared move from a fall of 300 basis points, through no move at all, to a rise of 300. Of everything on offer, that relationship is the one worth dragging rather than typing. The other is a button that reads the MACAULAY column straight into the MODIFIED boxes, so the error named below can be produced rather than merely described.

Where does this calculator stop, and who keeps the wording it leaves out?

Five blanks sit under this calculator, and each one is keyed to the cell that would have needed it filled. A list of rule names never shows when the missing wording starts to matter. A cell does.

The two holdings in the worked grid, the portfolio in the second panel, its benchmark and the two year reading of 6.25 per cent are invented.
Educational material. Not advice on any investment, tax, budget or market position.