How to Map the Exposures in a Fixed Income Portfolio

An exposure map is eight steps that end in a labelled grid, not in a single number. Write the base and its date. Record the benchmark. Difference the two MODIFIED durations. State the move and its shape. Compute the whole exposure. Split it into the benchmark part and the active part. Draw every unfillable row empty with its reason. Then say what has been established.

The word map is doing real work in that sentence. A measurement gives a number. A map gives a sheet on which some questions have answers written against them, some have a reason written where the answer would have gone, and every one of them has been asked out loud. A question nobody wrote down is a question the next reader will assume somebody answered. Most of the value in an exposure map sits in the second kind of row.

What is an exposure map, and what does it come out as?

Picture a stall outside one office building. At the end of the month the person running it adds up what is owed. The total is exact, because every slip is in the tin. Beside that total, the stall can also write how much is owed by the one customer who always settles late, because that customer is known by name. The stall has never kept a record of how the other customers pay, so it cannot write how many of them will settle late.

Three lines, then. The first two are filled from what exists. The third is left blank with a short note saying why. The guess gets used and the blank does not. A sheet with the third line blank and explained is the better sheet. A blank line with a reason beside it is the whole idea of an exposure map, and everything below applies that idea to a bond portfolio, in an order that leaves no room for skipping the awkward rows.

So what does the finished thing come out as? A grid. Rows are questions somebody will eventually ask about the portfolio. Cells hold either a figure with a label attached to it, or a reason written where a figure could not go. The map produces no verdict, no ranking and no single headline number, and the reason it produces none is that the questions it can answer and the questions it cannot both have to survive onto the same sheet.

Two habits make the difference between a map and a pile of arithmetic. The first is that the steps run in a set order. Each step consumes what the step before produced. The second is that a step which cannot produce a figure still has to produce something. The set order and the reasoned blank are the method. Everything else here is the specific eight rows a bond portfolio needs.

Step one: what single figure does everything else become a share of?

Write down the market value of the portfolio, and write the date it was struck on beside it. The dated amount is the base. Every later cell on the map is either a share of the base or a rupee amount worked out from it, so a map whose base is undated is a map whose figures cannot be checked against anything six weeks later.

The valuation date matters for a reason a household will recognise. When somebody says their savings come to a certain amount, the sentence is useless until it is clear whether they meant this morning or last Diwali. A portfolio is the same object with more zeroes on it. Step one produces a single dated rupee amount and nothing else. Making it a step rather than a heading is what stops it being skipped.

One decision inside step one shapes every weight further down the map. The two options look interchangeable and are not, so the table below sets them side by side.

Two holdings repaying the same face amount can be worth different amounts on the same afternoon, because they carry different prices. A weighted average built on the second row of that table is a different number wearing the first row's label, and nothing on the sheet will tell a later reader which one was built. Every weight discussed anywhere on this map is a share of market value, and the sentence carrying the weight says so rather than leaving it to a footnote.

Step two: what is the portfolio being measured against?

Write down the benchmark and the MODIFIED durationHow far a price shifts when the yield shifts a little, given as a rate of response and not as a span of years. Its construction from a schedule of payments is covered separately. it carries. Recording those two things is the entire step. The benchmark is not assessed, not questioned as the right one, and not adjusted. The step records which yardstick the rest of the sheet will lean on.

Step two gets left out more often than any other, and leaving it out does not fail loudly. The failure is quiet: the map looks complete and reports differences against nothing in particular. Every cell carrying the word active further down is a difference against something. If there is no benchmark, write that down as the answer to step two and stop expecting an active column. With nothing to difference against, the map has one column where it would otherwise have three.

Step three: what does the difference between the two figures have to carry?

Write down three things: the portfolio's MODIFIED duration, the benchmark's, and the portfolio's less the benchmark's. The third of those is the cell people copy out and quote, so it has to survive being copied. A cell that survives being copied carries a direction as well as a size.

A portfolio whose MODIFIED duration sits above its benchmark's is positioned longer. One sitting below is positioned shorter. Positioned longer and positioned shorter are opposite arrangements that produce identical sizes, and a cell recording only the size has thrown away half of what step three was for. Notice too that neither phrase is a forecast. Positioned longer describes how the sheet currently reads, not what anybody expects to happen next.

Step three is also where the labelling discipline of this sequence earns its keep. A MACAULAY durationThe average wait before the payments arrive, weighted by what each is worth today and counted in years. It is a different object from the MODIFIED figure, and swapping them skews a result by one plus the yield. and a MODIFIED duration are different objects that both get called duration in conversation. Difference one against the other and the answer is wrong in a way nothing on the sheet will reveal, so every cell on this map carries the word MACAULAY or the word MODIFIED against any figure of that kind.

Step four: what move is being priced, and what shape is it?

Write down how large the move is, counted in basis pointsThe unit rate moves get counted in. A single percentage point holds a hundred of them, so writing either unit where the other belongs scales an answer a hundredfold. and write its shape beside it. Then write the word assumption against both. Neither of them was observed. Nothing in the arithmetic below observed a move, forecast one or argued for one.

The shape used throughout this material is PARALLEL, meaning every point on the SPOT curveThe set of rates for money lent today and repaid at a single future date. Every rate in this subject area is marked as a SPOT or a FORWARD rate so the two are never merged. moves by the same amount at the same time. PARALLEL is not a description of how curves behave. The shape is a simplification, declared so that a single sensitivity figure is enough to price the move.

Every rupee figure this map produces is standing on the shape recorded at step four, and the map states that shape in the same breath as the figure rather than in a note beneath the grid. A move of any other shape asks something else and answers it differently, and is covered separately. No scenario of another shape is recorded for this portfolio. The missing scenario turns into an empty row at step seven rather than into a smaller number here.

Step five: what does the stated move cost the whole base?

Take the portfolio's MODIFIED duration, apply it to the size of the move recorded at step four, and apply that to the base recorded at step one. Write the result into the cell labelled WHOLE EXPOSURE. Two rules attach to the writing rather than to the arithmetic: do not put the figure into any cell that is not labelled, and do not put it anywhere without the move sitting beside it.

Step five produces a first order estimateAn answer built from a straight line laid against a curve. It is close for small moves and drifts as the move gets larger, in a direction settled elsewhere.. The estimate multiplies a sensitivity by the size of a move and stops there. The correction that would make it exact, and the direction in which the uncorrected figure tilts, both belong to convexityThe curvature a straight line sensitivity misses. It is covered separately, and this map takes the uncorrected estimate as it stands.. Convexity is covered separately, and the map takes the estimate as it stands.

Step six: how does one figure become two that add back to it?

Apply the benchmark's MODIFIED duration to the same move on the same base. The result is the BENCHMARK PART. Apply the difference from step three to the same move on the same base. The second result is the ACTIVE PART. Then add the two and check that they come back to the figure from step five.

The check looks like decoration and is not. The check cannot fail on correct inputs, and that is exactly what makes a failure informative: the only way these three figures come apart in practice is that somebody changed a base or a move between two lines of a spreadsheet. The addition is what proves the two parts are two views of one calculation rather than two separate calculations that happen to sit near each other.

Step six does a second thing as well, and that second thing is why the split earns a step of its own. The split shows a reader how much of the exposure arrived with the benchmark and how much arrived with somebody's decision. The two quantities are usually nothing like the same size, and a reader who has never seen them separated tends to assume they are closer than they are.

Step seven: what happens to a row that cannot be filled?

Attempt every remaining row. Where the inputs exist, fill the cell and label it. Where they do not, draw the cell empty and write the reason inside it, in the space where the figure would have gone. There is no third path. A row that is neither filled nor drawn empty has been deleted, and a deleted row is a map claiming to be complete.

The rows that remain are, awkwardly, the rows a reader most wants. The cost of a curve move of another shape. The holdings, split by industry. The borrowers, and how much of the base the largest of them accounts for. The money that has arrived or left, and when. Each of those is a fair question, and each gets attempted. Attempting a question is a different thing from answering it.

The reasons are not interchangeable, and writing the specific one inside each cell is what makes the blank readable. A cell reading nothing available says nothing. A cell saying that no split by industry is recorded here tells the next reader precisely which input would have to appear before that row could ever be filled, and that is a usable instruction rather than a shrug.

Step eight: what has the finished map settled, and what has it left open?

Write two short lists. The first names what the filled cells support. On any map built this way that comes to the cost of a stated move, in total and in the part attributable to positioning. The second names everything the empty cells leave open, spelt out rather than left for a reader to notice.

Then add one line that no step above produced, and none of them could have. The map says what a position costs under a stated move and says nothing at all about whether that position should be held. Somebody had already chosen to sit the portfolio longer than its benchmark, well before this sheet existed. The eight steps price that decision. The eight steps do not endorse it, do not argue for closing it, do not argue for widening it, and hold nothing that could support any of the three.

What do the eight steps produce when they run on one portfolio?

Here is the whole sequence run end to end on the portfolio this material records. Every figure below was written for teaching. Where each number lands, and what label it lands with, is the whole of the lesson.

Step one. The base is Rs 5,000 crore of market value, struck on the stated valuation date. Every figure below is a share of that amount or a rupee amount computed from it.

Step two. The benchmark carries a MODIFIED duration of 4.80. Recorded, not assessed.

Step three. The portfolio's MODIFIED duration is 5.20 and the benchmark's is 4.80, so the difference is 0.40. The direction goes into the cell alongside the size: the portfolio's figure is the higher of the two, so it is positioned longer.

Step four. The move is a rise of 100 basis points. Written the other way round, that comes to 1.00 percentage point. Its shape is PARALLEL. Both go on the sheet marked as an assumption.

Step five. A MODIFIED duration of 5.20 set against a rise of 1.00 percentage point works out at 5.20 per cent of the base, so the answer on a base of Rs 5,000 crore comes to Rs 260 crore. The figure goes into the cell labelled WHOLE EXPOSURE, with the move written beside it.

Step six. The benchmark's 4.80 against the same move on the same base works out at 4.80 per cent of the base, so Rs 240 crore goes into the cell labelled BENCHMARK PART. Applying the 0.40 difference to the same move on the same base works out at 0.40 per cent, so Rs 20 crore goes into the cell labelled ACTIVE PART. Adding the two, Rs 240 crore and Rs 20 crore come to Rs 260 crore, and the check closes.

Look at the proportion rather than at the two amounts. Almost the whole of what a parallel rise costs this portfolio arrived with the benchmark, before anybody chose anything. The part attributable to the positioning decision is one rupee in every thirteen of the total, and that ratio is the single most useful thing the split produces.

Step seven. Four rows come back empty, each carrying its own reason inside the cell. Step eight. The map establishes what a stated PARALLEL rise costs, in total and in the part attributable to positioning, and it establishes nothing else whatsoever. The grid below is the finished sheet.

Count the proportion on that sheet. Three cells answer something and four say why they cannot. The four blanks are what a reader would otherwise have quietly assumed somebody checked. A reader who expected the balance to run the other way has learned the more useful of the two lessons available here.

What rule runs through all eight of the steps?

Everything above rests on one rule, and it is worth stating separately because it is the rule people abandon under time pressure. Every step that touches a cell puts into it either a figure carrying a label or a reason carrying a border. A step never puts in a bare number and never leaves a gap. Step eight is the exception that proves the shape of it: step eight touches no cell, and what it produces is the reading of the cells the other seven filled or left.

A bare number is the fault this whole sequence exists to prevent. On the sheet above, two rupee figures sit thirteen times apart and look completely alike inside a cell. Strip the labels and nobody downstream can tell which of the two questions a figure was answering, and the sheet acquires an error that no arithmetic check will ever catch.

A gap is the worse of the two faults. The next reader fills it. An empty cell with a reason in it is closed: it says what is missing and what would have to appear for the row to be filled. An empty cell with nothing in it is an invitation, and somebody eventually accepts.

Who builds a map like this, and when does it get built?

The map is not an exercise. The map is the sheet somebody produces before a conversation, and it fixes which questions the conversation is allowed to treat as answered.

A lender's treasury desk builds it because the rate exposure sitting on the balance sheet has to be described to somebody who did not put it there. The description has to survive a challenge, and the fastest way to lose a challenge is to have quoted a figure that turns out to have been the chosen part when everybody read it as the whole. Step six and its labels exist for exactly that conversation.

An analyst covering a debt scheme builds it from the outside, as a way of reading somebody else's pack. Read that way, the empty rows are the interesting ones. The blanks say which questions the pack has not addressed, and that is often more informative than the figures it does carry. An analyst who can name the four missing rows can ask four specific questions instead of a vague one.

A household does a small version of it every time somebody asks how safe the savings are. The balance is knowable. Whether the money is locked up for three years is knowable. Whether the household could handle two months without income is knowable only if somebody has written down what two months of outgoings comes to, and most households have not. The discipline is identical at every scale: the question gets named, it gets answered where an answer exists, and where none does, the reason is written down rather than passed over.

One thing none of the three does with the sheet is decide anything from it alone. The map prices what exists. Deciding what to do about it is a separate conversation that starts where the map stops, uses inputs the map does not carry, and is covered separately.

Why is there no control to move here?

The single relationship a slider would show, the size of a stated PARALLEL move against what it costs, already sits under a control covered separately. A second control carrying the same idea would be decoration.

The subject here is the map itself, and the map is static. The map teaches an order and a recording habit, and neither of those becomes clearer when a number moves. The interaction is of a different kind: nine questions, several of which call for a prediction before the sheet reveals the figure. The gap between the prediction and what the split actually shows is what is worth carrying away.