Stating Limitations in Fixed Income Research

A limitation names one specific thing the analysis could not settle, at the place where the figure it bears on is printed, tightly enough that a reader can pick out the sentence to stop trusting. Three different sentences get filed under that heading, and only this one weakens the note. Cost is the test.

Here is the reason the sentence has to exist at all. Every analysis stands on inputs that were never obtained, assumptions that quietly decide the answer, and methods that own a slice of it. None of those three leaves any mark in the output. A price is a price, a ratio is a ratio, and a rate carries no memory of what was assumed to produce it. So a reader has exactly one route to any of it, and that route is a sentence somebody decided to write down.

The last of eight guides on writing fixed income research turns the practice of the previous seven back on the analysis itself. Everything below runs on invented material: an invented SPOT curve, an invented ten year bullet, an invented zero coupon claim built beside it, and Palash Cements Limited, an invented issuer. Every rate here compounds once a year. A defect found in the record behind this run is printed below in full, with the arithmetic that settled it. A worked correction teaches more than any description of one.

What exactly is a limitation, and what gets mistaken for one?

Three species of sentence live under that heading and they do three unrelated jobs. A limitation names a specific thing this analysis could not do. A disclaimer names what the writer will not be answerable for. A hedge softens a claim until nothing could contradict it. All three sound careful. Only one of them says anything about the work.

The separator is cost. A limitation makes the note weaker and more useful. The other two make it safer and no more useful. The separation is not a moral point, it is a mechanical one. Writing down that a figure could not be obtained hands the reader a reason to trust the headline less. Nobody writes that sentence unless they mean it. A disclaimer costs nothing to write and a hedge costs nothing to write. Costing nothing is exactly why so many of them get written.

Take a scooter changing hands on a Sunday morning. The seller who says the odometer was replaced two years ago and the reading starts from that day has cost himself something: the buyer will now pay less. The seller who says that no responsibility is accepted for any defect not disclosed has cost himself nothing and said nothing, and the seller who says that used scooters can always throw up surprises has said what the buyer already knew on walking over. Only the first sentence changes what the buyer would pay. Only the first one is a limitation.

Fixed income research does the same thing in more expensive language. Take a note working on the 220 basis point spread that Palash Cements Limited pays. Say in it that the whole of that spread has been treated as the cost of credit, when some part of it is the cost of thin dealing, and the headline figure in that note has just been weakened by its own writer. A note stating that fixed income involves risks has not.

Which four kinds of limitation are there, and why does the fourth decide the rest?

Sorting them by kind is not tidiness. Each kind gets written differently, sits in a different place, and fails in a different way when it is written badly. Three of the four are planned for. The fourth is the one that arrives without warning, and it is the one that decides whether a reader believes anything in the other three.

Kind one is a missing input. Something the analysis needed and could not get, named in the place the number would have gone. No observation of dealing exists anywhere in this material, so what could actually be sold cannot be stated, and a note pretending otherwise would be inventing a market. Kind two is an assumption that decides the answer. Not any assumption, but one where the answer moves substantially when the assumption moves. Kind three is a method that decides a slice of the answer. Writing it down requires the writer to have considered a defensible alternative and worked out what that alternative would have given, and the work is why kind three is the least written of the three.

Kind four is an error found after publication. Everybody expects to avoid the fourth kind, so nobody keeps a template for it, and it is the one that decides whether a reader believes the other three. The pairing is uncomfortable and it is worth sitting with. A writer who has never once printed a correction is either flawless or is not looking, and a reader cannot tell which from the outside. A writer who has printed one, in full, with the arithmetic that settled it, has given the reader something the other three kinds cannot: evidence that the checking is real.

Where in a note does a limitation belong?

Beside the figure it bears on. Not at the end, not in an appendix, and not on a slide kept back for questions. A figure travels and its qualifications stay behind, unless the qualification is physically attached to the figure. Physical attachment is the whole rule, and everything else about placement follows from it.

A number moves through an organisation in a particular way. Somebody reads the note. One sentence is what the person on the other end has time for, so one sentence is what gets copied into an internal message. The message gets forwarded. By the third hop the figure is being discussed by people who have never opened the note, and what travelled with it was the paragraph it sat in, not the ninth sheet at the back. A qualification printed on that ninth sheet is gone, and it went precisely at the moment the figure started being used.

The everyday version is a price card in a shop window. Rs 1,299/- in large figures on the front, and on the back of the card, in a place nobody turns it over to read, the words that the price applies to the smaller size only. Nobody is lying. The information is present in the document. The price travelled and the condition did not, so every single customer walks in expecting to pay Rs 1,299/- for the larger one.

Attaching the qualification to the figure is the same discipline as naming an assumption where it is used, covered earlier in this run, and it holds for the same reason. So the section at the end is not forbidden. The closing section may repeat what the paragraphs already said, and repeating it is often sensible for a reader skimming backwards. The closing section may not be the only place the limitation appears. A limitation that appears only at the end has been written for a reader nobody has ever met: somebody who reads a research note from front to back and quotes nothing.

One figure does the travelling in the drawing below, and it reads 3.67 per cent a year. Palash Cements Limited pays a spread of 220 basis points. Set the recovery assumption at 40 per cent, run the credit arithmetic backwards, and that rate is where it lands. The next block works it out properly and says what it is not. Here it is only a figure being carried about by people short of time.

How can a limitation that works be told apart from one that only sounds honest?

One question does it. Hand a reader this limitation, hand them nothing else, and ask them to point at the sentence in the note they should now stop believing. Can they? If the answer is yes, the sentence is doing its job. If the answer is no, the sentence is decoration, however sincerely it was meant, and sincerity is not the thing being tested here.

Two sentences, side by side. The first says that the analysis is subject to uncertainty. Reading it twice adds nothing that was not already known before the note was opened; the reader brought that in. The other one goes after a specific figure: 3.67 per cent a year, an implied default rate solved out of a recovery assumed at 40 per cent, with nothing in this material standing behind that assumption, so anybody who prefers 60 per cent should be reading 5.5000 per cent. One of those two sentences names which figure to distrust and supplies its replacement. Naming the figure and supplying the replacement is the entire job.

Notice what makes the second one work. Neither length nor humility has anything to do with it. The sentence names the input. It names the output that input decides. It gives the output under a different input. A reader can now do the substitution themselves without asking anybody anything, and that is what a limitation is for.

The substitution written out. Palash Cements Limited pays 220 basis points over the government rate, or 2.20 percentage points said the other way. How a spread becomes an implied default rate is covered under credit analysis and is not re-run here. The recovery assumptionA guess at how much of the money still outstanding a lender ends up collecting once a borrower stops paying. Whoever does the work picks the number, and nothing in this material pins it down. is chosen rather than observed, and choosing differently moves the answer a long way.

Three rows, one spread, and the answer doubles across them. A note that prints the middle row and calls it the default rate has printed a figure that belongs to somebody's guess. A note that prints the middle row and adds the outer two has printed a figure and shown how far it can be pushed. The second note is weaker and far more useful. Weaker and more useful is the pattern this whole guide keeps returning to.

And there is a second discipline hiding in that middle row. The unrounded reading is 3.6667 per cent and the record prints 3.67. Multiplying the printed 3.67 back by the 60 per cent that is lost on a failure lands on 2.2020 points rather than the 2.20 the working started from. Nothing is wrong. A display figure was carried back in as an input. The mistake is a different one, and it is worth naming wherever a reader is invited to check the arithmetic.

Now the demonstration that matters most. These three sentences are the best worked limitations this material contains. The three are not written here for the first time; every place that uses that 3.67 per cent carries them. Here they stand as specimens.

What does it look like when a printed figure cannot be rebuilt from the page it sits on?

The case that follows was found in the record behind this run, and that is the right way round for a guide about admitting things. The invented zero coupon claim used throughout this material is built so that the time to its single payment equals the invented ten year bullet's MACAULAY duration. The construction is the whole point of the pair: it forces the two instruments to share a MODIFIED duration of 6.5613 while behaving differently, and that behaviour is worked out earlier in this run. The construction also produced the defect.

The bullet's MACAULAY duration is 7.119062643353 years. Written to four places for printing, that is 7.1191 years. An earlier version of the record printed the maturity as 7.1191 years and the price as Rs 559.47/-. Both figures look ordinary. Set beside each other, they cannot both be right, and here is why.

Sit with the bottom row, because it carries something sharper than the defect itself. The two prices are Rs 0.0017/- apart. About a sixth of a paisa is nothing. Printed to the paise they are Rs 0.01/- apart. A whole paisa is a visible discrepancy, and a checking reader will chase it. The gap grew in the printing. Rounding does not merely lose precision, it can manufacture a difference that the underlying figures do not contain.

The reason is where the turnover point falls. Two place rounding switches at Rs 559.465/-, and that boundary sits between the two prices: Rs 559.4640/- is below it and Rs 559.4657/- is above it. Two figures 0.17 of a paisa apart end up on opposite sides of a line, and that line is not in the arithmetic anywhere. The line is in the display convention.

Now the part that makes this the right case for this guide, and it has nothing to do with bond arithmetic. The reader here is careful. The reader sees a maturity of 7.1191 years and a price of Rs 559.47/-, takes out the arithmetic, discounts Rs 1,000/- back over 7.1191 years at 8.50 per cent, and gets Rs 559.46/-. One paisa short. Nine times out of ten, that reader concludes that they made the mistake. The reader checks the exponent. Then the compounding. Then whether the material meant something by a convention they have not met. And the whole time the material was carrying a rounding it never mentioned.

Sending a careful reader hunting for a mistake of their own is the worst thing a teaching record can do, and notice that nobody was careless and no figure was wrong. Rs 559.47/- is a correct price for the instrument as constructed. Rs 559.46/- is a correct price for the instrument as described in print. The defect lives in the space between the two, and the only thing that closes it is a sentence.

Which gives a rule that runs through the whole of this material. Whatever a reader cannot put back together out of the figures printed alongside it should not have gone into print in the first place. Where the significant figuresHow many digits of a number are actually carried through a calculation, as opposed to how many are shown. A figure shown to four places may have been worked from twelve, and the two give different answers. shown are fewer than the ones used, either show more of them or say plainly which version produced what.

How is the correction written, and how was this one settled?

A correction has four parts. Writing three of them is honest about one page. Writing the fourth fixes a record.

The third row is the one worth copying, so here is how it worked. The price is exactly what is in dispute, so neither of the two disagreeing figures could settle the argument on its own. A third figure was needed, and the record happened to carry one: the convexity of the zero coupon claim. Convexity depends only on the time to the payment and the yield.

The record prints 49.0986, which matches the unrounded maturity and not the four decimal one, so the record was built on the unrounded figure and the printed maturity is the figure that was wrong. Look at the last column before moving on. At two places the two convexities are 49.10 and 49.10, identical, and the evidence disappears entirely. The check worked only because somebody had printed four places. A record kept to two would have had no way to settle its own dispute.

The argument for carrying more than one figure beside a result stands here in its strongest form: a second figure lets a reader check, and a third figure lets a reader adjudicate. A record carrying one number cannot be audited by anybody, including whoever wrote it.

What did the same correction reach, and what was the second one?

The fourth part is where a correction stops being a courtesy and starts being maintenance. Fixing the one guide where the defect was noticed does nothing for the six others carrying the same pair, and those six are where most readers will meet it. So the correction became a rule: every guide carries that maturity as the bullet's MACAULAY duration rather than as a rounded decimal, and wherever the four decimal figure is shown, Rs 559.46/- is printed beside Rs 559.47/- with the rounding that gave each. One of them is this guide.

Three separate people found this independently and not one of them accepted the printed figure. The behaviour is worth naming. Producing it is the reason the whole practice exists. Scepticism as an attitude is cheap. What counts is the specific habit of recomputing a printed number and refusing to let the mismatch go.

The second correction runs the other way, and no arithmetic was wrong in it at all. Running the other way is what makes it the more instructive of the two. A difference in MODIFIED duration once carried the word years in this material. Years belongs to MACAULAY duration, the one measuring time. A MODIFIED duration measures sensitivity instead: how far the price moves when the yield moves. Subtract one sensitivity from another and sensitivity is what comes out. The figure concerned was a difference of 0.40. A difference of 0.40 reads as a price move of roughly 0.40 per cent for each 100 basis points, and calling it 0.40 years long names the wrong quantity entirely.

Every number tied. Every sum reconciled. One word was wrong, and the word had been attached because it made a checking routine work smoothly. A wrong unit adopted because it made a check pass is still a wrong unit, and it is harder to find than an arithmetic slip precisely because everything around it reconciles. The provenanceWhere a figure came from and what was done to it along the way: which source, which convention, how many digits were carried and what was rounded when. A number without it cannot be checked, only believed. of that word was a convenience, and nobody had asked what quantity it was naming.

A note printing an erratumA published statement that something already issued was wrong, printed so that whoever holds the earlier version can find out. It sits with the original rather than replacing it quietly. of that second kind gets a particular kind of reader reaction, and it is not the one writers fear. Readers do not conclude that the writer is careless. Readers conclude that somebody is reading their own work with the same suspicion they bring to everybody else's, and that is exactly the signal a correction is supposed to send.

Which absences are written out here, and why print a list of them at all?

Ten things are missing from the material behind this run, and all ten are named below. A body of work whose own gaps are catalogued can be checked by anybody, and a body of work whose gaps are discoverable only by trying to reproduce a figure cannot be checked by anybody at all. Checkability is the entire argument for printing the list, and it is the same argument as putting a limitation beside its figure, moved up one level.

The difference is the one between a shop that keeps a list of what it does not stock and a shop where a customer finds out by asking. Both shops have the same stock. One of them can be checked from the doorway.

Every guide that reaches one of those ten names it in the place the number would have gone, rather than working around it quietly. The reason for the gap goes into the blank space itself, and nothing plausible gets invented to fill it. The discipline is a small one, and it is what keeps the whole set honest. The alternative is a guide that looks complete and is not.

What does a limitation never look like?

Four shapes turn up constantly and none of them is doing the work. A limitation never says that markets are uncertain. The reader knew that before they arrived. A limitation never says that past performance is no guide. A statement true everywhere carries no information about this note. And a limitation never says that the analysis depends on assumptions without naming one of them. The naming is the whole content of the sentence.

And a limitation never appears in identical words in every note the writer produces. The moment a limitations section becomes boilerplateWording carried unchanged from one document to the next until nobody looks at it. It was written once, for a reason, and then stopped being read because it stopped being different., the reader stops reading that section, and the one genuine limitation inside it becomes invisible. Repetition does not make a limitation weaker, it makes it unread, and an unread limitation and an unwritten one land in exactly the same place.

The everyday version is the safety announcement at the start of a journey taken forty times before. Every word of it is accurate and important. Nobody in the vehicle is listening, and if one sentence were changed today to say something that mattered specifically to today, nobody would catch it either. Unchanging wording costs exactly that, and the cost is paid by the writer rather than the reader.

Who actually reads the limitations, and what do they do with them?

Four different people read that section and each one is looking for something different. Knowing which four is worth having before the writing starts.

A lender reads it for the kind two limitations. Which recovery was assumed, and does the answer survive the opposite assumption? A lender who cannot find the assumed recovery in a credit note goes looking for it, and if the note does not name one, that lender now knows the writer either did not think about it or did not want to say. Both readings are bad and one of them is worse.

An analyst reading somebody else's note is looking for kind three, the method. Not because methods are interesting, but because a method deciding a large share of an answer is the fastest place to find a disagreement worth having. If the note does not say how it was worked, that analyst has to rebuild it from scratch to find out. Usually the note gets put down instead.

Somebody deciding whether to trust the writer at all reads for kind four. Has this person ever printed a correction? What did it look like? A record keepingThe duty on whoever publishes research to keep what was sent out, to whom, and when, for a stated period afterwards. What the period is and what must be kept is set by the authority and revised, so it is not written out here. duty sits behind that, and the duty belongs to the authority to state, but the reader's instinct runs ahead of any requirement: a writer who has never visibly corrected anything is a writer whose checking cannot be inspected.

And the household version, the one most people actually live. Somebody choosing between two fixed deposits finds that one of the two documents says the rate shown applies only to deposits placed before a certain date and only to sums above a certain size. The document naming the conditions is less attractive and it is the one that can be acted on. The other one has given a rate and left the conditions to be discovered after the money has moved. Nothing about that is unusual and nothing about it is dishonest. The difference is simply between a document written to inform and a document written to be filed.

How much limitation is too much?

A note that limits every figure has made no claim, and a reader can do as little with it as with a note carrying no limitations at all. Both ends are unusable and neither end is dishonest. Unusable without being dishonest is what makes this a judgement rather than a rule.

The judgement runs on materiality. Would this limitation change what a reader takes away from the note? A missing figure that would move the answer belongs in. A missing figure that would not move it does not belong in. A single line saying so beats printing a hedge about it and beats leaving it out silently. A reader who notices the gap unaided will otherwise spend an hour wondering what it hides.

And the direction of the error goes in wherever the direction is known. One limitation saying the figure is probably too high is worth several saying the figure is uncertain. Direction is the most useful thing a writer can hand over and it is the thing writers most often keep back, usually because saying which way an error runs feels like a second claim. Direction is a second claim. It is also the only part a reader can use.

Materiality and direction are the whole of the judgement, and they are why this cannot be reduced to a template. The two ends of the scale are easy to describe and neither of them is where a usable note sits.

Limitations are not a quantity. Adding them one at a time would draw a neat hump, rising and then falling, and nothing about the judgement behaves that way. How the implied default rate moves as the recovery assumption moves is the one relationship in this subject that does vary smoothly, and it is covered separately, alongside the arithmetic that produces it.