The Fallen Angel: What Crossing the Line Actually Costs
A fallen angel is a bond assessed out of the higher of the two broad credit quality groupings after having sat inside it. Nothing in the contract moves on that day. The list of people permitted to hold it moves. Any widening in the spread afterwards therefore carries two charges at once, and one price will never reveal how they split.
What exactly is a fallen angel?
The name misleads nearly everybody who meets it, so start with the object itself. A fallen angelA bond assessed out of the higher of the two broad credit quality groupings, having previously been inside it. is not a kind of bond. A fallen angel is a bond that has made a move. The bond sat inside the higher of the two broad credit quality groupingsOne of the two broad divisions a rating scale is split into by a line drawn across it. that a rating scale is split into, and it has since been assessed outside that grouping. The move across the line is the entire definition, and every difficulty below grows out of it.
Now hold the bond still and audit what actually altered. Palash Cements Limited, an invented cement manufacturer, has issued a five year bond carrying a 9.10 per cent annual coupon on Rs 1,000.00/- of face, priced throughout under annual compounding with one discounting period a year. On the morning after a crossing that coupon is 9.10 per cent a year. The face amount is Rs 1,000.00/-. The maturity date is the date it always was. The contract behind the bond reads word for word as it read the previous evening, and the position a holder takes in the queue if payment stops is the position it was already in. A bond can move across a line without moving at all, and everything below follows from that.
The shape of that event is familiar without the vocabulary attached. A tenant has paid rent on the first of the month for eleven years. A housing society rewrites its rules and the tenant is now in a category that requires a co-signer. Nothing was missed that morning. The household earns what it earned last week. A category moved, and consequences the tenant never negotiated are now attached to it. The category move is the whole event, and noticing how little of it concerns the tenant is the skill worth having.
So the honest framing is a question about arithmetic rather than a question about drama. If nothing in the contract moved, and the price of the bond then moves, what was the price responding to? The rest of this guide is one long answer to that question.
A bond becomes a fallen angel. Which of its terms changed?
Where is the line, and why is it drawn here without labels?
The line is drawn on a rating scale. The scale, how many steps it carries, what each step is called and which steps sit on which side of the line are set by the rating agencies in their published method documents and by the Securities and Exchange Board of India (SEBI) at sebi.gov.in. All of it is revised from time to time, and the naming of each side belongs to those sources.
There is a harder reason than tidiness. A scale written out from memory is not merely stale on the day the scale is amended. Such a scale presents somebody else's current wording as settled knowledge, so it was wrong the moment it was typed. The scale below is drawn with blank rungs for that reason, and the reason is printed inside the drawing itself.
The same discipline reaches the borrower. Palash Cements Limited holds no credit rating, so no crossing happens to it. The arithmetic below stands around a crossing instead, and every sentence in it stays true whether or not any rating exists.
Why is the line drawn without either side of it being named?
What actually changes on the day of the crossing?
The answer is the set of people permitted to hold the bond. Many holders operate under mandates written against a credit assessment: the mandate names a grouping, and a bond assessed outside that grouping stops being a thing that holder may keep. Which categories of holder may hold which categories of debt, and how a credit assessment is used inside a rule about who may hold what, are set by SEBI at sebi.gov.in and by the Reserve Bank of India at rbi.org.in. Both move over time, and the mechanism runs without any particular rule being quoted at all.
The mechanism is four short steps, and every one of them can be checked. Fewer permitted holders means fewer people who are able to buy. Fewer able buyers means a seller has to accept a lower price to find one. The yield is solved out of the price rather than announced, so a lower price on payments that have not changed is a higher yield. And a higher yield measured against an unchanged five year government SPOT rate of 6.90 per cent a year is a wider credit spread. Not one step in that chain required the borrower to have done anything.
Instinct already trusts the everyday version, so take that first. Ten shops in one mall are allowed to stock a product. A rule arrives and six of them are no longer allowed to. The product is identical, the manufacturer is identical, the shelf life is identical. The four remaining shops will not pay the wholesaler what the ten would have paid, and the wholesaler who needs to clear stock this week is the first person to find that out. Nobody's opinion of the product changed. The room simply emptied.
Notice the word missing from all of that. Nobody counted anyone. Nobody has counted how many holders were eligibleA holder is eligible for a security when its own mandate permits it to hold that kind of debt. Eligibility is a property of the holder, not of the borrower. before a crossing or after one, so the shrinking set of buyers is a mechanism rather than a quantity. A mechanism yields a direction, and a direction is enough to make the arithmetic that follows uncomfortable in the right way.
Fewer holders are permitted to hold the bond after the crossing. Why would that show up in the yield?
What if the spread stood at 320 basis points instead?
Care is needed at this point. No spread on this bond has ever been recorded moving, no second issuer stands beside it for comparison and no series of past spreads runs behind it. So the behaviour of a price on a crossing cannot be stated. The one move left is to supposeA supposition is a figure set out on its own face in order to work an argument. Nothing in the record supports it, and it is labelled at every use..
Suppose then, purely for the sake of the argument, that the spread on Palash Cements Limited's five year bond stood at 320 basis pointsOne hundredth of a percentage point, so 3.20 percentage points is 320 basis points and 1.00 percentage point is 100 basis points. rather than the 220 basis points used throughout the rest of this sequence. In the other unit that is 3.20 percentage points rather than 2.20 percentage points. The distance between them is a widening of 100 basis points, or 1.00 percentage point. The 320 basis point figure is a supposition, labelled as one at every use below, and no crossing is claimed to produce it.
| s | the recorded credit spread, in percentage points a year |
| y | what Palash Cements Limited promises, 9.10 per cent a year, annual compounding |
| g | the five year government SPOT rate, 6.90 per cent a year on the invented curve |
The supposition now sits on top of that recorded figure rather than in place of it, so the reader can always see which part came from the record and which part was supposed.
| s* | the supposed spread, in percentage points a year |
| s | the recorded spread of 2.20 percentage points a year, which is 220 basis points |
| w | the supposed widening of 1.00 percentage point, which is 100 basis points |
The supposition rests on the recorded 2.20 percentage points, so recall what that figure already settled. At an assumed recovery of 40 per cent of the amount owed, the loss given defaultThe share of the amount owed that is not recovered when a borrower stops paying. Loss given default is one hundred per cent less the recovery rate on the same base. is 0.60 of the amount owed, and 2.20 divided by 0.60 is 3.6667 per cent a year. Multiply straight back and 3.6667 times 0.60 returns 2.2000 percentage points. The arithmetic closes, and it closed long before any crossing was supposed.
The spread is supposed to stand at 320 basis points instead of 220. Before the arithmetic below, how many defensible implied annual default rates does that support?
How can one supposed spread support two different answers?
The supposed 320 basis points can be read the way the arithmetic invites. Every basis point of it is payment for default. Dividing 3.20 percentage points by a loss given default of 0.60 gives 5.3333 per cent a year. Checked backwards, 5.3333 times 0.60 returns 3.2000 percentage points, the supposed spread exactly. Nothing whatever is wrong with any step of that.
| pd | the implied annual default rate, per cent a year, on the exposure as base |
| s* | the supposed spread of 3.20 percentage points a year |
| L | the loss given default of 0.60 of the amount owed, from an assumed 40 per cent recovery |
Now read the identical number the other way. Suppose 100 of the 320 basis points is buying a smaller set of permitted buyers rather than paying for default. The mechanism set out above would produce precisely that. The credit part is then 2.20 percentage points, or 220 basis points, and 2.20 divided by 0.60 is 3.6667 per cent a year. Check that backwards too: 3.6667 times 0.60 returns 2.2000 percentage points. The second reading closes as well. And 3.6667 per cent a year is exactly what this borrower's recorded spread implied before any crossing was supposed. On the second reading the arithmetic reports no change in the borrower at all.
| a | the part attributed to something other than default, in percentage points a year, here 1.00, which is 100 basis points |
| s* | the supposed spread of 3.20 percentage points a year, unchanged |
| L | the same loss given default of 0.60 of the amount owed |
Set the two answers beside each other and look at the distance between them. 5.3333 per cent a year against 3.6667 per cent a year is a gap of 1.6667 percentage points a year. The same supposed price, the same assumed recovery, the same division performed correctly twice, and two statements about how often this borrower fails that are nowhere near one another. The gap was not produced by any disagreement about arithmetic. The gap was produced entirely by a disagreement about what the extra 100 basis points is buying.
| Δpd | the distance between the two implied annual default rates, in percentage points a year |
| 5.3333 | reading one, the whole supposed spread treated as credit, per cent a year |
| 3.6667 | reading two, with 100 basis points attributed elsewhere, per cent a year |
Both readings check back correctly against the supposed spread. What does that establish about which one is right?
Move the attribution. Watch the bar keep its length.
One control, and it moves nothing about the borrower and nothing about the price. The control moves only how many of the supposed 320 basis points are assumed to pay for a smaller set of buyers rather than for default. The bar below is the supposed spread and it never changes length at any position. The price stayed exactly where it was while the answer above it travelled a long way.
The control stops at 100 for a reason worth reading. Beyond that the credit part would fall below the 220 basis points this bond paid before any crossing was supposed. Falling below it would claim the borrower became safer on the day it was assessed out of a grouping, and no step in the arithmetic supports that.
With 0 basis points of the supposed 320 basis point spread attributed to something other than default, the credit part is 3.20 percentage points and the implied annual default rate is 5.3333 per cent a year, and the price has not moved at any position of this control.
The worked positions in plain words, so both survive with the drawing stripped out. Palash Cements Limited's five year bond carries a 9.10 per cent annual coupon on Rs 1,000.00/- of face under annual compounding. With 0 basis points attributed, the credit part is the whole supposed spread of 3.20 percentage points, the implied annual default rate is 5.3333 per cent a year and the check returns 3.2000 percentage points. With the full 100 basis points attributed, the credit part is 2.20 percentage points, the implied annual default rate is 3.6667 per cent a year and the check returns 2.2000 percentage points.
Slide that attribution across its whole range and the implied rate does not jump about. The implied rate walks down a straight line. Attribute nothing and the reading is 5.3333 per cent a year. Attribute 50 basis points and the credit part is 2.70 percentage points, so the reading is 4.5000 per cent a year. Attribute the full 100 basis points and the reading is 3.6667 per cent a year. The price never moved once, so the straight line says the answer is a function of the assumption rather than of the price.
Why can the price not settle between the two readings?
Because a price is one number. A price is the outcome of everything every participant brought to it, and it arrives with no breakdown attached. A price does not carry a receipt itemising how much of itself was paid for the chance of not being repaid and how much was paid for the difficulty of finding a buyer later. No quoted price separates those two parts, so both readings stand and neither can be chosen over the other.
The absence is the argument rather than an apology for it, so it is worth stating precisely. There is no second issuer to compare Palash Cements Limited against. There is no series of past spreads to see how this one behaved. There is no count of permitted holders before or after. There is no recovery study and no default study of any kind. Picking a reading anyway would be picking it out of preference, and preference dressed up as arithmetic is the most expensive mistake available in this whole sequence.
Three limits travel with any implied default rate, here and everywhere in this sequence, and all three of them are load bearing here rather than decorative.
First, the 40 per cent recovery is an assumption, and no recovery study stands behind it. Move it and the answer moves with it. Hold the recorded 220 basis point spread perfectly still and a 30 per cent assumed recovery implies 3.1429 per cent a year, 40 per cent implies 3.6667 per cent a year, 50 per cent implies 4.4000 per cent a year and 70 per cent implies 7.3333 per cent a year. One price, four answers, and the assumption did every bit of that work.
| Assumed recovery, of the amount owed | Loss given default | Spread held still | Implied annual default rate, per cent a year |
|---|---|---|---|
| 30 per cent | 0.70 | 2.20 points | 3.1429 |
| 40 per cent | 0.60 | 2.20 points | 3.6667 |
| 50 per cent | 0.50 | 2.20 points | 4.4000 |
| 70 per cent | 0.30 | 2.20 points | 7.3333 |
Second, the whole spread is being treated as compensation for credit. In a real market some part of a spread pays for liquidityThe ease of selling something without moving its price. Liquidity is a separate matter from whether the borrower pays, and the two are not distinguished inside a single quoted price. rather than for the chance of not being repaid, and that part carries no separate label in the price. Split 0.40 percentage points off the recorded 2.20 as the price of a difficulty in selling and 1.80 percentage points of credit remain, so the implied rate falls from 3.6667 to 3.0000 per cent a year. A single quoted price offers no way to separate the two.
Third, an impliedSolved backwards out of a price under a stated assumption, rather than counted from what happened or forecast from what might. default rate is what the price says. It is not a forecast and it is not a counted frequency of anything. Nobody counted a default to produce 3.6667 per cent a year, and nobody counted one to produce 5.3333 per cent a year either. Both figures were solved backwards out of one supposed spread and one assumed recovery, and reading either of them as the probability that Palash Cements Limited fails has misread the arithmetic that produced it.
Which single measurement would settle the choice between the two readings?
Which way does the error run if the whole spread is read as credit?
The error has a direction, and the direction is worth more than either number. The implied rate is a credit part divided by a loss given default that does not move. Every basis point that in truth pays for something else, counted as credit, enlarges the numerator and does nothing else whatever. So the error can run only one way: the rate comes out too high, never too low, and never in a direction that depends on the circumstances.
A rate computed from a whole widened spread is therefore a ceiling rather than an estimate. The ceiling is the largest implied annual default rate the supposed price will support under the stated recovery assumption, and every honest reading of the same price sits at or below it. A ceiling is a much weaker claim than an estimate, and it is the strongest claim the arithmetic is entitled to make.
If some of a spread pays for something other than default and all of it is read as credit, is the implied default rate too high or too low?
The error that gets made, and what it costs
A reader holding the supposed spread, the recovery assumption and one division does exactly what they were taught to do. 320 basis points at the top of a working note. 3.20 divided by 0.60 written underneath it. 5.3333 per cent a year circled at the bottom. Beside the circle, the words implied default rate, with default rate underlined and implied left unmarked. The division is right. Not one figure in that note is a slip.
The mistake is in the quantity the division was applied to. The whole supposed spread was fed in as though every basis point of it were payment for default, and nothing in the price ever confirmed that it was. The identical supposed price under a different and equally unsupported attribution gives 3.6667 per cent a year, so the note is carrying an answer standing 1.6667 percentage points a year away from another answer with exactly the same claim to being right.
The error costs a second thing too, quieter and worse. The reader walks away believing something happened to the borrower on a day when nothing did. A correct calculation carried the belief there, so it is hard to catch and easy to repeat to a colleague.
The repair is one line. Before dividing a spread, write down what the spread is assumed to contain, and note that nothing in the price confirmed it.
What may honestly be concluded from a wider spread?
Less than either reading claims on its own, and it is still worth having. Four statements survive the crossing intact.
A reader may conclude that the spread is wider, if it is. A wider spread is an observation about a price, and it needs no theory to support it. Because the set of permitted holders changed on the day and nothing on the bond did, a reader may conclude that at least part of the widening may be buying something other than protection against not being repaid. The direction of the error is fixed and known, so a reader may conclude that an implied annual default rate computed from the whole widened spread is an upper reading rather than an estimate. And a reader may conclude that going any further needs a measurement the record does not hold, and one worth refusing to invent for convenience.
The one sentence everybody reaches for first is exactly what may not be concluded. The borrower cannot be said to have become more likely to fail on the day of the crossing, and the arithmetic supports a reading in which the implied rate did not move at all. Nor may 5.3333 per cent a year be treated as the probability of anything. And a back check tests the division and never tests the attribution the division was run on, so it is no evidence for either side.
How does anybody actually use this?
A lending desk quoting five year money to a borrower sitting near a line uses it as a warning about its own quote. The desk has to produce one number, and that number will contain both things whether or not anybody separates them. A committee shown only the total will read all of it as a view about the borrower. So the working file carries the credit part and the attribution as two lines rather than one, and the quote is defended in front of that committee by naming both.
An analyst reading somebody else's price uses it as a writing rule. Both readings go into the note, the attribution used is stated in the same sentence as the answer, and the recovery assumed is stated beside it, so the reader can move either input and watch the answer move. A note that prints one implied annual default rate without printing the attribution it rests on has told its reader something the price did not say.
A holder whose mandate is written against a credit assessment has the sharpest use of all, and it is not the question most people expect. The question is not whether the borrower got worse. Whether the holding is still one the mandate permits is the question, and it is about a rule rather than about a company, so it routes to SEBI at sebi.gov.in and to the Reserve Bank of India at rbi.org.in rather than to any arithmetic above.
And a household holding a corporate deposit or a bond gets the plainest use of the four. A quoted rate is payment for at least two separate things, and when it changes the useful first question is which of the two moved. Most people never ask it. A wider rate therefore reads as bad news about the borrower when it may be news about the room instead.
A colleague says the crossing proves the borrower became more likely to fail. What is the strongest reply available?
Where the rules behind all of this actually live
Every arithmetic step above is free of any rule set except the compounding convention. The convention is annual throughout, and the sums state it because they cannot be reproduced without it. The rows below are the rule-set items this guide touches. Each of them is set elsewhere and each is revised.
- The rating scale a credit assessment is expressed on, and what each step of it means. SEBI, sebi.gov.in.
- What a rating agency must disclose when it changes an assessment it has already given. SEBI, sebi.gov.in.
- Which categories of holder may hold which categories of debt. SEBI, sebi.gov.in.
- How a credit assessment is used inside a rule about who may hold what. SEBI, sebi.gov.in.
- The manner in which a corporate bond is quoted and dealt in, and by whom. SEBI, sebi.gov.in.
- The capital treatment that applies to holding a credit exposure. The Reserve Bank of India, rbi.org.in.
- The valuation norm that decides the price at which a credit holding is carried. The Reserve Bank of India, rbi.org.in.
References
| Source | Named for | Where |
|---|---|---|
| SEBI | The scale a credit assessment is expressed on and what each step means, what a rating agency must disclose when it changes an assessment, how an assessment is used inside a rule about who may hold what, which categories of holder may hold which categories of debt, and the manner in which a corporate bond is quoted and dealt in | sebi.gov.in |
| The Reserve Bank of India | The capital treatment of a credit exposure and the valuation norm that decides the carrying price of a credit holding | rbi.org.in |
Palash Cements Limited and the government spot curve are invented.
Educational material. Not advice on any investment, tax, budget or market position.