Term Premium and Credit Spread: Two Parts of One Yield
A term premium is the extra rate a lender wants for tying money up for longer, over and above what they expect short rates to average. A credit spread is the extra rate a lender wants for lending to one borrower rather than to the government for the same period. Both sit inside one corporate yield, and only the credit spread can be computed from a pair of prices.
Palash Cements Limited, an invented issuer used throughout this sequence, borrows for five years at 9.10 per cent a year. Given that figure, what is the lender being paid for? The honest answer has at least three parts in it, and one of the three cannot be pulled out of any price at all. The part that cannot be pulled out is the term premium, and knowing why it cannot be is worth more than a figure invented to fill the gap.
The arithmetic is easier to feel than to read, so start with a household version. A neighbour asks to borrow money and offers to pay something for the trouble. Three things want settling. First, what money costs in general right now. The neighbour has nothing to do with that one. Second, how long the lender is being asked to be without it. Being without it for five years is a different inconvenience from being without it for six months, and that difference wants paying for. Third, whether this particular neighbour is likely to hand it back. The neighbour's habits settle that one, and the calendar has no part in it. Three separate questions, three separate charges, and one single number handed over at the end that has all three inside it.
The bond version is identical in structure and only differs in that two of the three charges arrive already welded together. The government borrows for five years too. The government is paid for the use of money and for the length of the commitment, and nobody charges it anything for the third, so its five year rate covers the first two of those three things. So subtracting the government rate from Palash Cements Limited's rate removes two charges in one movement and leaves the third standing on its own, clean.
The bundling is a wonderful piece of luck for the third charge and a disaster for the second. The subtraction that isolates the credit charge so beautifully is the same subtraction that leaves the time charge and the length charge stuck to each other, inside one number, with no seam anywhere on them. Term premiumThe extra rate a lender wants for committing money for longer, over and above the average of the short rates they expect over the same period. is the name for that second charge. Credit spreadA borrower's yield less the government SPOT rate for the same maturity, always quoted over something and for a stated number of years. is the name for the third. In practice one thing separates the two: whether a number can be put on either.
A bond, the source of a price, and the arithmetic that pulls a rate for one future stretch of time out of rates for two others are all covered separately, and are used here rather than re-derived.
What is a credit spread, and what is it paying for?
A credit spread is the extra rate a lender wants for lending to one particular borrower rather than to the government for the same length of time. Every word of that definition is doing work: one particular borrower, rather than the government, for the same length of time. Remove the last clause and the quantity stops meaning anything.
Here it is on the invented figures used throughout. Palash Cements Limited has issued a five year bond with a 9.10 per cent annual coupon on Rs 1,000.00/- of face amountThe amount printed on the bond that is repaid at maturity, and the base on which the coupon rate is applied., issued at par, so its price is Rs 1,000.00/- and its yield is 9.10 per cent a year. The invented government curve behind this sequence puts the five year SPOT rateThe rate for money placed today and returned at one stated future date, with nothing happening in between. at 6.90 per cent a year. Both figures are stated on an annual compoundingOne discounting period a year, so an amount is divided by one plus the annual rate once for each year until it arrives. convention, and a sum cannot be reproduced without knowing which convention was used.
| s | the credit spread, in percentage points a year, over the five year government SPOT rate, for five years |
| ycorp | Palash Cements Limited's yield, 9.10 per cent a year, solved out of a price of Rs 1,000.00/- on Rs 1,000.00/- of face, annual compounding |
| r5,spot | the five year government SPOT rate, 6.90 per cent a year, read off the invented curve, annual compounding |
Both sides of that subtraction are prices, so two people holding the same two prices reach the same answer and neither of them has an opinion in it anywhere. Nobody decided that 2.20 was a fair charge for lending to a cement maker. Nobody weighted anything. Two numbers came off two markets and one was taken away from the other. The only thing anybody can disagree with is one of the two inputs, and both of them are observable.
The absence of opinion rules out a great deal. The spread does not require a view about the borrower. The spread does not require a model. The subtraction asks nothing about the purpose of the money, how the last three years went, or what happens if the cement business turns. All of those matter enormously for whether 2.20 percentage points is enough, and none of them is needed to say that 2.20 percentage points is what it is. Measuring a thing and judging it are different operations, and only the first is done here.
The units rule, because this is where mixing them is commonest
A percentage point and a basis pointOne hundredth of a percentage point, so 1.10 percentage points is 110 basis points and 0.65 percentage points is 65 basis points. are the same quantity in two units. One basis point is one hundredth of a percentage point. So 2.20 percentage points is 220 basis points, 1.10 percentage points is 110 basis points, and 0.50 percentage points is 50 basis points. A spread is written in both units the first time it appears, and one figure in points never sits beside a neighbouring figure in basis points. A hundredfold error walks into a note that way and stays there.
And no spread is ever written as a bare number. A spread is always over something and always for a stated maturity. Two hundred and twenty basis points is not a fact until it is completed: 220 basis points over the five year government SPOT rate, for five years. Drop either half of that completion and the figure can be reproduced by nobody, checked by nobody, and compared with nothing.
Palash Cements Limited yields 9.10 per cent a year for five years, and the five year government SPOT rate is 6.90 per cent a year. Which of these states the spread the way the convention here requires?
What is a term premium, and what is it paying for?
A term premium pays a lender for tying money up for longer, over and above what they expect the short rate to average across the same period. The clause that makes that definition hard is not where a reader usually looks.
A term premium is defined against an expectation, and an expectation is not a price, so the difficult clause is over and above what they expect. The credit spread was defined against another rate that somebody was actually charging. The term premium is defined against a rate nobody is charging: an average of future short rates that has not happened yet, that exists only in the heads of lenders, and that changes as they change their minds. The difference between a price and an expectation decides everything else about these two quantities.
The abstraction is slippery and the everyday version is not. Take the household version again. A lender has money to place for two years. Two things are on offer. The money can go out for two years in one go at whatever the two year rate is. Or it can go out for one year, come back, and go out again for a second year at whatever the one year rate turns out to be by then. The second route has an unknown in it. Nobody today knows what the one year rate will be a year from now. If the two year route is taken anyway and it pays more than the rolling route is expected to average out at, then the extra is payment for not having the option to change course halfway. The extra is the term premium.
Notice what had to be supplied to say that sentence: a view about what the rolling route will average out at. Without it, the sentence has no arithmetic in it: there is a two year rate that can be seen, and a thing that cannot, and the premium is the first minus the second. Change the view about the second and the premium moves, with nothing at all having happened to any price. Two people looking at exactly the same curve, on exactly the same day, with the same figures in front of them, will get different term premiums if they hold different views about where short rates are heading. Neither of them is wrong on the arithmetic. The two people simply put different numbers into a slot the market never filled.
The disagreement is not a defect in the idea. A term premium is a real thing, and a lender really does want paying for commitment. The point is about what kind of quantity a term premium is: an expectationA view about what a rate will be at some future date, which is held by people rather than quoted by a market, and which therefore cannot be subtracted from a price to give an agreed answer. is an input somebody supplies, not an observation somebody records. A quantity defined against a supplied input inherits the disagreement in the input, and no amount of care in the subtraction takes that back out again.
There is a second thing worth noticing, and this one is easy to miss. The term premium and the credit spread are not merely different in how measurable they are. The two charges compensate for different exposures. The term premium is about time and about the option to change course, and it applies to the government exactly as much as it applies to a cement maker. The credit spread is about a particular borrower and applies to nobody else. A bond can carry a large term premium and no credit spread at all, and a long government bond on this invented curve is exactly that. Every bond is for some length of time and time is never free, so no bond can carry a credit spread and no term component.
Why is a term premium harder to pin down with a figure than a credit spread is?
How does one yield contain both at once?
Take Palash Cements Limited's 9.10 per cent a year apart into the things somebody is being paid for. There are three, and they stack.
The first is what short money is expected to cost on average across the five years. The expected average of short money is the base, and it belongs to nobody in particular. The second is a term premium for committing for five years rather than rolling over five times. The third is a credit spread for lending to Palash Cements Limited rather than to the government. Written as a stack, those three add to 9.10 per cent a year, and there is nothing else in the sum.
| ycorp | Palash Cements Limited's yield, 9.10 per cent a year, annual compounding, for five years |
| r̄short | the average short rate expected across the same five years, per cent a year, supplied by a view rather than read off a market |
| τ | the term premium for committing for five years, in percentage points a year |
| s | the credit spread over the five year government SPOT rate, in percentage points a year, for five years |
Now the move that makes the comparison necessary. The first two of those three terms are exactly what the five year government SPOT rate is. The government is the reference everything else is measured against, so it is paid for the use of money and for the length of the commitment and for nothing more. So the first two terms, added together, are 6.90 per cent a year.
| r5,spot | the five year government SPOT rate, 6.90 per cent a year, which is the sum of the first two terms and never their separate values |
| 6.90 | a single observed figure covering both the expected average of short rates and the term premium, with no seam between them |
| 2.20 | the credit spread that is left behind, in percentage points a year, which is 220 basis points over the five year government SPOT rate |
The subtraction removed two things in one movement and left one thing behind, so the credit spread comes out sharp and the term premium does not come out at all. This is the asymmetry the whole comparison turns on. Nobody was careless. The reference rate being subtracted is itself a bundle, and subtracting a bundle shows what was not in it without ever showing what was.
The bundle works like a shopping bill with three items on it where two of the items were rung up together under one heading. The total is known. The third item sits under a heading of its own and can be taken off, so its cost is known. The bill cannot give the split of the joint heading between the two items inside it. No amount of staring at the total will produce the split, and inventing one produces a number that looks like the others and is not like them at all.
Which parts of Palash Cements Limited's 9.10 per cent a year are bundled inside the 6.90 per cent five year government SPOT rate?
What is the slope of the curve, and is any of it a term premium?
Here is where most readers reach for a shortcut, and it is a good shortcut that goes to the wrong place. If a term premium is compensation for going long, and long rates are higher than short rates, then surely the gap between a long rate and a short rate is the term premium. The thought is a reasonable one, and working through it carefully teaches more than being told no.
The gap is a real quantity and costs one subtraction to compute, so start there. On the invented government SPOT curve used here, the ten year rate is 7.35 per cent a year and the two year rate is 6.25 per cent a year.
| r10,spot | the ten year government SPOT rate, 7.35 per cent a year, annual compounding |
| r2,spot | the two year government SPOT rate, 6.25 per cent a year, annual compounding |
| 1.10 | the resulting slope, in percentage points, which is 110 basis points across those eight years of extra maturity |
The slope figure is solid. The gap is a subtraction between two observed rates, exactly like the credit spread was, and anybody with the curve gets 1.10 percentage points and 110 basis points. There is no expectation anywhere in it. So far the shortcut is behaving well.
A rising curve is consistent with lenders expecting short rates to be higher later, with lenders wanting paying to commit, or with any mixture of the two, and the slope by itself cannot tell those cases apart. The shortcut fails at exactly that step. Suppose lenders expect short rates to average a great deal more over the next ten years than over the next two. Expectations alone would produce a rising curve with no term premium in it whatsoever. Suppose instead lenders expect short rates to be flat forever but want a solid extra for locking money away for ten years. A rising curve of the same shape comes out of that story too, made entirely of term premium. Both stories fit 1.10 percentage points perfectly. So does every mixture between them.
The slopeThe difference between two SPOT rates at two different maturities, which is observed by subtraction rather than inferred from any view. is therefore a total, not a component. The slope is the sum of two influences that cannot be seen separately, presented as one number that can be seen perfectly. A term premium is a share of that slope, and the share is exactly the thing the slope refuses to reveal. Recording the slope as the term premium is not a rounding error or an approximation that improves with care. Calling a slope a term premium is a category mistake: it takes a sum and reports it as one of its parts.
The street version is a rent again. Two identical shops, one on a one year lease and one on a ten year lease, and the ten year lease costs more per month. The difference in the rent is known exactly. The rent slip does not say whether the landlord charges more because he expects the street to get busier over ten years, or because he wants paying for giving up the freedom to relet. Writing the difference down and labelling it as payment for lost flexibility invents the label. The number was real; the label was somebody's contribution.
The ten year government SPOT rate is 7.35 per cent a year and the two year is 6.25 per cent a year. State the slope in both units, and then say how much of it is a term premium.
Is a FORWARD rate the term premium in disguise?
The second shortcut is more sophisticated than the first and fails for a related reason. If the curve mixes expectations and premium together, then perhaps the rate implied for a single future year, pulled out of the curve by arithmetic, is the market's expectation for that year, and the premium is what stands between it and the rate that can be locked in today.
The implied rate is exact and worth seeing derived, so derive it first. Money placed for two years at the two year government SPOT rate of 6.25 per cent a year grows by a factor of 1.0625 twice. Money placed for one year at the one year government SPOT rate of 5.90 per cent a year grows by a factor of 1.0590 once. Whatever rate would take the second outcome to the first across the second year is the one year rate, one year FORWARDThe rate for money placed at one future date and returned at a later one, obtained by arithmetic from two SPOT rates rather than from an opinion about the future..
| r2,spot | the two year government SPOT rate, 0.0625 as a decimal, annual compounding |
| r1,spot | the one year government SPOT rate, 0.0590 as a decimal, annual compounding |
| f1,1 | the one year rate, one year FORWARD, as a decimal, being the unknown in this line |
Only one quantity in that line is unknown, so divide both sides by the factor for the first year and take one away. Nothing was assumed and nothing was estimated: the step is allowed because both sides describe the same two years of growth on the same money, so they must be equal.
| 1.0625 | one plus the two year government SPOT rate, squared to give 1.12890625 of growth over two years |
| 1.0590 | one plus the one year government SPOT rate, being the growth already accounted for in the first year |
| 0.06601157 | the one year rate one year FORWARD as a decimal, which is 6.6012 per cent a year when rounded to four places |
The FORWARD rate is a price for a stretch of future time that can be taken today, not an opinion about that stretch of time, and it already carries whatever term premium sits in the curve it was pulled out of. Read the derivation again and see that no expectation entered it anywhere. Two observed SPOT rates went in and one rate came out. If the two year government SPOT rate contains a term premium for two years and the one year contains one for one year, then the difference between them contains the difference between those premiums, and that difference is precisely what the arithmetic extracted. Nobody removed anything. A FORWARD rate is a rearrangement of the curve, and a rearrangement cannot shed what the curve carried.
So the second shortcut collapses into the first. The gap between a FORWARD rate and a SPOT rate is another subtraction between two prices, and it inherits the same problem the slope had: it is a total made of expectation and premium, presented as a single figure. The FORWARD rate is a more elegant total. A total all the same.
The labelling rule, and why it is enforced in every sentence
Every rate written anywhere here carries the word SPOT or the word FORWARD, and this is where the reason the rule has to be that absolute becomes visible. A SPOT rate is the rate for money placed today and returned at one stated future date. A FORWARD rate is the rate for money placed at one future date and returned at a later one. Two completely different objects, and on the invented curve behind this sequence they land almost on top of each other.
The one year rate one year FORWARD is 6.6012 per cent a year. The three year government SPOT rate is 6.55 per cent a year. The distance between them is 0.051157 percentage points, or 5.1157 basis points, and that closeness is deliberate rather than accidental. Forwards land near spots on any smooth curve, and a reader who meets both figures without labels will merge them without ever noticing they did it. The two objects are separated by their labels and by nothing else at all.
| Object | What it prices | Rate a year |
|---|---|---|
| One year government SPOT rate | Money placed today, returned in one year | 5.90 |
| Two year government SPOT rate | Money placed today, returned in two years | 6.25 |
| Three year government SPOT rate | Money placed today, returned in three years | 6.55 |
| One year FORWARD rate, one year from now | Money placed in one year, returned in two | 6.6012 |
| Distance between the last two rows | Two unlike objects sitting close together | 5.1157 basis points |
The one year rate one year FORWARD works out at 6.6012 per cent a year. Is that what the market expects the one year SPOT rate to be a year from now?
With the whole government SPOT curve and Palash Cements Limited's yield in hand, which of the two compensations can be computed?
Why is the term premium row left empty?
Separating a term premium from the expected average of short rates needs a stated expected path of short rates. No market quotes such a path, and none is supplied here: no survey of what lenders expect, no measured series, no model output, no forecast of any kind. The absence is total, and the row below is empty rather than filled with something plausible for exactly that reason.
Filling it would have been the easiest thing in the world. A comfortable-looking figure in the box makes the table look complete and professional, every reader accepts it, and nobody checks. A figure in that box would have been supplied by somebody rather than produced by the market, would sit indistinguishable beside the 2.20 that two prices produced, and would travel onward in somebody's notes carrying the same authority as the figure that earned it.
Two rows, one filled and one deliberately empty, say more about what can be known here than any amount of prose about uncertainty. The empty box is a piece of teaching, not a gap in the work. The empty box tells a reader where the boundary of the evidence runs, and exactly what would have to arrive before the boundary moved.
Vagueness about what is missing would be its own kind of dishonesty, so be precise. The missing ingredient is not more curve. Adding maturities to the curve does not help. Every added point is another total made of the same two influences. The missing ingredient is not a longer history either. The past path of short rates is not the expected future path. The missing ingredient is a statement, from somewhere outside the price system, of what short rates are expected to average over each stretch of time. Given that, a term premium falls out immediately as a subtraction. Without it, no arrangement of the curve produces one.
A careful reader will already be forming a question, and the bundling settles that one too. If the term premium cannot be measured, is the concept useless? No, and the reason is worth holding. The concept names what kind of thing a rising curve is, and it names the claim being made when a long rate is said to be high because lenders want paying for commitment. The claim can be true. A true claim is still not a measurement, and any account that hands over a figure for it without an expected path has quietly slipped a view into a column of prices.
Predict before reading on. Somebody measures every borrower's spread against the same two year government SPOT rate, whatever maturity each borrower actually issued at. What will their ranking of borrowers actually be measuring?
What does it cost to confuse the two?
Everything above has been about what the two compensations are. Failing to hold the distinction costs something more concrete than muddled thinking. A borrower gets charged for something they had no part in, the arithmetic never complains, and the resulting figure looks exactly like a correct one.
Here is the confusion in its commonest form. Somebody has Palash Cements Limited's five year yield of 9.10 per cent a year. The government curve is open in front of them. The nearest government rate to hand is the two year government SPOT rate of 6.25 per cent a year. Somebody subtracts.
| 2.85 | the answer that comes out of the mismatched subtraction, in percentage points, which is 285 basis points |
| 2.20 | the credit spread that actually belongs to Palash Cements Limited, over the five year government SPOT rate, which is 220 basis points |
| 0.65 | the government curve slope between two years and five years, 6.90 less 6.25, which is 65 basis points and belongs to no borrower |
Sixty five basis points of that answer is the shape of the government curve, and a borrower has just been charged for it. Nothing about Palash Cements Limited changed between the correct subtraction and the mismatched one. The company did not become riskier. Its bond did not reprice. Somebody picked a different reference and the borrower's spread grew by 65 basis points, nearly a third of what the spread actually is.
Two decimal places in a rate can feel like housekeeping, so put the 65 basis points into rupees and it stops feeling that way. Suppose the mismatched figure is used the way spreads are actually used, to price something. Take the 285 basis points as though it were the credit spread and add it to the correct five year government SPOT rate of 6.90 per cent a year, giving 9.75 per cent a year. Discount Palash Cements Limited's five payments at that rate instead of at 9.10 per cent a year, on the same annual compounding convention, and see what happens to the price.
| P | the price today, in rupees, on Rs 1,000.00/- of face |
| C | the annual coupon, Rs 91.00/-, being 9.10 per cent of Rs 1,000.00/- of face |
| F | the face amount repaid at the end, Rs 1,000.00/- |
| y | the annual rate used to discount, as a decimal, on annual compounding |
| t | the year each amount arrives, running from one to five |
| Amount, on Rs 1,000.00/- of face | At 9.10 per cent | At 9.75 per cent |
|---|---|---|
| Coupon, year one | 83.409716 | 82.915718 |
| Coupon, year two | 76.452535 | 75.549629 |
| Coupon, year three | 70.075651 | 68.837931 |
| Coupon, year four | 64.230661 | 62.722488 |
| Coupon and face, year five | 705.831437 | 685.175943 |
| Price, in rupees | 1,000.000000 | 975.201707 |
Subtract the two totals and the mistake has a price: Rs 1,000.000000/- less Rs 975.201707/- is Rs 24.798293/- on every Rs 1,000.00/- of face. Twenty five rupees on a thousand is what 65 basis points of somebody else's curve slope is worth when it is carried through to a price. On a holding of Rs 1,00,00,000/- of face it is Rs 2,47,982.93/-, and nothing whatever about the borrower produced it. The whole difference was a choice of reference rate made in a moment of convenience.
Palash Cements Limited's five year yield is 9.10 per cent a year, and somebody subtracts the two year government SPOT rate of 6.25 per cent a year. Take the answer apart.
The error that gets made, and what it costs
A reader measures a corporate spread against whichever government rate is nearest to hand, and records the whole answer as credit. The subtraction is perfectly valid, so the arithmetic never complains. It just answers a different question from the one that was asked. Against the five year government SPOT rate of 6.90 per cent a year, Palash Cements Limited's spread is 220 basis points. Against the two year SPOT rate of 6.25 per cent a year it is 285 basis points. Both figures are correctly computed and only one of them is a credit spread.
An unusual reader does not make it. The ordinary reader does: somebody with a curve open at one maturity and a bond at another, and that describes most readers most of the time. Nobody sets out to mismatch maturities. The mismatch happens because the two figures arrive from different places, at different moments, and the maturity is the one attribute a rate does not carry in its own name unless somebody writes it down.
The cost is worse when the mistake is made consistently. Take a set of borrowers who issued at different maturities and measure every one of them against the same short government SPOT rate. The longer a borrower's maturity, the more government curve slope gets swept into its spread. The list will then rank the long borrowers as worse credits than the short ones, and it will do so in a column headed credit spread, with every figure correctly computed. The list is a ranking by maturity wearing a ranking by credit as a disguise. No arithmetic went wrong, so no arithmetic check will catch it.
The repair is one line. The maturity is matched before subtracting, every single time, and printed beside the answer so that the next reader can check it.
What are the honest limits on reading the spread as credit?
The credit spread has had a generous hearing so far, and exactness about what the 2.20 percentage points is and is not belongs here. Exactness matters most where the spread is turned into something that looks like a probability, and that is the commonest onward use it gets.
The reading uses two definitions carried in from earlier and not rebuilt here. A recovery rateThe share of the amount owed that a lender gets back after a borrower stops paying, stated on the same base as the amount owed. is the share of the amount owed that comes back after a borrower stops paying, and loss given default is one hundred per cent less the recovery rate on the same base. Assume a recovery of 40 per cent of the amount owed. The 40 per cent is an assumption, and it is labelled as one every time it is used. Loss given default is then 0.60.
| s | the credit spread, in percentage points a year, over the five year government SPOT rate |
| pimp | the implied annual default rate, in per cent a year, solved backwards out of a price rather than measured or forecast |
| L | loss given default, as a share of the amount owed, being one less the recovery rate |
Only one of the three quantities is unknown, so divide both sides by loss given default and read it backwards. The step is allowed because the relationship is a product of two numbers and dividing both sides by one of them leaves the other standing alone.
| 2.20 | the spread in percentage points a year, being 220 basis points over the five year government SPOT rate |
| 0.60 | loss given default, on an assumed recovery of 40 per cent of the amount owed |
| 3.6667 | the implied annual default rate, in per cent a year, across the same five years |
A relationship that only works in one direction has not been understood, so run back the other way as a check: 3.6667 per cent a year multiplied by 0.60 is 2.2000 percentage points, the spread the arithmetic started from. Three honest limits sit on that figure, and none of them is optional.
First, the forty per cent recovery is an assumption
No recovery study stands behind it, and no price contains it. Moving the assumption moves the answer with it while the price stands completely still. Nothing shows more plainly how much work the assumption is doing.
| Assumed recovery | Loss given default | Spread, points a year | Implied default rate a year |
|---|---|---|---|
| 30 per cent of the amount owed | 0.70 | 2.20 | 3.1429 per cent |
| 40 per cent of the amount owed | 0.60 | 2.20 | 3.6667 per cent |
| 50 per cent of the amount owed | 0.50 | 2.20 | 4.4000 per cent |
| 70 per cent of the amount owed | 0.30 | 2.20 | 7.3333 per cent |
Same price, four answers, and the only thing that changed was a number nobody observed. A figure that moves from 3.1429 to 7.3333 per cent a year on an assumption should never be printed without the assumption written in the same sentence.
Second, the whole spread is being treated as compensation for credit
In a real market some part of a spread pays for something other than default, and the commonest such thing is the difficulty of selling the bond quickly at a fair price. Every basis point of that read as credit makes the implied default rate too high. Splitting 0.40 percentage points off the 2.20 as payment for something other than default leaves 1.80 points, and dividing by the same 0.60 brings the implied annual default rate from 3.6667 down to 3.0000 per cent a year. There is no way on this platform to separate the two, and any account that quietly assumes the whole spread is credit has made a claim it did not declare.
Third, an implied default rate is what the price says
An implied default rate is not a forecast and not a measured frequency of anything. Nobody counted defaults to produce 3.6667 per cent a year. The figure was solved backwards out of one spread and one assumption, in two lines of arithmetic, in the open. Reading it as the probability that Palash Cements Limited fails is a misreading of the arithmetic that produced it, and the word implied is attached to the figure at every single use for exactly that reason.
How does a lender or an analyst use this split in practice?
What somebody actually does with a yield that has three things inside it
The split is not a classroom tidiness exercise. Splitting the yield is the first thing done to it, and the discipline is mostly about what gets written down rather than what gets computed.
- Pull the government SPOT rate for the same maturity, at the same moment.Same maturity, because a spread measured over a different number of years is a different quantity, as the 220 against 285 comparison above shows. Same moment, because two figures taken hours apart can differ for reasons belonging to neither the curve nor the borrower. How a benchmark government yield curve is constructed and published is set by the Reserve Bank of India at rbi.org.in.
Without this one figure nothing below is possible.
- Subtract, and write down the maturity beside the answer.Two hundred and twenty basis points on its own is not a record of anything. Two hundred and twenty basis points over the five year government SPOT rate, for five years, on annual compounding, is a record the next reader can check and reproduce. The maturity is the attribute a rate does not carry in its own name.
2.20 percentage points, 220 basis points, over the five year government SPOT rate.
- Record the government limb as well as the spread, never the total alone.A file holding only 9.10 per cent a year has thrown away the information every later question needs. A file holding 6.90 plus 2.20 can answer whether the next move came from the curve or from the borrower, and the file holding the total cannot.
Two figures in the note, not one.
- Say nothing about the term premium, and say why.The government limb is a bundle. An analyst who writes that a long rate is high because lenders want paying for commitment has made a claim about an expected path of short rates, whether or not they meant to. Either supply the path and show the subtraction, or leave the claim out.
The empty box is a finding, not an omission.
- Attach the assumption to any implied figure, in the same sentence.If the 2.20 is ever turned into an implied annual default rate, the recovery assumption is written beside it every time, because a reader who meets the figure without it will carry away a probability that nobody claimed.
3.6667 per cent a year, implied, on an assumed recovery of 40 per cent of the amount owed.
A household lending money privately runs a rough version of the same discipline without naming it. The first question is the reference rate, the bank's own rate on a deposit of the same length. The second question is the spread: what extra this particular person should be paying. Comparing a five year loan to a cousin against a one year bank deposit rate and calling the whole gap a judgement about the cousin is exactly the mistake set out above, made in a kitchen rather than at a desk.
What single thing would have to be supplied before a term premium could be stated here?
Named here, and deliberately not written out
Every item below is set by an authority, changes on its own schedule, and is therefore named rather than stated. Not one row is filled in. The arithmetic above this block uses no rule set except the compounding convention, and that convention sits inside the arithmetic itself: a sum cannot be reproduced without it.
| Item | Where it is settled |
|---|---|
| The valuation norm that decides the price at which a credit holding is carried | Reserve Bank of India, rbi.org.in |
| How a benchmark government yield curve is constructed and published | Reserve Bank of India, rbi.org.in |
| The compounding convention a published yield is stated on | Reserve Bank of India, rbi.org.in |
| What an issuer of corporate debt must disclose, and to whom | Securities and Exchange Board of India (SEBI), sebi.gov.in |
| What counts as a default for reporting purposes, and who decides it has happened | SEBI, sebi.gov.in |
A second market goes in as extra rows, and the arithmetic above stays untouched. Keeping the arithmetic and the rule sets apart is what that separation buys.
References
| Source | What it is named for | Where |
|---|---|---|
| Reserve Bank of India | Government securities and how a benchmark yield curve is constructed and published, the compounding convention a published yield is stated on, and the valuation norm applying to a credit holding, all named and none stated | rbi.org.in |
| SEBI | What an issuer of corporate debt must disclose and to whom, and what counts as a default for reporting purposes, both named and neither stated | sebi.gov.in |
Palash Cements Limited, its five year bond and the government SPOT curve used here are invented.
Educational material. Not advice on any investment, tax, budget or market position.