Spread Return and Price Return: Splitting One Price Move

A corporate bond's yield stacks two quantities: a government rate for the same maturity, and a spread sitting on top of it. Price return is the whole move in the price across a stated stretch of time. Spread return is the slice of that move the spread limb produced by itself, with the other quantity held exactly still, and both are found by pricing the bond twice.

Underneath that sits one small fact with large consequences. A bond has one price, and that price is produced by one yield. But the yield is a sum, and the two things summed into it move for reasons that have nothing to do with each other. So the bond can be priced again with one of them changed and the other frozen, and the arithmetic gives that limb's share without any formula being required. Every figure below is built that way: the sum is set out with all its terms showing, evaluated, and then evaluated a second time with exactly one input altered. A reader who watches a total assemble line by line can rebuild it. A reader handed the total cannot check it, and will not try.

What is a spread, and what does it sit on top of?

The instrument comes first. Palash Cements Limited, an invented cement maker, is the only company borrowing money anywhere in this record, and it has a five year bond outstanding. Its terms set the face amount at Rs 1,000.00/-. The terms write 9.10 per cent a year against that amount. The terms put one payment date in each year, five of them in all. Separately, an invented SPOT curve carries a five year node, and that node is entered at 6.90 per cent a year. Money handed over today and returned on one named future date, with nothing arriving in between, is priced at a SPOT rate. Every rate in this guide carries that word or the word FORWARD. Without the label, two different objects get read as one.

The subtraction follows, and it is all a spread is. Taking 9.10 per cent and removing 6.90 per cent leaves 2.20 percentage points. Percentage points divide into hundredths, and each hundredth carries a name of its own: a basis point. Counted in the big unit the same gap is 2.20 of them. Counted in the small unit it is 220. Neither figure is a second quantity. The subtraction is the whole construction. There is no third input and no adjustment.

A spread is always OVER something and FOR a stated maturity, and a spread quoted without both of those named is not a usable number at all. "220 basis points" on its own says almost nothing: over what, and for how long? Here the answer is over the five year government node and for five years, and the reason the maturities have to match is that a five year rate and a thirty year rate are simply different levels. Subtracting one from the other measures the calendar as much as it measures the borrower.

The household version runs like this. Two neighbours borrow for the same five years. One is a salaried government employee and gets a rate. The other runs a shop and gets a higher one. The gap between the two rates only means something because the two loans run for the same stretch of time. Set the shopkeeper's five year rate against the employee's one year rate and the resulting gap is partly about the shop and partly about four extra years, with no way to tell which part is which.

One more thing the subtraction will not do. A rating scaleThe ladder of steps a firm expresses a credit assessment on, together with what each step is meant to signify. Whichever firm publishes a scale defines it. and the meaning of every step on it belong to whichever firm publishes that scale, and what such a firm has to disclose about its method sits with the Securities and Exchange Board of India (SEBI) at sebi.gov.in. So Palash Cements Limited has no assessment of its creditworthiness anywhere in this record, and the spread above is a subtraction rather than a verdict.

Why does a spread need the clock it was quoted on named beside it?

Here is a fact most readers meet late and would rather have met early. A spread is a distance between two rates, and a rate is not a number until the frequency with which it is applied has been stated. Change that, and the distance changes with it, even though nobody touched either contract. Every price and every rate in this guide runs on one discounting tick a year: an amount owed in four years gets divided by one plus the yield, four times over, and that is the whole convention. Written beside every price, it makes the sum reproducible. Left out, it does not.

Restating the same two yields shows what happens. Palash Cements Limited's 9.10 per cent a year is what the money actually grows by over twelve months. Restated on a twice a year basis it answers a different question: what rate, applied in two equal slices, reaches the same place? The square root of 1.0910 is 1.04450946, so each slice is 4.450946 per cent, and doubled that is 8.9019 per cent. The identical treatment of the government node turns 6.90 per cent a year into 6.7849 per cent. Both restatements close back exactly: compound 8.9019 twice and 9.1000 comes out, compound 6.7849 twice and 6.9000 comes out.

Now subtract on the new basis. The spread is 2.1170 percentage points instead of 2.20, or 211.70 basis points against 220. Nothing about either bond changed. Nobody renegotiated anything. The clock alone shrank the gap by 8.3022 basis points, and it shrank it because the larger rate loses more in the restatement than the smaller one does. So a spread quoted with no convention attached is missing an input, in exactly the way a price with no convention attached is missing one.

Treat the restatement as a demonstration of a mismatch and nothing more. The restatement is not a second convention for Palash Cements Limited's bond. The bond has exactly one convention, stated in the paragraph above and used without exception in every sum below. The restatement shows why the convention has to travel with the figure. Two people quoting the same bond, one on each basis, would disagree by eight and a third basis points and both would be right, and neither could tell why without asking a question neither had thought to ask.

What is price return, and across what stretch of time is it counted?

Price return is the change in a bond's price across a stated period, written as a rate on the price the period started from. Three components, and all three have to be named. A starting price. An ending price. And the stretch of calendar between them. A move of Rs 15/- across a year and the same move across a week are not the same event, and the number on its own cannot say which one happened.

Price return says that a price moved and by how much, and it says nothing whatever about why. That is the measure doing its job rather than falling short of one. A thermometer reports the temperature and does not report the weather system. Everything that follows exists because a price return with no explanation attached is the normal thing a reader is handed, and the explanation has to be assembled separately.

Two things sit outside it. Price return excludes the coupon, so it is not what a holder earned over the period. A holder who received Rs 91/- in cash and watched the price fall Rs 15.9875/- had a different year from the one the price return describes; the coupon stays out because what is being split here is the price and only the price. Price return also has no view about which limb caused anything. Both of those are handled elsewhere on their own terms.

Where does the Rs 1,000.0000/- this period starts from actually come from?

A starting price nobody can rebuild makes every later figure unverifiable, so the opening Rs 1,000.0000/- is derived here rather than asserted. Palash Cements Limited's bond went out at par, meaning the price on the day it was sold was the face amount itself. Take five yearly coupons of Rs 91/-, take Rs 1,000.00/- of face amount at the end of the fifth year, discount every one of those six amounts at 9.10 per cent a year on one tick a year, and add them up.

Add the right hand column yourself. The column closes on Rs 1,000.0000/- with nothing left over: six figures each rounded to four places, and their printed total is the unrounded total to every place shown. A close that clean is unusual enough to be worth saying out loud. The clean close happens only because the bond was priced at par, and it does not survive a move to any other yield, as a later column shows.

Issued at par means the contracted rate and the yield on the day of issue are the same number, and that is a definition rather than a coincidence. Work it backwards: if the price is the face amount, then the sum repaid at the end equals the sum handed over at the start, and everything else the bond delivers is the yearly Rs 91/-. Rs 91/- arriving each year against Rs 1,000.00/- put down is a yield of 9.10 per cent a year. Rs 91/- on Rs 1,000.00/- of face is exactly what a contracted rate of 9.10 per cent means, so the contracted rate is 9.10 per cent too. One number, arrived at twice, forced by the price.

How is one limb's share of a price move found without using a formula?

The bond gets priced twice. Pricing it twice is the whole method, and stating it as a procedure earns its space. Most readers expect something more elaborate and then reach for something less reliable.

Start with a price and the yield that produced it, and remember that yield is a sum of two limbs. To find what the first limb did, change that limb, hold the second one exactly where it was, and discount the same remaining payments again at the new total. The difference between the two prices is the first limb's contribution, in rupees. Then run the identical procedure for the second limb. Two subtractions, no approximation anywhere, and every step reproducible on a pocket calculator.

The procedure is what an attributionSplitting a result into the separate causes that produced it, so each cause carries a figure of its own instead of the whole result carrying one label. is: not a story about a price, but a second and a third price, worked deliberately, and then differenced. The method needs no measure of sensitivity, no approximation and no assumption beyond the two prices themselves, and no other method appears anywhere below. A reader with a calculator can reproduce every rupee below, and a reader who cannot reproduce a figure has no way of telling a correct attribution from a confident one.

One line is held throughout. The SHAPE of the relation between a bond's price and its yield is established, with no number put on how steep that shape is. A reader handed a multiplier before they can compute the thing it multiplies has learnt a number rather than a mechanism, so measures that put a figure on the steepness of that relation are covered separately and come after the split. Where steepness might be quantified, the direction is named instead and both prices are printed in full.

What happens to the price when the spread alone widens?

Move the calendar on by one year. Four dates are left, the coupon is still Rs 91/- on each of them, and Rs 1,000.00/- of face amount still falls due at the end of the fourth. Now change one thing and one thing only: hold the government limb exactly at 6.90 per cent a year and widen the spread by 50 basis points, from 2.20 percentage points to 2.70. The yield is 6.90 plus 2.70, or 9.60 per cent a year. Everything else on the bond is untouched.

Discount the same five amounts at the new yield.

The right hand column adds to Rs 984.0126/-, one paise more than the total printed under it. Nothing is wrong. Five amounts were each rounded to four places and four of the five roundings went the same way, so the printed parts carry a residual of Rs 0.0001/- that the unrounded parts do not. The check runs on the unrounded figures, and the printed column is what those figures look like once they are rounded for reading. Compare that against the issue price further up, where six rounded parts closed on Rs 1,000.0000/- with nothing at all left over. Par is the well behaved case, and it is well behaved only because it is par.

Now the two figures this block exists for. The price went from Rs 1,000.0000/- to Rs 984.0125/-, a fall of Rs 15.9875/-. On the Rs 1,000.0000/- the period started from, that is 1.5988 per cent below the start, and that is the price return for the period. Because the government limb was held exactly where it was, every basis point of that 1.5988 per cent is spread return, and the split needed no formula at all: it needed a second price.

What if the spread never moved and the government limb widened the yield instead?

The comparison turns on the next question, and the arithmetic answers it with something slightly startling. The same bond, the same four remaining dates and the same Rs 1,000.0000/- starting price. Now the spread is frozen solid at 2.20 percentage points, exactly where it was written, and the government limb moves up 50 basis points instead, from 6.90 per cent a year to 7.40. The yield is 7.40 plus 2.20, or 9.60 per cent a year.

The yield is the same one the previous block reached. So it is the same five amounts, divided by the same 1.0960 raised to the same powers, adding to the same Rs 984.0125/-. Same price. Same Rs 15.9875/- fall. Same price return of 1.5988 per cent below the start.

The price cannot tell the two stories apart, and that is forced arithmetic rather than a curiosity: the two limbs enter the discounting only through their sum, so anything that changes the sum by 50 basis points changes the price identically. Nothing in the discounting has a slot for where the yield came from. The discounting has a slot for the yield and nothing else.

But the attribution is completely different, and the difference is the entire point. In the first story the spread limb produced the whole of the 1.5988 per cent and the spread return is 1.5988 per cent below the start. In the second story the spread limb produced none of it, so the spread return is 0.0000 per cent and the whole of the price return sits with the government limb. Two attributions with no overlap at all, sitting under one price and one price return.

What happens when the government limb falls instead of rising?

Same bond, same four remaining dates, same Rs 1,000.0000/- to start from. Now hold the spread solid at 2.20 percentage points and take the government limb down 50 basis points, from 6.90 per cent a year to 6.40. The yield is 6.40 plus 2.20, or 8.60 per cent a year, and the five amounts get divided by 1.0860 raised to whichever power their year calls for.

The column has a residual too, and it leans the other way: the five printed amounts add to Rs 1,016.3417/-, one paise short of the unrounded total rather than one paise over. Three such columns now stand in this guide and no two of them behave the same. Five or six figures rounded for reading may print a total above, below or exactly on the real one, and the only way to know is to keep the unrounded figures and check against those.

The price went from Rs 1,000.0000/- to Rs 1,016.3418/-, a rise of Rs 16.3418/-. On the same Rs 1,000.0000/- base the rise is a price return of 1.6342 per cent above the start. The spread never moved, so the spread return here is 0.0000 per cent, and attributing any part of that gain to Palash Cements Limited would be inventing a cause the arithmetic never produced.

Why do two moves of the same size not produce two price moves of the same size?

Put the last two blocks beside each other. Fifty basis points of widening gave 1.5988 per cent below the start. Fifty basis points of falling gave 1.6342 per cent above it. Both moves were the same distance, both started from the same 9.10 per cent a year, both ran on the same bond over the same four remaining dates. And the answers are Rs 0.3543/- apart in rupees, or 0.0354 percentage points.

Watch the shape rather than the arithmetic. Look back at the curve two blocks up. The curve falls from left to right, the ordinary thing a price does when a yield rises. But the curve does not fall in a straight line: it is steeper on the left and flatter on the right. So a step to the right, into higher yields, costs less than a step to the left, into lower yields, gains. The gain is always the larger of the two, and the reason is the bend in the line rather than anything about which limb moved.

The trap sits in that last clause, and the clause is worth pinning down. The two answers do not differ because one of them came from the spread and the other from the government limb. Sending the government limb up 50 basis points instead produces exactly the 1.5988 per cent the spread produced, as worked immediately above. The asymmetry belongs to the DIRECTION of the move, not to the limb that made it. Direction and limb are two independent questions and a reader who merges them will attribute things to a borrower on the strength of an arithmetic property of discounting.

What is left over when both limbs move inside the same period?

Everything so far has moved one limb at a time. Moving one limb at a time is a convenience rather than a description of life. Move both and a new quantity appears. Take the spread out 50 basis points to 2.70 points and take the government limb up 50 basis points to 7.40 per cent a year, both inside the same period. The yield is 7.40 plus 2.70, or 10.10 per cent a year, and the five amounts discounted at that rate add to Rs 968.3698/-. The bond fell Rs 31.6302/- from Rs 1,000.0000/-, a fall of 3.1630 per cent below the start.

The attribution done the obvious way runs like this. The spread limb on its own, with the government limb frozen, cost Rs 15.9875/-. A 50 basis point rise in either limb reaches the same 9.60 per cent a year, so the government limb on its own, with the spread frozen, cost Rs 15.9875/- as well. Adding the two answers implies the price fell Rs 31.9751/-, or 3.1975 per cent of the starting price. The bond fell Rs 31.6302/-, or 3.1630 per cent. The two limbs priced apart and added overshoot the real move by Rs 0.3449/-, or 0.0345 percentage points, and that leftover is a real quantity rather than a mistake.

A careful reader will hit a small printing question here. Two printed falls of Rs 15.9875/- add to Rs 31.9750/-. The two unrounded falls add to Rs 31.9751/-. A display figure is a rounded reading rather than an input, so the leftover is worked from the unrounded pair.

Why does it exist at all? Because the second move lands on a price the first move has already shifted. The government limb's 50 basis points, applied to a bond already yielding 9.60 per cent rather than 9.10, costs less than it would have at the lower yield, for exactly the bend in the line the previous block worked through. Neither limb produced the saving alone, so neither can claim it. Report the leftover as a cross termThe part of a combined result produced only by two changes happening together, so it cannot be handed to either change on its own. and leave it in the open, rather than pushing it quietly into one of the two limbs and making that limb look bigger or smaller than it was.

Here is the same thing at a wedding scale. Two households agree to split a hall's cost: one pays for the extra hour, the other for the extra fifty guests. Book both together and the caterer gives a discount that neither would have got alone. Somebody has to decide who that discount belongs to, and the honest answer is that it belongs to the combination. Attributions that refuse to write that line down end up assigning it to whichever party the person doing the sums was thinking about.

What does the whole exercise look like set out row by row?

Five cases, one bond, one period of four remaining years, and one convention throughout. The price column carries the point on its own, ahead of anything else: two different rows carry the identical price, and their attribution columns have nothing in common.

The table has no column for the last row's third part, so it goes here: a cross term of 0.0345 percentage points above, belonging to both moves together. And the three printed parts of that row add to 3.1631 per cent below against the 3.1630 the price actually made, one ten thousandth of a point out. The residual is the same class as the discount columns further up, and it is settled the same way, on the unrounded figures.

Rows two and three are the whole argument compressed into two lines: identical yield, identical price, identical price return, and attributions that agree about nothing. With the first column covered, no arithmetic below it identifies which row is which.

Who actually reaches for this split, and what do they do with it?

Three people, and each of them wants a different half of it.

A lender who already holds a credit exposureAnything whose value depends on one particular borrower carrying on paying. A bond is one, a loan is one, and so is money a customer owes. to a borrower wants the spread limb and does not much care about the other one. If the spread on a borrower's paper has widened, that is information about the borrower, or at least about how the borrower is being priced, and it is worth a phone call. If the whole government curve moved, nothing has been learnt about the borrower at all, and a phone call would be wasted. So the lender runs the split precisely to decide whether there is anything to look into.

A person reporting a period's return to whoever the money belongs to wants both limbs, separately, and in writing. The reason is uncomfortable and worth saying: a report that says a holding fell 1.5988 per cent invites the reader to conclude the wrong thing, whichever direction is wrong on that occasion. Split into a limb the reporter chose and a limb the reporter could not control, the same figure turns the conversation towards the decision rather than the weather. One thing the split does NOT license bears stating. Splitting a return is not the same as saying either limb was a good idea. A split reports what happened and never whether it should have.

A treasurer at the borrowing end reads the split backwards. Palash Cements Limited, refinancing in a year, cares whether the cost of borrowing has moved because rates in general moved or because its own spread widened. The two would call for entirely different responses. And a household knows this shape already. A neighbour's home loan rate went up. Was it the bank repricing everybody, or was it something about that neighbour? The two feel identical from outside the house and mean completely different things inside it.

What can be said about why this spread would move?

Nothing at all.

The requirements are worth setting out. Saying why a spread moved would take some record of that borrower being assessed, some study of how often borrowers like it have failed to pay, some study of how much was got back when one did, some run of spreads over time to show what normal looks like, and at least one other borrower to compare against. Not one of those five things exists anywhere in this record, and inventing any of them would introduce a fact that nothing else here supports.

So read the 50 basis points correctly. The 50 is an input, chosen because a round number makes the arithmetic legible, and it is not an estimate of anything, not a typical move, and not a forecast. Every figure that followed from it is exact, and every one of them is exact about a supposition. A split is arithmetic and not an explanation, and the difference matters most exactly where a reader is most tempted to forget it. What a spread implies about an expected lossWhat a lender works out it is likely to lose on average: how often borrowers fail, multiplied by how much goes missing when one does., and the sum that turns the one into the other, is worked through separately.

Palash Cements Limited, its five year bond and the SPOT curve behind it are invented.
Educational material. Not advice on any investment, tax, budget or market position.