G-Spread, Z-Spread and Option-Adjusted Spread Compared

Three measures, one bond, three answers. A single government SPOT rate subtracted from a bond's yield gives a G-spread. The one constant addition to every rate along the SPOT curve that pulls the discounted payments back onto the price is a Z-spread. A Z-spread with an embedded option's value stripped out is an option-adjusted spread. Nothing about the borrower changes between the three.

What is a G-spread, and what exactly gets subtracted from what?

The first measure is the one already in use before anyone names it. Palash Cements Limited, an invented issuer, has issued a five year bond carrying a 9.10 per cent annual coupon on Rs 1,000.00/- of face, and it was issued at par, on annual compounding, so its yield is 9.10 per cent a year as well. The invented government SPOT rateThe rate for money placed today and returned at one stated future date. One rate, one date, no reinvestment assumed along the way. for five years is 6.90 per cent a year. One basis point is one hundredth of a percentage point. One rate subtracted from the other leaves 2.20 percentage points, or 220 basis pointsOne hundredth of a percentage point. So 2.20 percentage points is 220 basis points. In the same way 0.50 percentage points is 50 basis points..

The bond's yield less the government SPOT rate is a G-spreadA bond's yield less the government SPOT rate for the same maturity, quoted in percentage points and in basis points. One subtraction, two inputs.. The whole method is one subtraction between two numbers. One operation on two inputs is both the reason a G-spread is used everywhere and the reason it is the crudest of the three measures set out here. Two figures, one operation, and anybody can repeat it in their head on a call. Nothing about it is hidden, nothing about it needs a solver, and two people handed the same pair of rates will never disagree about the answer.

Two rules travel with a spread from here on, and both have cost somebody a correction. The first is the units rule. A percentage point and a basis point are one quantity in two clothes, so a spread is written in both units the first time it appears and a figure in percentage points is never left standing beside a neighbour quoted in basis points. The second is that no spread is ever a bare number. A spread is always over something and always for a stated length of time. So 220 basis points here means 220 basis points over the five year government SPOT rate, for five years, and detached from those two facts it means nothing whatever.

Why is one government rate not enough for five dated payments?

Here is the crack in the G-spread, and it opens as soon as the bond's actual payments are set out. Palash Cements Limited's bond hands over Rs 91/- at the end of each of years one, two, three and four. At the end of year five it hands over Rs 1,091.00/-, the last coupon plus the face amount. Five dated payments arrive on five different dates.

Now what the government SPOT curve behind this walkthrough says about money for those different lengths of time. One year money is priced at 5.90 per cent a year, two year money at 6.25 per cent a year, three year money at 6.55 per cent a year, and five year money at 6.90 per cent a year, all on annual compounding. Four different rates apply to four different dates, and the G-spread used exactly one of them.

On a curve that is not flat, comparing a five year bond against a single five year rate has quietly mispriced the four payments that arrive earlier. The rupee that comes back at the end of year one is being judged against a 6.90 per cent a year benchmark when the curve says the honest benchmark for one year money is 5.90 per cent a year, a full 100 basis points lower. The G-spread never sees that. The measure flattens the whole shape of the curve into whatever the rate happens to be at the final maturity.

The shape is familiar from ordinary life. A household comparing two loan offers by looking only at the last instalment has done the same thing: the schedule matters, and squashing it into one moment throws away the part that differs. A shopkeeper quoted one average price for a mixed basket of goods is in the same position. The average is not wrong, exactly. The average is just answering a question with fewer moving parts than the one that was asked.

One labelling rule is enforced harder here than anywhere. Every rate in this guide carries the word SPOT or the word FORWARD. A SPOT rate covers money placed today and returned on one stated future date. A FORWARD rate covers money placed at one future date and returned at a later one, and it is not a separate opinion about the future at all: it is already sitting inside the SPOT curve and can be pulled back out of it by arithmetic. On the invented curve behind this walkthrough a one year FORWARD rate of 6.6012 per cent a year sits within 5.1157 basis points of the three year SPOT rate of 6.55 per cent a year. Two objects that read almost identically are kept apart by their labels and by nothing else at all.

What is a Z-spread, and why does it need the whole curve?

The repair is obvious once the crack is visible. Stop comparing against one point and compare against every point the curve carries. Discount the year one payment at the one year government SPOT rate, the year two payment at the two year rate, and so on, each payment against the rate that genuinely belongs to its own date. Then ask what single constant has to be added to every one of those rates before the discounted payments add up to the price actually being paid.

The constant that closes the gap is a Z-spreadThe one constant amount added to every government SPOT rate along the curve which makes the bond's discounted payments equal its price. A Z-spread is solved for, not subtracted.. The G-spread compares a bond against a point on the curve; the Z-spread compares it against the whole curve. That is the entire difference between them, and every disagreement in their answers flows from it.

The method changes as well as the inputs. A G-spread is subtracted. A Z-spread is solved. A number is tried, all five payments are discounted at the curve plus that number, the total is compared with the price, and the number is adjusted. Too high a total means the constant was too small; too low a total means it was too large. Narrowing repeatedly, the answer converges. No closed formula waits at the end of the search, and none is needed. The search takes a computer a fraction of a second and takes the reader one idea.

The gap of Rs 93.4222/- is the whole problem stated as money. The market is paying Rs 1,000.000000/- for a set of payments that would be worth Rs 1,093.422187/- if they were as certain as government payments. Something has to be added to every discount rate to push that value back to what is actually being paid, and the Z-spread is the smallest possible way of saying what that something is: one constant, applied everywhere.

Seeing a gap close teaches more than being handed the answer. Watch the search actually run. Add nothing and the payments are worth Rs 1,093.4222/-, a surplus of Rs 93.4222/- over the price. Add 100 basis points to every SPOT rate and the total drops to Rs 1,050.9251/-, leaving Rs 50.9251/-. Add 200 basis points and it drops to Rs 1,010.5996/-, leaving Rs 10.5996/-. Add 227.1722 basis points and the total is Rs 1,000.0000/- and the gap is gone. The Z-spread is not announced by anybody, it is the number that survives when the surplus is squeezed out.

What happens at the four year point the curve does not carry?

Here is the awkward fact, named where it bites. The invented government SPOT curve behind this walkthrough carries points at one, two, three, five, ten and thirty years. The curve carries nothing at four years. Palash Cements Limited's bond pays at four years. A Z-spread cannot be computed without a four year rate, so a four year rate has to be supplied, and supplying one is an assumption rather than an observation.

The honest move is to say so and then show how much the assumption is worth. A missing nodeOne dated point on a curve where a rate is actually given rather than filled in. The curve is only observed at its nodes; everything between them is constructed. is filled by interpolationFilling in a rate between two points that do exist. Interpolation is a construction rather than an observation, and different rules for doing it give different answers., and the obvious rule is the linear midpoint of the two neighbours the curve does hold. The three year SPOT rate is 6.55 per cent a year and the five year SPOT rate is 6.90 per cent a year, so the midpoint is 6.7250 per cent a year. The midpoint of 6.7250 per cent a year is the reading used for every other figure here, and it is stated in the same breath as each figure it produced.

All three defensible readings stand side by side rather than one being chosen quietly. Holding the four year point at the three year SPOT rate of 6.55 per cent a year, the Z-spread solves to 228.2453 basis points. At the midpoint of 6.7250 per cent a year it solves to 227.1722 basis points. At the five year SPOT rate of 6.90 per cent a year it solves to 226.1073 basis points. A four year rate outside that pair of neighbours would bend the curve back on itself between two points the curve genuinely carries, and a bend of that kind is a claim about shape that needs evidence of its own.

The entire range of defensible answers covers 2.1380 basis points. The gap between the Z-spread and the G-spread on this same unchanged bond is 7.1722 basis points. Read that twice, because it reverses the instinct almost everybody arrives with. The choice of measure moved the answer more than three times as far as the missing number did. Before anyone argues about interpolation rules, they should check whether they are even comparing like with like.

What is an option-adjusted spread, and what gets adjusted out?

Some bonds carry a right written into the contract that lets one side change when the payments arrive. The commonest is a right held by the issuer to repay early. The right to repay early is worth something to whoever holds it, and it is paid for inside the yield. A spread computed on such a bond is therefore partly a credit measurement and partly the price of the right.

An option-adjusted spreadA Z-spread with the value of an embedded option taken out, so that what is left compares bonds whose payments arrive on comparable terms. is a Z-spread with the value of that embedded optionA right written into a bond that lets one side change when the payments arrive. A right held by the issuer to repay early is the commonest. taken out. What remains compares two bonds whose payments arrive on terms that can genuinely be set beside each other. Some of what looks like credit compensation is the price of the right. Leaving the right inside the spread therefore makes a borrower look better paid for credit than they really are.

No bond used in this walkthrough carries an embedded option. Palash Cements Limited's bond repays once, at the end, on fixed terms, and nobody can bring the repayment forward or push it back. So on this bond the option-adjusted spread and the Z-spread are the same number, 227.1722 basis points on the midpoint reading. The difference between the two is the entire reason the option-adjusted measure exists, and a bond with no embedded option cannot show it.

How large that gap grows depends on the terms of the right and on how likely the issuer is to use it.

What is a Default Spread, and where does it come from?

Everything so far has measured a gap between two prices. Now comes a different kind of quantity altogether, and the difference between the two kinds is the sharpest distinction in this guide. A credit spread is observed. A default spreadThe portion of an observed spread that expected loss arithmetic attributes to default. A default spread is inferred from assumptions rather than read off two prices. is inferred.

The default spread is the part of an observed spread that expected loss arithmetic attributes to default, and expected loss arithmetic says that a spread charged per year equals a default rate per year multiplied by the share of the amount owed that is lost each time default happens. The share of the amount owed that is lost is the loss given default, written in those words and never shortened, and it is one hundred per cent less the assumed recovery rate on the same base.

The default spread can never be larger than the credit spread. The two are equal only if every single basis point of the observed gap is payment for default and nothing else. Assume exactly that and the default spread on Palash Cements Limited's bond is the whole 220 basis points, which is 2.20 percentage points. Now carve 0.40 percentage points off the 2.20 as payment for something other than default, such as the difficulty of selling the bond quickly. The default spread falls to 180 basis points, or 1.80 percentage points, and the credit spread has not moved at all. The credit spread is still the same subtraction between the same two rates, so it is still 220 basis points.

Think of a vegetable seller who buys a crate for Rs 400/- and sells it for Rs 520/-. The Rs 120/- gap is observed and nobody can argue with it. How much of that Rs 120/- is payment for spoilage, how much for the cart, how much for the hours, is inferred, and two honest people will divide it differently. Neither of them has made an arithmetic mistake. The gap is one object; the story about what fills it is another.

Credit Spread vs Default Spread: which of the two survives a disagreement?

The two words are used for each other constantly, and the substitution is invisible when it happens. Set the two measures side by side on one bond, on one day. The credit spread is 220 basis points, or 2.20 percentage points, and it is a subtraction. The default spread is at most 220 basis points, and it is a conclusion. One of the two survives any disagreement about assumptions and the other does not.

The practical test takes one question. If two people holding the same two prices can arrive at different answers, they are talking about the default spread, whatever word they used. That test costs nothing and it catches the substitution before it reaches a spreadsheet, which is the only place it is cheap to catch.

How to read a Credit Spread: which five questions must it answer?

A spread arrives in an email with no covering note. Before it is used for anything, five questions have to be put to it, and every one of them has changed an answer at some point.

The compounding row is not housekeeping. Every price in this guide is struck on annual compoundingOne discounting period a year, so an amount is divided by one plus the annual rate once for each year. A different convention gives a different price from figures that look the same., so an amount is divided by 1.0910 once for each year at a 9.10 per cent annual rate. On that convention Palash Cements Limited's five payments discount to Rs 1,000.000000/- exactly. Discounting to the price paid is what being issued at par means here. With the convention written beside the price a reader can reproduce the sum; without it, no amount of surrounding text will serve.

A spread that cannot answer all five questions is not yet a usable number; it is a rumour with a decimal point. None of the five takes more than a sentence to answer, and the person who sent the figure almost always knows all five. The five questions simply were not asked.

Which measure should be used, and what decides it?

Preference is where the argument starts and the comparison is where it ends. Answer this by what each measure is compared against, never by preference. Use the G-spread where a quick, checkable comparison against one government point is enough and the curve is not steep. Use the Z-spread where the payments are scattered across a curve that is not flat, and curves are rarely flat. Use an option-adjusted spread where the bond carries a right that changes when payments arrive. Palash Cements Limited's bond carries no such right.

The rule underneath all three is the one that actually protects a reader: never compare two spreads computed on different measures. The gap between the measures on this single bond is 7.1722 basis points, and a comparison that ignores it has read its own arithmetic as a finding about somebody else.

What three limits travel with every inferred figure here?

Three limits attach to the default spread and to anything derived from it. None of the three is optional.

First, a recovery assumption is an assumption. A recovery rate is chosen rather than observed, and moving the choice moves the answer with it. Holding the 220 basis point spread perfectly still, a 30 per cent assumed recovery implies a default rate of 3.1429 per cent a year, a 40 per cent recovery implies 3.6667 per cent a year, a 50 per cent recovery implies 4.4000 per cent a year and a 70 per cent recovery implies 7.3333 per cent a year. One price, four answers, and the assumption is doing that much of the work.

Second, the whole spread has been treated as compensation for credit. In a real market some part of a spread pays for not being able to sell the bond easily, and every basis point of that counted as credit makes an implied default rate too high. Splitting 0.40 points off the 2.20 as payment for something other than default, the implied rate falls from 3.6667 to 3.0000 per cent a year. Nothing in an observed spread announces which part is which, so the two cannot be separated from the price alone.

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, and reading it as the probability that Palash Cements Limited fails has misread the arithmetic that produced it. Carry the four decimals inside any multiplication and print 3.67 per cent only as a reading, because 2.20 divided by 0.60 is exactly three and two thirds, and multiplying the rounded 3.67 back gives 2.2020 points rather than 2.2000. A reader who checks with the printed figure and lands two ten thousandths away must be told in advance which rounding produced which.