Basis Risk: When the Hedge and the Exposure Diverge

Basis risk is what a holder is left carrying when the contract and the holding refuse to line up. The two part ways in three separate places: the day each of them ends, the number of units each of them covers, and the thing each of them is written on. Whatever survives that mismatch moves in either direction, and an amount that can move either way is a second exposure rather than a deduction.

A contract is a standard object. The contract ends on a day somebody else chose, it covers a number of units somebody else fixed, and it is written on a thing somebody else decided was worth writing contracts on. A holding is none of those things. A holding is whatever one particular party happens to have, for exactly as long as it happens to matter to that party. Two objects built to two different specifications were never going to sit flush against each other, and the places where they fail to are where the whole subject lives.

What is the holder actually carrying once the two sides stop lining up?

Almost everybody gets the category wrong at first meeting, and the wrong category then survives for years. So start by getting it right. The leftover amount is not a fee. Nor is it a charge, a haircut, or a percentage taken off the top by anybody. Nobody bills for it and nobody receives it. The leftover amount is a residualWhatever is still at risk once a contract has been written against an exposure. A residual is what did not get covered, not what got charged., and a residual is an exposure: an amount whose size is not settled in advance and which can land on either side of nil.

Now the part worth keeping. Before the contract was written, the holder was exposed to a price, being the price of one unit of the reference asset. After the contract is written, the holder is exposed to something else entirely. The first exposure was to a price, and the second one is to the distance between two prices. Those are not the same kind of thing at all, and swapping the first for the second is not a step up or a step down. The swap is a change of subject.

Here is why that change of subject is the harder one to live with rather than the easier one, and it has nothing to do with size. A price is public. Somebody publishes it, a great many people watch it, and when it moves a long way somebody writes about it. A distance between two prices is nobody's headline. No screen leads with it, no report opens on it, and it moves for reasons that are arithmetic rather than newsworthy. So the holder has exchanged an exposure that was easy to describe and hard to escape for one that is easy to overlook and hard to describe, and the exchange settles nothing about which of the two is smaller.

Step out of finance for a moment. The shape is familiar. A household is selling its old scooter next month and buying a bigger one the same week. Once both legs are decided, what scooters cost in general stops mattering. A rise lifts what the old one fetches and lifts what the new one costs, and a fall drops both. The gap between the two is what matters now, and nobody in the market reports that gap. The gap is the household's own to work out, it moves for its own reasons, and if it is never thought about it will still be carried on the day both trades happen. The holder below is in precisely that position.

The name for that distance, which way round the subtraction is taken, and how it behaves as a final date comes near are covered separately and completely, and taken here as finished work. The plain question from the holder's side is a different one: given that the two sides do not line up, what exactly is left with the holder, and can a number be put on it?

In which three places does a contract part ways with a holding?

Three, and naming them as three is the whole point of this section. A reader who carries away one blurred idea, that a hedge is somehow imperfect, has nothing they can go and check on a Monday morning. A reader who carries away three named joints has three questions with three answers, and each answer is a fact about their own arrangement rather than a mood about hedging.

The first joint is the date mismatchThe contract ending on a different day from the day the exposure stops mattering to the holder.. The contract ends on a day the exchange set, and the holding stops mattering on a day the holder's own circumstances set. Nothing coordinates the two. The second joint is the quantity mismatchA holding that does not divide into a whole number of contracts, so part of it is left uncovered or one contract too many is written.. One contract covers a fixed number of units, the holding is whatever this party has, and the second rarely divides by the first. The third joint is the thing itself. No contract exists on what this party actually holds, or none with a date it can use, so a contract on something adjacent gets written instead.

The three are independent of one another, and a single party can be carrying all three at once. That is not a warning, it is a structural fact about how the two objects are built: the dates come from one authority, the contract size comes from the same authority on a different schedule, and what contracts exist at all comes from whether anybody found it worth listing. None of the three constrains the other two. So a party whose dates line up perfectly may still hold a leftover quantity, and a party whose quantity divides cleanly may still be writing on something adjacent.

Which gives the practical instruction, and it is the only one of them that costs nothing to follow: the three are checked separately and written down separately. A single line in a note saying the cover is imperfect does not say which of the three to go and look at, so it tells the next reader nothing at all. The three are looked at in completely different places: a calendar, a contract specification, and a list of what is listed.

Which price is which, before any of this is worked?

The whole subject turns on a distance between two prices, and a reader who has the two confused will read the arithmetic backwards. So one paragraph of housekeeping comes before a single rupee is worked. Three different quantities get spoken about loosely as the price of the contract. The three behave differently, and each is named specifically at every appearance below.

The first row and the second row look identical on a screen and are not. Read them together. Rs 2,130.00/- is the price written into an agreement that was struck for twelve months when the spot price of the reference asset stood at Rs 2,000.00/-. Work it rather than take it: Rs 2,000.00/- multiplied by 0.065 is a carry of Rs 130.00/-, and Rs 2,000.00/- plus that carry of Rs 130.00/- is Rs 2,130.00/-, at a financing rate of 6.50 per cent a year on a reference asset that pays nothing at all while it is held. Rs 2,130.00/- never moves again. Rs 2,065.00/- is what somebody would quote today for a fresh contract ending six months from today, with the spot price still at Rs 2,000.00/-. The arithmetic runs Rs 2,000.00/- plus Rs 2,000.00/- multiplied by 0.065 multiplied by 0.50, or Rs 2,000.00/- plus Rs 65.00/-. Both figures are arithmetic on a spot price and a financing rate, and arithmetic on today's price is not a forecast of tomorrow's.

One consequence of keeping them apart is worth carrying, and it is settled elsewhere rather than rebuilt here. Suppose the spot price did move, from Rs 2,000.00/- to Rs 1,920.00/-, a gap of Rs 80.00/- struck on a base of Rs 2,000.00/-. The price quoted for a new twelve month contract does not fall by Rs 80.00/-. The financing applies to the new spot price just as it applied to the old one, so Rs 1,920.00/- multiplied by 1.065 is Rs 2,044.80/-, and the quoted price falls from Rs 2,130.00/- to Rs 2,044.80/- for a gap of Rs 85.20/-. The worth of a contract already struck is the other quantity, and that one moves one for one with the spot price: Rs 85.20/- with a full year still to run is Rs 80.00/- once that year of financing is taken back out of it, to the paisa. Both statements are true, and they are true of different things. Each one gets named every time it appears for exactly that reason.

What does a date mismatch leave behind, in rupees?

The date mismatch is the one of the three that these figures can carry all the way to a number. Before the arithmetic starts, one assumption goes on the record and stays there. The spot price of the reference asset is held stillAn assumption that a price does not move at all, stated openly so that everything else in the example becomes visible against it. at Rs 2,000.00/- across this entire instance, deliberately, and every rupee that appears below is therefore the mismatch and nothing else. If the spot price were allowed to move, the reader would have no way of telling which part of the answer came from the price and which part came from the two sides failing to line up, and separating those two is the only thing this instance exists to do.

The situation is ordinary. A holder has one unit of the reference asset, bought at a spot price of Rs 2,000.00/-, and knows already that the unit will be sold at six months. The reference asset pays nothing at all while it is held. Every figure here is therefore built from a price and a financing rate and nothing else. Somebody else set the date, so the contract available runs twelve months, and it was written at Rs 2,130.00/-. No six month contract is listed, and the twelve month one cannot be made to end early. So the position gets carried to six months and ended there.

At six months the holder does two things on the same day. The spot price is being held still, so the unit is sold at the spot price of Rs 2,000.00/-. And the contract position is ended, by closing a positionEnding a contract before its final date by taking the opposite side of the same contract, so the two cancel and nothing is left open. against a contract with six months left to run. The closing price is not Rs 2,000.00/-, and the reason is neither subtle nor surprising once it is said aloud: a contract with six months left still has six months of financing inside its price. Rs 2,000.00/- multiplied by 0.065 multiplied by 0.50 is Rs 65.00/-, so the position closes against Rs 2,065.00/-.

So put the two prices from that one day side by side. The residual is never anything more than that. The holder received Rs 2,000.00/- for the unit. The contract position closed against Rs 2,065.00/-. Rs 2,065.00/- less Rs 2,000.00/- is Rs 65.00/-, and that Rs 65.00/- is the residual on this arrangement, on a day when nothing whatever happened to the reference asset. The residual here is exactly the financing that had not yet run off the contract. A date mismatch and a residual are the same fact stated twice.

Mislabelling that Rs 65.00/- is the commonest way it ends up in the wrong line of a plan, so label it carefully. The figure is a difference between two prices on one day. A payoff is what an obligation settles at on the day it settles, and no obligation settled here, so the figure is not a payoff. A premium is what somebody pays to enter, and nothing was paid to anybody, so the figure is not a premium. A profit needs a purchase price and a sale price on the same thing, and these are two prices on two different things, being one unit today and one contract ending six months from today, so the figure is not a profit either.

The general form survives after every figure above is forgotten. Memorise it in plain words. Any position closed before the contract's final date takes the unexpired financingThe part of the carry still sitting inside a contract price on the day the position is closed, because the contract has not yet reached its final date. still sitting inside the contract price with it. The two prices meet on exactly one day, being the day the contract ends, and on every other day they are apart by whatever financing has not yet run off. A gap of that kind is not a defect in anybody's contract. The gap is what a contract price is made of.

Rs 65.00/- against what, and why does the base decide the answer?

Somebody is going to convert that Rs 65.00/- into a percentage, and the moment they do, the base they pick decides what the sentence means. A residual gets talked about carelessly at exactly this point, and carelessness here reads as a size claim when no size claim has been made.

Rs 65.00/- divided by Rs 2,000.00/- is 0.0325, so the same Rs 65.00/- is 3.25 per cent of the Rs 2,000.00/- of exposure. Rs 65.00/- divided by Rs 160.00/- is 0.40625, so the same Rs 65.00/- is also 40.625 per cent of the Rs 160.00/- posted as initial margin. Both are correct arithmetic on the same rupee amount. A ratio without its base attached is not a small claim or a large one, and it is not a claim at all. Each ratio here is given with the base it was struck on, in the same sentence.

Where does that Rs 160.00/- come from, and what is it? The Rs 160.00/- is initial margin of 8.0 per cent of the Rs 2,000.00/- of exposure, and the 8.0 per cent is an assumed rate rather than a real one. Either figure alone misleads, so the margin reads properly only beside the leverage it implies: Rs 2,000.00/- of exposure standing on Rs 160.00/- posted is 12.50 times. For a sense of the scale that ratio works at, a gap of Rs 80.00/- between a spot price of Rs 2,000.00/- and one of Rs 1,920.00/- is 50.0 per cent of the Rs 160.00/- posted, and the same Rs 80.00/- is 4.0 per cent of the Rs 2,000.00/- of exposure. One gap, two bases, two very different sentences.

None of that establishes how large a residual is. The arithmetic establishes that the answer to how large depends entirely on the base the division is taken against, and that any figure quoted without its base has already lost the argument. The real margin a party posts is not 8.0 per cent: it is set by the clearing corporation under the framework the Securities and Exchange Board of India (SEBI) publishes at sebi.gov.in, it varies by contract and by day, and it moves.

What does a quantity mismatch leave behind?

The shape first, and the shape is something everybody already knows from buying anything sold in boxes. Eggs come by the dozen and seventeen are needed. Somebody who never met the buyer decided the size of the box. One box leaves five short, two boxes leave seven spare, and seventeen eggs cannot be bought at all. The eggs are the entire quantity mismatch, and the finance version adds nothing to it except the word contract.

One contract covers a fixed number of units of the reference asset. A holding is whatever this particular party has. The second rarely divides evenly by the first, and whatever fails to divide is left over. Whatever is left over carries the original exposure in its original form, entirely uncovered, exactly as it was before any contract was written. That is the important sentence, because a reader who has just met a residual as a small distance between two prices will expect the leftover to be something exotic too, and it is not. The leftover is the plain first exposure, sitting there untouched.

And the mismatch runs in both directions, the part people forget. A party can write one contract too many just as easily as one too few. Do that and there is no leftover holding; there is an unmatched contract position, running the other way, with nothing at all behind it. Both are quantity mismatches, both are residuals, and the second one is the more awkward of the two because a position with nothing behind it is a position taken on its own whatever anybody meant by it.

Now the part that matters more than the shape. No figure appears anywhere in this section, and that is a rule rather than an omission. The number of units inside one contract is set by the exchange under the framework the Securities and Exchange Board of India publishes at sebi.gov.in. The number differs by contract, and it moves. A figure written out here would be wrong rather than stale on the day it changed, and a reader who memorised it would carry a wrong number into a real arrangement. How a holding divides once that number is known, and how the covered and uncovered parts are counted, is covered separately.

What happens when no contract exists on the thing actually held?

A mismatch in the thing itself is the version most readers met first, and for many people it is the only version they have ever been told about. A party holds one thing. No contract exists on that thing, or none exists with a date the party can use, so a contract is written on something adjacentA different thing used as the subject of a contract because no contract exists on what is actually being held. instead. The two things then get on with their own lives. Each price moves for reasons of its own, the distance between them moves as well, and that movement, all of it, is what the holder has taken on.

The everyday shape is easy enough. A party needs cover on the vegetables one particular grower brings to one particular market, and the only price anybody quotes is for a different grade sold in a different market three districts away. The choice is that or nothing, so the contract gets written against what is quoted, and from that moment the position turns on how those two things behave relative to one another. The relationship between the two prices is not a fact about the grower and it is not a fact about the other market. A relationship of that kind is a fact about the pair, and it has to be established rather than assumed.

Only one priced thing appears in these figures and there is no second one, so this form of the subject cannot be worked here. Say so plainly rather than stepping around it. There is one invented reference asset here, at one spot price, at one moment. There is no other priced thing anywhere in these figures, no second price series, and nothing whatever recording how any two prices have behaved in relation to each other. Every conclusion in a made up example would come from the numbers chosen to reach it, so the example would look convincing and teach nothing.

So name what a reader would actually need before this form could be worked at all. Naming it is more useful than a fabricated demonstration. Three things. Two priced things, being the one actually held and the one the contract is written on. Both prices recorded over the same stretch of time, on the same dates, so they can be looked at together. And some tested account of what drives the distance between them, established rather than assumed. Two prices that have moved together are not thereby joined. Missing any one of the three, there is nothing to compute and nothing honest to say about size.

The honest summary is short. The form worked above needs neither a second thing nor a history, being a date mismatch on one reference asset with the spot price held still. The commonest form needs two priced things and a record of how they have moved together, and no arithmetic on one asset can stand in for that.

Does the residual always cost the holder something?

No. A reader has usually decided by this point that it does, and the decision is wrong. Everything so far has described something left over after an arrangement was meant to tidy things up, and left over sounds like lost. It is not. A residual is a difference nobody fixed, so it lands in the holder's favour precisely as readily as it lands against them.

Go back to the worked instance and see how thin the certainty actually was. The Rs 65.00/- came out at exactly Rs 65.00/- because two things were pinned down: the spot price was held still at Rs 2,000.00/- by assumption, and the financing rate was taken as 6.50 per cent a year with exactly six months left to run. Change the day the holding actually stops mattering and the amount of financing still inside the contract price changes with it. Change the financing rate that applies on the closing day and it changes again. Neither of those is fixed in advance on the day the arrangement is put on, and neither of them is something the holder sets. The holder could as easily find the two prices closer together on the closing day as further apart.

And the direction turns on which side is carried. The residual in the worked instance sits between a price received and a price closed against, and whether that distance falls to the holder's advantage or against it depends on which side of the contract the holder is on. Nothing in the arithmetic assigns it a sign in advance. The sign is not a technicality, it is the definition doing the work: an exposure is something that moves, and a deduction is something that is subtracted, and treating the first as the second is the single error this whole section is here to prevent.

The sharper version is worth carrying into any plan. A party who treats a residual as a known cost will be wrong twice rather than once. Everybody has already imagined and half expects the occasions when it turns out larger than the figure written down. Almost nobody has imagined the occasions when it lands the other way, and those are the more damaging of the two. The second case is taken up below.

Why does no size for a residual appear here?

Two questions arrive together at this point in any honest treatment of the subject. How big is a residual like this, usually? And how often does it actually matter? Both are exactly the right questions to be asking. Both need a measurement these figures do not hold, and the reason is the same for both.

How big needs a run of both prices over a stated stretch of time, and a measure of the distance between them across that stretch. How often needs a record of occasions on which it mattered, with the occasions counted. The figures here amount to one price, for one thing, at one moment, together with one financing rate. There is no run of anything, no record of two things moving in relation to each other, and nothing whatever that happened. Not a thin version of those things. None of them.

So no typical size, no usual rangeThe honest form for an amount whose size is not settled in advance: two ends rather than one figure. and no frequency can be got from the arithmetic above, and estimating any of the three from a single worked instance would be guesswork. The instance above is one closing date on one reference asset with the spot price held still on purpose, and generalising from it would be the same error as reading a forecast out of a carry calculation.

A typical size quoted from nowhere is precisely the number a reader would remember, repeat and rely on. The restraint is the teaching, not a limitation of it. Everything else here can be checked by the reader against the arithmetic in front of them. A typical size could not be checked against anything, and it would outlive every careful sentence around it.

Which question about a residual has no answer in these figures?

Whether it is small enough to live with. The question is completely reasonable, and it is the one a reader asks next. Answering it would need a size, a size needs a run of two prices over time, and no such run exists in these figures. A missing measurement is not the same thing as caution, and the difference matters: a missing measurement can be gone and got, while a general reluctance cannot.

The shape of the question a reader now takes elsewhere is worth more than a number would be. Which of the three mismatches applies, and whether it is one of the three or two of them or all three. On what dates, being the day the contract ends against the day the exposure stops mattering. Over what quantity, being units held against units covered. And against what thing, being whether the contract is written on what is actually held or on something adjacent to it. Alongside those, what the Securities and Exchange Board of India sets down at sebi.gov.in about the dates a contract runs to, when a position that has not been closed goes to delivery, and the conditions on which a position is treated as a hedge rather than as a position taken on its own.

And the closing line. Readers usually expect it to cut in one direction, and it cuts in both. Being able to name a residual is not a reason to accept one. Naming one is also not a reason to abandon an arrangement. Naming it establishes what is still being carried and where to go and look, and what to do about it is a decision that needs facts belonging to the particular arrangement.

The reference asset, the holder, every price, every rate and every unit count here are invented.
Educational material. Not advice on any investment, tax, budget or market position.