Cross-Currency Basis: The Spread That Should Not Exist

Cross currency basis is the gap between what two routes to the same borrowed currency actually cost. Borrow that currency directly, or borrow at home and swap into it: both leave the borrower holding it, paying for it and owing it back. The two should come to the same figure. The two figures do not agree, and the difference is written into one leg of the arrangement as an addition to its rate.

Anybody who could take both routes would keep taking the cheaper one until the difference stopped being there. Two ways of arriving at the same place should therefore cost the same, and that pressure is the entire reason a difference between them is worth a name. The name records that the difference does not disappear, and a difference that refuses to disappear says something untrue about the institutions that would have to close it rather than about the arithmetic they would be using.

One thing to settle before the first block opens. A cross currency basis is a difference between two prices, so putting a size on one takes both routes priced on the same day, in two currencies, at a single exchange rate. Those are market observations, and a market observation is a different kind of thing from a mechanism. The mechanism is what follows: what a basis is, where it is written down, and why it is there at all.

Chitrakoot Cements Limited, an invented manufacturer, gives the reasoning somewhere to land. Under the one agreement this material carries, Chitrakoot Cements holds the fixed side: 7.20 per cent a year goes out of it, and what comes back in is whatever the floating benchmark reads for the period. Chitrakoot Cements is the party standing in front of a choice throughout, deciding which of two ways into a borrowed currency it is going to take.

What are the two routes, and what do they have in common?

Start with what a company actually wants. A company wants a quantity of some currency it does not have, for a stretch of time, and it is willing to pay for the use of that quantity. Call that a funded positionCurrency held because it was borrowed, with a date by which it has to go back to whoever lent it.. The company holds the currency, the currency is not its own, and there is a date on which the whole lot has to go back.

There are two ways to get there. The first is the obvious one. Go to somebody who lends in that currency, borrow it, take delivery of it, pay interest on it in that same currency, and hand it back on the last date. Nothing crosses a border twice, nothing is exchanged, and the whole life of the borrowing happens in one money.

The second way starts somewhere else entirely. Borrow at home, in the currency the company already deals in. Now enter a cross currency arrangement: hand the home currency over at the start and take the wanted currency back in exchange. Pay interest during the life of the thing in the currency that came in. Then, on the final date, swap the two amounts back the other way, so the home currency returns and the wanted currency goes out again. The home borrowing is repaid with the home currency that came home.

At the end of both routes the company is holding the same currency, has paid for the use of it, and owes it back on the same date. That is the whole reason a comparison between them is even meaningful. If the two routes ended in different places there would be nothing to compare, and any difference in cost would just be the price of a different outcome. The two routes do not end in different places. Both end in exactly the same one.

Here is the version anybody can feel. Suppose a household needs a sack of rice. One route is to walk to the shop and buy it. The other is to hand money to somebody who goes to the shop instead, carries the sack back, and charges for the trip. The rice on the table is the same rice either way. The cost of the errand is not, and that difference is the subject.

The finance version is harder than the rice version because the cost of the second route is not a fee somebody quotes at the door. The cost is spread out. The cost sits inside a rate, gets paid a little at a time across every period, and is mixed in with the interest that was going to be paid anyway. A reader who is looking for the cost of the errand will not find a line called the cost of the errand.

Why should two routes to the same place cost the same?

The argument deserves a full and fair hearing. It is a good argument, and what matters is the point where it stops holding rather than any silliness in it.

Suppose the swapped route is persistently cheaper. Somebody notices, borrows at home, swaps into the wanted currency, and then lends that currency out to whoever would otherwise have borrowed it directly. The borrower who spotted the gap pays the cheap cost and receives the dear one, and the difference stays with that borrower. Nothing about that sequence requires a view on where any rate is going. The sequence requires only that a person can do both halves and chooses to.

Now watch what repeating the trade does. Every time somebody takes the cheap route, demand for that route rises and its cost is pushed up. Every time somebody lends through the dear route, supply there rises and its cost is pulled down. The two costs walk toward each other. The walking goes on until nothing is left in the difference worth the trouble, and at that point the two routes cost the same and the loop stops turning by itself.

The convergence argument is the same reasoning that makes a forward price arithmetic rather than opinion, pointed at a funded position instead of at an asset. The same reasoning appears earlier in this material and does the same job there: it produces a number without anybody having to predict anything, purely from the fact that two ways of reaching one outcome cannot sit at different prices while somebody is free to take both.

Look hard at the last clause of that sentence. The hinge of the whole argument sits there. The argument does not say the two routes are equal. The argument says the two routes are pushed into equality by a person acting. Take the person out and the argument produces nothing at all. Every box in the loop below is an action, and an action needs somebody who is permitted, funded and willing to perform it.

What kind of quantity is a basis, and what is it measured in?

Most of the confusion around the subject begins here, and it is a confusion about units rather than about mechanism.

A basis is a spread. A spread is an addition to a rate. So a basis is measured in fractions of a percentage point. On its own a basis is not an amount of money at all. A basis is a rate, and a rate is an instruction about how to turn a principal into money, not the money itself. A claim that a basis is small says nothing about how much it costs, and a claim that the notional under an arrangement is enormous says nothing either.

Three things have to be supplied before any spread becomes a rupee figure. A principal for the rate to be applied to. A period over which it is held. And a method of counting the days in that period, turning the calendar into a fraction. Miss any one and there is no amount. Supply all three and there is exactly one amount, and it is not open to argument.

The one agreement in this material shows all three, and it is worth following there before imagining it happening for a cross currency basis. Under that agreement the notional is Rs 1,000 crore, or Rs 10,00,00,00,000/- written out to its last digit. Written into it as the fixed rate: 7.20 per cent a year. Standing against that, and for the opening period only, a floating benchmark reading of 6.00 per cent. The two sit 1.20 percentage points apart.

The arithmetic then runs. Rs 1,000 crore, with 1.20 percentage points applied to it, held for one whole period and counted as a full one, so the day count fraction is 1.0000 and changes nothing. The amount that drops out is Rs 12,00,00,000/-, or Rs 12.00 crore. Turned round, the arithmetic runs backwards just as cleanly: that Rs 12.00 crore divided by the Rs 1,000 crore it was built on gives 1.2 per cent back, the gap it started from.

The same figure arrives by the longer road, and it is worth seeing once. The fixed rate applied to that notional puts the fixed leg at Rs 72,00,00,000/-, or Rs 72.00 crore. The benchmark reading applied the same way puts the floating leg at Rs 60,00,00,000/-, or Rs 60.00 crore. Both obligations fall due on one date and settle in one money. Only that shared date and shared currency let a single difference of Rs 12.00 crore stand in for two separate leg amounts. The Rs 12.00 crore travels from the fixed side to Saranga Capital Limited, an invented counterparty holding the floating obligation under the same agreement.

Should the benchmark land somewhere else in a later period, the difference would be a different size and might well point the other way. No later reading is written into the agreement, so the arithmetic above belongs to the first period alone.

A cross currency basis becomes money by those same three steps, and not one of the three has a value behind it in this material. The size of the spread would be a statement about a market. The principal would belong to a cross currency arrangement, and the only agreement worked out above spans a single currency. The counting method comes with the arrangement, so it goes missing along with the principal. Three gates, all shut, and no amount can come out the other side of them.

Where does the spread actually sit inside the agreement?

A reader who has only ever met the basis as a market word will not recognise it when they are handed the document. The basis does not announce itself, and it does not look like a charge.

Here is where it lives. A cross currency arrangement defines a rate for each of its two legs. The basis is written into one of those two definitions, as an addition to the rate that leg pays. The whole of it is a few characters inside one clause.

Everything else in the document carries on unchanged. The two principal amounts still move at the start, one in each direction. Interest is still paid in each currency across the life of the thing. The two principal amounts still swap back at the end. The calendar is still the calendar and the payment dates are still the payment dates. Nothing about the shape of the arrangement is different because a basis is in it.

The basis is not a fee and it is not a separate payment. There is no clause that says a charge of so much is payable on signing. There is no invoice, no arrangement fee, no line itemA named row on a statement or a plan, with its own heading that lets somebody scanning down the sheet see it and account for it. anywhere with the basis written next to it. The basis is paid a slice at a time, on each date money moves, right through to the last of them, folded into a rate that was going to produce a payment anyway.

One practical consequence is worth stating plainly. A careful reader is exactly who it catches out. A reader who goes through a cross currency agreement looking for what it costs to use the swapped route, and who looks for a charge, will conclude honestly and wrongly that there is no charge. The cost is real, it is contractual, and it is sitting inside a rate definition three clauses above where the search was being made.

Why do the two routes not come to the same cost in practice?

Go back to the argument and read what it needs rather than what it concludes. The argument needs somebody who can take both routes, freely, in size, on terms as good as anybody else gets, and for as long as the gap lasts. Four requirements, quietly stacked, and each one of them is a place where the argument can give.

Take the first. Not everybody can borrow directly in every currency. Borrowing in a currency means finding lenders who hold that currency and who are prepared to lend it to that borrower specifically. Whether such lenders turn up depends on who the borrower is, where it sits and what they already know about it. The institutions that can do it easily in several currencies are a small set, and they are usually not the ones with the strongest reason to want that currency for their own business. The people who most want the currency and the people who can most easily get it directly are two different groups.

Take the second. Running the swapped route uses up balance sheet capacityThe finite room a firm has on its own books for carrying one more position before something else has to come off to make space for it. on the books of whoever takes it. A cross currency arrangement is a position with a counterparty on the other end of it, running for years, and it occupies room that is not unlimited. Notice what that does to the argument. The bigger the gap grows, the more of the trade somebody would want to do, and the more of that scarce room it would eat. The very trade that is supposed to close the gap gets harder to do at exactly the moment the gap makes it worth doing.

Take the third. The terms on offer are not the same for everyone in the queue. Two parties looking at the same pair of routes can face different costs on each of them, so a gap that is worth closing for one of them is not worth closing for the other, and the one who can see it may not be the one who can act on it.

Take the fourth. Demand for funding in one currency does not turn up conveniently matched by demand in the other. There can be a long line of parties wanting the same direction at the same time, with nobody wanting the opposite side of it, and a swap needs two sides.

A gap that persists is evidence about who is able to act, not evidence that the arithmetic behind the argument is wrong. That is the sentence to carry out of this block. The convergence reasoning is not broken. The convergence reasoning was always a conditional statement, and the conditions are facts about people and institutions rather than facts about numbers.

What does the sign of a basis mean, and which way round can it go?

A basis is written against one leg, so it carries a direction. Adding it to one leg is not the same statement as adding it to the other, and the sign is what records which of the two routes is the dearer one for the party taking it.

Both directions occur. Neither of them is the natural state that the other is a departure from. The point is worth saying because the vocabulary invites the opposite reading: people talk about a basis widening or being negative in a tone that suggests one direction is normal and the other is a disturbance. The mechanism does not support that. The gap comes from who can act and who wants what, and both of those can point either way.

Which brings up the discipline that matters more than anything else in this block. A basis quoted without its currency pair, its maturity, the leg that carries it and the date it was taken is a number nobody can use. A basis like that is unusable in exactly the way a rate quoted without its period is unusable, and for the same reason: the figure is meaningless until what it is attached to is known.

Work through the four. The currency pairThe two currencies a quote or an arrangement sits between, always named as a pair. A rate for one currency by itself settles nothing. matters because a basis is a statement about a relationship between two currencies and not a property of either one. The maturity matters because there is no single basis for a pair: a one year arrangement and a ten year arrangement are different arrangements with different pressures behind them. Which leg carries it matters because the same economic fact can be written on either side with the sign turned round. And the date matters because a basis moves, so an undated one is a photograph with no timestamp.

The sign and direction for any currency pair are statements about a market, and no market is recorded anywhere in this material.

Why is no size stated for any basis?

Measuring a basis means pricing both routes and taking the difference. Neither route is priced here, and nothing in this material could price them. There is no exchange rate. There is no rate in any second currency. Even for the currency that is here, the readings the floating benchmark will show after the opening period were never written down, so there is nothing to build a cost out of over the life of anything.

So the difference is explained without a number attached to it, and the reason is worth spelling out.

A basis is quoted in small units. A plausible small number is the easiest kind of figure in all of finance to invent, and the hardest kind for a reader to challenge. A wrong rupee figure in the crores announces itself; somebody notices it does not fit. A wrong figure in fractions of a percentage point looks exactly like a right one. An invented spread sits on the screen carrying the authority of everything around it, and a reader is given no means of telling it apart from a figure that came from somewhere real.

A reader has no way of separating an invented spread from an observed one once it is on the screen.

The list of what has to be fetched instead is short, and fetching it is an errand rather than a complaint. Both routes priced, on the same date, for the same currency pair, at the same maturity, from a source that states where its figures came from. Four conditions, each checkable by whoever does the fetching, and that checkability is the whole of what provenanceThe record of where a figure came from and how it was arrived at, kept alongside the figure so that somebody else can go back and test it. buys.

The two route ledger, and the row that stays empty

Here is the worked instance, built out of structure rather than out of prices. Chitrakoot Cements is standing in front of the two routes. Every row that can be described in words is described in words, on both sides, and the one row that would need a price is the one left blank.

Read down the ledger and notice how much of it is genuinely knowable. The currency borrowed, what changes hands on the first day, what is paid across each period and in which money, what has to go back at the end. Every one of those is a matter of what the arrangement says, and every one of them can be written out without a single market observation. Then the fifth row arrives and the ledger stops. Filling it needs two prices, and a price is not a description.

Why is the gap not fitted to a moving control?

A slider moves one input and redraws a picture. Nothing lets a reader feel a relationship faster, and many relationships in this subject are shown that way. The gap between two routes does not admit that treatment, and the reason follows.

The relationship worth showing would be how the gap responds to pressure on the people who would close it. Building a control for that means inventing a size for the gap and inventing a scale for the pressure, then drawing a line between two things that were both made up. A slider like that would assert that the gap has a size, that the size responds in a stated direction, and that it responds by a stated amount, and this material supports none of the three. It would also be the most persuasive object in view, because a thing that moves under a hand feels measured in a way that a paragraph never does. A ninth question sits below instead.

Who decides which of the two routes is even open to a borrower?

The gap is evidence about who can act. Conditions set by an authority are part of what decides who can act, so they are mechanism here rather than paperwork wrapped around a mechanism.

Two conditions bear on the routes directly. Sending money out of the country is not something a company simply decides to do. The first condition sets what may go out across the border, in what circumstances, and through which channel. The second is whether a party may enter a cross currency arrangement at all, and what underlying needThe commercial reason a party puts forward for entering an arrangement, as against entering it for its own sake. it has to show for wanting one.

Notice what those two conditions do to route two specifically. If a party cannot show the required commercial reason, route two is not expensive for that party. It is shut. And a route that is shut cannot be part of a comparison, let alone part of a loop that closes a gap. The four supports under the convergence argument have a fifth underneath them: whether the party in question is allowed to stand on any of the four.

Who ends up holding this problem, and on which piece of paper?

Abstract until now. Here is who actually meets a basis during a working week, described by the document each of them has open when it happens.

The first has a funding planThe document a finance team writes ahead of time, setting out where money will be borrowed and what that borrowing is expected to cost over the year. open. Somebody in a treasury team is filling in what next year's borrowing will cost, currency by currency. If the company is going to hold a currency it does not earn, the plan needs a cost for it, and the cost depends entirely on which of the two routes the company will actually use. Fill that cell in from the direct route and then take the swapped one and the plan is understated from the first day. The funding plan is the document where the failure below begins.

The second has a set of published borrowing costs open. An analyst is comparing what two companies pay to fund themselves, and one of them funds partly in a currency it does not earn. One of the two carries a route inside it that the other does not, so comparing their headline borrowing rates side by side treats the two as measuring the same thing when they are not. The analyst's job is not to compute the gap. The job is to notice that a gap belongs there and to say so, rather than presenting a comparison that quietly assumes it away.

The third has a credit file open. A lender assessing a borrower who funds through the swapped route is looking at a cost that has more moving parts than a directly borrowed one: a rate, a spread, an exchange back at the end, and a party on the other side of the arrangement who has to be there on the last date. None of that makes the borrower worse. The extra parts make the borrower's funding cost something with more joints in it, and every joint is a question the file should have an answer to.

The fourth has a fee demand open, and there is no company involved at all. A household paying for a child's education abroad faces the same two routes in miniature. Send rupees on the day and let somebody convert them, or hold the other currency in advance and pay from that. The fee is identical either way and the total that leaves the household is not. Anybody who has compared what two different channels take for the same transfer has already met the shape of this subject, without the vocabulary attached to it.

Every one of those four is filling in a cell that needs a figure no reference text can supply, and the useful thing to carry away is that the cell exists at all. Knowing there is a gap, knowing it has four qualifiers and knowing where it is written into a document is the difference between fetching the right figure and never asking for it.

Does a gap between two routes mean anything a reader can act on?

A reader who has followed the argument will arrive at a question, and it is a fair one. If two routes to the same place cost different amounts, is the difference something somebody can take? Is this an arbitrage?

The question has no answer in either direction. A claim that the gap is available to somebody, and a claim that it is not, both need a size, a sign and a date for the basis, and a claim without those has nothing to stand on.

Whether anybody should be inside an arrangement like this depends on facts about a particular person that no reference text holds. Which of the two routes is genuinely open to them. How much of their own balance sheet the swapped route would consume. Which conditions attach to them specifically, given who they are and what they do. And what happens to them if the party on the other side of the arrangement is not standing there on the final date. Understanding the machinery is a reading skill; it settles nothing about whose hands the machinery belongs in.

There is one more thing worth holding, and it is about the shape of the reasoning rather than about currencies. A gap between two routes is a fact about the world that has to be explained, and the explanation is almost never that everybody else has failed to notice it. Where a difference appears that theory says should not be there, the productive question is which of the theory's quiet assumptions is doing the work, and who exactly is being stopped by it. A question shaped that way has an answer somebody can go and check. The other one usually does not.

Chitrakoot Cements Limited and Saranga Capital Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.