FX Exposure: Five Positions and Three Ways to Add Them Up

This tool puts five currency positions on one screen and adds them up three ways. At Vindhya Commercial Bank Limited, invented, the same five rows report Rs 72 crore, Rs 192 crore or Rs 312 crore against a Rs 240 crore cap, being 30.0, 80.0 or 130.0 per cent. Only the middle answer is the bank's own.

Left alone the tool holds this bank's book at month 12, and here it is in plain text. The dollar row is Rs 360 crore of assets against Rs 216 crore of liabilities, a net longA balance held in a currency, producing a gain when that currency rises against the reporting currency. of Rs 144 crore. The euro row is Rs 84 crore against Rs 180 crore, a net shortA balance owed in a currency, producing a gain when that currency falls against the reporting currency. of Rs 96 crore. Sterling is Rs 96 crore against Rs 60 crore, long Rs 36 crore; the yen row is Rs 36 crore against Rs 60 crore, short Rs 24 crore; the dirham row is Rs 24 crore against Rs 12 crore, long Rs 12 crore. Assets of Rs 600 crore less liabilities of Rs 528 crore is Rs 72 crore net long, and adding the five row nets gives the same Rs 72 crore by a different route. The three long rows make a long sideThe sum of the rows whose assets exceed their liabilities, being Rs 192 crore on this book. of Rs 192 crore and the two short rows a short sideThe sum in absolute terms of the rows whose liabilities exceed their assets, being Rs 120 crore on this book. of Rs 120 crore. Added with their signs the book reads Rs 72 crore, taken as the greater of the two sides it reads Rs 192 crore, and added in absolute terms it reads Rs 312 crore, being 30.0, 80.0 and 130.0 per cent of the Rs 240 crore cap of limit L7. A 5.0 per cent adverse move costs Rs 3.6 crore, Rs 9.6 crore or Rs 15.6 crore on those three totals.

The open position itself is covered separately. The subject here is narrower. Five positions nobody disputes, in five currencies nobody disputes, on one day nobody disputes, and the single figure the institution reports is not read off the book at all. A rule produces it. Moving the selector on the tool sends the figure from a third of the cap to a live breach with not one rupee of position changing hands. Adding a column up feels like a step rather than a decision, and that is the trap this guide exists to spring: somebody at this bank chose one of three arithmetics, wrote the choice into a policy, and every report since has printed the output of that choice as though it were an observation.

What is actually printed on the report this tool starts from?

Here is the artefact. At month 12, Vindhya Commercial Bank Limited, invented, is holding balances in five currencies other than the rupee, all stated in Rs crore equivalent so nothing has to be converted twice. The report gives each row a currency, a side and an amount, and above the rows sits limit L7, the bank's own cap on the net overnight open positionThe single figure a bank reports across all of its currency positions at the close of a trading day., at Rs 240 crore.

Everything anybody argues about here is missing from that report: it lists five rows and a cap and says nothing at all about how the five become one. The drawing below ends on that gap, in the dashed box at its foot, and the missing line is the selector on the tool above.

Where each number on this tool is found

The tool prints the provenance of each figure beside the box it goes into, and the table gathers the same notes in one place. Each says only where somebody would go to fetch a number, never what it means.

Why does the way the rows are added change the answer more than fourfold?

Three arithmetics are defensible on those five rows, and the tool shows all three at once rather than requiring a choice first. The algebraic sumAdding the rows with their signs, so a short is subtracted from a long. is what a spreadsheet returns when the total is dragged down a column of signed numbers: 144 less 96 plus 36 less 24 plus 12 is Rs 72 crore, being 30.0 per cent of the cap. The greater of the two sidesThe rule that reports whichever of the long total and the short total is larger, and ignores the smaller one. takes the long side of Rs 192 crore against the short side of Rs 120 crore and reports Rs 192 crore, at 80.0 per cent of limit L7, and that is what this bank reports and what its policy records. The sum of the absolutesAdding the rows ignoring their signs, which assumes every currency moves against the holder at once. is 144 plus 96 plus 36 plus 24 plus 12, being Rs 312 crore and 130.0 per cent. A figure that size would be a live breach.

One book, one day, five rows, and utilisation reads 30.0, 80.0 or 130.0 per cent depending only on how they are added up. The spread between the smallest and the largest is Rs 240 crore, the entire size of the cap. Nothing in the market moved. Nobody dealt.

The temptation the algebraic sum represents is completely natural, so the household version is worth holding on to. A person has promised to pay a US dollar tuition instalment for a nephew studying abroad, and separately a cousin has promised to send roughly the same amount in euros next month. Writing those two down side by side and calling the position flat is very tempting. The position is not flat. A dollar exposure and a euro exposure sit side by side, and the two only cancel if those two currencies always move together against the rupee, by the same amount, on the same day. If they always did that, they would be one currency and not two.

Notice what each arithmetic quietly assumes about the world. The algebraic sum assumes perfect correlation between every currency: they all move together, so a short in one is genuine cover for a long in another. The sum of the absolutes assumes the opposite extreme: every one of the five turns against the bank on the same day, so nothing covers anything. The greater of the two sides sits between them and answers a narrower question. How much of the book would be uncovered if every currency moved the same way? The narrower question has an answer even without a correlation estimate, and since this case carries no correlation estimate, the middle rule is the practical one here.

What moves the reported figure without anything moving underneath it?

Two switches on the tool produce the two mistakes this subject is known for, and both are worth making on purpose. Move the arithmetic selector from the greater of the two sides to added with their signs, and the reported figure falls from Rs 192 crore to Rs 72 crore, from 80.0 per cent of the cap to 30.0. Nobody dealt. The euro short is being treated as cover for the dollar long. The short is cover only if those two currencies always move together against the rupee. Now move the second switch and fold the Rs 480 crore net investment in the overseas branch in as though it were a currency balance. The long side becomes Rs 672 crore and the reported figure 280.0 per cent of the cap, a breach getting on for three times the size of the limit, on an investment that will not be settled in any currency on any morning. Neither switch moves the line underneath: assets less liabilities across the five currency rows is Rs 72 crore net long before and after, and the tool says so on screen while the headline travels between 30.0 and 280.0 per cent.

The tool therefore proves its own arithmetic on screen rather than asserting it. Assets less liabilities gives each row its net and its side. The five nets add to Rs 72 crore net long, and total assets of Rs 600 crore less total liabilities of Rs 528 crore give the same Rs 72 crore by a route that never looks at an individual row. The long side less the short side gives it a third time, and the long side plus the short side gives the gross of Rs 312 crore. Every one of those lines is checked against a second calculation as the figures are entered, so the word beside it is an agreement and not a decoration.

Which position can be closed without moving the reported figure at all?

Take each row to zero in turn and hold the other four still, and the results do not run in the order of the row sizes at all. Closing FX1, the Rs 144 crore dollar long, drops the long side to Rs 48 crore and the reported figure to Rs 120 crore at 50.0 per cent. Closing FX3 takes it to Rs 156 crore and closing FX5 to Rs 180 crore. Closing FX2 or FX4, the two short rows, moves it by nothing whatever: the long side is still Rs 192 crore, and the long side is what the measure takes.

The smallest position in the book moves the reported figure and the second largest does not. Read that back against the sizes: Rs 12 crore of dirham is worth more to this control than Rs 96 crore of euro and Rs 24 crore of yen put together. The pair is Rs 120 crore of position closed for a movement of exactly nothing.

The behaviour is not a defect in the measure, and it is important to say so before anybody starts calling it broken. A greater-of rule is built to answer one question: how much would be uncovered if every currency moved the same way against the bank. Whichever side is larger is the answer to that question, and the smaller side is simply not the binding question today. The fault is never in the arithmetic. The fault is managing to a number without knowing which half of the book it is looking at.

What is the floor under this measure, and why can no amount of selling break it?

Suppose the treasurer is told to bring the reported figure down as far as it will go and may touch only the long side. Sell the dirham and the figure goes from Rs 192 crore to Rs 180 crore. Sell the sterling too and it goes to Rs 144 crore. Sell the whole dollar long, so the bank has no long currency position left at all, and it stops at Rs 120 crore and will go no lower. The short side, untouched by any of this selling, is Rs 120 crore. Once the long side falls beneath it the short side becomes the greater of the two, and the measure stops responding.

A greater-of measure carries a floorThe level below which the reported figure cannot fall while the other side is left unchanged. set by the side nobody is managing, and here that floor is Rs 120 crore, being 50.0 per cent of the cap. A desk instructed to halve the reported number can get there, exactly, by closing the entire long book. A desk instructed to take it to a quarter of the cap cannot get there at all, however much of the long book it sells, and the reason is arithmetic rather than market.

What shape does the reported figure trace as one position is turned up?

Turn the dollar long upward from zero and hold the other four rows still, and the measure traces a shape with a corner in it. From a dollar long of Rs 0 the long side is Rs 48 crore, made of the sterling and the dirham. A long side that size sits below the Rs 120 crore short side, so the reported figure stays flat at Rs 120 crore and 50.0 per cent however many dollars are added. At a dollar long of Rs 72 crore the long side reaches Rs 120 crore exactly, the two sides are equal, and that is the kinkThe point where the two sides are equal and the measure changes from flat to rising one for one.. Past that corner every further rupee of dollar long is a rupee on the reported figure: Rs 192 crore at the locked Rs 144 crore, the cap exactly at Rs 192 crore of dollar long, and Rs 276 crore at Rs 228 crore. Breach B2 reached that last figure on month 9 day 2 and was squared the following morning.

The measure is flat and then straight, and the flat stretch is exactly where the control is not watching this half of the book at all. A matching corner sits on the other side, and a treasurer adding short positions will meet it. Where that corner sits is solved under how much room is left.

Every solved point on this tool, written out

Here is the whole of the arithmetic above as plain rows, so none of it depends on a picture rendering.

How much more exposure would take this cap to its limit, and from which side?

A treasurer sizing the next deal asks how much room is left. The report says Rs 192 crore against Rs 240 crore, so the obvious answer is Rs 48 crore, and that is right on one side only. Rs 48 crore added to the long side takes it to Rs 240 crore and the reported figure exactly to the cap. The same Rs 48 crore added to the short side takes it to Rs 168 crore, still under the long side, and moves the reported figure by nothing. From the short side the cap is Rs 120 crore away: Rs 72 crore to draw level with the long side at Rs 192 crore, and Rs 48 crore after that.

The headroomThe distance between the reported figure and the cap, which here is Rs 48 crore on one side and Rs 120 crore on the other. in this control is asymmetric by a factor of exactly two and a half, and a reader who does not know which side is bindingWhichever side is larger, and therefore the only side a greater-of measure can see on a given day. will size the next deal wrong. Sizing a deal off the headline number alone gets one side of the book right and the other side wrong by Rs 72 crore. On a Rs 240 crore cap that error is 30.0 per cent of the whole limit.

What does a 5.0 per cent move cost on each of the three totals?

One scenario sizeThe 5.0 per cent adverse move used here, which is the invented bank's own working assumption and is never a statement about any currency. applied to three totals produces three losses, and the gap between them is the gap the three arithmetics carry: Rs 3.6 crore on the algebraic Rs 72 crore, Rs 9.6 crore on the reported Rs 192 crore and Rs 15.6 crore on the gross Rs 312 crore. Same move, same book, and the only thing that differs is which total the percentage is taken on.

Name the object every time that last figure is printed. Rs 15.6 crore here is a currency scenario loss with every currency moving at once, and the same digits appear elsewhere in this bank as a one day distributional measure on its Rs 3,600 crore trading book, a completely different object. Two numbers that look identical and answer unrelated questions are how a committee paper goes wrong quietly, and the discipline that prevents it is trivially cheap: write the figure with its object attached, always.

What can this tool not compute, and why?

Inventing an answer would be worse than refusing one, so a tool that computes three things exactly should be equally exact about what it cannot reach. There is no currency value at risk here: producing one needs a volatility for each currency and a relationship between them, and this case carries neither. The 5.0 per cent move carries no probability. The move is a stated scenario size and nothing more. And there is no intraday figure. Every row is a month 12 end of day position, and breach B2 shows why that matters: the position reached Rs 276 crore and was squared the next morning.

The honest boundary of a tool is part of the tool, and the three things above are refused rather than approximated. Notice how narrow the refusal is: it is not that these measures are wrong or unknowable in general, it is that this particular case carries no input from which they could be built.

Who actually picks this tool up, and what do they do with it?

Three people read the same five rows for three different things, and only one of them cares about the headline figure at all. Devendra Achar, head of treasury at this invented bank, is the risk owner when limit L7 is crossed, and he uses the tool the way a driver uses a wing mirror: to size the next deal. He has to know which side is binding before he quotes. Quoting against the wrong side either wastes the limit or breaches it. The limit monitor who produces the overnight position report uses it for something duller and more useful. A fall in the reported figure can come from an actual reduction of exposure, or from the short side quietly growing past the long side. The second lowers nothing and looks identical on the report.

The market risk committee reads it for a third thing, the one the headline never shows. A figure of Rs 192 crore at 80.0 per cent is a comfortable-looking number, and a committee that never sees the two side totals underneath it cannot tell a book of one large long from a book of five positions netting to the same figure. The two books are different institutions in every practical sense, and the same control reports them identically. The practitioner habit worth carrying away is simple: never accept the aggregate without the two side totals beside it. The aggregate is a rule applied to the sides and not a fact about the book.

The household version of that habit is exactly the same shape. A person with a home loan floating on one rate and a small deposit earning another will often say they are square because the two amounts match. The borrower is not square. The two rates do not move together, and one of the two sides is going to be the one that hurts. Which side is the larger, and which one is actually being managed, is the whole question this tool exists to ask.

Where does the method come from, and what binds an Indian bank?

The mechanism in this guide is jurisdiction free. A set of balances in currencies other than the reporting one, and a rule for turning them into a single number, is arithmetic and would work identically anywhere. Jurisdiction decides which rule an institution must use, how large the resulting figure may be, and what has to be reported to whom.