The Currency Hedge: An Exposure You Did Not Choose

A currency hedge offsets the exchange rate part of a holding priced in another currency, leaving the local asset return less what the offset charges. The two returns multiply rather than add: a 10.0 per cent local gain with a 4.0 per cent currency gain is 14.4 per cent, not 14.0. Anybody holding a foreign asset carries two positions, and only one was chosen.

One feature makes the currency position unlike the rest of a portfolio. Almost every other exposure arrives because somebody decided to take it. Somebody chose the equity weight, somebody chose the credit standing, somebody sat in a room and argued about a holding until it was bought. The currency position arrives without any of that. The currency position walks in attached to a purchase made for a completely different reason, it sits there for as long as the holding does, and in most reporting it never appears as a line at all.

The running example is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy weights are equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore and cash 10.0 per cent at Rs 50 crore, and the actual weights drift away from those between one rebalancing and the next.

The Anantara portfolio holds nothing priced abroad, so every foreign figure below is a stated hypothetical and is marked as one wherever it is used. Suppose the mandate permitted a foreign listed slice of Rs 30 crore. A slice of that size would be 10.0 per cent of the Rs 300 crore equity sleeveThe part of a portfolio set aside for one asset class. A share of a sleeve and a share of the whole portfolio are different numbers, so the base has to be named each time. and 6.0 per cent of the Rs 500 crore portfolio. The allocation stays at 60, 30 and 10 throughout.

Where does a currency exposure come from when nobody chose one?

From the purchase itself, and from nothing else. To buy an asset that is priced in another currency the buyer must first hold that currency, so the transaction converts rupees into it and then buys. The asset will one day be sold back into rupees at whatever rate exists then, so from that instant the holder carries the asset and carries the currency. A currency exposureThe part of a holding's rupee value that depends on the exchange rate rather than on the asset. A currency exposure exists whenever a holding is priced in something other than the currency the holder reports in. does not need anybody to decide to take it, and an exposure nobody decides on is the one most often left unmeasured.

There is a household version of this that most readers have already lived through. A household decides to pay for a child's course that is billed in another currency, one year ahead. The decision everybody discussed was the course: the institution, the subject, the sum. The decision nobody discussed is that the household now carries a position in that currency for a year, sized at the whole fee, and it will find out what that position was worth on the day the fee falls due. Nobody at that kitchen table would say they had taken a currency position. The household took one anyway.

The institutional version differs only in the digits. A committee spends an afternoon on whether the hypothetical Rs 30 crore slice belongs in the portfolio at all, and the minutes record a long discussion about the asset. The minutes do not record that the portfolio now carries a currency position of Rs 30 crore for as long as the holding lasts. The two are inseparable while the holding exists, and the second one has a size, a period and a result whether or not anybody writes it down.

How do a local return and a currency move actually combine?

The rule of thumb is what breaks, so here is the arithmetic, stated as arithmetic and not as a rule of thumb. Call the asset's result in its own currency the local returnWhat a holding returned measured in the currency it is priced in, before any exchange rate is applied to it. A local investor in the same asset would report exactly that number.. Call the exchange rate change over the same period the currency move. The rupee returnWhat the same holding returned once its value has been converted back into rupees. The rupee return combines the local return and the currency move, and a holder reporting in rupees receives it. is one plus the local return, multiplied by one plus the currency move, less one.

The two returns multiply, they do not add, and the habit of adding them is accurate to within a fraction of a point on small moves and visibly wrong on large ones. The reason is easy to see once it is stated plainly. The local return is itself money sitting in the foreign currency, so when that currency strengthens the gain strengthens along with the original sum. The starting rupees are not the only sum converted at a better rate. The winnings are converted at that better rate too.

One rupee committed to that holding becomes 1.10 in the foreign currency, and each of those units is worth 1.04 times as many rupees at the end, so the rupee has become 1.144. Draw that as an area and the missing piece becomes impossible to miss.

What is the cross term, and why does adding the two get it wrong?

The cross term is the local return earning the currency move as well. In the stated hypothetical the cross term is 0.10 times 0.04, giving 0.004, or 0.4 percentage points. Adding the two returns gives 14.0 per cent. Multiplying them properly gives 14.4 per cent. The 0.4 points is not a rounding error and it is not noise: it is a real quantity with a name, a size and a sign, and it belongs to whoever holds the asset.

A term with a sign has to be tested in both directions, so now run the mirror. The same asset returns the same 10.0 per cent in its own currency, and this time that currency loses 4.0 per cent against the rupee. One plus 0.10 times one less 0.04 is 1.056, so the rupee return is 5.6 per cent. The cross term is now minus 0.4 points. The cross term carries a sign, so it makes a strong year stronger and a weak year weaker, and it does not average away over time in the way people assume it does.

How wrong does adding get? The error depends entirely on the size of the move, and the pattern is worth carrying around. The cross term is 0.10 times the currency move whenever the local return is 10.0 per cent, so it scales linearly with the move. On a one point move it is 0.10 points and nobody would care. On a thirty point move it is 3.00 points, larger than most of the things a committee spends its afternoon discussing.

The sign matters as much as the size, and a straight line makes that clearest. Plot the cross term against the currency move and it runs through zero: negative on the left, positive on the right, and exactly zero only where the currency did not move at all. A flat currency is the only place where adding the two returns gives the right answer.

What does a currency hedge take out, and what does it put in its place?

A currency hedgeA separate position taken so that a currency move on a holding is met by an offsetting move somewhere else, leaving the local asset result as the part that still comes through. is a separate position taken so that the currency term on the holding is met by an offsetting term elsewhere. The general property is covered under hedging: a hedge separates exposure from ownership, works in both directions, and is paid for whatever happens. Applied to a currency, the hedge leaves the local asset return as the part that still comes through. The local asset return is what somebody thought they were buying when they argued about the asset.

Three things arrive together and a description that names one of them has described a third of an arrangement: what is offset, what is surrendered, and what is charged. What is offset is the currency term on a stated amount for a stated period. An offset that works downward works upward too, so the surrender is the 4.4 points the currency would have added in the year it does add them. The charge is a rate applied to the amount, paid in the flat year as much as in the violent one.

The offset does not reach zero, and bluntness about that is worth more than reassurance. The offset is written on a fixed amount over a fixed window, and the holding underneath it moves, so what remains at the end is a smaller mismatch nobody spends much time looking at. A smaller mismatch is a real improvement over the position it replaced, and honesty requires naming what it still is: a currency hedge exchanges a large exposure everybody understands for a small one almost nobody measures, and the small one is still there.

Set the two side by side in rupees, using a charge of 1.0 per cent that the reader supplies. Hedged, the slice returns its 10.0 per cent local result less the charge: Rs 3,00,00,000/- less Rs 30,00,000/-, or Rs 2,70,00,000/- and 9.0 per cent. The slice does that whichever way the currency went. Unhedged it returns 14.4 per cent in the strong year and 5.6 per cent in the weak one.

Notice the symmetry. The symmetry is the whole ethical content of the comparison. The unhedged strong year gave 4.4 points more than the asset produced, and the unhedged weak year gave 4.4 points less. Equal and opposite. There is no version of this where the currency is taken off the downside and left on the upside, and any description that implies otherwise has stopped describing arithmetic.

Why does a hedge placed at the start rarely cover the right amount at the end?

Most treatments skip the drift that follows, and not because the drift is difficult. Drift is merely unglamorous. The offsetting position is written on a notionalThe stated amount a contract is written against. The contract's payments are computed from the notional, and the notional does not change on its own as the underlying holding gains or loses value.. The notional is a fixed number agreed at the start. The holding underneath it is not fixed. The holding gains or loses in its own currency over the period, so by the end the fixed number and the moving value are no longer the same size.

Run the stated hypothetical. The offsetting position was written on Rs 30 crore. Over the period the asset gained 10.0 per cent in its own currency, so the slice is worth Rs 33 crore in local terms. CoverageThe share of a holding that a contract's notional is currently set against. Coverage is the fixed notional divided by the holding's current value, so it changes as the holding changes even though the notional does not. is the fixed notional over the current value, so it is 30 over 33, or 90.9 per cent. Rs 3 crore of the slice now carries the currency with nothing set against it, and nobody made a mistake: that is simply what a fixed amount does when it sits against a moving value.

The drift runs both ways, and the direction people find surprising is the other one. If the asset had fallen 20.0 per cent in its own currency the slice would be worth Rs 24 crore, and the notional of Rs 30 crore would then be set against more than the whole holding: coverage of 125.0 per cent. The position now points at Rs 6 crore of currency the portfolio does not hold. Over-coverage is a position in its own right, and it appears without anybody transacting.

Coverage is therefore a figure that gets restated rather than assumed. Coverage was 100.0 per cent on the day the position was written and is a different number on every day after that, so a monitoring report that carries the notional but not the current coverage is carrying the easier half of the pair.

What does the arrangement cost, and against what base?

No cost level is quoted below, and the reason is arithmetic rather than caution. A plausible invented rate would be read as a market figure by a reader moving quickly, and a rate mistaken for a market rate is worse than no rate at all. So the cost is a rate the reader supplies, applied to the notional, and the arithmetic that turns it into a portfolio figure is written out in full. The multiplier is the teaching, not the level, and the arithmetic is identical whatever the level turns out to be.

At a reader supplied rate of 1.0 per cent on a notional of Rs 30 crore, the charge is Rs 30,00,000/-. The same Rs 30,00,000/- is 1.00 point of the Rs 30 crore slice, 0.10 points of the Rs 300 crore equity sleeve and 0.06 points of the Rs 500 crore portfolio. Three numbers, one charge, and the only thing separating them is the base each was divided by. A reader shown 1.00 point and a reader shown 0.06 points have been told two very different stories about the same Rs 30,00,000/-, so the base has to be named every time.

How large does a foreign slice have to be before the currency matters?

Smaller than most people expect, and the arithmetic settles it rather than intuition. Take an 8 point currency swing: the distance between the currency 4.0 per cent stronger and 4.0 per cent weaker. The cross term moves with the currency, so on the slice that is not an 8 point swing but an 8.8 point one: 14.4 per cent at one end and 5.6 per cent at the other. On the hypothetical Rs 30 crore slice that is Rs 4,32,00,000/- against Rs 1,68,00,000/-, a difference of Rs 2,64,00,000/- on an identical asset result.

The base law applies to a hypothetical exactly as it applies to a record, so now set the swing against both bases. Rs 2,64,00,000/- is 0.528 points of the Rs 500 crore portfolio, rounding to 0.53, and 0.88 points of the Rs 300 crore equity sleeve. Both are correct. The two figures answer different questions, and a report that slides between them without saying which base it is standing on has told a reader something untrue about how large the exposure is.

Is half a point of the portfolio worth an afternoon? Set it against the record's own numbers for the stated year and decide. At a beta of 1.08 against a benchmark that returned 6.1 points above the 6.5 per cent risk-free rate, the record splits its gross excess of 1.6 points into 0.488 points that was simply carrying more of the market and 1.112 points of gross alpha that was everything else. The hypothetical currency swing is about 0.528 points, roughly half of that gross residual, arriving from an exposure nobody selected.

The answer depends entirely on how big the slice is, so scale it down and up. An 8 point currency swing on a Rs 10 crore slice moves the portfolio 0.176 points; on a Rs 75 crore slice it moves the portfolio 1.32 points, larger than the record's entire gross alpha for the year. A currency exposure on a small slice is a real number rather than a rounding, and it is usually the fourth largest thing in the report rather than the first.

What has to be recorded before any of this can be separated later?

Five things, and the reason to list them is that most reports carry three. The local return. The currency move. The notional set against it. The stated period. The charge. Without those five nobody can later say whether a foreign result came from the asset, the currency or the arrangement placed against it, and that is precisely the gap that turns a monitoring report into a description.

The recorded version looks unremarkable, and its plainness is part of the argument for it. Three lines and a total, in points and in rupees, for one stated period. AttributionTaking a result apart afterwards to say which decision or which condition produced which part of it. Attribution can only be done from figures that were recorded separately at the time. is not something clever done to a number later; it is something that becomes possible because somebody wrote three lines instead of one.

What does a currency position leave completely unchanged?

The question gets skipped, and skipping it is how a currency discussion swells until it seems to be about everything. A currency exposure, and any position placed against it, changes the rupee value of a foreign slice and the size of the swing around it. Nothing else about how the portfolio was built changes. The 60, 30 and 10 policy weights stand. The Rs 300 crore equity sleeve stands. The 28 names inside it stand, and so does the constraint that no single holding may exceed 5 per cent of the portfolio.

The point is not tidiness. A committee that treats a currency discussion as a reason to reopen the allocation has let a Rs 30 crore question reach a Rs 500 crore decision, and the arithmetic never asked it to. Keeping the boundary makes the currency question smaller and therefore easier to answer honestly.

How does anybody use this in a committee room, on a Tuesday?

Four numbers, prepared before the meeting and read before the holding is opened. The asset's result in its own currency. The currency's move over the same stated period. The notional, if any, set against the slice, and the fraction of the current value that notional now covers. The charge. Everything above is the reasoning behind those four, and the four are what an investment committee like Rukmini Deshpande's actually needs in front of it.

An analyst reading somebody else's report uses the same four in reverse. If only one combined figure appears, the first question is not whether the number is good but what the number is made of. A combined figure cannot be compared with anything, cannot be attributed, and cannot be used to size the next decision. A lender assessing a borrower whose revenue is priced abroad runs an identical check: the borrower's operating result and the borrower's currency result are separate facts, and a covenant written against the combined figure has been written against two things at once.

A household does the same work with a pen and no jargon. The fee is written down in the currency it is billed in, next to what it would cost in rupees today and the date it falls due. If somebody offers to fix the rupee amount now, the cost of fixing it is written down as well, with a note that this cost is paid whether the currency helps or not, and that fixing the amount takes away the good surprise along with the bad one. The written note is the whole arrangement, at the size of one kitchen table, and the committee runs the same arithmetic on Rs 30 crore.