Hedge Ratio vs Hedge Effectiveness: Set It, Measure It

A hedge ratio is how many units of a contract are held against each unit of the thing already held, and somebody sets it before anything moves. Hedge effectiveness is a measure taken afterwards of how much of the movement in the thing held was actually offset. One is an instruction. The other is a report. Only the simplest version of either can be worked here.

Two objects are involved and they are moving at the same time. There is a thing somebody holds, and there is a position written against it. Only two questions are worth asking about that arrangement, and the awkward part is that they belong to different days. How much of the second should be put against the first? The first question has an answer this morning. How much of the first did the second actually catch? The second question has an answer only after a stretch of time has finished running. The two moments have to be kept apart. The shared word hedge invites a reader to treat the two numbers as one figure looked at twice, and they are not one number.

The everyday version comes before any arithmetic. A household books the hall, the cook and the band for a wedding eleven months out, at prices fixed today. On the morning of the booking the share of the eventual bill that has been fixed in advance is exactly known: three of the four big items, say. The fixed share is a ratio, and it is available immediately. Nobody can state on that morning how well the fixing worked. The prices that were not fixed have not moved yet. Eleven months later somebody can add up what actually happened to the unfixed items and say how much of the swing the early booking took out. The second number is the measurement, and no amount of confidence in the booking produces it early.

What is a hedge ratio, for a reader meeting one for the first time?

A hedge ratio is a quantity decision. The ratio says how many units of a contract are written against each unit of the thing already held, and somebody has to choose it. The ratio is a number a person sets, not a number any market reports. Nobody publishes the hedge ratio the way a price gets published, because there is nothing out there to publish: the figure lives in whatever decision was taken by whoever holds the thing and whoever wrote the position against it.

A reader will meet the ratio written two different ways and should recognise both on sight. The first way is as a ratio proper, one unit of contract against one unit held, and that reads as one for one. The second way is as a percentage of the holding that has been covered, and for that same arrangement the percentage is 100.0 per cent. The ratio and the percentage are the same decision in two costumes, and neither one is more correct than the other. Somebody who writes half a unit of contract against each unit held has set a ratio of one against two, or 50.0 per cent of the holding covered, and again those are one decision described twice.

There is a condition on both descriptions that gets dropped constantly. The units on the two sides have to be the same units before the ratio means anything at all. Units of the reference asset held, against units of the reference asset the contract stands on. If the left side is counted in kilograms and the right side in contracts, and nobody has said how many kilograms sit inside one contract, then the phrase one for one is not describing anything. A ratio quoted without saying what is held and what has been written against it is not a figure, it is a shape where a figure should be.

Look at everything the ratio leaves out. The ratio carries no view about where the price of the thing held is going, no estimate of how much it might move, and no claim about how the arrangement will turn out. The number is a count. Somebody looked at what they hold, decided how much of it to write a position against, and that decision is the whole of the number. Everything uncomfortable about hedging sits downstream of it.

What is hedge effectiveness, and why can nobody quote one today?

Hedge effectiveness measures how much of the movement in the value of the thing held was offset by the position written against it. Nobody sets this number; it is worked out of movements that have already happened. That single word, already, carries the whole of the definition. Effectiveness is arithmetic performed on history, and where there is no history there is no number.

Three things have to exist before the measure can be taken. There has to be a stated period, with a start and an end. A proportion measured across three weeks and a proportion measured across three years are different figures, and neither one is the other. There has to be a movement in the value of the thing held across that period. And there has to be the matching movement in what the written position produced across the same period. None of those three exists on the morning a position is opened. Effectiveness therefore cannot be quoted in advance of anything. Somebody who states the effectiveness of a position they have just entered is describing a hope, and the word for that hope is the ratio they chose.

The shape of the answer is a proportion, and it is usually dressed up as a percentage. Movement that was offset, sitting on top; movement that there was to offset, sitting underneath. A measure of eighty out of a hundred says that four fifths of the swing in the thing held was taken out by the position written against it, and that one fifth of it came through anyway. The number is a description of what happened and it is never a promise about what happens next.

Which of the two is set and which is measured?

Put the two definitions beside each other. Four lines separate them, and every one of the four is about time.

The ratio is set and effectiveness is measured. The ratio exists before anything moves and effectiveness exists only after a stated period has finished. The ratio is an instruction to somebody about how many contracts to write, and effectiveness is a report to somebody about what happened. The fourth line is the one that catches people. Between the setting and the measuring sits everything the ratio could not know, so a ratio can be perfectly sensible while the effectiveness measured afterwards comes out low. The thing held and the thing the contract stands on may not be the same thing. The position may end before the date it was written to. The holding may grow or shrink while the period runs. None of that was visible on the morning somebody chose a number.

The error this contrast exists to prevent is a quiet one. Somebody hears both terms in the same meeting, notices that both are about hedging and both come out as percentages, and files them as one idea. Then a ratio of 100.0 per cent gets reported as though it were effectiveness of 100.0 per cent, and a decision taken in advance is handed on as a result. Treating the two as one number seen twice turns an instruction into evidence. No mistake available with either number costs more.

What does one unit against one unit produce at every settlement price?

Now to the single arrangement that can be worked here honestly, with the figures written out rather than asserted. Somebody is sitting on a single unit of the reference asset, paid for at the spot priceThe price for buying something for immediate delivery, rather than for delivery on some later date. Covered separately. of Rs 2,000.00/-. One unit of a contract on that same reference asset is written against it, on the short side, at the agreed price of Rs 2,130.00/-.

Where does Rs 2,130.00/- come from? Rs 2,000.00/- sitting for twelve months, with financing charged over that stretch at 6.50 per cent a year, costs Rs 130.00/-. Stacked onto the starting price, that cost produces Rs 2,130.00/-. Adding the financing cost is the whole construction, and it works out that cleanly here for one reason worth saying out loud: the reference asset hands its holder nothing across the year. There is no receipt arriving in the middle to subtract from the financing, so the carryThe cost of financing something across a period while it is held. Carry is the reason a price agreed for a later date differs from the price today. is the financing and nothing else. The agreed price is a cost worked out in advance, and it holds no opinion whatever about which way the reference asset travels next.

The ratio here is one unit of contract against one unit held. One for one, and 100.0 per cent of the holding covered. The ratio is a decision taken this morning, and it is the only figure in the whole arrangement that anybody chose.

Here is the table, and it has four rows and one answer. The unit cost Rs 2,000.00/-, so the change in the value of the unit held is taken from Rs 2,000.00/-. The payoffWhat a position produces at settlement, counted before anything paid to get into it is taken off. on the short position is taken from Rs 2,130.00/-. Rs 2,130.00/- is the price the position was written at, and a short position gains when the settlement price comes in below it.

The last column is the one to read first, ahead of everything else in the row. The combined figure is plus Rs 130.00/- on all four rows, and the settlement price has no influence on it whatsoever. Work one row to see why. At Rs 2,400.00/- the unit held has gained Rs 400.00/- against the Rs 2,000.00/- it cost. The short position was written at Rs 2,130.00/- and settles at Rs 2,400.00/-, so it produces minus Rs 270.00/-. Add plus Rs 400.00/- to minus Rs 270.00/- and plus Rs 130.00/- is what remains.

The reason the last column never moves is worth stating as arithmetic rather than as a slogan. The change in the value of the unit held is the settlement price less Rs 2,000.00/-. The payoff on the short is Rs 2,130.00/- less the settlement price. Add those two and the settlement price cancels itself out entirely. The remainder is Rs 2,130.00/- less Rs 2,000.00/-, or Rs 130.00/-. The settlement price drops out of the sum because it enters twice, added on one side and taken off on the other, so it wipes itself out, and that wiping out is the entire mechanism of an offset. Two conditions make the cancellation exact here: the contract is written on the same reference asset that is held, and the position runs all the way to the final date, where the gap between the two prices is nil by construction.

What is that Rs 130.00/- left over, and is any of it a gain?

The Rs 130.00/- left over is the carry, and it is the very same Rs 130.00/- that built the agreed price in the first place. Take 6.50 per cent of a year against Rs 2,000.00/-: what comes out is Rs 130.00/-. The agreed price was Rs 2,000.00/- with that figure sitting on top of it. Writing the contract against the holding simply delivered the figure back.

The Rs 130.00/- is not a profit produced by hedging. Work it plainly. Somebody who borrowed the Rs 2,000.00/- to buy the unit pays 6.50 per cent a year on that borrowing, and over the year the financing on Rs 2,000.00/- runs to Rs 130.00/-. So Rs 130.00/- was collected by the pair and Rs 130.00/- was paid on the money that bought the unit. The two are the same size, they point opposite ways, and the pair nets to nothing at all.

None of that is a disappointment. The flat column is the arithmetic behaving correctly, and seeing why is worth a minute. The agreed price was built as the cost of carrying the thing to a later date. A position written at exactly that price therefore hands back the cost of carrying the thing and not one paisa more. The cost of carrying something and the money an arrangement built on that cost hands back are the same rupees counted once, so an arrangement that returns the carry cannot also be a way of earning. Anybody who reports the plus Rs 130.00/- while leaving the financing out of the same sentence has overstated the result by the whole of it.

There is a labelling discipline sitting underneath all of this, and it is not pedantry. The minus Rs 400.00/- and the plus Rs 130.00/- in the second column are changes in the value of a thing somebody has bought. The plus Rs 530.00/-, the nil and the minus Rs 270.00/- in the third column are payoffs on an obligation. Changes in value and payoffs are different kinds of figure that happen to be denominated in the same currency. And no figure anywhere in the arrangement is a profitWhat is left of a payoff once every cost of getting there has been taken off it. Until those costs are named, a payoff is not a profit. until every cost of getting there has been named out loud.

Why does the offset reach 100.0 per cent, and what has that proved?

A reader who has watched the last column come out identical four times running will want to say that the offset was complete, and in the arithmetic worked here it was. The completeness is an artefact of four assumptions built into the case, and it is not a result. Naming the four is the only way to stop the figure travelling further than it can carry.

First, the contract is written on the same reference asset that is held, so there is one price on both sides of the arrangement rather than two prices that resemble each other. Second, the position runs to the final date, where the gap between the two prices is nil by construction. Third, one contract covers exactly one unit held, so nothing has to be rounded to a whole number of contracts. Fourth, the holding does not change size at any point while the period runs.

Be honest about which of the four can actually be tested here. Three of them cannot. The second one can. Closing the position with three months still to run leaves a gap between the two prices of minus Rs 32.50/-. Minus Rs 32.50/- is Rs 2,000.00/- taken at 6.50 per cent a year for a quarter of that year. On that path the last column is no longer identical on every row, the offset is not complete, and what is left over is a residual rather than a clean carry. The basisThe spot price less the contract price, read at one moment. The basis shrinks towards nothing as the final date approaches. Covered separately. that produces that residual is worked in full separately.

The other three assumptions cannot be relaxed here at all. Removing the first would need a second reference asset priced against the first, and there is exactly one reference asset here. Removing the third would need the number of units inside one contract, and an authority sets that number. Removing the fourth would need a holding that changes size on stated dates, and no such holding exists here either. The 100.0 per cent worked above is an identity rather than a measurement.

What would be needed before either number could be worked out on anything real?

Naming what is missing is the useful half of an absence, and it is worth more than a formula would be. Start with the ratio. The ratio most often reached for in practice is not one for one at all: it is the ratio that would have moved the two positions most nearly against each other. Computing that one is a statistical exercise performed on history. The computation needs a record of the price of the thing held, a record of the price of the thing the contract stands on, a measure of how much each of the two moved, and a measure of how far the two moved together, which most people call correlationA measure of how closely two things have travelled together over a run of days or months. Correlation is built from a record of both, and one reading of either gives none of it.. All of that is taken across a stated period. The answer changes when the window changes, so the whole computation is redone as the period rolls forward.

Effectiveness needs less machinery and the same missing ingredient. The measure needs the movements that actually happened, on both sides, across a period that has finished. Nothing more exotic than that, and nothing less than that either.

Both numbers need a history, and one reference asset at one price with one financing rate is not a history. A ratio estimated from behaviour needs two price series and a measure of how far the two moved together. Effectiveness needs a period that has already finished and an outcome inside it. Neither can be reached from a single morning.

The same absence is why no control sits anywhere here. A control would have to move one of two things. Moving the relationship between two priced things needs a second reference asset, and these figures carry exactly one. Moving a run of movements needs the history just described as missing. A four row table worked in plain text stands here instead, with the assumptions that produce it listed underneath rather than hidden inside a control.

One more obstacle sits in the same place, and it bites even before any statistics are attempted. One contract stands on a fixed quantity of the thing it references, that quantity is the exchange's to fix within what the Securities and Exchange Board of India (SEBI) requires and is published at sebi.gov.in, and it does not stay put. A holding almost never divides neatly into whole contracts. Even setting a ratio therefore runs into a figure that only the authority can state, so the ratio anybody can actually write lands near the ratio they wanted rather than on it.

How does anybody actually use these two numbers?

Take somebody who runs the finances of a mid sized business and holds a quantity of something the business will sell in nine months. The ratio is a planning number for that person and it turns up at the start of the period. The finance head decides what share of the holding to write a position against, writes it, and from that morning onward the ratio is a fixed fact about the arrangement: so many units of contract against so many units held, expressible as a percentage covered, quotable to anybody who asks. Every conversation about the ratio happens before the fact. A number still open to choice is the number a meeting argues about.

Effectiveness turns up in a completely different conversation, months later, and usually with a different person asking. Somebody reviewing the accounts wants to know whether the position written against the holding actually behaved as cover. The question is answered by measuring what both sides did across the period and putting one over the other. And the reason it gets asked at all is that whether a position is treated as cover for something held, rather than as a position standing on its own, depends on conditions set by an authority rather than on anybody's intention.

The everyday version is a street vendor who buys tomatoes every morning and has agreed a price for half of next month's supply. The ratio is the half, and the vendor knows it on the day the agreement is made. Whether the half turned out to cover the swing is a question the vendor can only answer at the end of the month, by comparing what the unfixed tomatoes did against what the fixed ones saved. A ratio is a plan and effectiveness is a receipt, and no receipt exists before the shopping is done.

What neither number can settle

A reader who has watched a combined figure come out the same at four different settlement prices will read that as a demonstration that this arrangement works. The sameness is the least surprising thing here: the contract and the thing held are one and the same reference asset, so the arithmetic was always going to cancel. A demonstration needs an outcome attached to it, and an identity is not an outcome.

Whether anybody should write a contract against something they hold, and what ratio they should choose, is a separate question. So is how well any such arrangement has worked: answering that needs an outcome attached, a record followed over time, chances put on the results and a spread of possible results to argue from.

Before anyone could answer that question properly, a short and unglamorous list would have to be in hand. The holding, and for how long it is held. The job the position written against it is meant to do. The collateral posted against that position, and by whom. The events on the day cash genuinely crosses. And the conditions an authority requires before a position counts as cover at all rather than as a position taken on its own. Understanding how these two numbers are built is not a reason to build either of them.

The reference asset and its holder are invented.
Educational material. Not advice on any investment, tax, budget or market position.