Relative Value: Trading the Gap Between Two Prices

Relative value is approach 4 of the eight this sequence describes. The position sits in the gap between two related prices rather than in either price: one instrument is bought and the other sold short, so whatever both prices do together cancels between the legs. Only the gap changing survives, rupee for rupee. The gap is expected to close. The gap is not locked.

Start with something visible from a bus window. Two vegetable sellers work the same market, one at the top of the lane and one at the bottom. The seller nearer the main road charges Rs 44.00 a kilo for tomatoes and the seller at the quiet end charges Rs 40.00. Everybody who shops there knows why: the road end catches the office crowd walking home, and that footfall is worth Rs 4.00 a kilo. The difference has held for as long as anyone can remember.

Now the crop fails somewhere and tomatoes get dear. Both sellers put their price up by Rs 20.00. One is at Rs 64.00 and the other at Rs 60.00. Something enormous has happened to tomatoes. Absolutely nothing has happened to the Rs 4.00. An arrangement in which only that Rs 4.00 mattered would have let the crop failure pass by entirely, and the day the road-end seller finally lost her advantage and the two prices came together would have been the only day that mattered at all.

A difference that survives whatever happens to both prices is the whole subject here, and it is worth holding in mind before a single rupee figure arrives. Relative valueA position in the gap between two related prices rather than in either price. is the deliberate construction of exactly that situation in traded instruments. Relative value is approach 4 of the eight approaches this sequence describes structurally. The construction comes first, then the risk it carries.

What is the fund actually holding when it holds a pair?

Here is the worked case the rest of this guide runs on, and every figure in it belongs to Nilgiri Absolute Return Fund, an invented open-ended vehicle run by Nilgiri Alternatives Advisors Private Limited. The fund holds a pair of related instruments. The two instruments are left unnamed. Naming them would draw attention to the instruments when the arithmetic between them is the whole point. One of the two trades at Rs 96.00. The other trades at Rs 100.00.

Three numbers now exist rather than two. There is Rs 96.00, there is Rs 100.00, and there is the Rs 4.00 between them. The Rs 4.00 was not typed by anybody, is not quoted by anybody, and does not appear on any screen. Subtraction produces it, and the fund has taken its position in it. The fund holds no view at all on Rs 96.00 and no view at all on Rs 100.00: what it holds is the gapThe difference between the two prices, which is the thing actually being held. between them.

A position in a difference is genuinely a different kind of object from the positions most readers have met, and how odd it is deserves a moment. Somebody who buys a share has a view on a price. If the price goes up they are better off and if it goes down they are worse off, and the whole story fits in one line. Here there are two prices and neither of them, on its own, decides anything. The fund could be told that the first instrument had gone to Rs 200.00 and it would still not know whether it had made money or lost it. The fund would have to ask a second question, and the second question is the only one it cares about: what happened to the other one?

Every consequence of holding a pair is built around a subtraction. The household version is a person who has borrowed at one rate and lent at another. The borrower does not care what interest rates in general do. The difference between the two rates they are actually sitting between is the only quantity that matters to them. Rates can double and the difference can be untouched. Rates can be flat and the difference can vanish overnight. The level and the difference are two separate facts and one of them is not the other.

How Relative-Value Strategies Work: what does the construction do?

Everything else follows from the construction, so the construction is the part to learn first. The fund buys 25,00,000 units of the cheaper instrument at Rs 96.00 and lays out Rs 24,00,00,000. In the same breath it sells short 25,00,000 units of the dearer instrument at Rs 100.00 and raises Rs 25,00,00,000. Each side of the pair is called a legOne side of the pair, either the instrument bought or the instrument sold short.. The same number of units on both sides is not decoration; it is the mechanism.

Watch what the matched sizing does. Suppose both instruments now rise by Rs 7.00 for reasons that have nothing to do with either of them in particular. The bought leg makes Rs 7.00 a unit. The leg that was sold shortSold without holding it first, so the position gains when the price falls and loses when it rises. loses Rs 7.00 a unit. A short position is the mirror of a bought one. On 25,00,000 units each, that is Rs 1,75,00,000 made and Rs 1,75,00,000 lost. Everything the two prices did in common has cancelled between the legs and left nothing behind, and that cancelling is the entire purpose of putting the trade on in two parts rather than one.

So the trade is deliberately built so that most of what will happen to it does not reach it. Think of a household that runs a small tuition centre and rents the flat above it out. If the whole neighbourhood becomes fashionable, the rent it collects goes up and the rent it pays goes up, and the household's real position is the difference between the two rents rather than either one. Somebody who set that up on purpose, matching one against the other, would have made themselves indifferent to the neighbourhood and interested only in the spread. The household is running the fund's construction on rents instead of on instruments.

Two things follow immediately and both are worth stating flatly. The first is that the fund has to be right about only one number rather than two. Being right about one number sounds like less work and is a different kind of work. The second is that the short leg carries the property short positions always carry: there is no ceiling on how far a price can rise, so the loss on a position sold short is not bounded above. The unbounded loss on a short leg is worked in full elsewhere in this sequence, and named here because it sits underneath the construction and does not go away just because a pair looks tidy.

Why does the level both prices move to make no difference?

Readers usually accept the claim in words and disbelieve it in their stomachs, so it is worth writing out twice at two different levels and letting the arithmetic settle it.

Take the pair exactly as it stands: bought at Rs 96.00, sold short at Rs 100.00, 25,00,000 units each side. Now suppose both instruments move up to Rs 110.00. The bought leg has gone from Rs 96.00 to Rs 110.00, a gain of Rs 14.00 a unit. A short loses as the price rises, so the leg sold short has gone from Rs 100.00 to Rs 110.00 for a loss of Rs 10.00 a unit. Fourteen less ten is four. The position made Rs 4.00 a unit.

Now run it the other way and suppose both instruments move down to Rs 80.00 instead. The bought leg has gone from Rs 96.00 to Rs 80.00, a loss of Rs 16.00 a unit. The leg sold short has gone from Rs 100.00 to Rs 80.00, a gain of Rs 20.00 a unit. Minus sixteen plus twenty is four. The position made Rs 4.00 a unit again.

Read those two paragraphs next to each other. In the first, prices went up by Rs 14.00. In the second, prices came down by Rs 16.00. The two moves are opposite events of roughly similar size, and a person holding either instrument on its own would have had two completely different years. The pair produced the identical answer both times, and the reason is not a coincidence: in both cases the gap went from Rs 4.00 to nil, and the gap going from Rs 4.00 to nil is worth Rs 4.00 a unit whatever level it happens at. The gap closing towards nil is called convergenceThe gap narrowing towards nil., and it is the event the position is set up for.

The word people reach for here is neutrality, and it is worth being careful with it. The construction is indifferent to what both prices do together. The construction is not indifferent to what the two prices do separately, and not indifferent to being asked for cash while it waits. Separate movement and the demand for cash are worked out below, and both are where the honest teaching sits.

What is the profit and loss actually made of?

Once the position is understood to be in the gap, the profit and loss stops being mysterious and becomes a single subtraction repeated. The gap went on at Rs 4.00. Whatever the gap is now, the position has made the difference between those two numbers, multiplied by the number of units. Nothing else enters. The profit and loss is the change in the gap, rupee for rupee, in both directions.

Here is the reverse case, and it belongs in the same breath as the first two or the lesson is only half taught. Suppose the bought leg falls from Rs 96.00 to Rs 94.00 while the leg sold short rises from Rs 100.00 to Rs 102.00. The bought leg has lost Rs 2.00 a unit. The leg sold short was sold at Rs 100.00 and the price went up, so that leg has also lost Rs 2.00 a unit. Both legs lost, a perfectly possible outcome, and the position is down Rs 4.00 a unit. Look at the gap: it started at Rs 4.00 and it is now Rs 8.00. The gap widened by Rs 4.00 and the position lost Rs 4.00 a unit. The same rupee-for-rupee rule is running backwards.

On 25,00,000 units, every rupee of gap movement is worth Rs 25,00,000. The conversion of one rupee of gap into Rs 25,00,000 is enough to read the whole position, and here is the ladder it produces.

One thing here is not symmetric, and it is the most useful fact in this guide. Reading leftwards, the line ends. The gap cannot go below nil, so the very best this position can do is Rs 4.00 a unit, being Rs 1,00,00,000, and there is no arrangement of prices that gives it more. Reading rightwards, the line does not end. There is no arithmetic ceiling on how far apart two prices can go. The gain on this position is bounded and the loss on it is not, and that is arithmetic rather than an opinion about anything.

Where is the boundary between this and a difference that can be locked?

A reader who has followed everything so far will now be one short step from a wrong conclusion, and the step is easy to take. If the position is only in a difference, and the difference is expected to come back to nil, does that not make it a sure thing dressed up in arithmetic? The answer is no, and the reason is the single most important boundary in the subject.

There is a precise sense of the word arbitrage in which a difference between two prices can be locked: the two things being compared are the same thing under the covers, the two positions settle into one another, and the difference is captured by construction rather than by waiting. Arbitrage in that precise sense is a distinct approach with a distinct mechanism, is approach 8 of the eight this sequence describes, and is covered separately.

One sentence carries the whole boundary. A difference that can be locked is arbitrage, and a difference expected to close is what this guide describes. The word doing all the work in that sentence is expect. The fund has looked at two instruments, formed a view about why they differ, and concluded that the Rs 4.00 should not persist. The conclusion may be well reasoned and it is still a conclusion. Nothing in the construction compels the gap to close and nothing sets a date by which it must. There is no basis on which to compute a probability that it will, so no honest probability can be attached to it.

What separates this from a book built to have nil net exposure?

There is a second thing relative value is often confused with, and the confusion is understandable because the arithmetic rhymes. A whole portfolio can be assembled so that its bought positions and its positions sold short cancel each other out at the level of the portfolio, leaving the book with no net direction at all. A book built that way is its own approach, approach 3 of the eight, and has its own treatment in this sequence.

The difference is what the two sides are. Here, the two sides are two instruments with a defined relationship to each other, and the difference between them is itself the trade. There, the two sides are two halves of a whole portfolio, and the positions in one half need have no relationship at all to the positions in the other. One is a claim about two named things. The other is a property of an aggregate.

The everyday version is worth a line. A person who buys a season ticket and sublets the same seat has one relationship and one spread. A person who spends as much as they earn has a balanced budget. A balanced budget is a property of the total and says nothing about any single item in it. Both are described as balanced and only one of them is a position in a difference.

For orientation: Nilgiri Absolute Return Fund itself runs a different approach again, long-short equity, and at its record date its gross exposure was 180.0 per cent of net assets against a net exposure of 80.0 per cent. Both figures are that fund's own, at that one date, and they show only that a book can have direction while a pair inside it does not. How exposure is measured and what those two percentages actually describe is covered separately in this sequence.

Risk one: how does a gap that widens turn into a loss?

Everything up to this point has described a construction. The construction carries three things, each with a consequence of its own.

The first is that the gap can widen instead of closing, and widening is a loss rupee for rupee on a position whose entire premise was that it would narrow. This is not an exotic outcome or a tail event. Widening is the ordinary other direction of the same straight line, and a position built to make Rs 4.00 a unit when the gap closes loses Rs 4.00 a unit when the gap doubles. The word for the gap growing is wideningThe gap growing, which on this position is a loss rupee for rupee., and there is nothing unusual about it.

The second is that the gap can stay open longer than the position can be held. A position is not a wish; it is funded, collateralised and reported every day, and it exists only for as long as somebody keeps funding it. A gap that takes longer to close than the position lasts has, from the position's point of view, not closed at all.

The third is that the position can be forced shut before the gap closes at all. Forced closure follows from the second and is worked out below.

Look at the middle path for a second. The middle path does not blow up, does not do anything dramatic, and never reaches the stage where anything is forced. The gap simply widens to Rs 12.50 and sits there, and while it sits there the position stands at minus Rs 2,12,50,000 and every report the fund produces says so. A gap that is merely wider than it was, and stubborn about it, is enough.

Risk two: what actually holds the two prices together?

Ask what the construction quietly assumes an answer to. Why should these two prices have any relationship at all? Somebody looked at two instruments and concluded that the Rs 4.00 between them was explained by something, and that whatever explained it had changed or would change. The relationship between the two prices is an assumption about the two instruments, not a law of arithmetic, and it is the single thing most worth examining before anything else on the position is examined.

Go back to the two vegetable sellers. The Rs 4.00 was footfall from the office crowd. Suppose the office moves. The Rs 4.00 does not narrow, it inverts, and it does so for a perfectly good reason that had nothing to do with tomatoes. Anybody who had built a position on that Rs 4.00 coming back would have been holding a view about pedestrian traffic without knowing it.

Whatever makes two traded prices related in the first place is a question about the instruments rather than about the approach, and is covered separately. The answer is never structural on this kind of position. The two legs do not settle into each other, are not two claims on the same cash, and are not held together by anything except the reasoning that put them next to each other. If that reasoning was wrong, or was right and then stopped being right, the gap has no obligation to do anything at all.

Risk three: why is the collateral struck on both legs added together?

Readers miss the collateral risk most reliably, and it is the one that decides whether a position survives long enough to find out whether it was right. The fund's positions sit with Marudhar Securities Private Limited, an invented prime broker. Who sets the terms, and what a prime broker is for, is covered separately in this sequence. One number matters here, and where it is struck.

The broker does not look at the Rs 4.00. The broker looks at the gross positionBoth legs added together, which is what the collateral requirement is struck on., both legs added together: Rs 24,00,00,000 bought plus Rs 25,00,00,000 sold short, being Rs 49,00,00,000. On this fund's own contracted terms, the maintenance marginCollateral the broker requires against the gross position while it is open. is 15.0 per cent of the gross value of a position. The requirement puts Rs 7,35,00,000 of collateral against a trade whose whole answer is Rs 4.00 a unit. The 15.0 per cent is a contracted term between two invented parties rather than a market rate, a standard or a requirement of anybody.

Read the two bars in the lower strip together: the fund puts up more than seven times what the whole gap is worth if it closes perfectly. That is not a criticism of anything, and it is not a sign that something has gone wrong. Such a multiple is what happens when the requirement is struck on the size of the position and the answer is struck on the difference inside it. The two are measured on completely different quantities, and the ratio between them is simply what the arithmetic produces.

Work the boundary out and it makes the point sharper still. The Rs 7,35,00,000 posted, spread over 25,00,000 units, is Rs 29.40 a unit. The position went on at a gap of Rs 4.00. So the posted collateral is exactly used up when the gap reaches Rs 4.00 plus Rs 29.40, being Rs 33.40. Rs 33.40 is a very long way from Rs 4.00, and here is the honest reading of it: it is the wrong place to look. An approach of this kind runs many pairs at once, and what actually gets called is the fund's collateral against everything it holds, not this one pair's own. The Rs 33.40 is an arithmetic point on a scale, not a forecast, and a single pair rarely travels anywhere near it.

What does the position look like at every gap at once?

Every number is now in place: the entry gap of Rs 4.00, the conversion of Rs 25,00,000 for every rupee the gap moves, the Rs 7,35,00,000 posted, and the Rs 33.40 at which that posted collateral is exactly used up. The control below puts them on one scale, walking the gap from nil to Rs 34.00 while both the reading and the drawing change together. The question below can be answered from the arithmetic alone.

How can a position be right about the gap and be closed anyway?

Here is the sequence that everything above has been building towards, and it is the part a reader is most likely to get wrong because it does not feel like it should be possible.

The gap is Rs 4.00. The fund expects it to close. Suppose the fund turns out to be completely right, and the gap does eventually close. On the way there it first widens to Rs 8.00. At Rs 8.00 the position stands at minus Rs 1,00,00,000. The minus Rs 1,00,00,000 is not a paper mark that can be politely ignored until things improve: it is a loss the broker recognises immediately and against which cash has to be sitting. The eventual answer and the cash demanded on the way to it are two different things, and only one of them decides whether the position is still open when the answer arrives.

One fact from the collateral section makes the sequence bite. The broker calls on the fund's collateral across everything it holds, not on this pair's own Rs 7,35,00,000. An approach of this kind runs many pairs at once. A demand can arrive because of positions elsewhere in the book entirely, and when it arrives the fund closes whatever it can close. The pair that was going to be right may very well be the first one out. The pair does not get a hearing on its own merits.

The everyday version is uncomfortably close to home. A household is certain that the flat it has bought will be worth more in five years, and it may well be. The instalment is due every month regardless, and if the household loses the income that pays it, the flat is sold in year two and the household's correct view about year five is worth precisely nothing to it. Being right about the destination and being able to stay on the journey are separate problems, and the second one is the one that ends positions.