Limits to Arbitrage: Why Mispricing Can Persist

Arbitrage is meant to remove a mispricing by making its correction profitable. Limits to arbitrage are the reasons it does not. A mispricing can widen before it narrows, the capital doing the correcting belongs to people who can withdraw it, and the withdrawal arrives exactly when the mispricing is widest. Being right about where a price ends up is therefore not enough.

One separation governs everything that follows. A position has a destination and a path. The destination is where the price finishes. The path is every level the price visits on the way. Textbook arbitrageCorrecting a mispricing by taking the position that pays when the gap closes, rather than by holding a view about the future. cares only about the destination. The assumption, made quietly, is that the person taking the position can wait for it. Every limit set out below is a reason that person cannot wait. The path is therefore decisive, and the destination is irrelevant to whoever ran out of room before reaching it.

The separation is familiar from ordinary life. The rent on a shop falls due monthly while a judgement about the street pays out in a decade. A household that has bought that shop on a twenty year loan may be entirely right that the street will be busy in ten years and still lose the shop in year three. Nobody in that story was wrong about anything. The household was wrong about nothing and was removed anyway.

What is arbitrage supposed to do in the first place?

The failure is only visible against a complete description of the success. State the job fully. A mispricingA price sitting away from what the information already available implies it should be. exists. Somebody notices it and takes the side of the trade that pays if and when the gap closes. Their trading is itself part of what pushes the price back, so the act of noticing and the act of correcting are the same act, and the reward for correcting is the gap itself. Nobody has to be public spirited for the mechanism to work, and that is what makes it elegant. Self interest does the tidying.

Notice how much is loaded into the third step. The reward does not arrive when the person is right. The reward arrives when the gap closes and the person still holds the position. Those are two different events, separated by an amount of time nobody controls. In a textbook the separation is invisible, because a textbook position has no funder, no monthly reckoning and no end date. In practice every position has all three.

How are Limits to Arbitrage and Market Inefficiency held apart?

Inefficiency and the limits get bundled together constantly, and keeping them apart is worth doing carefully. Market inefficiencyA state in which prices sit away from what a stated set of information implies about them. is a claim about a state of the world: prices are departing from what a stated information set implies. Limits to arbitrage is a claim about persistence: here is why nobody has fixed it. One says the price is wrong and the other says why the wrongness is still there tomorrow morning, and they are settled by completely different evidence.

The difference shows up in what would have to be examined. Settling whether prices depart from information means looking at prices and at the information. Eugene Fama set that exercise out in his 1970 review of efficient capital markets in the Journal of Finance. Settling whether a departure can persist involves no look at prices at all. The examination falls instead on the conditions facing whoever would have to correct it: how long they can hold, whose money they hold it with, and what happens to them if the gap widens first. Those conditions are facts about people and contracts rather than facts about prices.

Because they are separate claims, all four combinations are available, and walking the grid is what fixes the distinction in place. Prices can be wrong with nothing in the way, in which case the wrongness lasts about as long as it takes somebody to notice. Prices can be right while every obstacle to correction sits there fully assembled, waiting for a job. And prices can be wrong with the obstacles in place. That third combination is the only one producing something a reader can still see. Anything visible for long enough to be studied is drawn from that one cell of the grid, and that selection is doing more work than most readers realise.

What is noise trader risk, and why will hedging not remove it?

The first limit surprises people. It is a risk created by the very thing being corrected. Noise trader riskThe risk that a gap grows wider because of buying and selling that has nothing to do with value at all. is the risk that a mispricing gets worse before it gets better, driven by participants trading for reasons unconnected to value. Bradford De Long, Andrei Shleifer, Lawrence Summers and Robert Waldmann named and modelled this in 1990, and the uncomfortable result is that the presence of such traders does not merely add noise around a correct price. Noise trading adds a risk that the person correcting the price has to carry, and being unable to shed that risk changes what they are willing to do.

Why can it not be removed the usual way? To hedgeTo take a second position so that a loss on the first is offset by a gain on the second. anything requires a handle: something identifiable that moves with the risk. A second position can then be arranged to move against it. Interest rate risk has a handle. A currency exposure has a handle. Trading unconnected to value offers no handle at all. There is no measurable thing it responds to, and therefore nothing to take the opposite side of. A shopkeeper can insure against fire, theft and a bad monsoon, and cannot insure against the possibility that the street simply becomes unfashionable for two years. The second thing is real, expensive, and has no policy attached to it.

Whose money is the correcting actually being done with?

Here is the part of Andrei Shleifer and Robert Vishny's 1997 paper in the Journal of Finance that usually gets dropped, and it is the part that matters most. Their argument is not really about markets. The subject is an agency relationshipActing with capital belonging to somebody else, who judges the holder on the results they can see.. Correcting a mispricing at any scale takes capital, and at scale that capital comes from people who are not the person taking the position. The suppliers of capital cannot evaluate the judgement directly. If they could evaluate it, they would not need anybody to make it for them. So they evaluate the only thing available to them: the visible result so far.

Sit with how reasonable each party is being. The person supplying the capital is not being foolish. The supplier has handed money to somebody on the strength of an ability they cannot inspect, and a run of poor visible results is genuine evidence about that ability, even if it is weak evidence. The person taking the position is not being foolish either. Nobody in this arrangement is behaving badly, and the arrangement still produces withdrawal at the worst possible moment. The result is therefore very hard to design away.

Why does the withdrawal arrive exactly when the opportunity is largest?

The collision between the two halves is the whole finding. Put them together. A mispricing that widens is, by definition, a bigger gap than it was. A bigger gap is a bigger reward for whoever eventually closes it. So on the arithmetic of the opportunity, a widening gap is good news and the correct response is to commit more capital, not less. On the arithmetic of the visible result, exactly the same event is the worst news available. The position holding the old gap is now showing a loss. One event, read two ways, and the reading that controls the money is the one that says leave.

Work it on the Palash 100, an invented index. Somebody positions against the index at Q1 with the index at 118.0. The eventual level at Q4 is 104.0. The opportunity available at the moment they open is therefore 14.0 index points on 118.0, or 11.9 per cent. Now stand at Q2 with the index at 131.0. Nothing about the destination has changed, but the distance to it has grown: 27.0 index points on 131.0, or 20.6 per cent. The opportunity at the moment of withdrawal is 20.6 per cent against 11.9 per cent at the moment of commitment, so the capital is being taken away from an opportunity that has nearly doubled.

The relationship is perverse rather than unlucky, and the difference matters. Bad luck is a coin that lands wrong. A perverse relationship is a mechanism that reliably points the wrong way: the larger the opportunity, the more strongly the arrangement pulls capital out of it. Arbitrage capital is therefore systematically least available in exactly the situations where it is most needed. The tidy self correcting story requires the opposite.

Can somebody be right about every single fact and still be removed?

Yes, and here is the worked instance, built on the invented Palash 100 index and nothing else. The index sits at 118.0 at Q1. Somebody judges it to be above value and positions against it. Hold on to this: that judgement turns out to be correct, and it stays correct at every point in what follows. By Q4 the index is at 104.0. The destination is exactly where they said it would be.

Now work the pathThe route a price takes on its way to wherever it finishes, which matters whenever a position can be closed before it gets there. rather than the destination. From 118.0 the index does not go down. The index goes up, to 131.0 at Q2, the peak of the whole eight quarter record. The rise is 13.0 index points against the position on an opening level of 118.0, and 13 divided by 118 is 0.11017, so the position is 11.0 per cent under water. Only after that does the index fall to 112.0 at Q3 and 104.0 at Q4. Measured from 118.0 to 104.0 the position gains 14.0 points on 118.0, and 14 divided by 118 is 0.11864, so it finishes 11.9 per cent ahead. The full journey is 11.0 per cent against them before 11.9 per cent in their favour, and the against came first.

Add one ordinary condition and the whole thing collapses. Suppose the capital is withdrawn if the position falls 10 per cent. The 10 per cent figure is an illustrative condition rather than a market rule, set so the arithmetic has something to bite on. Ten per cent below the opening level of 118.0 is an index level of 129.8, and the index reaches 131.0. So the line is crossed during Q2, at the peak. The peak is simultaneously the worst moment in the whole record and the last moment before the judgement starts being vindicated. The position is closed down 10 per cent. The person was correct about every fact available and finished with a loss, and no error anywhere in their reasoning would have produced a different outcome.

Look at the two numbers side by side. Readers underrate them. The path cost 11.0 per cent and the destination paid 11.9 per cent. The two figures are less than one point apart. On every Rs 1,00,000/- committed, the worst point cost Rs 11,017/- and the eventual gain was Rs 11,864/-, a difference of Rs 847/-. There is no comfortable margin here separating being early from being wrong, and there is no version of this position where the arithmetic left room to be relaxed about the middle.

What are the other limits, once noise and agency are set aside?

Noise trader risk and the agency relationship are the two that carry the argument, but they are not the whole list, and the remaining three matter because each one is sufficient on its own. Any single one of them removes the ability to wait, and removing the ability to wait is the only thing that has to happen for the destination to stop paying. The three are not refinements of one obstacle but three separate ways of arriving at the same place. Remove any one of them and the other two still stand. Take them in order, because the order is roughly how often each one actually decides the outcome.

Start with the horizonHow long a position can be held open before something outside the holder's control requires it to be closed.. Most readers quietly assume they are safe there. The reasoning goes: I have no funder and no quarterly reckoning. None of this applies to me, and I can wait longer than the professionals can. The reasoning is wrong twice over. For a professional the closing date sits in the arrangement that supplied the capital, and it reflects the supplier's needs rather than the price's schedule. But it is also wrong for somebody deciding alone. A person deciding alone has not escaped the agency relationship; they have merely become both parties to it, and the party who supplies the capital in that arrangement still has a school fee, a rent and a hospital admission that arrive on their own timetable.

The Shleifer and Vishny point that gets lost most often is this. Their argument is not that professionals are impatient and amateurs are patient. The length of time a position can be held is set by whoever supplies the money, and nobody supplies their own money in a vacuum. A household with a two month reserve has a horizon of about two months for anything it might have to sell, no matter what it intends. Intent is not a horizon. The reserve is.

Next comes the cost of carryWhatever it costs to keep a position open for another month, paid whether or not the price moves at all., the least dramatic limit and quite often the decisive one. Keeping a position open costs something every month. The monthly cost is not a market movement, and it does not stop when the price goes quiet. The consequence is arithmetical rather than psychological. The eventual gain is a fixed quantity determined by where the price finishes. The carry is a running quantity determined by how long the wait took. A long enough wait converts a correct judgement into a loss without the price doing anything wrong at all, purely by charging rent on the waiting. The point stands on the shape of the two quantities rather than on their size.

The third limit is the plainest and the one nobody enjoys. The judgement can simply be wrong. Not early: wrong. A price that looks away from value may be exactly where it belongs, with the error sitting in the judgement rather than in the price. The possibility of simply being wrong is not a footnote to the argument but load bearing, for what it does to the other limits. Being early and being wrong produce the same evidence at the moment a decision must be made. No withdrawal rule can be written that keeps the early positions and closes the mistaken ones. If they were distinguishable, whoever supplied the capital would keep funding the early ones and stop funding the mistaken ones, and the whole argument would dissolve. The two are not distinguishable, so the argument stands.

Put all five limits in one place and the shape of the argument becomes easy to hold. Two of them describe risks the position carries, two describe constraints on how long it may be carried, and one describes the possibility that there was nothing there to carry. Notice that not one of them is a claim that anybody behaved foolishly. Every limit on this list survives a world in which every participant is careful, informed and acting entirely reasonably. The mispricing survives for exactly that reason.

Where does this argument most often get mangled?

How does any of this reach somebody who is not correcting prices?

A reader can put one question to their own waiting, and an invented case makes the question concrete. On 12 October Meera Sundaram sells Suvarna Chemicals Limited whole at Rs 4,60,000/-, booking Rs 60,000/- on a cost of Rs 4,00,000/-, and she keeps Kesari Logistics Limited, then standing at Rs 1,95,000/- against a cost of Rs 3,00,000/-. Her stated reason is that she will sell it when it gets back to Rs 3,00,000/-. Read as an arbitrage sentence, that reason says: I am right about where this finishes, and I intend to wait for it.

Devika Rao, advising at Palash Advisory Services Private Limited, does not need to argue about whether Meera is right. The useful question is not whether the judgement is correct but whether the waiting is actually available, and the answer sits entirely in facts nobody has to forecast. Meera's monthly outgo is Rs 55,000/- and her reserve is Rs 1,10,000/-, which is two months. Her horizon for anything she might be forced to sell is therefore about two months, whatever her intention says. And a running cost of the same shape as a cost of carry already sits in her arrangements: Rs 1,80,000/- of card borrowing at 36.0 per cent sits beside a Rs 2,40,000/- deposit at 6.5 per cent, so clearing the borrowing from the deposit would save Rs 64,800/- and forgo Rs 11,700/-, leaving the separation costing Rs 53,100/- a year, or Rs 26,550/- across the six months from 30 September to 31 March.

Set that beside what the holding did. Kesari Logistics fell a further 20.0 per cent over those six months, from Rs 1,95,000/- to Rs 1,56,000/-, a further Rs 39,000/- gone. So over one six month stretch the price took Rs 39,000/- and the running cost took Rs 26,550/-, and only one of those two numbers depended on being right about anything. The waiting had a price tag fixed in advance and payable regardless. A cost of carry is exactly that, and it is visible in an ordinary household arrangement long before anybody goes near a market. One case is not evidence that any rule works.

So the professional use of this material is a change of question rather than a change of holding, and it applies identically to somebody deciding alone. Nobody can settle in advance whether a view is correct. Ask instead what would close the position before the view has had a chance to be settled. The second question has answers made of reserves, monthly outgo, borrowing rates and the terms of whatever supplied the money. Reserves and terms are available today, they require no forecast, and they decide the outcome more reliably than the judgement does.

Why are the limits and this sequence's boundary one fact?

Here is where the sequence closes. Across the other nine guides in this sequence, the refusal to turn an explanation into an instruction has been supported by three separate reasons: that a documented effect is usually measured before costs, and the invented log's own turnover groups paid up to 4.1 points a year to costs; that a published effect is read by everybody who read the paper, so what it did before publication is not what it does afterwards; and that the limits which let a mispricing persist are the limits that stop anybody capturing it. The three have been presented as separate reasons. In fact they are one reason, and the identity below makes it visible rather than asserted.

The identity is the finding this whole sequence exists to reach. Follow it slowly. Suppose a mispricing is easy to capture. Then somebody captures it, their trading closes it, and it is gone. So it is not available to be seen, studied, written up or read about. Now suppose a mispricing is still there, visible, discussed, sitting in a paper somebody has published. The very fact of its still being there is evidence that capturing it is obstructed. An unobstructed mispricing would have been closed by the first person who noticed. The conditions that allow a mispricing to persist are the conditions that prevent its capture, and those are not two related facts but a single fact described from two ends.

Now watch the three reasons fold together. Costs are one of the limits. Dealing charges, spread and tax are precisely what a person waiting has to pay out of the gain: the cost of carry under another name. Publication is another. An effect everybody has read about is one where the noise trader risk and the funding pressure fall on many people at once, and that is what makes a widening gap widen further. And the limits are the limits. Three reasons that looked independent turn out to be three views of the single obstruction. The boundary therefore stops being a rule laid on top of the material and becomes a result taken out of it.

One consequence, the escape route most readers reach for, deserves saying plainly. A longer horizon does not solve this. A longer horizon feels as though it should work: if the problem is being closed out at Q2, then surely the answer is to arrange not to be closed out. But the length of the horizon is the thing under discussion, not the thing available to fix it. Andrei Shleifer and Robert Vishny's argument is precisely that the horizon belongs to whoever supplied the capital, and telling a reader to lengthen theirs is telling them to change a term in somebody else's contract or to stop needing money on the dates they need it. Patience is not a strategy here. Patience is the resource the arrangement removes.

What does a reader who has come through the other nine now have?

Take stock of what has actually been handed over. Fourteen named mechanisms from the second sequence. A preference structure from the third, with a measured coefficient of 2.2 and the same sixty people choosing cautiously in gains and taking chances in losses on one afternoon. The change that arrives when decisions are visible, from the fourth. The effect of all of it on a holding, from the fifth. And in this sequence: a benchmark, an anomaly catalogue, a bubble anatomy, a sentiment measure, an account of how information becomes price, and one specific constraint worked in detail. The inventory is a substantial amount of understanding, and the honest answer to what it converts into is: nothing, on average.

The answer is worth separating from the disappointment it might produce. A reader who understands why a price moved understands something real, and understanding is not a consolation prize awarded because the trading did not work out. Understanding is what lets somebody read a claim about markets and know what kind of claim it is, spot the step where an explanation is quietly converted into an instruction, and recognise the shape of a promise that cannot be kept. The sentence a reader must never be able to assemble is this effect exists, therefore trade it, and the reason the sentence fails is now a result rather than a rule.

Meera Sundaram, Devika Rao, Palash Advisory Services Private Limited, the Palash decision log, the Palash 100 index, the Vindhya index scheme, the Nilgiri mid-cap scheme, Suvarna Chemicals Limited and Kesari Logistics Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.