Mean Reversion: Why Rates Are Modelled to Return

Mean reversion is a pull toward a stated level at a stated speed. The pull acts on where the quantity is expected to be, not on where it actually goes. The randomness keeps working against it at every instant, so the pull moves the average and leaves every individual path free to sit far from the level for years.

Two sentences carry mean reversion, and they pull in opposite directions. The first is that the pull is real: written into a description of a quantity, it makes the average of that quantity genuinely bend toward a level, at a rate that can be written down and checked. The second is that the pull is weaker than almost everyone reads it as being. The pull changes the expectation, and it changes nothing at all about any individual path. A quantity described this way is free to sit a long way from its level, for a long time, without anything in the description having failed.

The quantity being described is a short rate written r with a time subscript, its starting level is 5 per cent, and every figure below follows from that and from three more numbers stated when they are used. All four numbers are assumed rather than read off a market, so the arithmetic below is exact and the parameters it runs on are only plausible.

What is mean reversion?

Mean reversionA pull toward a stated level at a stated speed. The pull is written into a description of a quantity rather than observed in one. is a property built into a description of a quantity. The description states two things: that there is a level the quantity is drawn toward, and how strongly it is drawn there. Nothing else about the description changes. The quantity still moves randomly, still has no memory of where it has been, still cannot be predicted from one instant to the next. The only thing added is a bias in the direction of movement, and the bias points at the level.

Here is the everyday version, and it is worth holding on to because it is exactly right rather than roughly right. A room has a thermostat set to a temperature, and a window that keeps blowing open and shut. The thermostat pulls the room toward its setting. The window pushes the room around. At any instant the thermostat is pulling and the window is pushing, and the temperature actually read off the wall is the outcome of both. The thermostat determines where the room tends, and the window determines where the room is. Ask what the temperature will be in an hour and the honest answer has two parts: a centre that the thermostat is responsible for, and a spread that the window is responsible for. Mean reversion is the thermostat, and the thermostat is never the whole story.

Now the version with symbols. A quantity that reverts has, at every instant, a drift whose size is the distance to the level multiplied by a speed. Far from the level, the drift is large. Near the level, the drift is small. Exactly at the level, the drift is nothing at all and the quantity is pushed around by randomness alone. The last case matters. A reverting quantity sitting exactly at its level is not held there. Nothing is pulling it, and it wanders off immediately.

The right-hand side behaves as two parts added together. Read it that way. The first part is the level, and the level never changes. The second part is the starting distance from the level, worn away by a decaying factor. At time nought the decaying factor is one and the two parts add back to the starting value exactly. As time runs on, the factor shrinks and the second part fades. A shrinking positive quantity added to the level cannot carry the sum past the level, so the expectation slides from the starting value toward the level and never overshoots.

What are the two parameters, and what does each one do?

The commonest confusion about mean reversion treats the two parameters as one. The two are independent. The long-run levelThe value the pull is directed at. Here it is 6 per cent. The level says where, and says nothing about how fast. says where the pull points. The speed of reversionHow hard the pull acts for a given distance. Here it is 0.5 a year. The speed says how fast, and says nothing about where. says how hard it pulls for a given distance. The level and the speed answer different questions and are set independently.

The thermostat again. The setting on the dial is the level. Whether the heating is a small electric bar or a proper central system is the speed. A dial can be set five degrees above the current room temperature and attached to a feeble heater, or set half a degree above it and attached to a powerful one. Both are perfectly ordinary rooms. The dial says nothing about the heater, and watching the heater run says nothing about the dial.

In the description used throughout, the level is 6 per cent and the speed is 0.5 a year. The units of the speed are what readers stumble on, so the speed number deserves a moment. A speed of 0.5 a year does not mean the quantity moves half a percentage point a year. A speed of 0.5 means that for every one percentage point of distance from the level, the quantity is being pulled at a rate of half a percentage point a year, at that instant. Distance is the input; a rate of movement is the output. The two are joined by the speed, and that is all the speed does.

Which is why the pull weakens as it works. The quantity starts at a full percentage point of distance and a pull rate of half a point a year. The pull rate was only ever the distance multiplied by a fixed number, so by the time half the distance has been closed the pull rate has halved too. The pull is self-limiting: closing the distance is exactly what disarms it. That single fact is the reason the level is approached and never reached, and everything in the next two sections is a consequence of it.

The second half of that block is the entire mathematical content of mean reversion, and it is worth saying in plain words twice. Over any one year, at a speed of 0.5, the remaining distance is multiplied by 0.606531. The multiplication holds in the first year, when the distance is a full percentage point, and holds just as well in the ninth year, when the distance is a scrap. Proportional decayShrinking by the same fraction over each equal stretch of time, rather than by the same amount. A fixed fraction of a positive number stays positive. has no end point built into it. A positive number multiplied by 0.606531 any number of times gives a smaller positive number every time, and never nought.

How long does the pull actually take?

The question has no answer in the form the reader wants. The quantity does not arrive, so there is no date by which it arrives. So the question has to be replaced with one that does have an answer, and the replacement is the half-lifeThe stretch of time over which the remaining distance halves. Here it is 1.386294 years, and it is the same length of time no matter where the measuring starts.: how long does the pull take to close half of whatever distance is left?

The half-life has a clean answer, and the reason it is clean is the proportional decay above. Because the remaining distance shrinks by the same fraction over equal stretches, the time taken to halve it is the same wherever the measuring starts. The pull takes as long to go from a full percentage point down to half a point as it does to go from a hundredth of a point down to a two-hundredth. The half-life is a single number describing the pull at every distance and at every date. A number that general is the honest answer to how long.

Put the locked numbers into it. The distance at the start is a full percentage point, from 5 per cent up to 6 per cent. After 1.386294 years the expectation stands at exactly 5.500000 per cent, and exactly half a percentage point is still to be closed. After a second half-life, at 2.772589 years, the expectation is 5.750000 per cent and a quarter of a point is still to be closed. After a third, at 4.158883 years, it is 5.875000 per cent. After a fourth, at 5.545177 years, it is 5.937500 per cent.

Look at what that ladder is doing. Each rung takes exactly as long as the one before it and covers half as much ground. Five and a half years of pull, at a speed most readers would call brisk, and the expectation has moved 0.937500 of a percentage point out of the one point it had to travel. The rungs get shorter without limit and the ladder never has a top.

What does the whole closure sequence look like, year by year?

Here is the worked instance in full. The starting level is 5 per cent, the long-run level is 6 per cent, the speed is 0.5 a year, and every row below is that one formula evaluated at a different horizon. No row is fitted to data. The last column is the one that carries the argument.

Every figure is in percentage points, and every one follows from the four assumed parameters by arithmetic alone. Ten years of a pull that closes half the remaining gapThe distance between where the quantity is expected to be and the long-run level. The gap starts at 1 percentage point here. every sixteen and a half months, and 0.006738 of a percentage point is still open. Ten years contains 7.213475 half-lives, and 7.213475 halvings of one percentage point leave 0.006738 of one. The arithmetic does not permit an arrival, at any horizon, at any speed, ever.

What does the pull act on, and what does it not act on?

Everything so far has been about the expectationWhere the quantity is expected to be, averaged over every outcome the description allows. The expectation is a centre, not a forecast of any one outcome.. The word expectation has been doing quiet work, and it is time to make the work loud. The formula that produced the whole table above computes an average across every outcome the description permits. The result is a centre of a distribution. A centre is not a forecast of what any particular run of the quantity will do, and nothing in it steers any particular run.

A pathOne possible realisation of the quantity through time, as opposed to the average across all of them. The pull bends the average; it does not steer any single realisation. is one realisation of the quantity through time: what actually happened, once. At every instant the pull contributes a small nudge toward the level, and at the same instant the randomness contributes a shove of its own with no regard for where the level is. The path is the running sum of both. The pull never gets to act alone, so nothing in mean reversion forbids a path from moving away from the level, and moving away again, and staying away for years.

The everyday version is a walk home along a road with a strong crosswind. The road is the level and the walker's intention to keep to it is the pull. The wind is the randomness. The walker is genuinely trying to get back to the middle of the road at every step, and the wind is genuinely pushing the walker off it at every step. Where the walker actually is at any moment is the sum of the two, and on a windy enough day most of the walk can be spent on the verge without the walker ever having stopped trying to get back. Someone watching and concluding that a return to the centre must be due has confused the walker's intention with the walker's position.

The last clause is the sharpest thing in mean reversion. The pull was asked to close a distance of one percentage point. The spread it is working against settles at one percentage point. At one year the expectation has moved 0.393469 of a point and the spread around it is already 0.795060 of a point. The randomness is more than twice the size of the progress.

The same thing happens on two actual runs. Two fixed sets of twelve driving values, constructed rather than sampled, are fed through the description above at the same parameters, and the two lines in the figure below are the result. Both runs spend the whole year below 6 per cent, so the pull is directed upward at every instant in both.

Neither run is misbehaving. Both are exactly what the description produces. The thick line in the middle is the only thing the pull controls, and it is an average across every run the description permits, not a route that any of them takes. There is no rule about runs, so a run that is further from the level after nine months than it was on day one has broken none.

Why is the pull put on rates and not on prices?

The modelling choice follows the quantity. Mean reversion is a claim, and a claim is put where it is true, not where it is convenient. So the question to ask of any quantity is whether there is a level it is held near by something outside itself. If there is, a reverting description is describing something real. If there is not, the description has invented a level and the model will keep insisting on a return to a number that has no standing.

A rate passes that test. A rate is a ratio rather than an amount, so it has no natural tendency to grow simply because time has passed. The machinery that sets it and the economics around it hold it inside a band, and although that band moves, at any moment there is a level the rate is near. A rate that has drifted far from that level is under pressure from something. Because there is a level, a description that pulls toward one is describing the quantity rather than decorating it.

A price fails the test, and the standard process this subject area is built on shows why. The process starts at Rs 100/-, drifts at 8 per cent a year and carries a volatility of 20 per cent a year. Its natural description is proportional: it grows by a percentage, not toward a number. Ask where it should return to and there is no answer. Rs 100/- is where it happened to start, not a level anything holds it near, and after a decade of compounding at 8 per cent the starting value is simply a fact about history. A description that pulls a price back toward Rs 100/- is not modelling the price. The description is asserting something extra about the world that nothing in the price supports.

There is a second, sharper reason. Suppose a description said that a traded price is pulled back to a fixed level with a known speed. Such a description is a statement about where a traded quantity tends, and statements of that kind sit uncomfortably close to the no-arbitrage argument the whole of this subject area is built on. A rate is not itself a traded amount in the same way; it is a parameter inside the description of what things are worth. The pull can be put on it without the same conflict. The distinction is developed properly under the zero-coupon bond and under the short rate set against the market model, and it is only flagged here.

Where does the half-life actually change a decision?

Someone reading the output of a rate description, rather than building one, uses the half-life in one blunt and useful way: it tells them how much of the answer is coming from the starting level and how much from the long-run level, at whatever horizon they care about. The starting level and the long-run level are two different inputs with two different sources of doubt, and the half-life is the exchange rate between them.

At a horizon of one year and a speed of 0.5, the expectation is 5.393469 per cent. The figure is 0.606531 of the starting level and 0.393469 of the long-run level. Change the starting level by a tenth of a point and the one-year answer moves by 0.0606531 of a point. At ten years the weights are 0.006738 and 0.993262, so the same change to the starting level moves the ten-year answer by 0.0006738 of a point. For most purposes such a shift is nothing. The half-life says at which horizon the answer stops being about where the rate is now and starts being about where it is being pulled.

The shifting weight has a plain consequence for where doubt should be spent. At short horizons the starting level carries the answer, and it is generally the best known of the inputs. At long horizons the long-run level carries the answer, and the long-run level is a parameter somebody chose, so it is generally the least well known of the inputs. A ten-year figure from a description with a 1.386294 year half-life is very nearly a restatement of the long-run level with some arithmetic around it. Knowing that stops a reader from treating a long-horizon output as though it were an observation.

The same logic reaches the price of a claim. Under the same parameters a rate that reverts gives a one-year discount factor of 0.949216, against 0.951229 for a rate frozen at 5 per cent. On Rs 100/- of principal that is Rs 94.9216/- against Rs 95.1229/-, a difference of about Rs 0.2013/-. A difference that size is the whole reason a rate gets its own description rather than being treated as a constant. The bond that makes that comparison is covered separately.

What does mean reversion not promise?

Mean reversion does not promise a return. Everything above establishes exactly that, and the reason it needs establishing is that the words themselves push the other way. Reversion sounds like coming back. A long-run level sounds like a destination. Put the two together and the natural reading is that the quantity will get there, and that reading is not supported by anything above.

The natural reading comes apart into the three things it assumes. The first assumption is an arrival, and there is no arrival: the remaining distance is multiplied by a fixed proportion over every equal stretch and stays positive forever. The second assumption is that the arrival is soon, and at these parameters half of it is still outstanding after 1.386294 years and 0.006738 of a percentage point is still outstanding after ten. The third assumption is that the pull acts on the particular quantity in view, when it acts on the average across every quantity the description permits, of which the one in view is a single instance.