Tracking Difference vs Tracking Error: Gap and Spread

Tracking difference is how far a tracker ended from its index over a period. Tracking error is how much that gap moved about within the period. Measuring it needs a whole series rather than two endpoints. A steady charge shows up almost entirely in the difference; the parts that vary show up in the error. Neither number answers the other's question.

The shape of the confusion is worth naming before anything else. The two measures sound like a pair, and the names encourage it: one says difference and the other says error, so a reader reasonably expects the second to be a sharper version of the first. It is not. One of them is a level and the other is a spread. One records where two things ended up relative to each other. The other records how steadily they travelled to get there. The two measures are not two grades of the same measurement. They are two different measurements built out of two different quantities of data, and knowing one of them reveals nothing at all about the other.

An everyday version makes the split obvious. Two people take the same train to work every day for a year. At the end of the year both of them have arrived, on average, four minutes after the office opens. The average is a level, and it describes where they ended up. But one of them arrives four minutes late every single day, like clockwork, and the other is twenty minutes early on some days and half an hour late on others, and the two extremes happen to cancel. Their manager, asked which one is easier to plan around, answers instantly. The average arrival time was never a description of the journey. It never revealed the difference between the two, and it never could.

One scheme carries the whole argument. Girnar Asset Management Limited, an invented asset manager, operates the Girnar Broad Market Index Fund, a tracker following a broad index. For the stated year that index returned 12.40 per cent, and because an index is not a thing anybody can hold, that figure carries no charge of any kind. The Girnar Broad Market Index Fund returned 12.12 per cent net, after a charge of 0.20 per cent of the scheme's assets.

Three things are settled elsewhere and are taken as given. A charge accrues daily against a scheme's assets at a steady rate. A scheme's published return is already net of that charge while an index return is net of nothing, so a level gap between the two is expected before any question of skill arises at all. Building a tracking difference and taking it apart is covered separately and is used here as starting material. Separating the two measures is what is left. Neither number is a verdict on whether any tracker is well run.

What does a tracking difference actually measure?

One subtraction, carried forward. A tracking differenceThe scheme's return for a period less the index's return for the same period, stated in percentage points. is the scheme's return over a period less the index's return over the same period. On the Girnar Broad Market Index Fund for the stated year, subtract 12.40 per cent from 12.12 per cent net and minus 0.28 percentage points is what remains. The 0.28 points decomposes into 0.20 points of charge, being 0.20 per cent of that scheme's assets, and 0.08 points of everything else, and the split is taken as starting material.

The whole of that figure is a statement about two moments: where the two stood at the start of the period and where they stood at the end. Nothing between those two moments enters the calculation, and nothing between them can be recovered from the result. Two returns go in, one number comes out, and the number is a level. The number is the finishing distance between two runners, and it says nothing about who led at the halfway mark.

A build that only runs one way has not been tested. The build runs backwards as well. Starting with the index at 12.40 per cent, taking off the 0.20 points of charge leaves 12.20 per cent, where a perfect tracker would have finished. Taking off the 0.08 points of residualThe part of a shortfall left over once the named cause has been accounted for, here everything other than the charge. leaves 12.12 per cent, exactly what the scheme returned net. The two endpointA value at the start or the finish of a period, as opposed to a value from somewhere inside it. figures and the two components agree in both directions, so the arithmetic is checked and not merely plausible.

What does a tracking error measure, and what does it need first?

The requirement is the distinction. Start with what a tracking error needs. A tracking errorA measure of how much the gap between a scheme and its index varied across the sub-periods inside a window, rather than where it finished. takes the gap between the scheme and the index for each sub-periodA shorter stretch inside the measurement window, such as a day, a week or a month, over which a separate return is computed. inside the window and measures how much that seriesA run of values in order over time, rather than a single value describing a whole period. of gaps varied about its own middle. Not where it finished. How much it moved.

A difference needs two endpoint returns and an error needs a whole series of them, so the second is not a refinement of the first but a different measurement made out of different data. That is why one cannot be upgraded into the other by thinking harder. Where the series is absent, the second measure is not merely imprecise, it is not available, in the same way that how bumpy a road was cannot be worked out from the odometer reading at the end of the drive.

Two more things ride on that requirement and both matter. First, the sub-period length is part of the measurement rather than a detail of it: gaps measured daily and gaps measured monthly on the same underlying behaviour produce different figures, so a tracking error quoted without its sub-period length is not fully specified. Second, the spreadHow widely a set of values sits about its own middle, as distinct from where that middle is. is measured about the series' own middle. A scheme that is consistently behind by exactly the same amount every sub-period therefore has a large gap and almost no spread at all. The combination is not a contradiction. A large gap held steadily is the ordinary condition of a well run tracker.

What are the four combinations of gap and spread?

Put the two measures on two axes and four corners appear, and every one of them is occupied by schemes that actually exist. Small gap and steady tracking. Small gap and a gap that moved about wildly. Large gap and steady tracking. Large gap and a gap that moved about wildly. Every one of the four occurs, and one coordinate therefore cannot be read off the other.

Two of the four are the ones worth sitting with. The third combination, a larger gap held very steadily, is what a well run tracker carrying a real charge looks like, and it looks worrying to anyone who has been told that a bigger gap means worse tracking. A bigger gap does not mean worse tracking. A scheme that gives up a fixed slice every day will end further behind than one that gives up less, and it can do so with almost no wobble at all along the way.

The second combination is the one that surprises people. A scheme can finish the year almost level with its index having been well ahead of it in March and well behind it in September, with the two excursions happening to cancel. Its ending gap is tiny. Its journey was nothing of the sort. Handed only the ending gap, a reader would describe that scheme as a close tracker, and would be describing something that never happened. A household whose bank balance is the same on the last day of every month can nonetheless run out of money in the third week of each one. The month-end figure is true and it is also completely uninformative about the month.

Why does a steady charge land almost entirely in the gap?

Because it repeats itself unchanged. A charge accrues at a constant rate against the assets, so it takes very nearly the same small slice on Monday as it did on Friday and as it will next March. Feed that into a series of sub-period gaps and it does exactly one thing to the series: it moves every value down by about the same amount. The charge shifts where the series sits and barely touches how far the series wanders. An expensive tracker can therefore still be an extremely steady one.

The everyday version is a landlord who raises the rent by a fixed amount each month. The household's monthly surplus is lower by that amount every single month. The month-to-month swing in that surplus is what actually causes the trouble, and the swing is driven by the electricity bill and the wedding invitations and the school fees, not by the rent. The rent moved the level. The rent did not add a single rupee of variability. A charge on a scheme behaves in exactly that way.

The sign of the tracking difference for a tracker is almost never a surprise either. Nobody can hold an index and nobody pays anything for the privilege, so an index carries no costs at all. A scheme's published return is struck after its own charge. The scheme is therefore expected to finish behind before anybody asks a single question about how well the portfolio was managed. A tracker finishing behind its index by roughly its charge is the ordinary result, not the alarming one.

Why does the part that varies land almost entirely in the spread?

Because those parts do not repeat themselves. The 0.08 points of residual on the Girnar Broad Market Index Fund is made of things that are lumpy by their nature. Cash sits in the scheme and its level changes. Money arrives on some days and leaves on others, and never in a smooth trickle. The index itself changes its constituents at particular moments, and following it through those moments has a cost that lands on those days and not on the days in between. Every one of those items varies from one sub-period to the next, so most of whatever variabilityHow much a quantity moves about from one period to the next, as opposed to how large it is on average. a tracker shows comes out of the residual rather than out of its charge.

The oddity appears when this is set next to the sizes. The charge is 0.20 points, much the larger of the two components, and it contributes almost nothing to the spread. The residual is 0.08 points, much the smaller, and it contributes almost all of it. Size and variability are simply different properties, and the component that dominates one can be nearly absent from the other. A street vendor's rent is the biggest single line in the month and the most predictable; the day's takings are smaller and are the entire reason some weeks are frightening.

Which question does each measure actually answer?

Say each question out loud and the assignment becomes obvious. How far behind did the scheme end is answered by the tracking difference. How reliably did the scheme follow along the way is answered by the tracking error. Neither one answers the other's question, and a reader holding only one of them cannot infer the other.

The split has a consequence for how a factsheet is read. A reader who intends to hold for the whole period cares only about where the holding finishes, wants the difference, and can treat the error as close to irrelevant. A reader whose sale may fall on a date not of their own choosing cares how tightly the scheme hugged the index, wants the error, and will not accept the difference as a stand-in. Both readers are being sensible. The two readers are simply asking different questions and need different data.

Why is only one of the two usually available?

Availability, and nothing more interesting than that. A tracking difference needs two published returns, and two published returns are exactly what a scheme and an index reporter routinely put out. A tracking error needs a return series for both sides at some agreed sub-period length. A series is a great deal more data, and it is not routinely published beside a headline return. So the difference is the one that reaches the reader, not because anybody decided it was the better measure but because it was the one that could be computed from what was lying around.

The reading habit that follows is simple: when only a difference is available, the honest course is to state what is not known rather than to treat the number at hand as a full description. That sentence sounds like a small piece of etiquette and it is actually the whole discipline. A note that reads minus 0.28 points for the stated year, and no series available so nothing is known about how steadily it tracked is a complete and honest record. A note that reads minus 0.28 points, tracked closely is a fabrication wearing the clothes of a summary.

What happens to the journey when the destination is held still?

A single published figure can never say how the gap behaved along the way. The display below pins the average of a drawn path of sub-period gaps at exactly minus 0.28 percentage points, the recorded tracking difference for the Girnar Broad Market Index Fund for the stated year, and allows one change only: how much that path swings either side of it. The ending story never changes. The journey changes completely. The dashed line is the entire figure a factsheet would have supplied, and it stays exactly where it is while everything else moves.

The default setting is a flat path: a scheme sitting at exactly minus 0.28 points in every sub-period. The little square to the right of the chart is the same quadrant shown earlier. The horizontal coordinate is the one held fixed, and the dot moves straight up as the path widens while never moving sideways at all.

Why can no tracking error be computed from two annual returns?

Because the inputs are absent, and an absent input is a fact about the measurement rather than an apology for it. The Girnar figures come to one annual return for the scheme, one annual return for the index it follows, and no series of any kind. A tracking error would need the return of both for each sub-period inside the same window, and it would need the sub-period length the measurement was run at. The same underlying behaviour produces a different figure at a different length. Neither of those is available. Producing a tracking error figure from two annual returns would mean inventing the series it is built from, and no honest figure can come out of that.

Inventing a missing input is exactly the fault worth spotting in other people's figures. A measurement with a missing input is not a measurement with a rough answer; it is not a measurement. The honest move when an input is missing is to name the absence and stop, and the dishonest move is to produce something that looks like an answer because a blank space is uncomfortable. A blank space is a finding. The blank identifies precisely which question cannot be answered.

What survives when the measurement cannot be made?

More than a reader expects. Of the 0.28 percentage point shortfall on the Girnar Broad Market Index Fund for the stated year, 0.20 points is a charge that accrues at a constant rate and therefore varies almost not at all. The remaining 0.08 points is made of cash levels, the timing of money arriving and leaving, and the cost of dealing when the index changes, every one of which varies by its nature. So whatever this scheme's tracking error actually was, most of it must have come out of the 0.08 rather than out of the 0.20.

The exact status of that sentence is the discipline itself: it locates the variability without measuring it, and a reader who converts it into a number has done the very thing the distinction guards against. Knowing which room the noise is coming from is not the same as knowing how loud it is. Both are useful. Only one of them has been established here, and saying which one is what separates an honest note from a confident one.

The whole build sits below in one place, with the check that runs the other way.

The two checks agree with the build in both directions, and the last row is the one worth staring at. The row is not blank because the work was left undone. The row is blank because two annual returns cannot produce a series, and a row that says so is worth more than a row filled in with something plausible.

The rupee version makes the size easier to feel. Take a holding of Rs 1,00,000/- at the start of the stated year and apply each figure to it as an illustration. The index path adds Rs 12,400/- and the scheme's net path adds Rs 12,120/-, a difference of Rs 280/- for the year, the 0.28 points expressed in money. Split that the same way and 0.20 points is Rs 200/- and 0.08 points is Rs 80/-, and Rs 200/- plus Rs 80/- is Rs 280/- exactly, the same check as the table run in rupees. On Rs 5,00,000/- the difference for the year is Rs 1,400/-. Two cautions travel with those figures. The charge is levied against the scheme's assets through the year rather than against the opening amount, so Rs 200/- illustrates the size of the charge component and is not a statement that exactly that sum was taken. None of those rupee figures says anything at all about how steadily the gap behaved.

Girnar Asset Management Limited, the Girnar Broad Market Index Fund and Sohail Merchant are invented.
Educational material. Not advice on any investment, tax, budget or market position.