Tracking Error in Fixed Income: Sources and Interpretation

Take the portfolio's return for one period, subtract what the benchmark returned over the same period, and keep the difference. Collect a run of those differences. Tracking error is how widely that set is spread, in the usual measure of spread: a width, not a cost, not a ceiling, not a prediction. Producing one needs the run of periods behind those differences, and no such run has ever been recorded here.

Underneath that sits a distinction the four preceding sections have been setting up, and the distinction is where nearly every misreading of the measure begins. Two portfolios can carry exactly the same difference in MODIFIED duration against exactly the same benchmark and still produce dispersions nowhere near each other. A sensitivity is built out of what is held today, a dispersion is built out of what happened over a stretch of past periods, and no arithmetic carries a reader from one to the other. They are not two views of one quantity. The two take different inputs, they answer different questions, and they are quoted in different units.

What is tracking error the spread of, exactly?

Start with one period. The period could be a month, a week, a day, whatever the run under examination is made of. In that period the portfolio earned some return. Over the identical period, the benchmark earned some return of its own. Subtracting the second from the first gives one number, the active return for that period, and it sits happily on either side of nothing: some periods it is positive, some periods it is negative.

The same subtraction repeats for the next period, and the next, until the periods run out. The subtractions leave a set of differences, one per period. Tracking error is the standard deviationA single number saying how far, typically, a set of numbers sits from its own average. Small when they cluster together, large when they scatter. of that set of differences, and the phrase to hold on to is of that set of differences. Not of the portfolio's own returns. Not of the benchmark's. Of the gap between them, period by period.

The distinction between the gap and the two returns behind it is not fussiness. A portfolio can be violently volatile on its own account and still track its benchmark almost exactly. The benchmark was being thrown around by the same forces at the same moments, so the difference between the two stayed small every single period. Equally, a placid portfolio can wander away from a placid benchmark in a slow, steady drift, and the differences will be small but consistent. The measure has nothing to say about how much anything moved. The measure only ever looks at the gap.

The last move is bookkeeping. A stretch of weekly periods and a stretch of monthly periods would otherwise produce widths that cannot be set beside each other, so the result is normally annualisedRestated to the size it would come to over twelve months, so that measures taken over stretches of different lengths can be put side by side. before anybody quotes it. The rescaling changes the number quoted and changes nothing about what was measured.

What is tracking error not?

Four readings of this measure turn up constantly and three of them are simply wrong. The fourth is a subtler kind of wrong: it treats the width as a description of the position rather than of the difference the position happened to produce. Since the wrong readings are more common than the right one, they are worth taking one at a time.

Everything in that table has the same root: the measure describes a set of differences that came out of the past, and each wrong reading wants it to describe something else entirely, whether a rupee amount, an agreed limit, a future period or a holding list. The measure is disciplined about what it covers precisely because it covers so little. The measure takes a run of periods that already happened, looks only at the gap in each one, and reports how scattered those gaps were. Anything beyond that is being read into it.

Why will a sensitivity not turn into a spread?

A sensitivity and a dispersion sit close enough together that the mix-up is almost guaranteed. The earlier material set out how to measure a rate exposure and how to split it into the part the benchmark would have suffered anyway and the part attributable to sitting longer. The split produces the sharpest number in the whole sequence. When the discussion then turns to the gap between a portfolio and its benchmark, the sharp number is sitting right there, and it gets picked up.

Look at what each object is made of. The active part of a rate exposure starts from two MODIFIED durations as they stand today and a rate move somebody has stated. Nothing about last month enters it. Tracking error starts from a run of periods that are over, and nothing about today's positioning enters it beyond whatever effect it already had. One is a conditional statement about a single assumed event, the other is a summary of many events that already ran, and printing them in adjacent columns of the same table suggests a relationship no arithmetic supports.

The fifth row is the whole difficulty in one line. The left column has a number in it. The right column does not and cannot. When one of two adjacent fields is filled and the other is empty, the filled one migrates, and that migration is the failure set out further down.

Where does the difference in a bond portfolio come from?

A width is the summary at the end. Before it there are causes, and in a bond portfolio there are six worth naming. The six are laid out below in the order a reader tends to meet them rather than in order of size. Their sizes cannot be ranked from anything recorded here.

The right hand column read downwards shows the shape of the problem: the first row asks for four things that all exist here, and the five rows beneath it ask for records this platform has never held. That is not a gap to be apologised for, and it is not unusual. Most of what feeds a tracking error figure is holding level detail, so a description of the sources is available to anyone while the figure itself is available only to whoever holds the underlying records.

The last point on the diagram deserves saying out loud. Even where a full return series exists, the width that comes out of it is one number covering all six sources at once. Pulling it apart into how much came from duration positioning and how much from what was held by industry is a separate exercise with separate inputs, and it is not something the width itself reveals.

How many of those six can be sized from what is recorded here?

One. Not two, and not one and a half. The first source can be given a figure because everything it needs is written down, and each of the remaining five needs a record kept by whoever holds the underlying positions. The number that comes out of the sizeable one is the number that gets misused later, so it is worth working slowly.

The portfolio holds Rs 5,000 crore of market value at a MODIFIED duration of 5.20, and the benchmark stands at a MODIFIED duration of 4.80. The gap between the two is 0.40. Suppose a rise of 100 basis points that is the same size at every point on the curve. THE WHOLE EXPOSURE is what the market value would give up on a first order estimate. The figure comes to 5.20 per cent, or Rs 260 crore. THE ACTIVE PART is the slice attributable to the difference in MODIFIED duration alone. The figure comes to 0.40 per cent, or Rs 20 crore. Both readings were built in the mapping walkthrough earlier in this sequence and neither is rebuilt here.

Since the relationship between the size of the stated move and the size of the active part is a straight one at this level of estimate, the same four inputs give a rate rather than a single amount: each basis point of a rise that is the same size at every point on the curve separates the portfolio from the benchmark by Rs 20,00,000/-, and that per basis point figure is the honest form of what this material knows. The Rs 20 crore is that rate multiplied by a move somebody picked.

Now the five that cannot be sized, each with its own reason rather than a shared shrug. Source two needs to know where along the curve the portfolio sits, and no list of holdings and maturities exists here. Source three needs the market value split across industries, and no such split exists here. Source four needs the borrowers named with an amount against each, and there is no such list. Source five needs a cash position on each measurement date, and none is recorded. Source six needs two valuation sources, and this platform holds one set of figures with no second source to strike them against.

What goes in the cell where the figure belongs?

Nothing goes in it, and the cell is drawn anyway. An account that never mentions the figure teaches the reader nothing about why it is absent. A drawn cell with a reason inside it teaches exactly that, and it also states precisely what would have to turn up before the cell could be filled.

The list of what would have to turn up is short. A return seriesA period by period record of what a holding earned, one figure per period, built around every rupee that came into it and went out of it. for the portfolio, covering a stated run of periods. A matching one for the benchmark, covering the identical periods. And underneath the first of those, a record of money arriving and leaving. A return for a period cannot be struck honestly without knowing what came in and what went out during it. None of the three is recorded here, and how a return series is built around flows of money is settled elsewhere.

Strictness matters here in particular because a dispersion is the one figure a reader could not check against anything else. Every other number here ties to something: the Rs 260 crore whole exposure ties to the market value and the MODIFIED duration, the Rs 20 crore active part ties to the difference between two MODIFIED durations, and a reader who suspects either can rebuild it in a line. A width has no such anchor. An invented one would sit in its cell looking exactly like a real one, and nothing alongside it would contradict it. Within two hops of being quoted, it would be somebody's fact.

How is a tracking error figure read when one exists?

Suppose one arrives. The figure comes as a width quoted in the same units a return is quoted in, and it has almost certainly been restated for a year. The width describes how far apart two streams of return have been over the exact run of periods it was worked out on, and over no other run. The last clause is not a technicality. A different run gives a different answer, from the same portfolio, without anybody trading anything.

There is a second question to ask of any such figure, and it separates two objects that share one name. One kind of tracking error is worked out from returns that already happened; the other is estimated from what is held right now, using a model of how the things held tend to move together. The first is a summary of history. The second is a projection, and it depends on modelling choices somebody made. The two answer different questions, they can differ by a lot on the same portfolio on the same day, and a figure quoted without saying which of the two it is cannot be used for anything.

Can a wide positioning difference sit beside a narrow spread?

A wide positioning difference can indeed sit beside a narrow spread, and the reason is worth working through slowly. The portfolio sits 0.40 longer in MODIFIED duration than the benchmark. The 0.40 is fixed until somebody changes it. The difference it produces in any given period, though, depends entirely on what rates did in that period.

Use the rate that came out of the sizing above. Each basis point of a rise that is the same size everywhere on the curve moves the portfolio Rs 20,00,000/- away from the benchmark. So a period in which the curve moved by 100 basis points produces a gap of Rs 20 crore, and a period in which it moved by 10 basis points produces a gap of Rs 2 crore, from an identical position. Run a stretch of quiet periods and the differences are small and the width is narrow. Run a stretch of violent ones and the same position throws off differences ten times the size, so the width is ten times as wide, and nothing about the portfolio changed between the two stretches.

A width therefore describes a stretch of time as much as it describes a portfolio, and the run of periods must be quoted alongside the figure. Comparing two tracking error figures worked out over different stretches also tells almost nothing about the two portfolios. Most of what separates the answers may be the difference between the two stretches.

What does this look like away from bonds?

Two neighbours leave for the same office at the same minute every morning and walk. One of them turns left at the second junction and rejoins the main road further along; the other stays on the main road the whole way. The turning adds a hundred metres to the walk. The hundred metres is a fixed, knowable, one number fact about the difference between the two routes, and it could be measured with a wheel this afternoon.

Now ask a different question: how differently do the two of them arrive? Over a week, some mornings they reach the door together. Some mornings one of them is four minutes behind. Arrival depends on the crossing, the school run, whether the tea stall queue spilled onto the pavement. So the answer is not a hundred metres and never will be. The turning is the position and the spread of arrival gaps is the dispersion, and knowing the first tells remarkably little about the second.

Push it one step further and the point sharpens. Take a week of school holidays, when the roads are quiet: the two neighbours arrive within a minute of each other every day, and the spread is narrow. Take a week of heavy rain: the same turning, the same hundred metres, and now the gaps run to eight or nine minutes. Nobody changed the route. The week changed. And when somebody hands over a number describing how differently they arrived, the first thing to ask is which week it came from.

The drawing leaves out the whole difficulty in miniature. The seven bars are there, the average is there, and the single number saying how far the seven typically sit from that average is not worked out. Seven mornings were recorded, so for the two neighbours it easily could be. For the portfolio described here it cannot be. The equivalent of those seven mornings has never been written down anywhere on this platform.

Who actually uses this, and how?

Three people reach for a tracking error figure, and each of them wants a different thing from it.

The person running the money uses it as a check on whether the positioning that was intended is the positioning that showed up. Somebody decides to sit 0.40 longer in MODIFIED duration than the benchmark, expecting that decision to produce differences of a certain size. If the width that comes back is far wider than that decision could plausibly account for, something else is separating the two sides, and the six sources tell them where to look: an industry concentration nobody meant to build, cash sitting idle, a borrower held far heavier than intended. The width does not identify the culprit. The width says only that the search is worth starting.

The person deciding where to put money uses it to tell two arrangements apart when the return figures look similar. Two arrangements measured against the same benchmark over the same stretch, one reporting a narrow width and one a wide one, were run very differently, whatever their returns did. The narrow one stayed close; the wide one wandered. Neither is better in the abstract. The point is that the reader of a regulatory returnA form of stated figures a regulated body has to file on a timetable, in a layout it does not choose. or a factual disclosure can see the difference at all, and can then ask whether the wandering was intended.

The person supervising a mandate uses it as an agreed boundary, and this is the one use that changes the measure into something it is not on its own. Where a written mandate says the width shall stay under some agreed level over a stated run of periods, the figure stops being merely descriptive and becomes a test. Note carefully what happened there: the ceiling came from the mandate, not from the measure. The width itself never contained one. The second row of the wrong readings table makes exactly that point.

The arithmetic is not really about bonds, so here is a household version. When one household has a home loan on a floating rate and the neighbour has one fixed for five years, the difference in what the two pay each month is not a fixed number. The difference is small in months when nothing moves and large in months when a great deal moves, from an unchanged gap between the two loans. Describing that with a single figure without saying which run of months it came out of would be wrong, and a tracking error figure asks for the same discipline.

What can never be said about this portfolio?

The list of what cannot be said about this portfolio is longer than the list of what can. No run of returns for either side exists here, so how widely the portfolio's returns have differed from the benchmark's cannot be said. Neither an industry split nor a borrower list exists here, so no part of a difference can be attributed to either. Whether the width would come out narrow or wide depends on stretches of time this platform does not record at all.

The remainder is bounded and exact. Somebody positioned this portfolio at a MODIFIED duration of 5.20 against a benchmark at 4.80, on Rs 5,000 crore of market value. Under a rise of 100 basis points that is the same size at every point on the curve, that positioning is worth Rs 20 crore of separation on a first order estimate, and the arithmetic stops there. Whether the positioning was right, and whether it should be closed or widened, are judgements no arithmetic settles, and what rates do next is nobody's to know. A rating is a different object again, produced by different work, and a fair value is a third. Neither follows from a width, and no price target follows either.

A reader could reasonably ask about one more absence. A rate move that is not the same size at every point on the curve would separate the two sides differently again. No such move is carried here, so that separation cannot be worked out. A twist in the curve is settled elsewhere in this subject area. The rupee figure it would do to this portfolio is one nobody on this platform is in a position to write.

Why there is no control to drag here

A control here would have had to manufacture the one figure with no record behind it. Any movable control would need to feed a run of period returns into the measure and hand a width back, and that width would be fiction the moment it appeared on the screen. A slider looks as though it computed something, so reading a made up width off one is worse than reading it in a sentence. The empty cell in the figure above does the work a control would have done badly, and the nine questions ask the reader to perform the separating instead.

The portfolio, the benchmark it is measured against and the two neighbours walking to work are invented.
Educational material. Not advice on any investment, tax, budget or market position.