Return Series Measures: Average, Compound and Spread

One record of monthly changes has four summaries and they answer four different questions. Over the ten invented months of the Vasant unit the plain average is 2.00 per cent a month, the ten months multiplied together come to 16.86 per cent in total, the compound rate is 1.57 per cent a month, and the months are spread 9.96 per cent around their own average.

Two invented columns of monthly readings run through this material, the Nakshatra unit and the Vasant unit, ten readings apiece, and the Vasant column is the one summarised four ways below. One convention holds throughout: with these columns, the size of a month's move and the return it produced are one and the same figure, and neither word is given a second meaning anywhere.

What goes into this calculator, and what comes out of it?

The arithmetic takes one thing: a list of monthly changes, written as percentages. Ten of them here, but the count does not matter to the arithmetic. Four numbers come back, and the whole reason this needs a guide rather than a single line is that the four are not four attempts at the same answer. The four are answers to four separate questions, and the trouble starts when one of them is quoted as though it had answered the others.

Here is the everyday shape of it first. A tea cart outside an office keeps a note of how each day went against the day before. At the end of ten days somebody asks how the ten days went, and there is no single honest reply. One reply gives what a typical day looked like. Another gives what the ten days came to when laid end to end. A third gives what one steady day, repeated ten times, would have had to look like to land in the same place. A fourth gives how wildly the days differed from each other. Four questions, four numbers, and choosing which one to say out loud is already an answer to a question the listener did not ask.

The calculator below does exactly that with the Vasant unit's ten invented monthly changes. Same list in, four numbers out, each tagged with the question it settles.

What does the plain average actually answer?

Add the ten monthly changes and divide by ten. On the Vasant unit that gives 2.00 per cent, and on the Nakshatra column beside it the same arithmetic gives 1.00 per cent. Nothing subtle is happening. The subtle part is which question the answer belongs to.

The plain average answers exactly one question: if a single month were drawn from the ten at random, what should it be expected to say? That is a real question and 2.00 per cent is a correct answer to it. Over a hundred such draws, each month replaced after it is read, the readings centre on 2.00 per cent. No other summary answers that particular question. The plain average is not a mistake and never was.

The plain average does not answer what the ten months came to when they were laid end to end. The reason lies in what an average actually promises. Ten times the average is the sum, and that is arithmetically the whole of what averaging means. So ten changes averaging 2.00 per cent add to 20.00 per cent. The plain average says nothing at all about what those ten changes multiply to, and a run of monthly changes multiplies.

What comes out if the months are multiplied together instead?

Turn each month into a factor. A change of 3.00 per cent becomes 1.0300. A change of minus 13.00 per cent becomes 0.8700. Then multiply all ten factors together in the order they occurred. The answer is 1.16856953, a total of 16.86 per cent over the ten months.

The multiplied total is the number a reader almost always means when they ask how the record did, and it cannot be got by adding the ten monthly changes. Adding them gives 20.00 per cent. Multiplying them gives 16.86 per cent. Same ten months, same arithmetic rules, different question, and the two answers are 3.14 percentage points apart on a record only ten months long.

Why multiplication rather than addition? Because the second month's change is applied to whatever the first month left behind, not to the level things started at. A stall that takes twelve hundred rupees on Monday and does thirty per cent better on Tuesday is not working from twelve hundred any more. Tuesday's rise is measured against Monday's takings. Every month in this list is measured against the month before it, so the months chain. The cumulativeBuilding up as it goes, so that any one reading already carries everything before it inside itself rather than standing alone. Vocabulary carried in from earlier work. figure after any month is the product of every factor up to that point.

What single monthly rate would have produced that same total?

Now ask a different question of the same ten months. Forget that they differed. Suppose one steady monthly change had happened ten times over and had landed in exactly the same place, at 1.16856953. That steady change is the tenth root of 1.16856953, less one, and it comes to 1.57 per cent a month.

The tenth root is the compound rate, and the compound rate is the per month version of the total rather than a rival to it. Ten months of 1.57 per cent, chained, land at 16.86 per cent. Ten months of 2.00 per cent, chained, land at 21.90 per cent, and the record did not do that. The compound rate of 1.57 per cent sits 0.43 of a percentage pointThe unit used when subtracting one percentage from another. Two rates standing at 2.00 and 1.57 are 0.43 percentage points apart, and reporting them as differing by 0.43 per cent would be a claim about something quite else. below the plain average of 2.00 per cent, and it is never the other way round.

The Nakshatra column does the same thing on its own numbers. Its plain average is 1.00 per cent a month. Its ten factors multiply to a total of 8.84 per cent, and the steady rate that reaches that total is 0.85 per cent a month. Lower again, by a smaller amount, and that smaller amount will turn out to be the whole subject of the next two sections.

Why is the compound rate never the higher of the two?

Because giving a percentage back does not undo having lost it. The fall is worked out on a larger baseThe level that a percentage is worked out on. Ten per cent of a larger base is a larger amount than ten per cent of a smaller one, even though the percentage is written the same way both times. and the rise that follows is worked out on the smaller one the fall left behind, so the rise has less to work with.

Set percentages aside entirely for a moment. A wedding caterer quotes for four hundred guests. The number drops by a quarter, to three hundred. Then it goes back up by a quarter, and a quarter of three hundred is seventy five, so the count stands at three hundred and seventy five and not back at four hundred. Nobody finds that surprising when it is guests. The same arithmetic holds for monthly changes, and it stops being obvious the moment the words per cent appear.

The more the months differ from each other, the wider the gap between the plain average and the compound rate, and a record whose months are all identical has no gap at all. That is not a rule of thumb, it is arithmetic, and the next section measures it.

How much of that gap does the spread account for?

The ten months of the Vasant unit sit 9.96 per cent apart from their own average, measured the ordinary way. Squaring that spread gives the varianceThe spread with the square left on it. Take each figure's distance from the average, square each distance, average the squares, and stop there rather than taking the square root back off., and half of the variance comes to 0.50 per cent.

Here is something that looks like a coincidence and is not. The plain average of 2.00 per cent, less that 0.50 per cent, lands on 1.50 per cent. The true compound rate is 1.57 per cent. The spread accounts for very nearly the whole of the gap between the two summaries, and the months differing from each other is where the gap comes from.

The caveat matters more than the result. 1.50 per cent is not 1.57 per cent. Half the squared spread is 0.50 and the true gap is 0.43, and those two are 0.07 apart. Half the squared spread is a good approximation to the gap and it is not an identity, so it is close enough to explain where the gap comes from and not close enough to be quoted as the gap. Anybody who writes down 1.50 per cent and calls it the compound rate has quoted an approximation as a result.

What do the ten log changes add up to?

There is a second way through the same arithmetic, and it exists for one reason. Take the logarithmThe exponent that answers a question of this shape: how many times over must one fixed number be multiplied by itself before it lands on this one? Treated in its own right elsewhere and taken as known here. of each month's factor rather than the factor itself. The ten numbers that come out are 2.9559, then 16.9743, and so on down to minus 3.0459 for the last month. Log changes add plainly, the way any numbers add. The ten of them come to 15.5780 per cent, carried in the neighbouring material as 15.58 per cent. Their average is 1.5578 per cent, carried as 1.56 per cent.

Log changes exist because they turn multiplying into adding, and that is the whole of the reason rather than a convenience of notation. Take the exponentialThe return trip. Hand it a logarithm and it gives back whichever number that logarithm was taken of in the first place. of 15.5780 per cent and out comes 1.16856953, precisely the answer that chain of multiplications gave. Ten multiplications have collapsed into a single sum, and the two roads meet on every decimal place the arithmetic carries.

The same trick shows up once more in a place worth noticing. The compound rate of 1.57 per cent is the exponential of the average log change of 1.5578 per cent, less one. So the compound rate is nothing more mysterious than an ordinary average taken in the world where multiplication has become addition, and then carried back.

Which of the four should a summary lead with?

Now the part that decides whether any of this reaches anybody usefully. Somebody has to write one line about the ten months, and the order the four numbers appear in is not a matter of taste.

A reader asking how the record did means the total over the record, so lead with the total. The compound rate is the same fact stated per month, so put it next to the total and save a reader working it out. Without the spread a reader cannot judge either of the first two, so put the spread third. A total of 16.86 per cent from months sitting 9.96 per cent apart is a different animal from the same total out of months that barely moved. And put the plain average last, or leave it out.

Why last? Not because it is wrong. The plain average is exactly correct and answers a real question. The plain average goes last because it is the narrowest of the four and the one most often quoted as though it had answered the others. Quoting 2.00 per cent a month on a record that compounded at 1.57 per cent a month is not a lie and is still misleading, and both halves of that sentence have to be held at once.

Field notes, in the sense of where each number is found rather than what it means. When somebody hands over a record and asks for these four, these are the places to look.