Simple, Compound and Log Return: The Same Ten Months

A simple change measures one movement against the level it started from. A compound rate is the single change which, repeated in every period, would reproduce what a whole record actually did. A log change is the logarithm of one plus the simple change, and it adds where the others multiply. On the Vasant unit's ten invented months the three read 2.00 per cent, 1.57 per cent and 1.56 per cent a month.

The ten monthly changes of the Vasant unit and the ten of the Nakshatra unit, two records built for teaching, arrive here whole. Their totals across the whole ten months, and how far the months within each record sit from one another, are set out under the summaries a record of changes can carry. Everybody already has the plain arithmetic of a levelThe running value of something at a point in time, as opposed to how much it moved. A level is a where; a change is a how far. and a movement. Three ways of writing down that same movement follow, each defined on its own terms before any of them is set against another, and all three are then run over the same ten invented months to show in figures how far apart three correct answers get.

One definition first. For these invented units a month's change and that month's return are the same number: the unit moved by so much over the month, and that movement is what every form in this guide is a way of writing. Nothing is being redefined between one form and the next. The movement is fixed; only the notation is on trial.

What are the three forms, in one sentence each?

A comparison that starts contrasting before it has finished defining teaches nothing, so all three forms get a definition of their own before any one of them is set against another. Read the three slowly. The definitions are shorter than they look.

A simple change is the movement divided by the level it started from. If something sat at a hundred and finished the month at a hundred and three, the movement was three, the baseThe level a change is measured against, which is normally the level at the start of the period. Change the base and the same movement becomes a different percentage. was a hundred, and the simple change was 3.00 per cent. Every part of that sentence matters and the part people drop is the base. The same movement of three against a base of fifty is 6.00 per cent, and against a base of two hundred it is 1.50 per cent. A simple change is therefore never just a movement. A simple change is a movement together with the level it is being judged against, and that level is always the one at the beginning of the period.

A compound rate is the single change which, repeated in every period of a record, would reproduce exactly what that record actually did. A compound rate is not an average of anything in the ordinary sense. A compound rate answers a rebuilding question: if the ten months this record actually had were replaced by ten identical months, what would each of those identical months have to be so that the finish is the same? The rebuilding question is a strange one on first encounter, and its shape is worth holding on to. The months are not being summarised. The months are being replaced by ten copies of a single month, chosen so that nothing about the destination changes.

A log change is the logarithm of one plus the simple change. A logarithmAsk what power turns one fixed number into another, and the answer is a logarithm. The natural sort, used everywhere here, takes the base that calculus prefers. takes a multiplier and hands back something that can be added. So the simple change of 3.00 per cent becomes the multiplier 1.0300, and the logarithm of 1.0300 is 2.9559 per cent. The log change is close to the simple change and not equal to it, and it came out lower. Both of those turn out to matter later, and neither is an accident.

An everyday version holds all three forms at once. A tea cart trades for a week. One answer describes the week by its average day. A second describes the week by the one steady day that would have produced it. A third rewrites each day in a form where the seven days simply add up, and nobody asks for that third answer until the day they need to add. All three describe the same week and none of them is the correct one. Correctness is not the axis: what separates the three is which question each one answers.

Which of them adds across months, and which multiplies?

Now the contrast, and it is a single mechanical fact with a very long shadow. Simple changes multiply across periods and log changes add, and that one difference is the whole practical distance between the two forms. Everything else in this guide is a consequence of it.

Take the multiplying side first. A simple change of 3.00 per cent turns a level of 1.0000 into 1.0300. Chaining a second month onto the first does not mean adding the second month's percentage to it. The second month becomes its own factorOne plus a change, written as a multiplier. A rise of 3.00 per cent is a factor of 1.0300 and a fall of 3.00 per cent is a factor of 0.9700., and the two factors multiply. The Vasant unit's second month was 18.50 per cent, so its factor is 1.1850, and 1.0300 multiplied by 1.1850 is 1.22055. Two months in, the level has moved 22.06 per cent, and 22.06 is not 3.00 added to 18.50. Chain all ten months this way and the level lands at 1.16856953.

Now the adding side. Turn each of those same ten factors into its logarithm and ten log changes come out, and those simply add. The first two are 2.9559 per cent and 16.9743 per cent, and they add to 19.9302 per cent. A logarithm turns multiplication into addition by construction, and the sum is therefore neither a coincidence nor an approximation of the multiplied answer. The sum is the multiplied answer written in a different alphabet. All ten added come to 15.5780 per cent.

Adding simple changes across periods is the commonest arithmetic mistake made with any record of changes, and it is a quiet one. Nothing about the numbers signals that anything went wrong. The Vasant unit's ten monthly changes added together come to 20.00 per cent. Multiplying the ten factors gives what the record actually came to, 16.86 per cent. Both answers look reasonable, both are computed from the same ten figures, and only one of them is what happened.

What do all three give on the same ten months?

Here is the whole record, with every form worked out beside it. The ten monthly changes of the Vasant unit, in the order they happened, are 3.00, then 18.50, then 2.50, then 17.00, then minus 3.00, then minus 13.00, then 6.50, then minus 1.00, then minus 7.50 and finally minus 3.00 per cent. Nothing is sorted and nothing is dropped.

The level column is the ten factors already multiplied together, month by month, so its last entry is the whole record. The factors themselves are one plus each simple change and they are printed in full in the strip above.

The bottom row carries all three answers at once. The simple average of the ten months is 2.00 per cent. The compound rate is 1.57 per cent a month. The average log change is 1.56 per cent. Three defensible answers to what a month on this record was worth, differing by nearly half a percentage pointThe unit for the distance between two percentages. Moving from 2.00 per cent to 1.57 per cent is a fall of 0.43 percentage points, and calling that a fall of 0.43 per cent would mean something different. on one entirely unremarkable record of ten months.

The compound rate is the one form here that is genuinely new, so it deserves slow attention. The ten factors multiplied together come to 1.16856953. A product of 1.16856953 says the record finished 16.86 per cent above where it started. Now the rebuilding question: what one monthly change, repeated ten times, lands on 1.16856953? The answer is 1.57 per cent a month. Raising 1.0157 to the tenth power gives 1.1685694, and the last two places differ from the record's 1.16856953 only because the rate itself has been rounded to two decimals; carried out further the rate is 1.5700008 per cent and the tenth power lands on 1.16856953 on the nose. So the compound rate is not an average of the ten months at all. The compound rate is a substitute for the ten months, chosen so the destination is preserved.

Why does the compound rate come out below the simple average?

The direction here is worth feeling rather than reading, so ask the prediction before doing the arithmetic. The ten months average 2.00 per cent. Is the compound rate above that or below it?

The compound rate sits below the simple average, it sits below on every record whose months are not all identical, and the reason is that a fall is taken off a bigger starting level than the one the following rise gets added to. Take a level of 100.00 and put the Vasant unit's worst month through it: a fall of 13.00 per cent takes 13.00 units away and leaves 87.00. Now put a rise of exactly the same 13.00 per cent through what is left. Thirteen per cent of 87.00 is 11.31, not 13.00, so the level climbs to 98.31 and finishes 1.69 below where it began. The two percentages were identical and they averaged to precisely nothing, but the level did not come back.

The household version is one everybody has met. A shopkeeper takes a fifth off a price and then, a month later, puts a fifth back on. The price does not return. The reduction came off the full price and the increase went onto the reduced one, so the two moves were never the same amount of money even though they were the same percentage. Nothing about compounding is being smuggled in there. The shortfall is arithmetic about which base a percentage was taken of.

Which gives the exact condition under which the gap disappears. The gap between the simple average and the compound rate is created entirely by the months differing from each other, so a record whose months were all identical would show a gap of exactly nothing. Test it: ten months of exactly 2.00 per cent give a level of 1.21899442, and the one steady month that rebuilds 1.21899442 over ten months is 2.00 per cent, precisely the simple average. On the Vasant unit's actual months the two read 2.00 per cent and 1.57 per cent, a distance of 0.43 percentage points, and that distance is a direct report of how far the ten months sit from each other.

What does the log form buy that the others do not?

So far the log form has looked like a curiosity that sits near the compound rate. The log form earns its keep here, and the demonstration takes one line.

The ten log changes summed come to 15.5780 per cent. Now the exponentialThe operation that undoes a logarithm. Feed it a log change and it hands back the factor that log change came from. of that one sum: out comes 1.16856953, the multiplied factor, to every decimal place the arithmetic carries. Not close to it. It. A chain of ten multiplications has turned into one addition and one undoing, and that swap, rather than any question of taste in notation, is why anybody bothers with the log form at all.

Why does anybody care about turning multiplication into addition? Because almost every tool that summarises a set of numbers is built to add. An average adds and divides. A running total adds. A fitted line adds. A measure of how far readings sit from each other adds squared distances. Given a column of factors, any of those tools will do the wrong operation politely and return an answer that looks fine. Given a column of log changes, adding is exactly the right operation, and adding is the one the tool already knows.

Now the caution, and it is the sharpest distinction in the whole comparison. The average log change is 1.5578 per cent and the compound rate is 1.5700 per cent. The average log change and the compound rate are two different quantities that happen to land near each other on this record, and they must never be quoted as one number. Rounded to two decimals the pair reads 1.56 and 1.57, still two numbers. Rounded to one decimal the pair reads 1.6 and 1.6, one number, and a real distinction has been rounded out of existence by a formatting decision. The average log change is the mean of the ten logarithms. The compound rate is what comes out after undoing that mean. Taking the exponential of 1.5578 per cent gives 1.015700, a factor of 1.5700 per cent, and that is the exact relationship between the pair: one is the log form of the other, not a rounding of it.

Where does the log form break down?

Every form has a range it works over and the honest thing is to state it rather than to assume it away. A log change needs one plus the simple change to be above nothing, so a simple change of minus 100.00 per cent has no log form at all, and anything worse than that has none either.

Follow why. A simple change of minus 100.00 per cent turns a level into nothing, so the factor is 0.0000, and no power that anything is raised to ever reaches nothing. The logarithm is not merely awkward at that point. There is none to write down. And as the simple change approaches minus 100.00 per cent from above, the log change does not politely approach some finishing value: it dives, without limit. At a simple change of minus 50.00 per cent the log change is minus 69.3147 per cent. At minus 90.00 per cent it is minus 230.2585 per cent. At minus 99.00 per cent it is minus 460.5170 per cent, and it keeps going.

Now the other end of the range, and it is the reassuring one. Near a simple change of nothing the two forms track each other almost exactly, and they stay close over the whole range this record occupies. On the Vasant unit's ten months the worst reading is minus 13.00 per cent, whose log change is minus 13.9262 per cent, so nothing on this record comes anywhere near breaking and every figure in this guide is safely inside the form's range. The limit is real even though it does not bite here, and stating it is the difference between knowing a form's range and hoping it holds.

When does the choice between them actually matter?

A distinction that cannot be switched off becomes pedantry, so what follows is when this one is material and when it is not, both stated as figures rather than as a preference.

Over a single period, and for a small change, the three forms very nearly agree. The Vasant unit's first month was a simple change of 3.00 per cent. Its log change is 2.9559 per cent. The compound rate over one month is the month itself, so it is 3.00 per cent. The three answers sit within 0.0441 of each other, so all three agree to one decimal place and only part company in the second. Where the subject is one month of one record, arguing about which of the three is meant is a way of not saying anything.

Over many periods, or where the changes are large, the three separate and the separation is not small. The same record's second month is a simple change of 18.50 per cent whose log change is 16.9743 per cent, and there the two are 1.5257 apart on a single reading. Run all ten months together and the plain average and the compound rate come out 0.43 percentage points apart every month. The choice matters exactly when the periods are many or the changes are large, and both of those conditions are about how far the readings sit from each other rather than about which form is fancier. On this record the spreadA single figure for how far a set of readings sit from each other. Computing it is covered separately. of the ten months is 9.96 per cent, and it is that width, not the average, that opens the gap.

The panel above shows one more thing that no static figure can: the widening is smooth and one directional. Shrink the changes and the three collapse onto each other; stretch them and they fan out; and the plain average is always the one on the outside. Nothing about that ordering is a property of this particular record. The plain average is at or above the compound rate on every record there is, and it is exactly equal only when every period is identical. Saying which form is being quoted therefore costs one word and closes the whole question.

Which form should be used for which job?

Choosing a form is the part somebody actually has to do on a Tuesday. An analyst summarising a column of monthly readings, a lender reading a set of period changes on an account, a student totting up a set of practice figures and a household comparing one month against the whole year are all doing the same three way choice, and it resolves the same way every time. Pick the form by the question, not by habit, and then name the form out loud.

Use the simple form when the question is about one period. One period is exactly what the simple form measures. Ask what the Vasant unit did in month four. The answer is a rise of 17.00 per cent. There is nothing to convert, nothing to compound and nothing to argue about, and reaching for a log change to answer a one month question is showing off rather than working.

Use the compound rate when the question is about a whole record stated per period. The compound rate is the only one of the three that reproduces the record by construction. If somebody quotes 1.57 per cent a month for the Vasant unit's ten months, that figure can be tested on the spot: 1.0157 raised to the tenth power must land on the record's own 1.16856953, give or take the rounding carried into the rate. If it lands somewhere else entirely, one of the two figures is wrong, and that has been established in ten seconds. Neither of the other two forms can be checked against the record that way, and being checkable is worth a great deal.

Use the log form when the changes have to be added, averaged, or handed to any tool that assumes adding. Adding is where the log form's one property earns its keep, and it is the reason the form is everywhere in quantitative work rather than in ordinary conversation. Convert at the start, do the adding work in log changes, and convert back at the end with a single exponential.

And then there is the rule that costs one word. On these ten invented months 2.00 per cent, 1.57 per cent and 1.56 per cent are all correct answers to slightly different questions, and a reader handed a bare number cannot tell which question was asked, so always say which form is being quoted. The word that fixes it is simple, or compound, or log. Nobody has ever regretted writing it and plenty of people have spent an afternoon reconciling two figures that were never supposed to match.

What does the same arithmetic give on the other record?

One worked example is a demonstration and two is a pattern, so here is the identical treatment on the Nakshatra unit's ten months. The Nakshatra months run 1.00, then 6.00, then minus 4.00, then 11.00, then 1.00, then minus 9.00, then 6.00, then 1.00, then minus 4.00 and finally 1.00 per cent. Same order, same period, same three forms.

The pattern holds on both records and the size of the gap tracks how wide the months are. The Nakshatra unit's months run from minus 9.00 to 11.00 per cent while the Vasant unit's run from minus 13.00 to 18.50, so the Nakshatra unit is the narrower record and its gap is the smaller one, 0.15 against 0.43. Nothing about the second record needed a different method, a different order or a different caution. Working on both records unchanged is what it means for all of this to be arithmetic rather than a convention.