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Exponential percentages: a useful proposed terminology

Imagine that the price of something increases by 25%. Later on, the price returns to the original value. How much did it decrease? Well, someone could say 25%. But it dropped by 20% only because you must count the percentages from the new, higher price.

For example, the price went from $100 to $125 and back to $100. Those $25 in the decrease are just 1/5, or 20%, of the $125 price before the second step.

Is there a way to describe the situation with two numbers that will be equal? Of course, there is. You may say that the price increased 1.25 times and then decreased 1.25 times. Alternatively, you may use my new terminology and count the logarithms.

You may say that the price increased by 22.3 E% (exponential percent, or expo-percent, or E-percent for short) and then decreased by 22.3 E%. That's because ln(1.25) = -ln(0.8) = 0.22314... i.e. exp(0.22314) = 1.25, exp(-0.22314) = 0.8.

The advantage of this terminology is that these exponential percentages may be easily added and inverted. While the new terminology has all the virtues of the multiplicative notation, it is actually an additive one. For very small changes (in the limit when they go to zero), the exponential percentages agree with the ordinary percentages because exp(x) = 1+x + o(x) where o(x) is negligible for very small x.

However, when the percentages are large (with either sign), the nonlinearities become important. For example, the Dow Jones Industrial Index grew from 100 sometime in the 1910s to 800 in the 1970s and 10,000 in 2009. You could also say that it grew by 208 E% between the 1910s and the 1970s and by 253 E% between the 1970s and 2009.

To compute the actual prices from these numbers, you need to exponentiate. For example, exp(2.08)=8 and exp(2.53)=12.5 give you the multiplicative increases. However, the exponential percentages may be added in a simple way - because exp(a+b) = exp(a) exp(b) and exp(-a) = 1/exp(a), as many of you know. ;-)

So 2.08+2.53 Ef = 4.61 Ef = 461 E%. And exp(4.61) is approximately 100, which gives you the multiplicative increase between the 1910s and 2009. I used Ef = 100 E% and you have surely understood me.

I think that the advantage of using E% is that it would be clear that the numbers in front of E% would describe the actual "rate" that must be exponentiated to get the multiplicative increase or decrease - like in continuous compounding.

In particular, in the case of personal banking, E% would immediately tell you that the figure in front of it describes the interest rate in compound interest: it must be exponentiated to calculate the annual percentage yield which is written without any "E". For example, the 22.3 E% annual interest rate adds 25% of the original amount (yields) to your account after one year.

It's clear that this terminology, kind of common in cosmology where we talk about the e-foldings (we usually talk about the "e-folding time" which is the proper time of a world line along which the linear dimensions of the Universe are expanded 2.718... times), could be useful in economics, finances, biology, demographics, and lots of other places.

Related units:

  • One e-folding, symbol: Ef. That's the process when a quantity gets multiplied by (or divided by) "e", analogous to doubling or tripling (or halving or thirding, whatever is the right word: but with the base "e" instead of "2" or "3")
  • One exponential percent, one expo-percent, one e-percent, or 1 E%. Here, 1 E% = 0.01 Ef.
  • One exponential permil, one expo-permil, one e-permil, or 1 E‰. It's equal to 1 E‰ = 0.001 Ef = 0.1 E%.
  • One exponential basis point, one expo-basis point, one e-basis point, 1 Ebp, or 1 E‱. It's equal to 0.0001 Ef = 0.01 E% = 0.1 E‰.
  • One exponential part per million, one expo-ppm, or one Eppm. It's equal to 1 Eppm = 0.000001 Ef. Similarly for billions etc.
Incidentally, one could also add subscripts to change the base from "e" to something else. For example, 1 E_{2}f would be one doubling = ln(2) Ef = 0.693 Ef, E_{2}% would be 1/100 of it, and so on. For example, the CO2 warming without feedbacks is 1.2 °C/E_{2}f.

Also, it would be inconvenient to write the logarithms of the prices in the grocery stores because many consumers can't exponentiate too quickly and they would have a hard time to buy the bread or anything else. Consequently, too many people could die of hunger because of that.

On the other hand, I do think that e.g. the stock market indices would benefit if their natural logarithms were being released instead of the current form of the indices. The daily change of the index in E% would automatically give you the "relative" change.

For example, DJIA says 10,609.65 which means down by 100.90 or -0.94% (from 10,710.55 on Thursday). In the new notation, the improved DJIA index would be the natural logarithm of the previous number (multiplied by 100, to get from Ef to E%). On Friday, it closed at 926.95 from 927.89 on Thursday. It decreased by 0.94 E%. You can simply subtract the numbers and you obtain the e-percentage decrease.

(For bigger changes, you shouldn't forget that the figure in front of E% must be exponentiated first.)

There are obviously many cases in science where the logarithms are kind of more natural than their exponentials, see e.g. Dualities vs Singularities where you must take the logarithms of the radii of tori (and the coupling constants) before you may use linear algebra to obtain some fun results.

See also a similar proposal for the units of evidence.

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snail feedback (2) :

reader The Master said...

Hi Lubos,

Exponential percentages is an interesting idea, but E% may become less sensitive than usual % which may be observed from

\delta E%A =(\delta %A)/%A,

here A is some price, \delta %A is small change of price A in usual percentages, \delta E%A - is corresponding change in exponential percentages, %A is some treated area of changes. It becomes less sensitive for high %A.

reader Luboš Motl said...

Dear quantense, thanks, yes, I realize that, of course. In my opinion, it's clearly an advantage.

If something increases by 9,000 percent, it really doesn't matter whether it is 9,000 or 9,001 percent.

The increase corresponds to 449.98 and 449.99 E% - they only differ by 0.01 E% rather than 1% in the exponentiated notation.

And it is sensible because 9000% is just huge, and there is absolutely no qualitative difference between 9000% and 9001%.


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