JDN 2457415

For today’s post I’m taking a break from issues of global policy to discuss a bit of a mathematical pet peeve. It is an opinion I share with many economists—for instance Miles Kimball has a very nice post about it, complete with some clever analogies to music.

I hate when we talk about percentages in asymmetric terms.

What do I mean by this? Well, here are a few examples.

If my stock portfolio loses 10% one year and then gains 11% the following year, have I gained or lost money? I’ve lost money. Only a little bit—I’m down 0.1%—but still, a loss.

In 2003, Venezuela suffered a depression of -26.7% growth one year, and then an economic boom of 36.1% growth the following year. What was their new GDP, relative to what it was before the depression? Very slightly less than before. (99.8% of its pre-recession value, to be precise.) You would think that falling 27% and rising 36% would leave you about 9% ahead; in fact it leaves you behind.

Would you rather live in a country with 11% inflation and have constant nominal pay, or live in a country with no inflation and take a 10% pay cut? You should prefer the inflation; in that case your real income only falls by 9.9%, instead of 10%.

We often say that the real interest rate is simply the nominal interest rate minus the rate of inflation, but that’s actually only an approximation. If you have 7% inflation and a nominal interest rate of 11%, your real interest rate is not actually 4%; it is 3.74%. If you have 2% inflation and a nominal interest rate of 0%, your real interest rate is not actually -2%; it is -1.96%.

This is what I mean by asymmetric:

Rising 10% and falling 10% do not cancel each other out. To cancel out a fall of 10%, you must actually rise 11.1%.

Gaining 20% and losing 20% do not cancel each other out. To cancel out a loss of 20%, you need a gain of 25%.

Is it starting to bother you yet? It sure bothers me.

Worst of all is the fact that the way we usually measure percentages, losses are bounded at 100% while gains are unbounded. To cancel a loss of 100%, you’d need a gain of infinity.

There are two basic ways of solving this problem: The simple way, and the good way.

The simple way is to just start measuring percentages symmetrically, by including both the starting and ending values in the calculation and averaging them.
That is, instead of using this formula:

% change = 100% * (new – old)/(old)

You use this one:

% change = 100% * (new – old)/((new + old)/2)

In this new system, percentage changes are symmetric.

Suppose a country’s GDP rises from $5 trillion to $6 trillion.

In the old system we’d say it has risen 20%:

100% * ($6 T – $5 T)/($5 T) = 20%

In the symmetric system, we’d say it has risen 18.2%:

100% * ($6 T – $5 T)/($5.5 T) = 18.2%

Suppose it falls back to $5 trillion the next year.

In the old system we’d say it has only fallen 16.7%:

100% * ($5 T – $6 T)/($6 T) = -16.7%

But in the symmetric system, we’d say it has fallen 18.2%.

100% * ($5 T – $6 T)/($5.5 T) = -18.2%

In the old system, the gain of 20% was somehow canceled by a loss of 16.7%. In the symmetric system, the gain of 18.2% was canceled by a loss of 18.2%, just as you’d expect.

This also removes the problem of losses being bounded but gains being unbounded. Now both losses and gains are bounded, at the rather surprising value of 200%.

Formally, that’s because of these limits:
lim_{x rightarrow infty} {(x-1) over {(x+1)/2}} = 2

lim_{x rightarrow infty} {(0-x) over {(x+0)/2}} = -2

It might be easier to intuit these limits with an example. Suppose something explodes from a value of 1 to a value of 10,000,000. In the old system, this means it rose 1,000,000,000%. In the symmetric system, it rose 199.9999%. Like the speed of light, you can approach 200%, but never quite get there.

100% * (10^7 – 1)/(5*10^6 + 0.5) = 199.9999%

Gaining 200% in the symmetric system is gaining an infinite amount. That’s… weird, to say the least. Also, losing everything is now losing… 200%?

This is simple to explain and compute, but it’s ultimately not the best way.

The best way is to use logarithms.

As you may vaguely recall from math classes past, logarithms are the inverse of exponents.

Since 2^4 = 16, log_2 (16) = 4.

The natural logarithm ln() is the most fundamental for deep mathematical reasons I don’t have room to explain right now. It uses the base e, a transcendental number that starts 2.718281828459045…

To the uninitiated, this probably seems like an odd choice—no rational number has a natural logarithm that is itself a rational number (well, other than 1, since ln(1) = 0).

But perhaps it will seem a bit more comfortable once I show you that natural logarithms are remarkably close to percentages, particularly for the small changes in which percentages make sense.

We define something called log points such that the change in log points is 100 times the natural logarithm of the ratio of the two:

log points = 100 * ln(new / old)

This is symmetric because of the following property of logarithms:

ln(a/b) = – ln(b/a)

Let’s return to the country that saw its GDP rise from $5 trillion to $6 trillion.

The logarithmic change is 18.2 log points:

100 * ln($6 T / $5 T) = 100 * ln(1.2) = 18.2

If it falls back to $5 T, the change is -18.2 log points:

100 * ln($5 T / $6 T) = 100 * ln(0.833) = -18.2

Notice how in the symmetric percentage system, it rose and fell 18.2%; and in the logarithmic system, it rose and fell 18.2 log points. They are almost interchangeable, for small percentages.

In this graph, the old value is assumed to be 1. The horizontal axis is the new value, and the vertical axis is the percentage change we would report by each method.

The green line is the usual way we measure percentages.

The red curve is the symmetric percentage method.

The blue curve is the logarithmic method.

For percentages within +/- 10%, all three methods are about the same. Then both new methods give about the same answer all the way up to changes of +/- 40%. Since most real changes in economics are within that range, the symmetric method and the logarithmic method are basically interchangeable.

However, for very large changes, even these two methods diverge, and in my opinion the logarithm is to be preferred.

The symmetric percentage never gets above 200% or below -200%, while the logarithm is unbounded in both directions.

If you lose everything, the old system would say you have lost 100%. The symmetric system would say you have lost 200%. The logarithmic system would say you have lost infinity log points. If infinity seems a bit too extreme, think of it this way: You have in fact lost everything. No finite proportional gain can ever bring it back. A loss that requires a gain of infinity percent seems like it should be called a loss of infinity percent, doesn’t it? Under the logarithmic system it is.

If you gain an infinite amount, the old system would say you have gained infinity percent. The logarithmic system would also say that you have gained infinity log points. But the symmetric percentage system would say that you have gained 200%. 200%? Counter-intuitive, to say the least.

Log points also have another very nice property that neither the usual system nor the symmetric percentage system have: You can add them.

If you gain 25 log points, lose 15 log points, then gain 10 log points, you have gained 20 log points.

25 – 15 + 10 = 20

Just as you’d expect!

But if you gain 25%, then lose 15%, and then gain 10%, you have gained… 16.9%.

(1 + 0.25)*(1 – 0.15)*(1 + 0.10) = 1.169

If you gain 25% symmetric, lose 15% symmetric, then gain 10% symmetric, that calculation is really a pain. To find the value y that is p symmetric percentage points from the starting value x, you end up needing to solve this equation:

p = 100 * (y – x)/((x+y)/2)

This can be done; it comes out like this:

y = (200 + p)/(200 – p) * x

(This also gives a bit of insight into why it is that the bounds are +/- 200%.)

So by chaining those, we can in fact find out what happens after gaining 25%, losing 15%, then gaining 10% in the symmetric system:

(200 + 25)/(200 – 25)*(200 – 15)/(200 + 15)*(200 + 10)/(200 – 10) = 1.223

Then we can put that back into the symmetric system:

100% * (1.223 – 1)/((1+1.223)/2) = 20.1%

So after all that work, we find out that you have gained 20.1% symmetric. We could almost just add them—because they are so similar to log points—but we can’t quite.

Log points actually turn out to be really convenient, once you get the hang of them. The problem is that there’s a conceptual leap for most people to grasp what a logarithm is in the first place.

In particular, the hardest part to grasp is probably that a doubling is not 100 log points.

It is in fact 69 log points, because ln(2) = 0.69.

(Doubling in the symmetric percentage system is gaining 67%—much closer to the log points than to the usual percentage system.)

Calculation of the new value is a bit more difficult than in the usual system, but not as difficult as in the symmetric percentage system.

If you have a change of p log points from a starting point of x, the ending point y is:

y = e^{p/100} * x

The fact that you can add log points ultimately comes from the way exponents add:

e^{p1/100} * e^{p2/100} = e^{(p1+p2)/100}

Suppose US GDP grew 2% in 2007, then 0% in 2008, then fell 8% in 2009 and rose 4% in 2010 (this is approximately true). Where was it in 2010 relative to 2006? Who knows, right? It turns out to be a net loss of 2.4%; so if it was $15 T before it’s now $14.63 T. If you had just added, you’d think it was only down 2%; you’d have underestimated the loss by $70 billion.

But if it had grown 2 log points, then 0 log points, then fell 8 log points, then rose 4 log points, the answer is easy: It’s down 2 log points. If it was $15 T before, it’s now $14.70 T. Adding gives the correct answer this time.

Thus, instead of saying that the stock market fell 4.3%, we should say it fell 4.4 log points. Instead of saying that GDP is up 1.9%, we should say it is up 1.8 log points. For small changes it won’t even matter; if inflation is 1.4%, it is in fact also 1.4 log points. Log points are a bit harder to conceptualize; but they are symmetric and additive, which other methods are not.

Is this a matter of life and death on a global scale? No.

But I can’t write about those every day, now can I?