How to calculate the decline of decline

June 3, 2009

Analysts and strategists assessing whether there’s an economic recovery on the way are increasingly referring to “second derivatives”. It usually means a measure, say production, has declined, but not by as much as it did last month, or quarter.

Are second derivatives a strong basis for optimism? If you have to perform differential calculus to make a point, it may be a sign of desperation.

Equities markets continue to factor in a recovery, with the FTSE 100 up about 30 percent from its six-year low of March 9.

Yet the economic data does not support this view.

Look at production figures from most major economies over the last few months, for example. You will see them in decline, month by month. It will be a series like:

200, 188, 173, 160, 151

The data won’t necessarily form the neat curves we know and love from our knowledge of differential calculus. But the principles are the same.

Produce another series of figures, with each number in the series representing the difference between one month’s data and the previous month. This will be a series of  negative numbers.

-12, -15, -13,  -9

This is the first derivative.

Then produce another series, which is the series of differences between these differences. This is the second derivative.

-3, 2, 4

Here, you will notice that there are some positives. It’s on this basis that you might say things are getting better. Or, at least, that they’re getting worse at a lower rate.

Of course, if works in reverse if you’re looking at a series where bigger figures point to a weaker economy, such as unemployment numbers.  There, you’re looking for the numbers to turn negative to support an optimistic stance.  

In some cases you might have to look harder for the green shoots. UK unemployment, for example, appears to be going up at a rate that is, if anything, increasing.

But maybe, if the rate at which it is increasing is decreasing, there’s some cause to be optimistic. So you just have to look at the third derivative. 

Expect to see a rate of change in the frequency with which these expressions are used.

(Reuters photo: Toru Hanai)

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