In Leaps and Bounds!

Sometimes news can make your head spin.

Just this week, Japan’s GDP numbers were announced and they report that the GDP has expanded by an annualized 1.9% and about 0.5% on a quarter-on-quarter basis.

Previous to these announcements, Japan was struggling to avert a recession, which is not surprising considering how it has been struggling to barely keep itself afloat for the last decade.

So you will think that a positive growth should really be a good news for Japan, right? Wrong. Instead of rejoicing in the growth numbers the economists seem worried. Well, we are all too familiar with their quibbles and quarrels but this time around the reason for their worry has got nothing to do with economics! 

Rather a queer phenomenon seems to have inflated these growth numbers: leap-year effect.

What could be this leap year effect? Well, as the name suggests, it has got to do with an extra day in February. A Jan-Mar quarter in 2016 will have to be adjusted for this one day before we compare it with a Jan-Mar quarter of 2015. But how much can just one extra day of spending, (not of earning, mind you) really change the growth numbers? 

A lot apparently, as the Japanese economists seem to point out and this is where the story starts to get interesting.

According to some estimates:

“The effect of the extra day from the leap year may have magnified first-quarter growth by about 0.9 percentage points on an annualized basis and by 0.3 percentage points on a q-o-q basis” Reuters

“Excluding the impact of leap year, which added an extra day to February, GDP probably expanded around 0.2 percent [on a q-o-q basis, instead of the reported 0.5%]” Reuters

Economists are infamous with their craft, but a 0.3% error on a 0.5% estimate is just too much trickery for even the economists!  So, what is really going on in here?

To understand the intricacies, let’s just brush up our concepts on GDP calculations. A typical year is divided into four quarters. While a calendar year starts with Jan-Mar quarter, a fiscal year starts with the Apr-Jun quarter. A quarter-on-quarter growth rate is measured as the GDP growth in the current quarter over the preceding quarter. For example, a q-o-q growth rate for Q4 2015-16 (fiscal year), would be the growth that happened in the months of Jan-March 2016 over that happened in Oct-Dec 2015. 

From this q-o-q growth rate, an annualized growth rate is derived. This annualized growth rate is a hypothetical number which extrapolates the q-o-q growth rate over the next three quarters to arrive at an annual figure, with the assumption that the next three quarters will behave exactly same as the current quarter.

With these definitions under the belt, let’s proceed to frame our problem here. Consider an economy where within a quarter, each day is exactly identical in terms of economic activities. People wake up, work, produce, consume, save and sleep in exactly the same amount and fashion each day of that quarter. Once every day in a quarter is identical, we can calculate "per day GDP" by simply dividing the GDP in a quarter with the total days in a quarter. We can then postulate that this per day GDP is what would have been counted for 29th Feb, and then simply deduct this from the quarterly numbers to get the leap-year adjusted number for GDP in the Jan-Mar quarter.

But for a moment think about this: what really changes on 29^th Feb? If you are a regular employee, do you get paid more? Do you pay an extra rental for the apartment? Do you pay extra on the monthly travel pass, membership cards? No, not really. But then do you pay for an extra day of gas, electricity bills? Do you buy groceries for that extra day? Yes!

So arguably, there are some portions of GDP which are insulated from the extra day effect, and some which will get increased. A per-day GDP estimate has both these portions embedded. In order to disentangle the portion which really gets affected by an extra day, we follow a rough fraction, say, 25%. Empirically, this number will be different, but for our immediate purpose this is a start.

So now our job is almost done. All that we have to do is to calculate a per-day GDP number and 25% of this is what we will deduct from the Jan-Mar GDP levels to adjust for the leap-year effect. Here are the sample calculations based on the actual numbers for Japan.

#	Item	Value	Comments
a	GDP in Oct-Dec 2015	4.4515	Japan's GDP in trillion $
b	Q-o-Q Growth Rate	0.50%	Official Figures
c	GDP in Jan-Mar 2016	4.4737	a*(1+b)
d	Annualized Growth Rate	2.02%	(1+b)^4-1
e	GDP per day in Oct-Dec	0.0484	a/(31 Oct+ 30 Nov+31 Dec)
f	GDP Increment on Leap Day	0.0121	25% of e
g	GDP in Jan-Mar without Leap Day	4.4616	c-e
h	Modified Q-o-Q growth rate	0.23%	(h/a)-1
i	Modified Annualized Growth Rate	0.92%	(1+h)^4
j	Difference in q-o-q growth	0.27%	b-h
k	Difference in annualized growth	1.10%	d-i

Well, the numbers do turn out to be similar to what Japanese have been suggesting: the difference in GDP growth rate on a q-o-q basis is around 0.27% and a difference of 1.1% on an annualized basis! Voila! Who would have thought that an innocuous day could make such a difference! 

For a country like Japan, where the q-o-q growth numbers are <0.5%, this makes a huge difference. What about India? Well, the leap year effect will definitely be there, but will it matter enough? Well that depends on how much portion of GDP really gets affected by an extra day. For our above calculations that portion was 25%. However, this will vary, country by country - and this number is really the crucial number here. The higher it is, the higher will be the leap year adjustment. 

All that said and done, the basic dilemma remains: why should we work when we don't get paid extra! Well, unsurprisingly, we are not alone in this thought. In-fact there is a Petition to make leap day a holiday!