BackUnderstanding exponential growth
Fast growth is neither a necessary nor sufficient condition for growth to be called exponential. For that to happen, it should follow a specific pattern
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Commentators like to use the term exponential growth to describe a quantity that is growing rapidly. The fascination with the term possibly has its origins in the theory of computation, where algorithms that take an exponential amount of time (as a function of input length) belong to a different class altogether from algorithms that can compute in a time that is a polynomial function of input length.
Nevertheless, the use of the term exponential growth is often incorrect, since it is simply used for systems that grow fast without necessarily being exponential. In fact, fast growth is neither a necessary nor sufficient condition for the growth to be characterised as exponential. For that to happen, the growth should follow a specific pattern. In this piece, we will try and understand a simple method to identify and characterize exponential growth.
One of the sectors that is currently hot in the start-up world is technology for finance, usually abbreviated as Fin-Tech. As a recent special report in The Economist (mintne.ws/ 1CH6mec) mentions, Fin-Tech comprises several classes of technology, all of which aspire to make finance easier and more efficient. One of them is blockchain technology, which forms the basis of crypto-currencies such as Bitcoin, and is being used to maintain land title registries in Honduras (mintne.ws/1Mar75b), an idea worth pursuing in India as well).
Then, there are peer-to-peer lending aggregators (such as Lending Club or Zopa) that use aggregate data from multiple non-traditional sources to make better lending decisions. Companies such as TransferWise use innovative technologies to significantly cut down costs of cross-border fund transfers, currently a lucrative source of income for banks.
The class of Fin-Tech that is of immediate interest to us in India, however, is innovation in payments. Finance minister Arun Jaitley mentioned it in his budget speech (mintne.ws/ 1N2qnjZ) about increasing the market share of electronic payments. An earlier piece in Mint (mintne.ws/1PDKMun) showed what it costs the Reserve Bank of India (RBI) to enable offline payments by means of currency notes and made a case for subsidising electronic payments. Recently, there has been a proposal (mintne.ws/1Iw96Yk) by the finance ministry to further subsidize electronic payments.
The good news is that things are already moving on the payments front. The Reserve Bank of India has received bids for new payment banks (which only accept deposits and enable payments), and the licences are scheduled to be handed out in another couple of months. A number of companies have introduced mobile wallets (such as Paytm and Shmart) to ease payments. Large banks such as HDFC Bank Ltd and Axis Bank Ltd have introduced mobile apps (Chillr and Ping Pay, respectively) that allow quick and seamless payments to people in the user’s contact list, without requiring bank account details.
Going by data released by RBI, some forms of electronic payments are actually seeing exponential growth. What does it mean and how do we detect that?
A quantity is said to grow exponentially in time if the increment in a given time period is proportional to the level at the beginning of the time period (mathematically inclined readers might recall that the solution to the equation dx = kx dt is given by an exponential function). The constant of proportionality, which determines the rate of growth, actually does not matter. So if you have a quantity (the population of a country, say) that grows at 1% each year (value at end of each year is 1% more than the value at the beginning), it is actually showing exponential growth. If the revenues of a start-up are doubling each year (or growing at 100%), that is exponential growth, too.
We will now look at growth in several kinds of payments, based on data released by RBI, and analyse whether the said growth is exponential. Figure 1 shows the growth in the amount transacted using National Electronic Fund Transfer (NEFT) system, which is used to transfer money between bank accounts (this is still old finance, since it requires you to log into your bank account, add a payee using full details, and then transfer the funds, which don’t necessarily get credited immediately to the payee bank account).
You might notice that the volumes are steadily growing over time. People with a mathematical background might recognize that this curve looks like an exponential (a quantity that grows exponentially, when graphed, resembles an exponential curve), but those without such a background need not despair, for it is not hard to detect this. In order to determine whether a quantity is showing exponential growth, all we need to do is to graph it with a logarithmic Y-axis (most standard statistical and graphing tools allow this, including Microsoft Excel: mintne.ws/1J0sfqb).
When we plot the above graph using a logarithmic Y-axis, we get this figure 1A.
If a quantity grows exponentially, its logarithm grows linearly. Hence, if we plot the quantity with a logarithmic Y-axis, we should see a straight line. What we see here in figure 1A is something that is approximately straight, but not quite; the curve is better approximated by two straight lines, one until the beginning of 2012, and another with a lower slope from then onwards. In other words, the volume of NEFT transactions is still growing exponentially, though at a lower rate than it was a couple of years ago.
Next, let us look at Immediate Payment Service (IMPS), a technology that allows instant payments over mobile phones. The protocol for IMPS is still not straightforward, and involves entering a one-time authorization code. Yet, as figures 2 and 2A (same graph with linear and logarithmic Y-axes) show, this mode of payment is actually showing exponential growth.
Again, we see it is close to exponential growth, but not quite; there are three regimes in the figures above. The platform saw fairly slow growth until 2013, after which it started growing rapidly. A year after that, though, while growth still continued, it is not as rapid as it was in the preceding year. The right edge of the above graph can be seen to be a straight line, so we can see that IMPS is also growing exponentially, but not with the same rate as it was in 2013.
It is clear from the above graphs that the best way to determine if something is growing exponentially is to use a logarithmic graph, so we will dispense of the linear graph.
Figure 3 shows the volume of payments made using mobile wallets (classified by RBI under prepaid payment instruments).
There was rapid takeoff in the second half of 2012, but the system stalled during 2013. Since 2014, however, we see that there has been exponential growth.
Volume of transactions involving debit and credit cards have been growing slowly, but exponentially as figure 4 shows.
Mobile banking transactions are growing exponentially, too, and rather quickly, as figure 5 shows.
We are in for exciting times as far as payments are concerned. It appears that there is regulatory will to enable electronic payments, and it is coming at a time when different sections of the payment world have shown exponential growth. Regulatory support is key for the continued growth of this sector, though.
We will end by looking at the growth of a decidedly low-tech payment mechanism, the humble cheque. Figure 6 shows the value of payments made using paper clearing mechanisms (which includes the cheque). This is one mechanism that is clearly in decline. And that is not necessarily a bad thing.
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