There’s a lake. A tiny algal colony has formed on its surface. It’s observed that this algal colony doubles in size every day, and if left untreated, it’s expected to cover the total surface of the lake in 60 days.
Now, on which day will the algal colony cover half of the surface of the lake? Take your time to think because the answer might just surprise you. As Kit Yates writes in The Maths of Life and Death—Why Maths is (Almost) Everything: “A common answer that many people give to this riddle, without thinking, is 30 days.” And that’s wrong simply because “the colony doubles in size each day; if the lake is half-covered one day, it will be completely covered the next day”.
So, the answer to the question—when will the algae colony cover half of the lake’s surface—is rather surprisingly, fifty-nine days, just one day short of the day when the algae will cover the lake completely. Now, this is an excellent example of exponential growth—a phenomenon most of us find difficult to appreciate—simply because most change, at least in our heads, is linear.
In a way, this is evolutionary. As Yates puts it: “Typically, for our forebears, the experiences of one generation were very much like the last: they did the same jobs, used the same tools and lived in the same places as their ancestors. They expected their descendants to do the same.” But things started to change a few centuries back, and most human lives now change majorly within a single generation.
Another definition I found while Googling states: “The growth of a system in which the amount being added to the system is proportional to the amount already present: the bigger the system is, the greater the increase.” This makes slightly more sense but still doesn’t make things very clear.
In fact, let’s try and understand this through a few examples. As Yates writes: “The more infected people we have in the early stages of a disease outbreak, for example, the more people they will infect and the more the number of cases will rise.”
This is something that many politicians across the world did not understand when the covid pandemic started to spread. Like algae in the lake, the initial numbers were low. As the then US President Donald Trump tweeted on 9 March 2020: “So last year 37,000 Americans died from the common flu. It averages between 27,000 and 70,000 per year. Nothing is shut down. Life and the economy go on. At this moment, there are 546 confirmed cases of CoronaVirus [sic], with 22 deaths. Think about that!” By the time his presidency ended on 9 March 2021, 24.5 million covid cases had been recorded in the US. This led to 400,000 deaths.
It was like the algae colony in the lake on the 59th day: by the time it was big enough to be a problem, it was too late to do anything about it. Even on the 55th day, only 3% of the lake would have been covered by algae.
A similar dynamic works in Ponzi schemes as well. A Ponzi scheme is an investment fraud where to lure investors, higher returns are offered compared to other investments available in the market at that time.
The trouble is that the money that comes in is not invested anywhere to generate returns. An illusion of a proper investment scheme is created by using money being brought in by new investors into the scheme to redeem investments of older investors who have already invested in the scheme. Of course, for the scheme to keep going, newer investors need to keep coming in so that the money being brought in by them is higher than the money that needs to be paid to the older investors and the money going into the pocket of the promoter of the scheme.
In that sense Ponzi schemes in order to keep running need to keep growing exponentially. In The Maths of Life and Death, Yates talks about a Ponzi scheme which, after 15 rounds of investment had 10,000 people. As he writes: “Fifteen rounds further on, however, and one in every seven people on the planet would need to invest to keep the scheme going.” This is why almost all Ponzi schemes collapse under their own weight, and most people don’t see that coming because of the initial small size of the scheme.
This dynamic also works when people take on loans at very high interest rates through informal mechanisms. In a recent interview, the filmmaker Hansal Mehta talked about borrowing money at 7% per month to complete a movie. He didn’t mention the amount he had taken on.
But Rs 50 lakh borrowed at 7% per month can amount to Rs 1.13 crore at the end of 12 months, Rs 2.54 crore at the end of 24 months and Rs 5.71 crore at the end of 36 months.
This is how fast the outstanding amount rises. Of course, no lender is going to wait for two years or three years for such a loan to be repaid.
Now, in these examples, even though the starting numbers were on the lower side, the growth rate was steep, so things escalated very quickly and came to an end. But at the same time, exponential growth can take time to come into play, which is why many people fail to appreciate the power of compounding.
The best—though very stretched example—of this, is provided by Peter Lynch in his book One Up On Wall Street, which was first published in 1989. In this example Lynch talks about the Indians of Manhattan who “in 1626 sold all their real estate to a group of immigrants for $24 in trinkets and beads”. Then Lynch goes on to say that: “For 362 years the Indians have been the subjects of cruel jokes because of it.” Further, 362 years after 1626 means 1988—the year in which Lynch was perhaps writing the book.
After this, Lynch carries out what can be best described as a thought experiment and assumes that if the Indians of Manhattan had invested $24 in 1626 at 8% interest per year, their investment would have done very well over the long-term. Now, the question is, when he was writing the book in 1988, how much would that $24 be worth.
Dear reader, do you want to take a guess? No, no. Don’t open the Excel sheet. Lynch writes that the investment would be worth $30 trillion in 1988. Yes, you read that right. I ran the numbers. The value comes to around $30.17 trillion. (This is an excellent example of why the human mind cannot always comprehend exponential growth).
Now, let’s carry this thought experiment forward by assuming that the investment continued to grow at 8% even after 1988. In 2022, the investment would be worth $413 trillion, which would be much more than even the global gross domestic product (GDP) of $164.2 trillion (in current US $ terms) during the year. (Of course, if that investment had really happened, then the global GDP would have been much, much higher).
Take a look at the following chart that plots the growth of $24 invested in 1626 at 8% per year over the centuries. This is what exponential growth looks like. It starts slow and then takes off after a point of time. In this case, the take-off happens sometime in the 1960s and 1970s, only a few centuries later, but a similar and not-so-dramatic dynamic can play out over a much smaller time frame as well.
Of course, this example is a thought experiment at best. It would have been almost impossible for anyone to have stayed invested this long and at the same time ensure that there were ways of making 8% per year continuously.
Stock markets have been around for a few centuries. But the companies listed on these bourses keep changing, as many companies do not maintain their best performance or simply collapse. Further, the Manhattan Indians, could have possibly invested in gold. Also, as time went by, the number of claimants to this money would have gone up, and different people would have wanted to do different things with it and not allowed it to grow as much as it did.
Nonetheless, there is a very important lesson that this example offers, which is that when it comes to money or investing, it takes time for exponential growth to kick in, but when it does—money in absolute terms—grows by leaps and bounds, in a short period of time (short is used in relative terms here).
Now, let’s look at something which is actually implementable. Let’s take the case of that very boring form of investing—the public provident fund (PPF)—available to everyone. In my limited personal experience, I find people not taking this very seriously, given that there is nothing to do after investing.
In fact, I was the victim of a similar kind of thinking and didn’t start a PPF account for many years until I did the basic math at the heart of it all and decided to start one sometime in 2009 or 2010.
PPF has an initial lifespan of around 15 years. Currently, a maximum of Rs 1.5 lakh can be invested in it in any given year. One also gets a tax deduction for the investment. But under Section 80C, there are many other things that one can take a deduction for. So, let’s forget the tax deduction and just concentrate on the investment. Also, it is tax-free on maturity.
Let’s consider a 25-30-year-old individual who starts investing Rs 1.5 lakh in PPF at the beginning of every year and earns 7.1% per year on it (as is the current rate of interest). At the end of 15 years, the investments would be worth Rs 4.1 million.
Now, this is where the fun starts. At this point in time, the individual would be around 40-45 years of age. The PPF account can be extended for five years at a time, with or without contribution.
Let’s assume this individual continues to contribute and extends the account every five years. How much would the investments be worth 15 years later? Rs 15.5 million. In the first 15 years, the investment grew to Rs 4.1 million. But in the next 15 years, it grew by more than Rs 11 million to Rs 15.5 million. That’s exponential growth for you. It takes time to kick in.
If the investment continues for five years more, the investment would be worth Rs 22.7 million. So, individuals starting a PPF account at 25 are likely to have this amount by the time they turn 60 and are about to retire. The earlier you start, the more you benefit. (Isn’t so much of writing on money just about repeating the right cliches)
A similar logic works for the Employees Provident Fund or any other provident fund which guarantees a certain rate of return every year. Or as I had said a couple of weeks back, quoting the film lyricist, writer and poet Gulzar, and as he wrote in the title song of the 1979 film Gol Maal: “Paisa kamaane ke liye phir paisa chaahiye (to earn money you need money again).”
Of course, the government may not continue to pay 7.1% per year in the years to come. That risk remains. Nonetheless, if the government does reduce the interest rate on PPF further, then it is likely to be in an environment where the rate of return on other fixed-income instruments (like fixed deposits, bonds, etc.) will also fall. So, investing in PPF will still make sense.
The important point here is to stay invested, which most of us find difficult to do. We like to tinker. In fact, a similar logic applies to investing in mutual funds as well. As I had said two weeks back, the data from the Association of Mutual Funds in India (AMFI) shows a little more than half of the money invested in equity mutual funds stays invested for a period of more than two years.
This doesn’t let the accumulated corpus reach a stage where it would be able to benefit from exponential growth. For that, one has to stay invested for at least a decade or even more. Some readers said that this must be because investors must be moving money from underperforming schemes to well-performing ones.
Well, this is an issue that needs a detailed piece, which I might do at a future point of time, nonetheless, moving in and out of schemes chasing performance also comes at a cost. Also, everyone cannot be chasing performance. There are people out there who do take their money out of equity mutual funds in two years and then don’t reinvest the money back into these schemes.
Finally, it is worth remembering something that Pulak Prasad writes in What I Learned About Investing from Darwin: “What is needed to become a successful investor is not intellect, a commodity, but patience, which is not.” And that’s something worth thinking about, along with the fact that the entire financial services industry benefits from you getting in and out of investments. They benefit from activity at your end. And they don’t want you, dear reader, to realise that smart money can also be dumb at the end of the day.