Albert Einstein is – falsely – held to have said that the eighth wonder of the world is compound interest. In an idyllic past that many of us still remember, deposit accounts at banks earned (above-inflation) interest; we were actually paid for the money we set aside to the bank.
Compound interest captures this amazing feature of money already earned to earn you even more in the future. Over long time periods it produces truly amazing results.
A hundred dollars at 3% real interest becomes $103 at the end of the year. The second year’s interest income is then based on $103 rather than the initial hundred you put into the account, giving you 9 cents in addition to the $3 interest that your initial hundred earned you. (I know this sounds negligible but bear with me). That extra interest earned gives you even more interest in the next period, as the bank bases the 3% payout on your outstanding balance of $106.09 in the third year.
Over long time periods, this initially microscopic earning increase translates into unfathomable riches, most of which is generated towards the end of whatever periods you’re measuring. Here’s an example to keep showing the power of that paltry 3%:
| Total Wealth ($) | Interest Income ($) | |
| Year 0 | 100 | 3.00 |
| Year 1 | 103 | 3.09 |
| Year 2 | 106.09 | 3.18 |
| Year 3 | 109.26 | 3.28 |
| Year 4 | 112.55 | 3.38 |
| Year 5 | 115.93 | 3.48 |
| Year 10 | 134.39 | 4.03 |
| Year 20 | 180.61 | 5.42 |
| Year 30 | 242.73 | 7.28 |
| Year 50 | 438.39 | 13.15 |
| Year 75 | 917.89 | 27.54 |
| Year 90 | 1,430.05 | 42.90 |
| Year 99 | 1,865.89 | 55.98 |
| Year 100 | 1,921.86 | 57.66 |
In the beginning, say during the first decade, you were only earning the original 3% plus a few cents. No biggie. At the end, you’re netting $40-50 more every year than you did in the early days – and the total sum has stunningly grown to almost $2,000.
Had you, instead of keeping the interest accumulating on the account, been taking out $3 to consume in every time period, your sum wealth in year 100 would be no more than the $100 you put in; of course, over this century you would have enjoyed $3 worth of additional goods and services every year, for a sum total of $300 over your century-long lifetime. Not bad, but it dwindles before the $1,922 dollars your compounding interest-generating account would have yielded you.
Whenever financial advisors talk about “nest eggs” and the importance of saving for retirement, this is the effect they refer to: letting even small amounts of money work for you over and over and over, makes a huge difference down the line.
This is because compounding interest is an exponential function.
It is no secret that most of us are starstruck before exponential curves and that we usually don’t grasp their truly awesome meaning. It’s a good thing, then, that they are remarkably rare and limited in important ways.
Epidemiology 101
Contrary to what almost everyone has said during the last three months – from media talking heads to presidents and epidemiologists and your neighbor – the growth of pandemics is not exponential. The number of people infected and number of dead don’t follow “exponential” curves. They follow S-shaped curves.
An S-shaped curve is initially like an exponential curve, before it stagnates and levels off. What we’re arguing over is when exactly this exponential phase will end, and what we might be able to do to bring that fortuitous event closer.
The difference between compounding interest, which is unboundedly exponential, and the current pandemic is that diseases have very strict upper bounds. They can only grow so much. At some point, there are no more people to infect, and way before that, most people encountered by an average carrier are already infected, making its capability of infecting others virtually zero. Not to mention the social distancing that all of us are currently engaged in.
I’ll happily leave it to those medically trained to assess the medical applicability of “herd immunity” or such strategies. What any level of immunity indicates, however, is that the S-kink happens earlier and faster than we initially thought. What I am objecting to is mindlessly describing the disease as spreading “exponentially,” and the mayhem of conclusions that thereby follows.
It is farcical that people who until a few weeks ago couldn’t even spell the word “exponential” – let alone explain what it meant – ran straight across this intellectually hysterical spectrum and made the opposite mistake: drawing exponential curves until they ran out of paper or whiteboards.
The otherwise brilliant podcast Radiolab is a case in point. In an episode on March 27 titled Dispatch 1: Numbers, managing editor Soren Wheeler tried to make sense of infection rates and deaths by sharing two important illustrations of the power of exponential curves. First: the penny that doubles every day for a month – would you rather take that, or a million dollars? Most people intuitively grab the million, but when you run the doubling of a penny for 31 days straight (yes, do take out that calculator), you end up with over $10 million. Insane.
Second: the ever-expanding shed. Imagine that the shed Wheeler works from (10×10 feet), doubles every day. Fifth day: the shed has turned into the size of Wheeler’s house. A week later, it’s an average-sized city and at the beginning of next month, it’s the size of New Jersey. Then growth happens ridiculously fast. Nine days later, this small shed has engulfed the United States and a few days later the entire world. Stunning.
While these stories are impressive to get across the idea of exponential growth, almost nothing in the world is like that – particularly not the coronavirus. What’s so stunning about the shed and the doubling pennies is that their growth is uncannily fast (doubling in a day is a lot) and that there is literally no end or limitation to them. In the real world, exponential processes decay as they face real constraints. They run out of fuel. They level off. They face boundaries like air resistance or sparsely populated areas.
The examples Wheeler provides are crazy because they are unreal.
Almost nothing in the real world is truly exponential. F1 cars and spaceships and missiles and sprinters like Usain Bolt are logarithmic: they accelerate fast at first, at decreasing rates and level off rather quickly. Avalanches move exponentially only for brief moments of their run before they decelerate and stop. Bacteria or animals or human beings can breed like rabbits, but only given the rare occasion of infinite food and space – and in the human case: want. At even early stages in our economic development we rather invest in the quality of our children than mere quantity. In any case, world human population growth has been falling since 1968.
Physically speaking, the reason for the elusive exponential is simple: acceleration, per Newton, is an increasing net force operating on the same mass. But where can additional force come from? And it nevertheless quickly runs into obstacles like air resistance and fuel availability. We briefly observe exponential power growth in the chain reaction of nuclear power – indeed, that’s what makes it such an astonishing and powerful energy source.
Some other things do seem exponential. Moore’s law is one, where the number of transistors of microchip doubles every 18-24 months while costs are plummeting, making computing power faster and cheaper at exponential rates. The science writer Matt Ridley asks how much longer this fundamental law of computing power can continue, as everyone has since Gordon Moore himself did so in 1965. Ridley writes, “the atomic limit is in sight” – transistors are now smaller than 100 atoms across. If, at long last, Moore’s law ends, we should demote its exponential status to a peculiar time frame of 50-plus years. Boundedly exponential.
The improvement in economic standards of living is a good candidate for a truly – and limitless – exponential process. That’s the reason compounding interest does not run out of steam. The bank, making more loans at higher rates than your deposits, to productive businesses in a therefore growing economy, can afford to keep paying you more and more interest.
Relevant for this government-induced corona recession, not even the Great Depression – so great of an economic calamity that we capitalize its spelling – did much harm to the long-run trajectory of American economic growth. After the miseries of the 1930s and 1940s, economic growth returned to trend.
The spread of coronavirus, and the unstable numbers of fatalities that come with it, does look exponential – but they are only exponential over some range. What smart people in suits and lab coats try to figure out is what that range is, and which of our myriad of heedless political actions are helping.
Some of them have been outright stupid, as presidential announcements closing in-bound flights to the U.S., causing millions of Americans to rush through crowded airports, planes and immigration halls – contrary to all medical advice that aim to give the virus less opportunity to jump to other people. Or the butchering of tests. Or the countless regulations that prevent even donations of crucially needed hospital equipment to reach the health care workers who can aptly wield them.
This pandemic has so far taught us at least one thing entirely contrary to what average opinion would have you believe: the curves we observe every day are not boundlessly exponential; they are S-shaped.
They are, thankfully, only exponential over some range.
Made available by the American Institute for Economic Research.
