The Miracle of Compound Interest

One of the most memorable math lessons my grandfather taught me was "the miracle of compound interest." This is a lesson that everyone should know, but many people do not.

A great introduction to the lesson that my grandfather used was what might be called "The Parable of the Doubling Penny" which provides a memorable challenge to intuition.

The Doubling Penny

If you start with $0.01 and double your money every day for a year, how much money would have at the end of the year? $10, $100, $1000? Take a guess!

Mathematically, we are starting with 0.01 and multiplying it by two 365 times:

What does this equal if you do the calculation?

$751,533,626,487,626,648,569,070,089,221,440,991,301,881,302,634,686,635,501,610,374,244,697,292,922,233,352,950,032,061,665,551,888,458,711,040

A quick search says that the value of all assets in the world is about $500 trillion. This crazy 107-digit figure from doubling the penny, which can be (approximately) represented more compactly as $7.515336\times 10^{107}$, is about $1.5030673\times 10^{93}$ TIMES larger than the value of all the assets in the entire world ($V$):

Generalization

Now, let's generalize the "doubling penny" scenario mathematically:

• $A$ - the final amount
• $P$ - the original principle
• $r$ - the rate
• $n$ - the number of compounding cycles

In the doubling penny scenario, $P=0.01$, $r=2$, and $n=365$. We can plug in different values and still get pretty staggering results.

Let's say we start with $100 and add 3% a day for a year.

How about a more modest scenario? Daily compounding would be great for multiplying your savings. But, finding this kind of rate would be extremely rare. Right now savings bonds are paying something like 5% a year. Let's say we start with $1000 and earn 5% a year for 10 years. How much would we have at the end?

This realistic scenario doesn't seem all that consequential. But, don't worry, the real world still provides astronomical cases.

The US national debt

The current US national debt is about $35 trillion. 30 year US bonds currently pay about 4.5% per year. Apparently this interest is paid twice a year. So, let's say the semi-annual rate is 2.25%. If the US Government doesn't add to the principal debt every year (they add trillions to the debt every year recently but let's ignore that for simplicity). At the current interest rate, what will the debt be after 100 years?

• P=$35 \text{ trillion}
• $r=1.0225$
• $n=200$

Almost 3 quadrillion! (I know, it's hard to read all those digits!) In current dollars, 3 quadrillion would be about 6 times more than the total value of every asset on earth! Obviously, this amount of US dollars is non-sensical if we consider the value of the dollar to be constant. But, the value of the dollar is not constant. Thanks to arbitrary expansion of the money supply, unsustainable government spending (but I repeat myself), and fractional reserve lending (but I repeat myself again), the value of the dollar in real terms will decline to nothing in the current system.

Currently the US sovereign debt of $35 trillion represents a staggering 7% of all assets on the planet (using the $500 trillion figure, maybe that estimate should be reviewed). All things remaining equal, in 100 years, the value of the dollar should go down by about 99%. At this pace, $100 in 100 years will have the value of $1 today. A tiny bag of pistachios is $5 today. In 100 years, everything continuing as it is, we would expect this handful of pistachios to cost $500.

Indeed, based on the price of gold 100 years ago ($20.67 per ounce) and the price of gold today ($2180), a 99% reduction of the value of the dollar over the next 100 years would be exactly what we would expect to see given the last 100 years.

Opinions on an uncertain future

A steady decline would be OK I guess, like slowly boiling a frog. A cheap meal used to cost $0.05, now maybe $5.00, in 100 years, $500 🤷‍♀️. But, history has little precedent for this experiment in fiat credit expansion and debasement. The debasement of the Roman denarius and the decline of the empire comes to mind. On the other hand, there are many cases of dramatic collapses of fiat currencies. In our current circumstances, perhaps a dramatic collapse is more likely than a slow decline. Without proper education, politicians can easily convince ignorant people (the apparent majority) that giving away free money can reduce costs and that raising the minimum wage can improve conditions for workers. The government can just create money and spray it everywhere. The people even demand it! Wars? No problem. A billion dollars in advertising for an experimental pharmaceutical product? Let's go! We can even let pharmaceutical companies charge arbitrarily high prices, spend money to protect their monopolies with "intellectual property" and then have the government pay the bill. Just print it up! There is no limit to all the great reasons to create new money from nothing!

The best case is that in 100 years a cheap meal will cost $500. But, the current system is already so bloated that we might see a dramatic collapse before we see people accustomed to paying $500 for a happy meal. Good luck out there!