Posted in Science & Nature

## Exponential Growth

Imagine that you have won a strange lottery where they give you two options of payment: they can either pay you one million dollars up front, or they can pay you one cent on the first day, then double the amount you have every day for a month (i.e. 1 cent on day 1, 2 cents on day 2 etc.). Which would you choose?

It may seem obvious that the \$1 million up front is far better than accumulating a few cents every day. But by the end of the month (day 31), you would actually have accumulated \$5.37 million. How did this happen?

The secret to this extraordinary increase is the power of exponential growth. If you double a number constantly at a regular interval, it grows at a staggering rate. Let us look at the above example again.

On day 1, you have 1 cent. By day 10, you already have 2(10-1) = \$5.12. Now we can see that instead of mere cents, we are gaining \$5 in one day.
By day 15, you have \$163.84. Now the doubling nets you another \$163.
By day 20, you suddenly have \$10,485.76.
We pass \$1 million at day 28 where we have \$1.34 million.
Day 29 you have \$2.68 million and you can see how we end up with \$5.37 million – over five times the amount we would have received compared to the first option.

This shows the sheer power of doubling. It is an important principle to grasp as we see exponential growth all around us in life. Nuclear chain reactions undergo exponential growth to power nuclear reactors. Positive feedback in speakers undergoes doubling amplification, resulting in the sharp screeching sounds. Compound interest follows exponential growth, allowing investments to give substantial returns over time (or result in crushing debt). Bacteria divide in two each time, resulting in a rapid population boom.

Understanding exponential growth also helps us make sense of scary situations such as pandemics. Viral infections are spread from one person to multiple people, represented by a basic reproduction number (R0). In the case of the COVID-19 (2019 coronavirus) pandemic, the R0 was between 2 and 3, meaning that left unchecked, the number of infected individuals would essentially double every few days.

Although this seems obvious, if you didn’t know about exponential growth, it would be terrifying to hear that one day you have 8 cases in a country, but in a fortnight, there are over 1000 cases, with each day presenting increasing numbers of newly infected patients. The media preys on this effect by providing anxiety-inducing headlines. But in reality, the headlines might as well read: “virus continues spreading in predictable exponential fashion“.

Another strength of knowing about exponential growth in a pandemic is that it lets us predict what would happen without any intervention. The number of cases would explode in a matter of weeks, resulting in catastrophic numbers of unwell people taken off the workforce, accompanied by mass casualties. Hospitals would be completely overrun, crippling the nation’s healthcare system and resulting in even more deaths as the infection runs rampant.

Therefore, efforts to reduce the spread of the virus through social distancing and effective quarantining are vital to reduce the rate of exponential growth, flattening the curve and making the number of cases more manageable for the healthcare system to deal with.

Posted in Science & Nature

## Compound Interest

When is the best time to invest? Is it when you have sufficient income and savings that you feel that you have a surplus to invest with?

The correct answer is much simpler: yesterday, with the second best time being today. Because of the magic of compound interest, investing early is the best strategy possible.

Thanks to a simple mathematic rule, compound interest rewards early, small investments more than late, large investments.

The way compound interest works is that after a given time interval (e.g. year), interest (as a percentage of the original investment) is paid out. The next year, interest is paid out again but as a percentage of the new amount. As an example:
1000 x 1.08 = 1080 (end of year 1)
1080 x 1.08 = 1166.40 (end of year 2)
1166.40 x 1.08 = 1259.71 (end of year 3)
…until end of year 10

If we use mathematical shortcuts and convert this into a formula, we can express it as:

(A = future value, P = present value, r = interest rate as decimal, n = number of periods/years)

For example, if we invest \$1000 (PV) at an interest rate of 5% (r=0.05) for 10 years, then:

\$1000 x 1.08^10 = \$2158.92,

meaning we have earned \$1158.92 over 10 years. Taking it further, in 30 years our investment would have grown to \$10062.66 – ten times our original investment.

Because the formula uses exponents (or powers) for the time, your investments grow exponentially with time. This means that the earlier you invest, the greater your returns become disproportionately. This is why within 10 years, we have more than doubled our initial investment despite a reasonable interest rate and not doing anything else.

A rule of thumb for calculating how long it will take your investment to double is to divide 72 by the interest rate in % (e.g. 7). This is the number of years it will take for your investment to double (e.g. 72/7 = 10.3 years).

On top of this, if we invest small amounts every year, then we can benefit even more from the exponential growth of our investment. For example, just by adding in \$100 every year, we end up with an additional \$564.55 of investment earnings at the end of 10 years – a 50% increase in returns.

Unfortunately, mathematics works both ways and compound interest also applies to certain loans, such as credit cards. This means that your debt will grow exponentially unless you aggressively pay it back, making it seem impossible to pay off your credit card debt sometimes.