An Eighth Wonder

Mike made a proposal to his friend that I will pick the garbage every day from your home for one month. In return, I want double the money that I had received the previous day. On the first day, he got 1$. How much money will he get at the end of one month?

Mike received 1 Billion$ at the end of one month! This is the power of doubling.

The human brain is evolved to think in a linear way. It is counter-intuitive to think in an exponential way. If someone asks you, "What is the result of 9+9+9+9+9+9+9+9? " The answer comes to your mind within a couple of seconds. If someone asks you, "What is the result of 9*9*9*9*9*9*9*9?" The answer won't come as easily as it comes in the above case. You need to think harder to arrive at the answer. Exponential thinking is complex than linear thinking.

The best outcomes in life come through exponential thinking(Compounding). When the small changes are compounded for a long period of time, it delivers outstanding results. Compounding depends upon the runway(the length of time) and the rate at which changes are occurring.

Understanding both the power of compound interest and the difficulty of getting it is the heart and soul of understanding a lot of things. Charlie Munger

Is it possible to do exponential thinking within a couple of seconds?

Yes, of course. Before deriving the mental shortcut to do exponential thinking within seconds. Let's first understand the core idea behind the compound effect.

Suppose you have invested 100$ in a company that is giving the return of 10% per year. How much money do you get at the end of one year?

Yes, you guessed it right. 110$

The money earned at the end of one year = P + P * R * T where P is Principal, R is Rate, and T is the number of times in a year.

= P (1 + R * T)

The money earned at the end of the second year = P * (1 + R * T ) * (1 + R * T)

= P * (1 + R * T) ^2

= 100 * (1 + .1 )^2

= 121$

Did you notice something?

The money earned at the end of 1st year starts generating money. An extra dollar you earned in the 2nd year is the result of 10$ that you earned in the 1st year. 100$ has generated a revenue of 10$. 10$ has generated a revenue of 1$ in the subsequent year. That is why you get 121$ at the end of two years.

Money makes money. And the money that money makes, makes money. Benjamin Franklin

The money generated at the end of n years = P * (1 + R *T)^N

The shortcut to do exponential thinking in the head, one should know after how time the amount gets doubled?

Let's derive it from the above result.

=> 2 * P = P * (1 + R)^N ; T is 1 because of yearly interest rate

=> 2 = (1 + R)^N

Taking natural log to both sides

=> ln(2) = ln((1 + R)^N)

=> 0.6931 = N * ln(1 + R)

For small R, ln(1 + R) ~ R (In Investing, R is less than 30%)

The equation becomes,

=> 0.6931 = N * R

=> 69.31 = N * R (Taking R in the form of percentage in the shortcut)

The number of years taken to double the amount is = 69.31 / R

69.31 can be rounded off to 72 because 72 has more multiples than 69.

Therefore, the number of years taken to double the money = 72 / Rate of return. This is known as the rule of 72.

Key insights of the Rule of 72

1. As the Rate of return increases, the no. of years taken to double the money decreases. For example, earning with an interest rate of 8% will double the money in 9 years whereas earning with an interest rate of 10% will double the money in 7.2 years.

2. The number of years taken to double the money is independent of the initial investment. For example, suppose the money will be doubled in 5 years then It doesn't matter whether you put 1k$ or 1 Million$, after 5 years the amount will be increased to 2k$ and 2 Million$ respectively.

Applications of the Rule of 72

A) Suppose you have invested 10k $ in a company that gives a 9% rate of return per year. You hold the stock for the next 40 years. How much money will you have at the end of 40 years?

As per the Rule of 72,

Years taken to double the money = 72 / 9

= 8 years

Total number of doubles in 40 years = 40 / 8

= 5 years

The amount of money at the end of 40 years = 2^5 * Initial Investment

= 32 * 10k

= 3,20,000 $

B) Suppose you have invested 800$ in the stock of some company. After 10 years, the money will be increased to 4800$. What is the rate of annual return?

The money will be increased to 6 times over the span of 10 years.

2 doubles will increase the money to 4 times whereas 3 doubles will increase the money to 8 times. Roughly, in 2.5 doubles the money will be increased to 6 times.

2.5 doubles over a span of 10 years.

Years took to double the money = 10/2.5

= 4 years

As per the Rule of 72,

Rate of return = 72/N

= 72/4

= 18 %

The company's stock gives a return of 18% per annum.

C) Why does one shouldn't hire a fund manager?

A simple thought experiment to understand the whole idea.

Without hiring the Fund Manager:

Rate of Interest = 9%

Years = 40

Years took to double the money = 72/9 years

= 8 years

Total number of doubles in 40 years = 40/8

= 5 doubles

Amount of money you will receive at the end of 40 years = 2^5 * Initial Investment

= 32 * Initial Investment

With the help of a Fund Manager:

Fees Charged = 1%

Rate of Interest = 9% - 1% = 8%

Years = 40

Years took to double the money = 72/8

= 9 years

Total number of doubles in 40 years = 40/9

= 4.5

Amount of money you will receive at the end of 40 years = 2^4.5 * Initial Investment

= 22.62 * Initial Investment

Loss incurred by hiring the fund manager = (32 - 22)/32 * 100

= 31.25%

1% fees of Fund Manager incur the loss of 31.25% over the span of 40 years. Sometimes figures are deceptive!

Intentions don't matter. Actions do. That's why being ethical is hard. Naval Ravikant

The compounding effect can also be applied outside the investing domain.

1. A 3% increase in GDP will double the economy in 24 years whereas a 2% increase will double it in 36 years.

2. An increase in inflation from 2% to 3% causes the money to lose half of its value in 24 years instead of 36 years.

3. Paying 16% interest on a credit card the amount you owe will double in 4.5 years.

All the great things in life whether in health, wealth, or relationships come through compounding. Tiny changes extended over a long period of time generate true success and progress. Time plays an important role in compounding. Invest deeply when you find the right people and the right thing to do. Inculcate the habit of exponential thinking to become successful in life. Always choose habits over willpower. Curate an ecosystem that will nudge towards the desired behavior to reap the benefits of compounding.

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Photo by Alberto Restifo on Unsplash