BC COMPOUNDING
Saved by @SAnngeri

1/

Get a cup of coffee.

In this thread, I'll walk you through the benefits of turning capital quickly.

The math behind turning capital is beautiful. It leads us to the number "e", which plays a vital role in so many different fields -- from astrophysics to biology.

 2/

Imagine we have $1M.

Also, we know an extraordinarily generous bank where we can deposit this $1M.

This bank will pay us interest. And not a paltry 1% or 2%, but a hefty *100%* per year!

(I know, I know. But humor me, will you?)
3/

So, if we deposit our $1M at this bank, it will earn that 100% interest and become $2M in 1 years' time.

But that's *only* if the interest is *compounded annually*.
4/

What if the interest is *not* compounded annually?

For example, suppose it's compounded half yearly?

"Compounded half yearly" means: instead of paying us 100% interest *once* a year, the bank pays us 50% interest *twice* a year (ie, once every 6 months).
5/

That is, we get to "turn" our money twice -- each time earning 50% on it.

Here's a quick picture of how that looks.

The first 6 months will turn $1M into $1.5M (ie, +50%). And the second 6 months will turn this $1.5M into $2.25M (another +50%).
6/

So, compounding half yearly works out *better* for us than compounding annually.

Half yearly means we end up with $2.25M after 1 year.

Annually means we end up with only $2M in the same 1 year.
7/

This is a *fundamental* fact about turning capital.

In general, the more quickly we can turn our capital, the higher our return will be.
8/

For example, the picture below shows how much we end up with if we turn our capital even faster.

As the table shows, *quarterly* is even better than *half yearly*.

*Monthly* is even better than *quarterly*.

*Daily* is even better than *monthly*.

And so on.
9/

But there also seems to be a "law of diminishing returns" at work.

For example, doubling our turns from annually to half yearly netted us $250K ($2M -> $2.25M).

But doubling our turns *again*, going from half yearly to quarterly, only netted us ~$191K ($2.25M -> $2.44M).
10/

This also is a fundamental property of capital turns.

As we crank up our turns more and more, the incremental benefit we get becomes less and less.

We still get a positive benefit each time. But the benefit gets smaller as we go:
11/

If we dig a little deeper, we can come up with a formula.

Suppose our capital turns N times per year.

Our formula then predicts how much each $1 of our money will become after 1 year -- as a function of N.

Here's the formula and its simple derivation:
12/

Jacob Bernoulli was one of the first mathematicians to study this formula in the context of capital turns and compounding money.

Bernoulli's work dates all the way back to 1690!

That's when he published a paper about this formula in the journal Acta Eruditorum:
13/

In the paper, Bernoulli proved that the formula *converged* to a limit.

Here's what that means:

As we keep increasing the number of turns per year N, the number (1 + 1/N)^N (which is what each dollar of our money becomes after 1 year) bumps up against a hard ceiling.
14/

Bernoulli proved that this ceiling had to lie somewhere between 2.5 and 3.

That is, *no matter how fast* we turn our capital (even if we turn it every nanosecond, or a billion times faster than that), we cannot grow $1 into $3 or more in 1 year. The ceiling prevents us.
15/

We now know that the ceiling is ~$2.718 -- between $2.5 and $3 as Bernoulli knew it had to be.

*If* we turn our capital *infinitely* often, we will grow $1 into roughly $2.718 in 1 year. Not more.

There's a strong law of diminishing returns at play.
16/

This magic number (2.718281828...) is called "e".

e has a special place in mathematics.

It's one of the most important universal constants. And it turns up in all sorts of surprising places -- from probability distributions to the shapes of galaxies!
17/

For those who'd like to nerd out even more, here's a sketch of Bernoulli's proof that the limit e has to exist and that it has to lie between 2.5 and 3.

(Don't worry if you don't get this math. I promise that you won't need it to make sense of the rest of this thread!)
18/

As I said, e plays such a vital role in so many areas beyond capital turns and compounding.

For example, the formula for the Gaussian probability distribution (known more commonly as the "bell curve") depends strongly on e:
19/

Another fun "curve" whose formula involves e in a big way: the famous Gateway Arch in St. Louis!
20/

Oh, and what if our bank paid us a more realistic interest rate -- say, R% per year?

Well, *if* we can turn our capital infinitely often, each $1 of ours becomes e^(R/100) dollars in 1 year.

I set R=100 just so the answer simplified to e dollars. But other Rs work too.
21/

I can't resist pointing out one more place where e showed up unexpectedly: in Google's IPO filing!

https://t.co/If8ikJzU5t

That's e billion dollars, folks!
22/

If you want to learn more about the rich history behind e, and some of the math surrounding it, I recommend the book "e: The Story of A Number" by Eli Maor. It's a great read!

Amazon link: https://t.co/pVlJE1Rzap
23/

For more on the investing principles behind capital turns, inventory turns, and working capital management: https://t.co/CeEcrX6JY8
24/

I also very much enjoyed this Focused Compounding episode, where Andrew and Geoff discuss some of these concepts in their usual illuminating way (h/t @FocusedCompound).

~45 minute video: https://t.co/ZYDmvCOp5n
25/

If you're still with me, thank you very much!

Capital turns are a key concept in compounding.

In addition to investing insights, I hope this thread also gave you an appreciation for the rich history and the beautiful math behind this topic.

Enjoy your weekend!

/End

More from 10-K Diver

10-K Diver
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@Tom_Voltz1 1/

That’s a possibility.

Buffett has this concept of “owner earnings”: how much cash can be safely taken out of a business each year without hurting it?

Take out more than this, and we risk hurting the business’s competitive position, earning power, unit volume, etc.

2/

At the same time, leaving capital in a terrible business is not a great idea either. It’s like throwing good money after bad.

In fact, directing capital away from textiles (a bad business) and into insurance (a better business) is one reason for Berkshire’s success.

/End
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10-K Diver
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1/

Get a cup of coffee.

In this thread, I'll walk you through the importance of understanding *correlations* between bets.

For example, a portfolio of *correlated* stocks can have very different performance characteristics compared to a portfolio of *uncorrelated* stocks.


2/

Imagine we put $100K into a stock.

In the next 1 year, the stock could go down 30% (our worst case scenario).

Or it could go up 50% (our best case scenario).

Or it could give us a return somewhere between these extremes.

Say all such returns are equally likely.

3/

A "probability density diagram" can be used to visualize such scenarios.

On the X axis, we take all possible outcomes (in this case, -30% to +50%).

And on the Y axis, we plot the likelihoods of these outcomes.

Like so:


4/

Take any 2 outcomes on the X axis -- say, R1 and R2.

Now, calculate the area under the probability density diagram between R1 and R2.

Our probability of getting a return *between* R1 and R2 is exactly this area.

Like so:


5/

This is called a "uniform" distribution -- *all* returns between -30% and +50% are *equally* likely.

But we can also imagine other "non-uniform" scenarios, where *some* returns are more likely than others.

Here too, probability density diagrams can help us.

A few examples:
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10-K Diver
10-K Diver
@10kdiver
1/

Get a cup of coffee.

In this thread, I'll help you understand Geometric Expectations.


2/

Imagine we have an investment opportunity.

If the investment goes well, we get to double our money.

But if it goes badly, we'll end up losing three fourths of it.

There's a 50/50 chance of either outcome.

The question: is this a good bet or not?

3/

A simple way to approach this question is to calculate the bet's "expectation".

For every $1 we bet, there are 2 possibilities:

1. The lucky case, where we double our money and end up with $2, and

2. The unlucky case, where we lose (3/4)'th to end up with just $0.25.

4/

Each of these outcomes is equally likely.

So, on *average*, every $1 we put in turns into ($2 + $0.25)/2 = $1.125.

This is a *positive* 12.5% return.

Such bets are called *positive expectation* bets. On average, we expect to *make* (rather than *lose*) money on them.

5/

Suppose we could take many turns betting at these odds.

In any one turn, we could of course lose money.

But because our bet has positive expectation, *over time* we expect to make money.
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10-K Diver
10-K Diver
@10kdiver
1/

Get a cup of coffee.

Let's talk about the Birthday Paradox.

This is a simple exercise in probability.

But from it, we can learn so much about life.

About strategic problem solving.

About non-linear thinking -- convexity, concavity, S curves, etc.

So let's dive in!

2/

Suppose we came across a "30 under 30" Forbes list.

The list features 30 highly accomplished people.

What are the chances that at least 2 of these 30 share the same birthday?

Same birthday means they were born on the same day (eg, Jan 5). But not necessarily the same year.


3/

What if it was a "40 under 40" list?

Or a "50 under 50" list?

Or in general: if we put M people on a list, what are the chances that *some* 2 of them will share the same birthday?

4/

Clearly, this is an exercise in probability.

To solve it, we'll assume 3 things:

1. No Feb 29 birthdays,

2. Each person on our list is *equally likely* to be born on any one of the other 365 days (Jan 1 to Dec 31), and

3. The birthdays are all independent of each other.

5/

Another way to state the problem:

We have M people, and a 365-sided fair die.

Each person is allowed to roll the die once -- and is thereby assigned a number between 1 and 365 (both inclusive).

What are the chances that *some* 2 people will get assigned the same number?
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10-K Diver
10-K Diver
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1/

Get a cup of coffee.

In this thread, I'd like to share with you a simple way to measure and track your progress towards financial independence.


2/

Financial independence is an end goal.

To define this goal properly, you need to answer 2 key questions.

(i) When would you consider yourself financially independent? And,

(ii) In how much time do you want to get there?

3/

Typically, financial independence means:

a) You won't have to work another day in your life for money, and

b) You and your family will still be able to live comfortably to the end of your days.

4/

Let's say you want to reach financial independence in the next 10 years.

How much money do you need to amass by that time?

Well, suppose your family's expenses are ~$60K per year now.

If inflation runs at ~2% per year, 10 years from now, your annual expenses will be ~$75K.

5/

Let's say you want your expenses to be under 3% of your portfolio at that time.

This is called the "3% Rule".

It means your portfolio's size should be about $2.5M (as 3% of $2.5M = $75K).

For more on the 3% Rule and such:
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