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?
1/
Get a cup of coffee.
In this thread, I'll help you understand Geometric Expectations.
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?
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.
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.
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.
In fact, as we take more and more turns, our chances of making money approach 100%, and our chances of losing money approach 0%.
In this sense, positive expectation bets are *sure things*.
They're "good" bets to make.
For example, suppose we place 100 bets at these odds.
Let's say we bet $1 each time. So, we *put in* $100.
In this case, there's a ~93.34% chance that we'll make money.
And on average, we expect to end up with $112.50 (our +12.5% expected return).
In the scenario above, we did not let our 100 bets *compound*.
We bet the *same* $1 at each turn.
We did not take the winnings from our first turn and roll them into our second, and so on.
But if we did that, our money would grow even faster, right? At +12.5% per turn?
So, if we start with $1, and take 100 turns betting at these odds, rolling over our winnings from each turn to the next, we'll most likely end up with something like $1 * (1.125^100) = ~$130K, right?
That's what compounding $1 at 12.5% a hundred times gets us.
Sadly, no.
If we roll over our winnings 100 times, our chances of making money *drop* to a tiny ~0.04%.
That is, with probability more than 99.95%, we'll end up *losing* money.
How is this possible?
Well, let's think about what happens to our $1 as it goes through this "compounding" process 100 times.
At every "lucky" turn, our wealth gets multiplied by 2 (doubled).
But at every "unlucky" turn, it gets multiplied by 1/4 -- as (3/4)'th is lost.
So, each unlucky turn cancels out 2 lucky turns!
So, if we are to make money overall, the lucky turns will have to outnumber the unlucky turns at least 2 to 1.
That is, we need to get lucky at least twice as often as we get unlucky -- or we'll end up losing money.
The chances of getting lucky that often are pretty slim -- because getting lucky or unlucky is a 50/50 thing.
Over 100 turns, we're most likely to get lucky 50 times and unlucky the other 50 times.
But to make money over 100 turns, we need to get lucky at least 67 times.
So, it all comes down to whether we "roll over" our bets or not.
Without roll over, our expected return is +12.5% per bet.
But with roll over, our expected return per bet turns *negative* -- approaching -29.3% as we take more and more turns.
Key lesson 1: Positive expectation bets, when rolled over, can lead to *negative* expected growth rates!
Each turn may carry a positive expectation. But the compounded effect of many turns can carry a negative expectation.
So, there are really 2 kinds of expectations.
Our normal idea of expectation -- the return we expect to make without roll over -- is called *arithmetic* expectation.
But there's also *geometric* expectation. This is the return we expect to make with roll over.
And as our example above showed, a bet can carry positive arithmetic expectation but negative geometric expectation.
Let's take a general bet -- and derive formulas for its arithmetic and geometric expectations.
In general, a bet can have N possible outcomes:
With probability p_1, our money gets multiplied by f_1.
With probability p_2, it gets multiplied by f_2.
And so on.
Like so:
Here are the formulas for the arithmetic and geometric expectations of such bets, along with a couple of examples:
Key lesson 2: Even if a bet has positive arithmetic expectation, it may be a bad idea to roll it over repeatedly.
For roll over to be a good idea, the bet must have positive *geometric* expectation.
The famous Kelly Criterion is a strategy that transforms an *arithmetic* advantage into a *geometric* one.
The idea is: if a bet has positive arithmetic expectation but negative geometric expectation, we only roll over *part* of our winnings, not *all* of them.
For our "double or lose three fourths" example above, Kelly would only roll over (1/6)'th of our wealth from each turn into the next turn's bet.
This achieves an expected growth rate of +10.4% per turn -- much better than negative 29.3%!
For more: https://t.co/LjuaCg3mGm
1) Get a cup of coffee.
— 10-K Diver (@10kdiver) May 24, 2020
In this thread, I'm going to walk you through "The Kelly Criterion".
Another observation:
Whenever a bet has a non-zero chance of total loss, its geometric expectation is 0.
This is a Russian roulette type situation. Its (arithmetic) expectation may be positive, but if we keep pushing our luck and rolling over, we'll eventually blow up.
To learn more about the origins of geometric expectations, I recommend reading Daniel Bernoulli's paper:
https://t.co/vN6yNvmk9J
But if that's too much for you, here's a wonderful thread by @breakingthemark that summarizes the paper's main ideas: https://t.co/fdCd3X0E9B
I just re-read Bernoulli\u2019s 1738 paper \u201cExposition of a New Theory on the Measurement of Risk\u201d which is the foundational paper of Expected Utility Theory.
— breakingthemarket (@breakingthemark) December 17, 2020
It\u2019s Amazing
It\u2019s so wildly different than EUT that its hard to believe this was its beginning.
Let\u2019s see if you agree.
I also very much like this article by @TaylorPearsonMe that beautifully ties together many key ideas that come into play in such repeated betting scenarios: ergodicity, antifragility, barbell strategies, the Kelly Criterion, etc.
https://t.co/FHefGhuUWf
And speaking of ergodicity and repeated betting, Prof. Sanjay Bakshi (@Sanjay__Bakshi) has created a wonderful set of slides exploring these ideas:
https://t.co/2OsvrmvxeX
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1/
Get a cup of coffee.
In this thread, I'll walk you through 2 probability concepts: Standard Deviation (SD) and Mean Absolute Deviation (MAD).
This will give you insight into Fat Tails -- which are super useful in investing and in many other fields.
2/
Recently, I watched 2 probability "mini-lectures" on YouTube by Nassim Taleb.
One ~10 min lecture covered SD and MAD. The other ~6 min lecture covered Fat Tails.
In these ~16 mins, @nntaleb shared so many useful nuggets that I had to write this thread to unpack them.
3/
For those curious, here are the YouTube links to the lectures:
SD and MAD (~10 min): https://t.co/0TwubymdE6
Fat Tails (~6 min):
4/
The first thing to understand is the concept of a Random Variable.
In essence, a Random Variable is a number that depends on a random event.
For example, when we roll a die, we get a Random Variable -- a number from the set {1, 2, 3, 4, 5, 6}.
5/
Every Random Variable has a Probability Distribution.
This tells us all the possible values the Random Variable can take, and their respective probabilities.
For example, when we roll a fair die, we get a Random Variable with this Probability Distribution:
Get a cup of coffee.
In this thread, I'll walk you through 2 probability concepts: Standard Deviation (SD) and Mean Absolute Deviation (MAD).
This will give you insight into Fat Tails -- which are super useful in investing and in many other fields.
2/
Recently, I watched 2 probability "mini-lectures" on YouTube by Nassim Taleb.
One ~10 min lecture covered SD and MAD. The other ~6 min lecture covered Fat Tails.
In these ~16 mins, @nntaleb shared so many useful nuggets that I had to write this thread to unpack them.
3/
For those curious, here are the YouTube links to the lectures:
SD and MAD (~10 min): https://t.co/0TwubymdE6
Fat Tails (~6 min):
4/
The first thing to understand is the concept of a Random Variable.
In essence, a Random Variable is a number that depends on a random event.
For example, when we roll a die, we get a Random Variable -- a number from the set {1, 2, 3, 4, 5, 6}.
5/
Every Random Variable has a Probability Distribution.
This tells us all the possible values the Random Variable can take, and their respective probabilities.
For example, when we roll a fair die, we get a Random Variable with this Probability Distribution:
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IMPORTANCE, ADVANTAGES AND CHARACTERISTICS OF BHAGWAT PURAN
It was Ved Vyas who edited the eighteen thousand shlokas of Bhagwat. This book destroys all your sins. It has twelve parts which are like kalpvraksh.
In the first skandh, the importance of Vedvyas
and characters of Pandavas are described by the dialogues between Suutji and Shaunakji. Then there is the story of Parikshit.
Next there is a Brahm Narad dialogue describing the avtaar of Bhagwan. Then the characteristics of Puraan are mentioned.
It also discusses the evolution of universe.( https://t.co/2aK1AZSC79 )
Next is the portrayal of Vidur and his dialogue with Maitreyji. Then there is a mention of Creation of universe by Brahma and the preachings of Sankhya by Kapil Muni.
In the next section we find the portrayal of Sati, Dhruv, Pruthu, and the story of ancient King, Bahirshi.
In the next section we find the character of King Priyavrat and his sons, different types of loks in this universe, and description of Narak. ( https://t.co/gmDTkLktKS )
In the sixth part we find the portrayal of Ajaamil ( https://t.co/LdVSSNspa2 ), Daksh and the birth of Marudgans( https://t.co/tecNidVckj )
In the seventh section we find the story of Prahlad and the description of Varnashram dharma. This section is based on karma vaasna.
It was Ved Vyas who edited the eighteen thousand shlokas of Bhagwat. This book destroys all your sins. It has twelve parts which are like kalpvraksh.
In the first skandh, the importance of Vedvyas
and characters of Pandavas are described by the dialogues between Suutji and Shaunakji. Then there is the story of Parikshit.
Next there is a Brahm Narad dialogue describing the avtaar of Bhagwan. Then the characteristics of Puraan are mentioned.
It also discusses the evolution of universe.( https://t.co/2aK1AZSC79 )
Next is the portrayal of Vidur and his dialogue with Maitreyji. Then there is a mention of Creation of universe by Brahma and the preachings of Sankhya by Kapil Muni.
HOW LIFE EVOLVED IN THIS UNIVERSE AS PER OUR SCRIPTURES.
— Anshul Pandey (@Anshulspiritual) August 29, 2020
Well maximum of Living being are the Vansaj of Rishi Kashyap. I have tried to give stories from different-different Puran. So lets start.... pic.twitter.com/MrrTS4xORk
In the next section we find the portrayal of Sati, Dhruv, Pruthu, and the story of ancient King, Bahirshi.
In the next section we find the character of King Priyavrat and his sons, different types of loks in this universe, and description of Narak. ( https://t.co/gmDTkLktKS )
Thread on NARK(HELL) / \u0928\u0930\u094d\u0915
— Anshul Pandey (@Anshulspiritual) August 11, 2020
Well today i will take you to a journey where nobody wants to go i.e Nark. Hence beware of doing Adharma/Evil things. There are various mentions in Puranas about Nark, But my Thread is only as per Bhagwat puran(SS attached in below Thread)
1/8 pic.twitter.com/raHYWtB53Q
In the sixth part we find the portrayal of Ajaamil ( https://t.co/LdVSSNspa2 ), Daksh and the birth of Marudgans( https://t.co/tecNidVckj )
In the seventh section we find the story of Prahlad and the description of Varnashram dharma. This section is based on karma vaasna.
#THREAD
— Anshul Pandey (@Anshulspiritual) August 12, 2020
WHY PARENTS CHOOSE RELIGIOUS OR PARAMATMA'S NAMES FOR THEIR CHILDREN AND WHICH ARE THE EASIEST WAY TO WASH AWAY YOUR SINS.
Yesterday I had described the types of Naraka's and the Sin or Adharma for a person to be there.
1/8 pic.twitter.com/XjPB2hfnUC
1/ 👋 Excited to share what we’ve been building at https://t.co/GOQJ7LjQ2t + we are going to tweetstorm our progress every week!
Week 1 highlights: getting shortlisted for YC W2019🤞, acquiring a premium domain💰, meeting Substack's @hamishmckenzie and Stripe CEO @patrickc 🤩
2/ So what is Brew?
brew / bru : / to make (beer, coffee etc.) / verb: begin to develop 🌱
A place for you to enjoy premium content while supporting your favorite creators. Sort of like a ‘Consumer-facing Patreon’ cc @jackconte
(we’re still working on the pitch)
3/ So, why be so transparent? Two words: launch strategy.
jk 😅 a) I loooove doing something consistently for a long period of time b) limited downside and infinite upside (feedback, accountability, reach).
cc @altimor, @pmarca
4/ https://t.co/GOQJ7LjQ2t domain 🍻
It started with a cold email. Guess what? He was using BuyMeACoffee on his blog, and was excited to hear about what we're building next. Within 2w, we signed the deal at @Escrowcom's SF office. You’re a pleasure to work with @MichaelCyger!
5/ @ycombinator's invite for the in-person interview arrived that evening. Quite a day!
Thanks @patio11 for the thoughtful feedback on our YC application, and @gabhubert for your directions on positioning the product — set the tone for our pitch!
Week 1 highlights: getting shortlisted for YC W2019🤞, acquiring a premium domain💰, meeting Substack's @hamishmckenzie and Stripe CEO @patrickc 🤩
2/ So what is Brew?
brew / bru : / to make (beer, coffee etc.) / verb: begin to develop 🌱
A place for you to enjoy premium content while supporting your favorite creators. Sort of like a ‘Consumer-facing Patreon’ cc @jackconte
(we’re still working on the pitch)
3/ So, why be so transparent? Two words: launch strategy.
jk 😅 a) I loooove doing something consistently for a long period of time b) limited downside and infinite upside (feedback, accountability, reach).
cc @altimor, @pmarca
4/ https://t.co/GOQJ7LjQ2t domain 🍻
It started with a cold email. Guess what? He was using BuyMeACoffee on his blog, and was excited to hear about what we're building next. Within 2w, we signed the deal at @Escrowcom's SF office. You’re a pleasure to work with @MichaelCyger!
5/ @ycombinator's invite for the in-person interview arrived that evening. Quite a day!
Thanks @patio11 for the thoughtful feedback on our YC application, and @gabhubert for your directions on positioning the product — set the tone for our pitch!
