My last post highlighted that growth is sustainable only if you are employing a strategy that manages risk relative to your bankroll. This seems very simple but I notice it being overlooked constantly in public discourse.
In this post I want to dip my toe into looking at the formula to build a typical thought process for applying Kelly.
First, I will respond to "shut up and multiply” as a rational tenet. More specifically, I am countering the opening statements in this off-shoot about comparing saving 400 lives with certainty rather than 500 lives with 90% certainty. Only multiplying omits the fact that any decision under uncertainty carries variance and even a positive expectancy bet will bankrupt you if you employ the incorrect betting strategy.
Let's begin by looking at an actual Kelly equation to build our mental model and then we can return to apply it.
A good formulation for the binary solution is1
f = P - (1-P)/B
which is really saying the probability of success (P) less the probability of losing (1-P) scaled down by the payout of the bet (B) is the fraction of bankroll to wager (f). I am aware that I am not winning a Pulitzer prize for this description, but let's get a takeaway from why this equation matters and then I can try to apply it to the above problem.
Takeaway 1: In a binary outcome, the fraction of your bankroll wagered only requires estimates for two parameters - P = probability of success, and B = return on your wager if you win.
Let’s apply the logic from my previous post.
What is our effective bankroll? If we are pontificating about 400 certain lives saved vs 500 lives saved with 90% probability. It *really* matters if there are only 500 people in existence or 8 billion. From here on I will only refer to it as a wager in dollars because the lives framing is not helpful.
What is the distribution? The problem is defined as a binary outcome. This allows us to understand the distribution we are dealing with and choose the binary formula for Kelly stated above.
What is the variance? Well we get the variance from the probabilities in the binary solution so we don’t even need to explicitly compute it here.2
What is the probability of success, P? The problem clearly states that we win $500 only 90% of the time so we can assert that P = .9.
What is the return on our wager if we win, B? We need to be aware that choosing between 400 guaranteed dollars and 500 uncertain dollars is equivalent to a statement that you are wagering 400 dollars at -400 American odds or a payout of 1 to 4. This sets our B variable.
We have collected all the relevant variables.
Simply, P = .9 and B = (500 - 400) / 400 = .25
.9 - (1 -.9) / .25 = .5
This is saying that we should bet (up to) 50% of our bankroll with the probability of success at 90% and the return on a win at 25%. That is a really high fraction! On the other hand this is an unrealistically amazing bet. What it doesn’t say is shut up and always take the bet.
Now that we see that was very simple!
Allow me to go on a tangent here to help build intuition for whether a return seems unrealistic. As a simple rule of thumb, you can explicitly use the Kelly formula to find out how fast you become very wealthy.3 If it isn’t measured in decades to centuries, then you are probably calculating something wrong or being scammed.
Using the example above, if you were offered a bet which could scale up with your bankroll daily with a starting bankroll of $1000, then you would have a mean of $574M after one year. Ie after 365 bets where you repeatedly wager 50% of your bankroll you quickly become very wealthy. For fun let’s say you keep it up for another year, then you approximately have wealth equal to the entire world.4 I did 1000 simulations of 365 bets and got the following (in millions)
min = .0002
25th = 38
median = 709
75th = 11,185
maximum = 37,193,990 (Yes that is 37 trillion dollars in 1/1000 outcomes)
OK you get it, that is an unrealistic bet.
In my next post, I want to look at a more realistic example to get an intuition for what a reasonable bet size would be under real world possible expectancy. This is an unrealistically good return for risk and since I anchored you to this first problem, let me reiterate that full Kelly is maximum risk neutral betting aggression. We will go over the math justifying the aggression statement.
I omit A from this because I am lazy and it doesn’t affect the intuition or this example
Ah yes, I was following the previous post’s step-by-step instructions. Back to binary.
We are assuming there isn’t a minimum here.
Well only $330t vs $510t in world net worth according to McKinsey estimates as of November 2021. What’s a couple hundred trillion between friends?