Investing Fundamentals: Probabilities (Free Tool)
“If you don’t get this elementary, but mildly unnatural, mathematics of elementary probability into your repertoire, then you go through a long life like a one-legged man in an ass-kicking contest.”
– Charles T. Munger
If you’re like me when I started learning how to invest, I knew I should know how to use probabilities in my process but I thought it would come later. I seemed to find so many investors who talked about its importance, but few actually demonstrated how to use them.
Thanks to Mohnish Pabrai and his book The Dhando Investor, he gave a real-world example of how he actually used probabilities. Once I understood it, I simplified it, created a Spreadsheet (linked below), and now use them all the time in my fund. This specific method is named weighted averages. I promise it’s easier than it sounds. If you’re just here for the spreadsheet, use the link below:
Spreadsheet link: Weighted Averages of Probabilities
What are weighted averages?
Weight is a word describing importance. If something was weighted heavily, then it would be of higher importance. When something was weighted lightly or weighted low it would be of lower importance. With probabilities, this is expressed as a percentage.
Averages are the arithmetic definition, you add up a series of numbers and divide them by the amount in the series. The average of 3 + 5 is 4. That’s because there are 2 numbers, and so the formula would be:
(3 + 5) / 2
= (8) / 2
Weighted averages is the method of assigning importance to a series of possible scenarios that give you the average outcome you could expect over time.
A simple weighted average example
Often times coin-flips are used for examples to show simple probabilities. We will assume that the coin we are using for our example is a fair coin, meaning that it isn’t a trick coin and that it has two scenarios: 50% odds for heads and 50% odds for tails.
Let’s say someone was going to bet you $1 for every time a coin flip lands on heads, but you would lose $1 every time the coin flip lands on tails.
If you did this flip just once, you have a 50% chance to get $1 and a 50% chance to lose $1. Right now, that makes this a gamble.
But what if you were offered to do it 1,000 times?
You would expect to win about half of those 1,000 times, or 500 times and win $500 dollars.
You would also expect to lose about half of those times or about 500 times, and lose $500 dollars.
That means you would not expect to win or lose any money. You could say the expected return is $0.
You can come to this mathematically by using weighted averages.
Multiply the weight (probability) of winning by the amount you win and subtract from it the weight (probability) of losing multiplied by the amount you lose.
([win probability] * [win $]) - ([lose probability] * [lose $])
Win: 50% * $1 = $0.50
Lose: 50% * $1 = $0.50
$0.50 - $0.50 = $0.00
This means that the amount you would expect to win is $0.00 (after doing this many times). In the world of investing, this would not be a good investment.
Let’s make the example more interesting.
Weighted averages on a worthy investment
Using the same example of a coin flip, let’s change the amount that we win and lose. Before we said you win $1 and lose $1. Now, let’s say that you win $2 and lose $1. It is still a fair coin and so the percentages are still 50%/50%.
Rather than doing the math to see how this would turn out, let’s do the formula first.
Win: 50% * $2 = $1.00
Lose: 50% * $1 = $0.50
$1-$0.50 = $0.50
Now the formula shows that we expect to get $0.50, on average, for each flip. If we did this 1,000 times, that would be 1,000 * $0.50 = $500.
Let’s go through the example without thinking about the formula to verify it makes sense conceptually.
500/1,000 of those times we would expect to win. Each time we won, we expect to get $2. Therefore, we expect to win $1,000 dollars.
500/1,000 of those times we expect to lose. Each time we lose we expected to lose $500.
Therefore we would expect to win $1,000 and lose $500, leaving us with $500. The formula checks out.
This is a subtle but important lesson about investing. Often time we aren’t given the opportunity to repeat an option 1,000 times immediately, but it is possible to believe that throughout our life we may get 1,000 opportunities for a bet with good odds.
A final simple example
So far, our examples have only modified the price, and not the weight or the probability. Let’s continue from our last example, but this time let’s say we have an unfair coin, and it has a 75% chance to land on heads and a 25% chance to land on tails. Our formula looks like this:
Win: 75% * $2 = $1.50
Lose: 25% * $1 = $0.25
$1.50-$0.25 = $1.25
1,000 coin flips would result in 1,000 * $1.25, = $1,250, or $750 more than the previous example. This hopefully makes more sense, but often in investing, it isn’t as simple as a “win $X or lose $X”, there is often a range of options.
Dealing with more than two scenarios
So far we have only dealt with two outcomes: heads or tails; winning or losing. Let’s write out the same formula but in a slightly different way. We need to generalize it so it doesn’t only apply to win-and-lose scenarios. It should apply to any number of different scenarios and probabilities.
To generalize this, can remove the word “win” and “lose” and simply write out how much we expect to get out of each probability. The formula becomes:
([probability] * $amount]) + ([other probability] * $[other_amount])
If we use our last example, this becomes:
(75% * $2) + (25% * -$1) =
($1.5) + (-0.25) = $1.25
The main difference is that if we expect to lose money, we put a negative sign in front of the value. This becomes important when we introduce additional scenarios.
Let’s say we were using a four-sided die as used in some board games with values 1, 2, 3, 4. Let’s say our bet is winning $2 if it lands on an odd number (1,3) or losing $1 if it lands on an even number (2, 4). How do we write this?
(25% * $2) + (25% * -$1) + (25% * $2) + (25% * -$1) =
($0.50) + (-$.25) + ($0.50) + (-$.25) =
You can see it starts to look a little complicated (this is why we use a spreadsheet). But now we can deal with many different scenarios. What if we didn’t have a losing scenario, but simply different amounts we won? What if we got $2 for odds and also won $1 for evens? How much could we expect to win on the average roll?
(25% * $2) + (25% * $1) + (25% * $2) + (25% * $1) =
($0.50) + ($.25) + ($0.5) + ($.25) =
We would expect to get $1.5 for each roll of the die. If we repeated rolling this die 1,000 times, we would expect to win $1,500.
A real investing scenario
We’ve gotten through the hard part. We know what a weighted average is and how to use the formula. Now, how do we apply this to actual investing? This is a real example that my fund invested in last year.
We were following the company Seritage Growth Properties (SRG), and their management released a document on September 14, 2022 saying that they were going to dissolve the company and give the proceeds to their shareholders. They announced that they expected to return $18.50 — $29 per share to the shareholders over 18–30 months. At the time of the announcement, SRG was selling for $12.25 per share.
My fund used the spreadsheet above and put in four different scenarios, multiplied the probability by price per share, and then added all those values up.
|Price per share||$18.50||$23.50||$29||$6.50|
The result was $22.40. If this was repeated 1,000 times we would expect to get an average of $22.40 per time that goes through.
You might ask “how did you come up with those probabilities?” To which I respond with a seemingly-snarky quote:
“It is better to be roughly right than precisely wrong.”
– John Maynard Keynes
The answer is we estimated. We didn’t do any sort of fancy calculations or mathematics to come up with the numbers. We had been following the company for some time, so we felt like we had a rough idea of how good the management was, and then we played it conservatively. We gave only a 10% chance that they would get their best-case scenario and a 5% chance that the stock would actually lose half its value. We gave them a 30% chance that they would do the worst of their efforts.
We played with the numbers until we felt satisfied that it was a roughly-right representation of the scenario.
We could have done more scenarios, we could have done fewer—as long as all the probabilities added up to 100%.
I won’t go into it here (but you can see it in the spreadsheet), but we did another analysis on the length of time estimates they gave (18–30 months) to determine how fast we expected to get the money.
The weighted averages method is a simple way to use probabilities with investment decisions. A weight is a probability you assign to a specific scenario to get the average expected outcome if the scenario were to repeat many times. You can use any number of scenarios to come to your answer.
You can use my spreadsheet to do these calculations: Weighted Averages of Probabilities.
Let me know in comments below any thoughts you might have.