Did you know that gambling drives investment theory?

It’s a not often talked about fact that many of the recent technological advances came about thanks to the “adult” industry.  Online payments, streaming content, and the huge increase in internet bandwidth are all thanks to adult industry.  Back in 2002 when I first started out in high frequency trading, the people setting up our servers wondered why we needed so many.  We told them we were an adult entertainment company, and they didn’t even bat an eye.  They just asked for free access…

In the same way, much of investment theory comes from gamblers.  Fancy pants traders, quants, and hedge funder managers spend much of their time dealing with probability theory, which was invented to quantify games of chance.

Today, I’m going to talk about one such technique that bridges the gap between gambling and investing: the Kelly Criterion (which I usually refer to as just plain old Kelly).  Kelly answers the oldest wagering question: “how much should you bet?”

In recent years it’s been picked up by the finance industry as a whole, and is simple enough that it could and should be used by all investors.

## So What is This Kelly Thingamajig?

We’ve already said that the Kelly Criterion tells you how much you should bet.  Note, it doesn’t make a guess at it.  It’s a proven fact.  (there are few caveats though).

I want to play a hypothetical game with.  You place a wager, and I flip a coin (we’ll assume it’s fair and I don’t have magician fingers).  If the coin comes up heads, I will pay you double your wager.  For tails I keep your wager.

How much will you bet?

Before you can answer, you need more information.  We need to yank out some details from the rules of the game and your situation:

• What’s the largest amount you can afford to wager?  We call this your bankroll.
• How much do you get if you win?  These are your odds.
• How much in your favor is the game?  This is your edge. (if the game is not in your favor, why in the heck are you playing?)

So the Kelly Criterion is simple:

optimal bet = edge/odds * bankroll

In this case let’s say your bankroll is \$100.

We’ve already been told you when you win you get paid double your bet, so your odds are 2/1=2.

Edge is trickier.  It’s (chance of winning * winning payout – chance of losing * loss amount).  For you that’s 0.50*2-0.50*1= .05

Now we plug and chug: Optimal bet = edge/odds*bankroll = .5/2*\$100 = \$25.

Your first bet in the game should be \$25.  You have to recalculate your ideal between before every round.  If you win the first toss, your new bet is .5/2*\$125=\$31.25, if you lose it’s \$18.75.

## Is Kelly Really the Best?

Yes…sort of.  The caveat I mentioned about is Kelly criterion only applies if you are going to play a game over and over again.  If you’re playing once and are never gambling again, then it’s not much help.

However, if you’re in it for the long haul, Kelly is the optimal approach.

Don’t take my word for it, look through the mathematical proof.

Or…we can pick a few approaches and play the game a bunch of times?

• Fixed Bets – The easiest thing to do it bet the same amount every time.  Say \$50.
• Kelly Criterion
• Double Kelly Wagering –  Maybe you like some of the ideas of Kelly, but want to make more money faster.  Why not double the bet?
• Half Kelly Wagering – Kelly too conservative?  Why not put down half the bet?

I wrote a python script to do the heavy lifting here.  If you want to make sure I didn’t mess up my code, send me a tweet or a BiggerPockets message and I’ll send you all 50 lines of code.  Otherwise, you’re just going to have to trust me.

Interestingly enough, this approach of picking some methods, trying them out, and tallying the results is called a Monte Carlo simulation and is commonly used in computer science, statistics, and finance.  Guess where it came from?  Yep, gambling.  It’s named after the Monte Carlo casino.

Let’ s see how our approaches are doing after 10, 50, 100, and 1000 plays:

#### 10 Plays

Here’s our bankroll after we’ve played the game 10 times:

Double Kelly is off to a strong start.  Heck, it’s made more than triple what Kelly has.  Fixed and Half Kelly are sucking wind a mile back.  Maybe there’s something to being aggressive?

#### 50 Plays

Double Kelly got hit hard.  Turns out being more aggressive is risky and has sever downsides.

Kelly is now squarely in the lead trailed by half and then double.

#### 100 Plays

Only 100 plays and it’s not even a contest anymore.  Kelly has made 15 times more than the runner up (half).

#### 1000 Plays

This chart is worthless…and that’s why it’s great.  Don’t you love exponents?  The scale is so big that Kelly is the only one that is even visible.  If you found someone with more money than there is on earth who would agree to play this game 1000 times, the Kelly approach would yield you a billion times more than the runner up (half kelly again).

Are you convinced yet?

## When in Doubt, Be More Conservative Than Kelly

In that last couple of results above, you’ll notice that half Kelly outperformed double Kelly.  After 100 iterations, half Kelly did 1.5 times better than double Kelly.  After 1000 iterations, half did ~40 trillion times better than double (no, I can’t wrap my mind around that scale either).

You want to err on the side of being more conservative, not more aggressive.

There is a chance you could wind up with a short burst of extraordinary returns, but that’s not the norm and you will always be overcome by more conservative approaches.

## How Does This Apply to Real Estate Investing?

You might recall when we first started out that I said there is a very thin line between investing and gambling.  Well, you can turn every investment decision into a Kelly problem (some are easier than others though).

Let’s look at a house flipping example.

Say you can buy a house for \$100,000.  You expect that when it’s all said and done you can clear \$20,000 after rehab costs and commissions.  However, you can’t be certain of that.  When you get into a property, unanticipated issues might crop up, the cost of materials could increase, or maybe the market turns against you.  For the sake of this example, we’re going to say there’s a 20% chance you lose \$10,000.

But wait…this ignores the “cost of carry.”

Assume it takes 6 months to flip the house and you get a 100% interest only loan at 5% APR and property taxes are 2% a year.

Financing costs = 100,000 * .05 / 2 = \$2,500.

Property taxes = 100,000 * .02 / 2 = \$1,000.

To convert this into Kelly terms we could say we’re wagering \$13,500 to win \$16,500, and we have a 80% chance of winning and a 20% chance of losing.

Odds = \$16,500/13,500 = 1.22

Edge = .8*1.22-.2*1= .776

Remember, optimal bet = edge/odds*bankroll.  Time for some algebra!  \$13,500=.776/1.22*bankroll.  That means your bankroll should be at least \$21,224.  That’s on a “no money down” investment!

We’re not even including your labor.  How many hours will you have to invest?  How much is that time worth?

Can you make this investment with less than \$21,224?  Absolutely, but remember, when you’re not sure, always err on the side of being too conservative.

## Wrap It Up: Investing the Kelly Way

In many ways, gambling is like adult entertainment.  They’re both on the seedy side of life, they both are looked down on, and they both drive progress in a number of industries.

I hope you have a decent understanding of one popular gambling/investing approach: the Kelly Criterion.

Go try it for yourself.  Calculate your optimal bet for an investment you’re considering.  What percentage of your bankroll should you put on the line?  Drop me a line and let me know.

Photo: conorwithonen

During Kenny’s decade in finance he bought many single family rentals in rural areas, as a hobby. Along the way, he talked some brave souls into joining him as investors and recently retired from finance to take his hobby to the next level. Find more by and about Kenny on his personal blog and his recently created twitter account!

• Ha! Thanks Karen.

1. Great post, Kenny – I completely agree.

As a former avid Hold’em player (one of these days, when RE stops “doing it” for me, “I’ll be back”, as Arnie says), if has often occurred to me that basic poker strategies like bluffing, semi-bluffing (negotiating), minimizing losses, maximizing wins (profits), calculating bet sizes (investment amounts) VS pot size (potential returns), as well as a great many additional concepts, all influence my deal analysis significantly – it becomes very easy, once you’re familiar with those concepts in a different context, to evaluate deals based on your existing portfolio parameters, your risk tolerance as derived from it, and the amount you’re required to invest VS all of the above, as well as projected returns, etc – you’ve just given me an idea for a future article, thanks! 😉

• Good points all around Make sure and let me know when your article comes out!

2. Awesome post! It’s like you got sick of splitting hairs on investing vs. betting and just said “OK. If you’re going to bet, use Kelly.” 🙂

• Lol, sort of.

3. Man, right after you produce some of the best thinker articles this site has had, you go and do another great article.

This one was just plain fun though, and very well explained. Thanks Kenny!

• Thanks for the kind words Joe!

4. Thanks for this one Kenny. You have again inspired me to face my quant limitations :))

• Cheers Doug! You going to work through a specific investment?

• I am VERY excited about this application in Real Estate Development projects. Simulation seems a lot more rigorous and robust than linear regression. While I barley got by with a C in Econometrics my last semester of undegrad…I do remember looking up and seeing that big Epsilon….and thought….man I can drive a Mack truck through there.
Thanks for NNT and reinvigorating my desire to address something that really stymied me.

take a look at this when you get a chance….cutting edge to me…and WAY out of my league but I am determined to master it and USE it: http://goo.gl/YfO3Bg

Its seems they used Thermodynamics to predict city growth…I think the follow on article will be mind bending and very fruitful http://goo.gl/fPTiO1. I have not joined to get my hands on that one yet.

• Interesting. Lots of things to check out here.

5. Great and entertaining post @Kenny.
Couple questions, in terms of real estate investment shouldn’t your bankroll for the Kelly include all your cash and all the equity in your portfolio? In your Monte Carlo Kelly didn’t surpass Double Kelly up until 50 hands or turns so wouldn’t a long term holder (not a flipper) who anticipates having fewer than fifty properties be better ‘betting’ the double?

Lot’s to think about, especially in relation to Mandlebrot’s “The Misbehavior of Markets” (http://amzn.to/IX07sa on Amazon) which I’m reading right now.

• Thanks for the comment Giovanni.

Yes, your bankroll should include your equity position. In my example we didn’t have any.

The only reason double Kelly outperformed is because the sequence started with a good string of heads. This is not normally the case. If you redid the first 10 throws 1000 times and averaged the results after every coin flip, Kelly would beat half Kelly, and half Kelly would be double Kelly.

In the stats world we would say the expected value is highest for Kelly. What we saw here was just variance.

I’ve not had a chance to read that book. Is it too technical for an audiobook?

Cheers,

Kenny

• Kenny thanks for the details on Kelly. On the book I would say it’d be very good via audio, the only thing is there are visuals of how the math works in the book that I find helpful, so maybe the audio and ebook versions both?

6. Kenny – Fantastic post! I really enjoy your post, knowing that you put a decent amount of time and real data into them. Having a math and comp sci background, I love looking at various algorithms and assist in my investing and analysis.

• Best of luck Tom! What asset classes you looking at?

7. Thanks for the post Kenny! Was wondering if you ever dabbled in blackjack or any other advantage play? Seems like it’d be right down your alley.

• Dabbled yes…gone through the effort of learning a system more complicated than the one in Beat the Dealer? No.