Kelly criterion question

[prunes]

I play around with the Kelly criterion calculators online sometimes. A minor question I have that I never see addressed is this: Kelly's formula is based on parimutuel odds, but these odds don't translate well to the stock market where situations often aren't binary. That is, if I think a stock is a double and I'm wrong, chances often are that I won't lose all my money. So how do you properly estimate the odds as an input to Kelly's formula in this situation?

 

Maybe someone can work through an example for me. How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

[twacowfca]

I play around with the Kelly criterion calculators online sometimes. A minor question I have that I never see addressed is this: Kelly's formula is based on parimutuel odds, but these odds don't translate well to the stock market where situations often aren't binary. That is, if I think a stock is a double and I'm wrong, chances often are that I won't lose all my money. So how do you properly estimate the odds as an input to Kelly's formula in this situation?

 

Maybe someone can work through an example for me. How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Your odds is the upside (100) divided by the downside (30).  The probability as stated is 75%.  The full Kelly bet is 67% of your capital at risk. It would be foolish to make a full Kelly bet under almost all circumstances for many reasons.

 

I like to use the Kelly  calculator at albionresearch.com

 

The Kelly criterion can be used to maximize the geometric mean (the average percentage return) of the growth rate of capital in a theoretical sense, but even then the volatility is crazy.  With full Kelly betting, there is a 1% chance that you will draw down 99% of your capital in a lengthy series of bets or a 50% chance that the draw down will be 50%.  In real life, the series of good bets available is not often lengthy, and money lost is not often quickly recovered.

 

Sadly, if you overestimate the probabilities or the odds in your favor, as is often the case, The overbetting that results from using what you think is a full Kelly proportion guarantees that the capital in a series of bets will periodically crash forever to a very small fraction of the original capital.

 

The power of Kelly comes into play when the probability of success after an accurate assessment is extremely high, like 99%.  Then, a half Kelly bet might be appropriate.  For example, Warren put almost all of his insurance company assets into high quality common stocks in 1973- 1974 when they were selling for about one third what a private buyer would have paid to own them outright.  :)

[bathtime]

Great reply!

[twacowfca]

I play around with the Kelly criterion calculators online sometimes. A minor question I have that I never see addressed is this: Kelly's formula is based on parimutuel odds, but these odds don't translate well to the stock market where situations often aren't binary. That is, if I think a stock is a double and I'm wrong, chances often are that I won't lose all my money. So how do you properly estimate the odds as an input to Kelly's formula in this situation?

 

Maybe someone can work through an example for me. How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Your odds is the upside (100) divided by the downside (30).  The probability as stated is 75%.  The full Kelly bet is 67% of your capital at risk. It would be foolish to make a full Kelly bet under almost all circumstances for many reasons.

 

I like to use the Kelly  calculator at albionresearch.com

 

The Kelly criterion can be used to maximize the geometric mean (the average percentage return) of the growth rate of capital in a theoretical sense, but even then the volatility is crazy.  With full Kelly betting, there is a 1% chance that you will draw down 99% of your capital in a lengthy series of bets or a 50% chance that the draw down will be 50%. 

 

Sadly, if you overestimate the probabilities or the odds in your favor, as is often the case, The overbetting that results from using a full Kelly proportion guarantees that your capital in a series of bets will periodically crash forever to a very small fraction of the original capital.

 

The power of Kelly comes into play when the probability of success is extremely high, like 99%, by an unbiased assessment.  Then a half Kelly bet might be appropriate.  For example, Warren put almost all of his insurance company assets into high quality common stocks in 1973- 1974 when they were selling for about one third what a private buyer would have paid to own them outright.  :)

 

 

Interestingly, there is another opportunity hiding in plain sight that has about a 90% probability to have a positive return in the short term with the downside risk almost certainly limited to a very low single digit percentage.  We added to our BRK holding recently paying just a hair above 110% of their Q1 BV/SH reported this weekend.  BRK closed Friday about 3% above that threshold, below which Warren said he would buy back BRK shares aggressively.  At that price level, there is very little downside risk and enormous upside as a potential long term hold. 

 

This entry point and timing is quite asymmetrical for risk/reward.  If the market tanks, Warren's more or less aggressive repurchasing of BRK's shares could result in a positive return or limit any loss to very low single digits for the next three months. As the end of Q2 draws near, 110% of BRK's Q2 book could be estimated and the position could be reassessed.  Almost all scenarios with positive price action for the market during the next three months should lead to returns ranging from flat to very good.

 

Owning BRK at the current price is like having a free put on the stock with a strike price 3% below the market price that will reset every three months with a new strike price based on 110% of the BV at the end of that quarter. 

[tombgrt]

Exactly my thoughts twacowfca. I bought more BRK on Wednesday and Friday this week.

[netnet]

... Warren put almost all of his insurance company assets into high quality common stocks in 1973- 1974 when they were selling for about one third what a private buyer would have paid to own them outright.  :)

 

I'll have to go back and look this up but I think you may be wrong on this.  I suspect WB did not "put almost all in"; rather he put in a substantial amount.

[twacowfca]

... Warren put almost all of his insurance company assets into high quality common stocks in 1973- 1974 when they were selling for about one third what a private buyer would have paid to own them outright.  :)

 

I'll have to go back and look this up but I think you may be wrong on this.  I suspect WB did not "put almost all in"; rather he put in a substantial amount.

 

If I'm not mistaken it was the great majority of the assets that were invested in stocks, about 85% or so, leaving only a small cushion of cash to pay current claims and expenses.

[Jurgis]

I'm trying to get my head wrapped up around Kelly and its (in)applicability for investing.

 

One thing that annoys me a lot is that almost everyone uses Kelly incorrectly for non-100%-downside calculations.

 

I'll pick on twacowfca above, but they are not alone. There are lots of others (e.g. https://dqydj.com/optimal-asset-allocation-with-the-kelly-criterion/ )

 

If you are trying to evaluate a situation where loss is 1-a and gain is 1+b with probability p (and q = 1-p), correct Kelly is given in Wikipedia ( https://en.wikipedia.org/wiki/Kelly_criterion ) and Ed Thorp's paper ( http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/KellyCriterion2007.pdf - OK, the man might be a genius, but his paper writing style is pretty lousy  ;)).

 

The right formula is f* = p/a - q/b

Rewriting it, it is f* = (pb - qa) / ab

While most people make a mistake by using

fbad = (pb - qa) / b

 

I guess the mistake comes from assuming that Kelly's criterion informal specification:

fKelly = expected net winnings / winnings if you win

applies to situation where loss is not 100% (i.e. a != 1 )

Actually it doesn't.

If you think about it, fbad makes no sense. Assuming a = 0, it becomes fbad = p, which makes no sense: if you can't lose, you should not be betting just p part of your portfolio. You should be betting infinity levered portfolio which is what the correct formula gives. ;)

Correct formula also reduces to correct result if a = 1.

 

So Kelly's for the scenario above is even more extreme:

How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free.

 

What does this show?

 

- People are bad at math.  ;D (Actually, I am too, I have trouble following a lot of things in Thorp's paper or even some things in Wiki article)

- People are bad at estimating returns and probabilities.

- Kelly's as base makes almost zero sense for stocks. For example, scenario above - Kelly's answer is extreme. Yeah, I know, possibly it was a fake example. But let's take a bit modified example from https://dqydj.com/optimal-asset-allocation-with-the-kelly-criterion/ p=85%, d=5%, a=2% and I am not comparing against bonds, but just saying that stocks may return 5% and lose 2% long term. Then Kelly is... drumroll ... 39.5. Wow. 39.5 leverage. Even 1/4 Kelly is almost 10x leverage. But then this assumes that stocks only drop 2%, which might be right on average for 10 year periods, but they could drop 50% in between... Of course, if you use d=5% and a=50%, the you'd get negative Kelly and would not invest in stocks period. (I know I know this is wrong way to use Kelly, don't beat me ;)).

- It also makes no sense to apply Kelly that relies on a long series of bets to a single bet that is held 10+ years. You might be able to adjust Kelly or your process to assume you rebet every day, but that introduces other issues (e.g. what is expected win/loss for a one day bet on a stock?).

- Applying Kelly to stocks is way more complicated than people expect. Unlike a bet which is resolved in single outcome (win/lose), stock price is continuous and possibly infinite sequence. What does it mean that stock will lose 30%? Tomorrow? In a year? Forever? Do you care if it loses 30% next year if your holding expectation is forever? OTOH, you cannot say that stock has zero chance to go down just because you plan to hold it long term?

- There are Kelly's adjustments to Brownian random walk single stock situation and even more complex correlated multi stock portfolios (see Wiki/Thorp for some), but these become even more math intractable compared to original Kelly's. I'm not sure there are many people (if any) that apply these correctly on blogs and forums. There might be quant hedgies that have programmed them correctly (I am sure Thorp did ;)), but I'm pretty sure they don't publish their algos as open source programs. ;)

 

On the lark: does anyone understand and has implemented a single stock Kelly's where you are trying to decide how much of the stock to buy compared to another asset with probabilistic return? Restrictions: the stock return should not be based on past returns and volatility (cause IMO this is worthless), but could be based on Brownian random walk model with directional drift that incorporates some volatility measure. The return of alternate asset could be based on Brownian random walk or just mean/stdev.

[alpha asset strategies]

 

So Kelly's for the scenario above is even more extreme:

How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free.

 

Jurgis-

 

I think your math is incorrect with regard to the above example; isn't the correct answer around 67.50% of your bankroll (based on Kelly Criterion)?

[Jurgis]

 

So Kelly's for the scenario above is even more extreme:

How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free.

 

Jurgis-

 

I think your math is incorrect with regard to the above example; isn't the correct answer around 67.50% of your bankroll (based on Kelly Criterion)?

 

The answer 67.5% is based on the bad formula (see fbad in my post). 67.5% is actually incorrect answer.

[alpha asset strategies]

 

So Kelly's for the scenario above is even more extreme:

How much would Kelly say to bet if I thought that a stock had 100% upside, 30% downside and I thought I had 75% chance of being correct?

 

Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free.

 

Jurgis-

 

I think your math is incorrect with regard to the above example; isn't the correct answer around 67.50% of your bankroll (based on Kelly Criterion)?

 

The answer 67.5% is based on the bad formula (see fbad in my post). 67.5% is actually incorrect answer.

 

Are you sure about that?  This website agrees with my math:

 

http://www.albionresearch.com/kelly/default.php

 

I was always taught that Kelly could never be more than 100% of bankroll.

 

 

[Jurgis]

Yes, I am sure about that. This is exactly the error everyone makes when applying Kelly's in situations where the loss is not 100%. Please read my post, please read Wiki page and read Thorp's paper. Don't rely on people who just parroted wrong formula for the situation where it doesn't apply (or applies incorrectly) based on the way they thought they understood Kelly. (BTW, the leverage is assumed to be free, so you are welcome to change any levered result to unlevered 100%).

 

You get no levered results for original Kelly's because with a 100% loss risk, levered bets always lead to 100% loss eventually. Even if your loss chance is just 0.0001%.

[alpha asset strategies]

I need to investigate this further.  It appears that the formula that I was using - as well as the indicated website - assumes a binary outcome (ie. only 2 possible outcomes - either win or lose everything).  Most investments obviously have more than 2 possible outcomes.  Is that the reason for the discrepancy? 

[scorpioncapital]

"Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free."

 

What about volatility? Leverage may be free (unlikely) but at between 2-3 to 1 leverage, even if your ratios and certainties play out in the end as you predicted, you could get a temorary dip that gets you a margin call. So intuitively it does make sense to have some other variable, a cap for possible draw-down.

 

[alpha asset strategies]

I've been thinking more about this discussion.  Virtually every single investment has at least a minimal possibility of ruin - 100% loss.  For example, even Berkshire Hathaway has a tiny possibility - probably much less than 1% - of going bust.  The same could even be said of investing in T-Bills (ie. in the event of some unthinkable worldwide catastrophe in which the US Government collapses).  As such, from a practical standpoint I cannot think of any financial investment where Kelly would suggest an investment of 100% or more of bankroll.  After all, isn't Kelly structured so that it guarantees to protect against financial ruin (assuming all assumptions / probabilities are accurate)?  Please correct me if I'm wrong?

 

This may be turning into more of a theoretical vs. practical discussion.  However, from a practical standpoint, I just cannot "wrap my head" around Kelly ever suggesting a 100% (or more) of bankroll investment into a financial investment due to the possibility of a Black Swan event (which would lead to financial ruin).

 

 

[Guest Schwab711]

"Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free."

 

What about volatility? Leverage may be free (unlikely) but at between 2-3 to 1 leverage, even if your ratios and certainties play out in the end as you predicted, you could get a temorary dip that gets you a margin call. So intuitively it does make sense to have some other variable, a cap for possible draw-down.

 

The Kelly Criterion has assumptions that fail to be met when applied to investing. As a result, we should use another methodology. I would guess that methodology would be outstandingly complex and nearly impossible to use.

 

I bet Jurgis purposefully picked a very interesting example.

[writser]

Agreed. Amongst other things the Kelly formula doesn't take into account the time horizon of your investments (i.e. the opportunity cost), it assumes your bets are uncorrelated (which probably doesn't hold for your portfolio) and it makes an assumption about the utility of money that probably doesn't apply to individuals who want to retire safely. It's interesting to play around with but I don't use it in practise.

[LC]

"Answer: You'd invest 2.25 of your bankroll. I.e. you'd have to lever 2.25 on 1 if leverage is free."

 

What about volatility? Leverage may be free (unlikely) but at between 2-3 to 1 leverage, even if your ratios and certainties play out in the end as you predicted, you could get a temorary dip that gets you a margin call. So intuitively it does make sense to have some other variable, a cap for possible draw-down.

 

The Kelly Criterion has assumptions that fail to be met when applied to investing. As a result, we should use another methodology. I would guess that methodology would be outstandingly complex and nearly impossible to use.

 

I bet Jurgis purposefully picked a very interesting example.

 

Jurgis didn't pick the example, prunes did (see May 4 above)

 

Kelly criterion works best when odds can be calculated with very high accuracy, and the game continues such that there is always another round to play. Investing doesn't really meet those criteria.

[Jurgis]

I'll answer to everyone and not post-by-post. Sorry if I miss your point and don't answer it. Feel free to remind me.

 

Volatility and low probability of 100% loss - yeah, sure you are right. But that's an argument against any leverage.

Also please read up on Thorp. He used Kelly's, modified Kelly's and way-modified Kelly's for investing with great results - mostly arbitrage, which is more calculable, but still. He used large leverage (over 1.5x) in some cases. And arbitrage can blow up too, so he's a great example of someone who successfully used levered results from Kelly's even with possibility of loss. So there's at least one example that this can be used successfully. I'm not saying you should use it, but I'm trying to understand if/when I should use it and how. ;)

 

In terms of volatility, correlations, portfolio sizing, etc. that people brought up, please read Wiki and Thorp. I'm feeling this conversation is a bit just skimming the surface if we are not even acknowledging and understanding readily available material. There are formulas that account for volatility and correlations (see the "Application to the stock market" section on Wiki and corresponding parts of Thorp's paper). I mentioned this already in my first post.

 

BTW, this also addresses somewhat the questions about time and about the fact that in betting you make repeated bets (and in stock investing you might not).

 

Now, arguments can be made that these formulas are artificial and that Brownian drift plus volatility coefficients are hard to estimate, and that correlations are hard to estimate, etc. I don't necessarily disagree with these arguments. What I am looking at is not mindlessly applying the Brownian motion Kelly's, but rather if using these formulas and putting in actual numbers in them would provide some insight into stock selection or allocation that I did not have before. Also I am looking if they can be modified further for the exercises I am interested in, i.e. where I don't use historic drift and volatility, but rather use my future estimates based on some kind of fundamental insights about a business.

 

BTW, there is an actual example in Thorp's paper of allocating money into BRK (page 29, example 7.3). There's even a real case study (section 8, page 31) of allocating money into BRK, SP500, Tbills and BioTime (something that client owned and wanted to keep). Thorp's paper is a pain to slug through and I've only done cursory read so far, but for me this might be interesting to go through in detail. I'll see if I can allocate time to do it sometime in the near future. :)

 

So far for me I feel that thinking about Kelly, its variations and applications is a time well spent. For other people it might not be. :)

 

Peace, Brownian motion and partial derivatives

[scorpioncapital]

Also definition of loss may not be fully captured. In the above thread the question of loss on t-bills was interesting. If you define loss as getting paid back, chance of loss is probably 0%. If you define it as purchasing power, chance of loss is much higher, maybe 100% at current prices. But it seems the observer gets to decide how they define loss, as long as they are themselves clear what a loss really is. Opportunity cost was mentioned as another type of silent loss.