It's worth noting that for a finite number of bets (the number being known to the bettor) the Kelly Criterion is not the optimal strategy. Intuitively, the closer you are to the end the less worried you should be about going bust (as you won't be missing out on more of these great odds).
No, you're just bringing up a separate issue from the one implied by the parent.
Your own risk tolerance, the values of your life, the opportunity costs outside of this one -- those are real complexities but we ignore them to answer a question about how best to maximize your bankroll within the current situation.
That seems contrary to the criterion... Expected value dismisses risk and the fact that once someone loses their money, it is much harder to gain it back.
If you have 100 dollars..and you lose 50% of it on a fair coin bet, you now have $50.
Say you play again..You win. Youll only have $75 ($50 times 150%.)
Thats the basis of risk analysis. That losing and winning arent equal for finite bankrolls just because their probabilities are equal.
I'm unable to make heads-or-tails (har har har) of the Gwern blog post. Admittedly, my knowledge of statistics and machine-learning is lacking, but is there summary I can work backwards from?
The first grey box with text in it _is_ the summary.
The actionable takeaway for me is to try and solve this problem with some machine learning approaches, since it's simple enough that we compare to exact solutions, but also interesting enough.
The takeaway from a social point of view is that even people working in finance are not very good at playing these games individually. (The companies they work for might still be doing much better as a collaborative effort, and with more simulations and backtesting used.)
There was a great moment in financial history when only a few people had figured out how to use computers to price things, and they made a lot of money. Thorpe did very well at blackjack because the casinos didn't know that a winning strategy against them was possible.
Now everything financial has been analyzed to death and nobody can make money with technical analysis.
Technical analysis is playing heads-up against math PhDs who get paid $250k a year to try to beat you. Yeah, you're not going to be in for a good time.
This article is more about how you manage risk and learning what your personal risk tolerance is. If you take more or less risk than you "should", then all else equal, you'll wind up with less money.
Besides, these are not "just" math PhDs, but math PhDs backed by some of the largest private equity trading firms in the world, armed with multiple datacenters of 5000 blades each, running machine learning algos and playing "what if" scenarios 24 hours a day, to build rulesets and aligned structured positions so they can scrape news feeds and react in milliseconds.
Anyone who thinks they can sit at home and day-trade with these sharks without being fleeced is a fool. But then, anyone who is a fundamentals trader and is still in this outrageously irrational market is also a fool.
One of the first things I learned as a trader: "The market can remain irrational longer than you can remain solvent."
This is fascinating, and really makes you consider the volatile nature of markets from a different perspective. That is, it seems to imply that your return is less about the choice of companies to invest in and more about your general exposure, which seems consistent with the rise of index funds and diversification in general.
Or in other words, don't pick your stocks, pick your leverage. Interesting read, only thing missing was a deeper dive into the tools used to produce the graphs, or even better, a link to a Github which would reproduce them from the raw data.
Plugging in an average stock market performance (over last 20 years):
Return: 8%
Standard Deviation: 17.72%
Risk free rate: 3% (guess)
This gives f = 1.59.
This means the optimal strategy is to be putting 159% of your money in an ETF index?
This strategy is in stark contrast with the typical 70%-30% allocation of investments. How do we reconcile this? Intuitively, the function of your age needs to be included in this equation (I think somebody commented on this already).
Maximum expected value isn't the same thing as best distribution of results. The levered portfolio does better in expectation, but at the cost of being in for a really bad time in a bear market.
Plus, many investors can't stomach the losses of a 100% equities portfolio, let alone something levered up.
>How do we reconcile this? Intuitively, the function of your age needs to be included in this equation (I think somebody commented on this already).
Age is a proxy for funding amount. A lot of investment goals aren't just "maximize the amount of money I get". They often care about things like "I need to spend $X/yr in retirement". So your investment goal should also put some weight on how likely it is to sustain that, and a higher bond percentage at higher investment amounts will do that.
You can find FDIC insured bank accounts at 1.25% nowadays. There's also I-bonds at 1.9% inflation-linked, with a purchase limit of $10k/yr and a 1-year redemption lock-out.
3% is more accurate for the cost of margin loans or the opportunity cost of pre-paying a mortgage.
yah, 3% is a high assumption for the risk-free rate. but if it were 0%, you'd be losing money to inflation over time (treasuries are only proxies for the theoretical construct that is the risk-free rate).
no single security provides the risk-free rate but perhaps a (small) portfolio of securities could get you the risk-free return.
I'm not sure how applicable to criterion is to investing. The main differences are you typically don't lose your entire 'bet', and bets aren't decided in a single moment but the outcome is slowly revealed over time.
It prompts very different behavior, to extend the coin flip metaphor: If you could slow down time and see the flip turning bad, you could quickly withdraw your bet. Then you'd be much more likely to bet a higher fraction.
I was thinking about this while reading. The Dot Com and Sub Prime mortgage crashes took a lot of people buy surprise, but not everyone. Moving into T-bills, cash, or precious metals at the beginning of those downturns would be enough to make the 2x Kelly and 3x Kelly perform better.
Criterion can tell us what was behind the biggest blowups in finance, why levered ETFs are generally a bad idea, and how aggressive investors can maximize their wealth without risking ruin.
That is not necessarily true regarding leveraged etfs.
Leverged ETFs when used as a substitute for being fully invested can be more safe than being fully invested in the non-leveraged version. The idea being one puts 1/3 of their capital in the 3x S&P 500 ETF and the 2/3 in T-bills. The maximum loss, no matter what, is still only 33%.
Ruin (if it is defined to mean 100% loss of capital) is impossible with a leveraged etf [it just means your wealth asymptotically approaches zero], but it is possible with margin debt.
That only helps you if your rebalancing period is less frequent than it should be, letting your leverage drift up on good days and down on bad days. If the market goes down by 16.6%%, your levered ETF goes down by 50%, and your portfolio is now 80% treasuries and 20% 3x levered ETF.
If you rebalance daily in order to maintain your desired stock exposure, you end up owning the S&P 500 as your buy and sell orders of the levered ETF counteract the ETF's buy and sell orders on the underlying index to maintain its leverage. All you're doing is paying higher fees, trading costs, the spread on T-bills vs institutional margin costs, and realizing capital gains and losses pointlessly.
What you want is to buy call options on the S&P 500 index. The market is far more efficient here and way better for you.
Gwern details this extensively here: https://www.gwern.net/Coin-flip well worth checking out.