Can you Apply Kelly's Formula to Investing ?

Kelly's formula (sometimes called "fortune's formula") is widely touted as applicable to Investing, and in particular to investing via Equity Options.

At the risk many readers will see this as wrong-headed, I will say: While I have no doubt Kelly's formula is CORRECT, I think applying Kelly's formula to Investing is not useful - and not really needed.

Here's why.

Kelly's formula, as defined, gives the optimal % of your capital to place in a bet. "Optimal" in the sense that it will lead, in the long term, to a higher end balance - i.e. to maximum cumulative gain.

It seems reasonable to use it to decide on the "optimal" % of capital to use in a given trade, or a given investment class etc. after all, each trade is a "bet" of sorts.

However, it seems to me for retail traders this is problematic, and maybe completely bogus.

Note - I am NOT claiming Kelly's formula is wrong/not useful. I am just making the point that applying it to investing may be an awkward fit, and we should expect results to not be as good as "promised" by the formula.

Kelly's formula in Stock Investing

Kelly's formula deals with a (long enough) series of IDENTICAL bets, where :

• We Know The chances of a win/loss and they have to be the same in each round (it assumes wins are re-invested, so you can't do many rounds in parallel)
• We Know The Gain and Loss of the bet - assumed to be identical each round
• The rounds are independent - winning/losing in round X does not affect at all the chances of winning/losing in round X+1 or X+N

   In investing, NONE of these assumptions really hold. 

1. The chances of winning or losing trades are not really known
2. Gain/Loss is not known in advance, and is not the same for each round
3. Trade rounds chances of win/loss are not Independent.

• Win/Lose chances: People might convince themselves that they can estimate the chance of a win/loss of a trade, but if we could do THAT, we'd be sitting on a yacht in the Caribbean long ago, having made mostly successful trades. 
• The chances of winning or losing a trade are not fixed and change each trade, even for a trade done soon after the previous one, and on the same equity. Each trade has its own attributes, its own max-win and max-loss, and its own context in terms of market situation (bull market? earning season? FED interest change? etc. etc. ) even if we use the "average historical win/loss ration" as an estimate - THAT also changes as we do more trades...
• Gain/Loss Amount is not really known: Gain/Loss for many/most trades is NOT really known for a trade. Sometimes we have an estimate of the maximum gain or maximum loss, but in many cases not even that. If you buy a stock and it went up, if you sell today, you know your gain. But if you keep it for tomorrow, you almost surely will have a different gain. Same if you hold a losing Stock - the Loss is not really known. How MUCH will stock go up/down until we exit? Shall we exit early on a winning trade, or hold on longer to a losing one in order to stick to the fixed gain/loss we assumed in the formula up front?
• Loss is not usually 100% : Kelly's Formula assumes that when you lose, you lose 100% of your bet. This is rarely the case in Equity trading (tough it might be in options trading - see below). For situations with PARTIAL loss, we need to use a modified Kelly's formula See Alon Bochman's Excellent explanation.
• Gain/Loss amount is not fixed: It is rare to get the exact same level of gain/loss in several separate trades. We might use Stop-loss and limit orders to "set" max gain/max loss numbers, but as is well known, this is not a guarantee (think a stock that gaps up/down overnight), and are we willing to commit to hold a stock until it hits one of these triggers, as long as it will take, just to have a known Value of Gain/Loss? 

Given the above, people have come up with various attempts to fit Kelly's formula square peg into Trading's round hole: 

Can we use historical results as the Win/Lose chances and Gain/Loss?

A common suggestion I see is to use our historical averages to set the values of Win/Loss probability and Gain/Loss ratio for Kelly's formula.

• We look at OUR history as traders (assuming we have a long enough history to use as a sample) we calculate our win/lose ration (# of trades won/lost) and our average gain/loss; and we ASSUME that the NEXT trade we are considering has these attributes: That our chance of winning it or losing and how much we'll win lose are as per our average history. Moreover, we are really assuming that as we will make the next 10, 100 or 1000 trades, they will all keep having these attributes. To me, that seems a bad set of assumptions. For example, if you calculate your averages today, and then re-calculate them in (Say) 3 months, you are likely to get different numbers - violating the Kelly assumption of a sequence of identical "bets". If your Historical record was mostly during a bull market, and the market enters a correction,  your averages will be wrong - and it will take many trades to bring the averages close to current day reality.
• A second problem with this is that while you get an estimate for chances of winning and average Gains/Losses, this does not mitigate the other factors discussed above - trades not being independent, amounts not fixed, and the effect of our decision as to when to start and how long to hold each trade open. 

in effect, we are asking: "if this trade was one of a long series of identical ones, what would Kelly's % be ? " knowing full well this is NOT the case.

It gets worse.

• Time is not accounted for in Kelly's formula. "Bets" take no time (and all bets take the same time to execute). But Investing rounds take time, and the time they take is very consequential (and, again, it is not fixed)
• Even supposing Kelly's formula was directly applicable to investing, it needs a long sequence of trades, with gains re-invested, to show its benefits. But in most cases, to win on a trade we need to stock to move - and that can take sometimes a day, sometimes weeks. Assuming 1 week per trade, we'd need several years before we saw the benefit. 
• The time we hold a trade open affects all other variables - chance of win/loss and the amount of gain/loss we can have in that "round". Compare day-traders with a trader who buys stocks and waits for them to grow, or even waits for them to become long-term investments - which lowers the tax, and therefore changes the win/loss values. 

So, it seems equity trading does not meet the conditions to have Kelly's Formula work its magic.

Options are different (or are they?) 

At first glance it seems Option trading is a better fit. In option trading we can use a series of structurally similar trades (e.g. a series of vertical spreads), with known durations, known (Maximum) Gain/Loss amounts (and loss frequently being 100% of the amount traded), and even with an estimate of the Win/lose probability. Hurray?

Not so fast.

• Can we do a series of identical trades (Bets)? Even if all the above information was really known (and it isn't - see below), we still do not have a sequence of Identical trades. Even doing weekly vertical spreads on the same underlying equity does not really make for identical trades - The prices move, volatility changes, etc. etc. The old saying goes that you can't enter the same river twice - well, you can't trade the same trade twice, either.
• Do we really know the duration? While we have firm expiration dates when a trade is started, in many cases we may want to close out earlier. If we are holding a short option position, we may be forced out earlier due to early assignments, or a margin call. So, in fact, the expiration dates are the MAXMIUM duration and actual trade duration is not really fully known. and what do we do with Calendar (diagonal) trades? do we use the earlier or later expiration?
• Do we really know Gain/Loss ? While we know the MAXIMUM gain/loss, in many trades these are not the actual results. e.g. a Long Call has (in theory) an infinite gain. Plugging infinity into Kelly's formula is a surefire way to screw it up, so we will have to invent the value of Gain to use. In Many trades (e.g. Vertical spreads) the trade can end up with a gain or loss different than the maximum values if the underlying price does not move a lot.
• Can we Reinvest our gains/Losses? if we "roll" an option for a Debit/credit - is that a gain/loss? is that a new "round"?
• DO we know the chances of winning/losing? option trading calculators frequency show a "Probability of win" value. Is that good enough? what they show is the probability of profit under multiple assumptions, that mostly are almost guaranteed to change during the life of the trade. The main assumption is that the price of the underlying will move in accordance to the statistical analysis of how it HAS moved in the past for some period the calculator uses. Even then, market sentiment and external events (e.g. War in Ukraine, or a Federal Reserve meeting) will mean the value is, at best, a guesstimate. As for Equities, there are proposals to use our historical Win/Loss ratio (and average Gain/Loss) as the Values to plug into the Kelly's formula. is that better? if we are considering a risky trade (assuming a large potential gain), is it better to plug in the more conservative historical averages?
• Implied Volatility While Theory of option calculations tell us what option prices SHOULD BE given known price of underlying, price movements attributes, time to expiration etc. etc, the actual price of options is almost never the same as it should be in theory. Implied Volatility is the fudge-factor value options calculators generate that would make the the result generated by the calculator match reality, and represents the effect of trader psychology and sentiments. By definition, IV is only known retroactively, and changes from trade to trade.

So? is it all useless? 

• Kelly's formula is reportedly being used by big institutions. They have on staff lots of math/finance PhD's  (which I'm not) who may read this and laugh at me, but I'll also note that these institutions do millions of trades, and in many cases do High-frequency trading (where each 'round" takes only milliseconds) and so may have sequences of trades long enough, and maybe similar enough to each other for Kelly's formula to maybe be useful. 
• It is clear that the basic idea of using no more than X% of the capital in each trade is a good idea. Every trader knows this as "money management". My point is just that deciding this X% by Kelly's formula where we estimate (cough, cough.... guess) the chances of winning etc. and assuming we'd get optimal  results is optimistic and misguided - but see final paragraph for a reasonable alternative.
• Kelly's Formula, when applied correctly, will lead to optimal results, but will have a lot of Up/down swings that are hard to take psychologically for many traders. Therefore, is is common to see recommendations to use "Half Kelly" strategy - where you actually trade 50% of the amount/% recommended by Kelly's formula. This generates smaller returns, but at a much reduced volatility.

Given the above, I doubt anyone can really find the optimal X% for trading using Kelly's formula in trading.   

The good news is I think we can do just fine without using Kelly's formula.

if you play around with a Kelly's formula calculator - and be sure to use one that accounts for the fact that in most cases a loss is a PARTIAL loss - You will quickly see that typical results are around 5%-20% (for say 51%-52% Win probability, and 30%/30% possible gain/Loss).

so, we can simply take that as a "poor man's Kelly %" estimate, with no calculation using guessed-but-assumed correct values.

Applying the "Half Kelly" advice to reduce Up/down swings, we get the result that a good % of capital to invest is 5%-10% in each "round".

We have to make peace that while we are not doing the "optimal" - which we concluded is not really knowable - we ARE doing better than blindly deciding how much to "bet" each time. Moreover, just sticking to the principle of only having a (smallish) fraction of our total capital at risk each round guarantees we will never go to 0.

Important Note: Common wisdom says "do not invest more than X% in one Trade." 

 I believe this is flawed advice. 

 It is certainly better than investing a lot of your capital in one trade , but not as  good as it is presented. The hidden assumption behind this advice is that the trades are independent. if I invest (say) 5% in each Trade and do 10 trades, than I will lose  some, and win some, and if I manage to win in more trades than I lose (or manage to win in fewer trades but with a bigger cumulative gain ) than overall I win.  But as we know trades placed around the same time are NOT independent. If we trade at the start-of-correction week (as in March 2020) we might well lose all 10 of our  trades - and hence 50% of our capital - despite each trade only risking 5%.    

A better/safer policy is to only invest a maximum of X% of our capital IN EACH ROUND.