How to write a tweet that gets over 300k views; and why diversification is probably good

 Well that blew up:

https://x.com/investingidiocy/status/2032438612961165409

At eight words this is almost certainly* my most viewed and liked tweet ever (although I have nearly 25k followers, so thats 18,000 or so that didn't like it) . Short, pithy, funny; I should retire from my Xmaxxing game right now (just kidding; there are still plenty of gamblers, crypto nuts and MAGA idiots waiting patiently to be educated; and I have a million more dad jokes to inflict on the unsuspecting public).

* X doesn't provide statistics to non premium users, but I am pretty sure it's up there

Obviously the point I was making was about diversification. Buying 50 stocks is better than buying 2 or 3 because of said diversification, unless certain - quite challenging - conditions are met (I'll be spending much of this post quantifying exactly what those conditions are). 

Now it was very interesting to read the negative replies (the positive ones were just lovely, thanks). They generally fell into variations of the following:

• "Something something asymmetry" (which a few kind people explained to me; obviously as a trend follower, ex-professional options trader and occasionally lottery ticket buyer I didn't know there could be such a thing as an asymmetric bet)
• "You just have to choose the right three stocks". A couple of people were even kind enough to share the list of stocks that will definitely go up in the future.
• "If you are so smart, why aren't you as rich as Druck?"
• What the tweet means, but didn't actually say, is: "Yes, you should buy 50 stocks and ensure 2 or 3 of them are home runs"
• Again what the tweet meant, but didn't say, is: "you should use a Taleb barbell type strategy and build a portfolio mostly consisting of boring stuff; plus a few carefully selected lottery tickets"
• Also what Druck apparently does, but again wasn't in the tweet, is use stops which means it's fine  and dandy to have concentrated positions.
• "Risk and reward you idiot". Thank you for kindly explaining the basic premise of modern portfolio theory (MPT) in such a succint way, I weirdly hadn't come across this idea before in over a quarter of a century of studying and researching portfolio optimisation.
• "Diversification might make sense once you are rich, but not if you are trying to build wealth."

Whilst some of the above points might seem facile to the cognoscenti who are reading this post, several of them actually raise some very interesting points about a core argument in finance: the trade off between skill (if you have enough, then diversification is bad), and luck (if you don't have enough of this, then diversification is always better); as well as the role of risk and access or not to leverage.

Hence this post - the ultimate diversification-good-or-bad post! I'll whizz through quite quickly from 1st principals some stuff that will be familiar to most of you (though definitely new to many of the people who replied above if they are reading this) ; and end up in more novel territory I have been considering a lot lately - the best way of quantifying and comparing uncertain outcomes; which feeds into optimal back testing practice.

Some irrelevant stuff

Let's first dismiss some of the problems above as mostly irrelevant:

• "Something something asymmetry": All bets in finance are to an extent asymmetric, eithier good (positive skew) or bad (negattive). Arguably long only bets in stocks which have limited downside but unlimited upside fall into this category (though a distribution of stock returns will have fat negative tails for return periods of less than a year; beyond that you get saved by autocorrelation). But an asymmetric diversified portfolio of bets will outperform a non diversified asymmetric portfolio unless certain conditions are bet: asymmetry doesn't affect the basic argument here. Apart from lottery tickets and maybe options (which I don't think Druck uses, and the persistent buying of which will lose you money in the form of paying the variance premium) the degree of asymettry isn't known in advance, so we are left with the problem that unless we can forecast the future to some degree of accuracy diversification is always better.
• Also what Druck apparently does, but wasn't in the tweet, is use stops which means it's fine to have concentrated positions: Using stops will just add positive skew (asymettry) to strategy returns; plus if assets have trends at the appropriate horizon this will be at no cost or even improve the average return. A diversified portfolio of bets with stops will outperform a non diversified portfolio of bets with stops; so again, this doesn't change the basic question we're trying to answer. As someone whose diversified trend following system with implicit stops trades over 200 assets, I am clearly putting my money where my big mouth is on this one.
• "If you are so smart, why aren't you as rich as Druck?": Putting aside that Druck isn't actually putting his entire fund into three stocks, as discussed in the many of the replies to the post, my immediate response to this is if Druck is so smart, why isn't he as rich as numerous other people I could mention who do use massive amounts of diversification? One data point doesn't really prove anything. It could just be that he is highly skilled, or very lucky; two explanations I will explore at length in the post proper. 

With that in mind we are left with three interesting arguments to debate:

• "All" you need to do is pick better stocks; eithier as part of a small concentrated portfolio, a larger one or a barbell setup.
• More risk means more return
• Diversification makes more sense once you are rich

Risk 'n' Leverage 

One key point we need to address is whether we are working in an environment with leverage or without it; and if our assets volatility is sufficiently high that will not matter. With leverage diversification will improve returns by more than without.

To explain you first need to know that my favourite formula in finance is that Kelly optimal risk is equal to your Sharpe Ratio; if your SR is 0.5 then your optimal risk is 50%. My next favourite formula, which you can use to derive my first favourite formula, is the Gaussian approximation* for geometric return AKA CAGR: 

m - 0.5s^2 

where m is the arithmetic mean and s is the standard deviation. My third favourite formula is that risk will be reduced by sqrt(N) if you increase the number of assets in your portfolio by N, and all your assets are independent; higher correlations will lead to smaller reductions. As a rule there are diminishing benefits to diversification, as it gets harder to find assets with low correlations, especially if your universe is just US stocks. But there will always be benefits from diversifying as long as correlations never get to 1.

* By the way with positive skew the CAGR will be higher than it is for Gaussian; with the reverse true for negative skew, but to reiterate doesn't change the overall findings we will get.

Let's make some assumptions for now just to illustrate the effect of having more leverage. We have a choice between N=3 assets, and N=3*(4^2) = 48 assets (close enough to fifty). The correlation of those assets is 0.5 (seems reasonable enough for US stocks drawn from multiple sectors, but we'll explore other values later). We also assume we give those assets equal weights. 

From favourite formula number three the maximum reduction in standard deviation will be 4 if they are independent, but they are not so it's more like 1.14 (the ratio of 1.40 and 1.22 which are the improvements from N=1 going to N=48 and N=3 respectively). For example if the standard deviation of each stock is 30%, then it will be 30%/1.22 = 24.5% for a portfolio of 3 stocks and 30%/1.4 = 21.4% for 48 stocks.

Let's also suppose for now we can't predict future Sharpe Ratios, and the expected SR is 0.25 on each asset, and the risk free rate is the current 3.75%. Then each stock will have an excess return of 0.25*30% =  7.5% which equates to a total return of 11.25%. That seems optimistic, but we're just playing with numbers for now.

An unleveraged portfolio of 3 stocks will have an average return of 11.25% and a standard deviation of 24.5%; 48 stocks will come in at 11.25% and 21.4% respectively. The Sharpe Ratios on those should be 1.22*0.25 = 0.306  and 1.4*0.25 = 0.35. Check you get those numbers if you calculate the SR manually.

There is a messy spreadsheet you can play with here: https://docs.google.com/spreadsheets/d/12Y06FAxSW6Ii-dDg4TM1H9g-ZBHTUt6mi1a8Htskeuo/edit?usp=sharing Make a copy, do not ask for edit access. This won't allow you to calculate the reduction in risk from diversification, I used python for that. Here is the messy python code: https://gist.github.com/robcarver17/85238b6d538d6e3acedccfc1cc5a76c8

What about a leveraged portfolio? The optimal leverage to run these at will be risk targets of 30.6% and 35.0% for 3 and 48 stocks. Assuming for the moment we are mad enough to do that, we will end up with leverage ratios of 30.6/24.5 = 1.25 and 35/21.4 = 1.64; and after subtracting borrowing costs we get returns of 13.1% and 16.0%, excess returns of 9.37% and 12.2%; and Sharpe Ratios of 0.306 and 0.35 since SR is unaffected by leverage.

Now using my 2nd favourite formula, we can work out the expected geometric returns. They come in at:

- Unleveraged 3 stocks: 8.25%

- Unleveraged 48 stocks: 8.95%

- Leveraged 3 stocks: 8.44%

- Leveraged 48 stocks: 9.87%

 

In simple terms, we can't 'eat' risk reductions in the form of improved annual returns without leverage. But lower risk does improve CAGR by 70bp. With leverage we can convert all of our lower risk into higher returns, and thus boost CAGR even more: by over 140bp.

Note the maths above for unleveraged traders would be better if:

• The individual assets were riskier; at over 50% the optimal leverage would be below 1 even for 48 stocks and we would improve CAGR 'as if' we had leverage.
• The per asset Sharpe Ratios were lower; if below 0.15 then the risk target even on the larger portfolio would be sufficiently low that again the optimal leverage would be below 1 even for 48 stocks

The fact that diversification improves CAGR even for unleveraged traders is probably one of the most important yet not widely known results in finance.

Of course the key number in all of the above is the average correlation between stocks; if it's 0.7 then the CAGR improvement falls from 70bp to 42 bp unleveraged; and from 143bp to just 52bp leveraged. A more realistic scenario is if you have a choice between picking three relatively undiversified stocks* and 48, then the larger portfolio will inevitably include several assets that have a higher correlation, bringing your average correlation up, and the diversification benefits down. However it's impossible for the benefit of diversification to go to zero if we assume identical standard deviations and Sharpe Ratios.

Risk 'n' Reward you idiot

You can hopefully already see that the 'you take higher risk therefore you get more reward' without diversification just isn't relevant here. It is true - and a trusim of the efficient frontier in CAPM which assumes constant SR - that you can get an improvement to CAGR by adding riskier assets if you are an unleveraged investor and the SR is sufficiently high that you can put more risk on without going past the optimal Kelly point (if you are leveraged you don't care since you can always achieve your required risk target even if assets have a very low standard deviation). 

But a larger portfolio of riskier assets will still have a higher geometric return than a smaller portfolio of assets with the same risk! 

Basically these people are mixing up the two key kinds of risk in MPT: specific and systematic risk. The choice of the universe of possible assets sets a given risk 'background' systematic level. In certain circumstances you can sometimes but not always improve CAGR by choosing a universe of assets with higher risk. But you can always improve it further by subsequently diversifying your portfolio so you hold more of the universe, and thus reducing your specific risk. 

Taking a higher risk through not diversifying means you are not being rewarded for taking more risk; you only get that reward if you trade riskier assets.

Diversification is only for rich people, take one

You can hopefully also see that the 'diversification is only for rich people' argument also doesn't apply. One of the trusims of financial economics is that rich people should have safer portfolios. In practice that might mean running their portfolio risk at half Kelly rather than full Kelly (where risk target = SR). But changing your risk target, whilst it will for example reduce the benefit from using leverage, doesn't affect the improvements you would get from diversifying!

Eg If I rerun the numbers above with a 50% instrument risk (so leverage doesn't help, simplifying the argument) I get these numbers:

- Unleveraged 3 stocks: 8.44% CAGR with 30.6% risk

- Unleveraged 48 stocks: 9.87% CAGR with 35% risk

It still makes more sense to diversify, even with a higher risk target. 

If I now set the maximum risk target as a rich person trying to protec their wealth at 20%, then the advantage of diversification is diminished but it is still there:

- Unleveraged 3 stocks: 7.87% CAGR

- Unleveraged 48 stocks: 8.75% CAGR

Again the logic is clear: just because a poorer person can take higher risk doesn't mean they shouldn't diversify. Not diversifying means you won't benefit from the CAGR boost that comes from the lower risk that brings. A poor person who can't use leverage should probably invest in a universe of riskier stocks (assuming they don't end up above kelly optimal risk in doing so), but a diversified basket of such stocks will be better than just three.

There are more subtleties with this poor vs rich that we will come on to later in the post.

A first stab at calculating the skill premium required

With the above back of the envelope figures we can work out the skill we would need as stock pickers to overcome. To close the 70bp CAGR gap for an unleveraged investor between N=3 and N=48, with the initial assumption of a correlation of 0.5 and SR=0.25; would require to to lift your SR from 0.25 with 48 assets to 0.2735 with only three assets. That is a SR difference of 2.35bp.

Actually the premium is slightly larger; if you had 45 assets with a SR of 0.2485; and three assets with a SR of 0.2735, then your SR on your three asset portfolio would be 0.2735 and on your 48 asset portfolio it would come out at 0.24. You need to be able to find three assets with a SR that is about 2.5bp higher than the average of your universe. Or to put it in terms more people would understand, you need to find three assets whose expected average annual return is 0.75% higher than the rest of your universe. That doesn't sound that hard. If correlations were 0.7 rather than 0.5; then you just need an advantage of 0.45%.

The role of modelled uncertainty

45bp to 75bp of extra annual returns doesn't sound that hard. But in this exercise so far we've completely ignoring the role of uncertainty. We don't actually know any of the numbers I have thrown about in these formula, except for N of course (assuming we can count to three, and keep counting to 48). In practice standard deviations and correlation estimates are broadly predictable enough we can treat them as forecastable (and it's only really if we are using leverage all the way up to the Kelly optimal that we should care that much about the exact figures). But forecasting Sharpe Ratios is much harder. And that is to ignore the fact that financial returns aren't really drawn from Gaussian distributions with linear correlations. We need to think about the effect all this will have on our results, and we will do so for the rest of the post.

For now though, let us indeed assume that financial returns are drawn from linearly related Gaussian distributions with stable distributions, whose parameters we know. However we can't assume that this will gurantee outperformance from a given portfolio with superior theoretical risk adjusted returns. This is because luck will still play a part in the form of modelled variance. Expected performance will only be achieved in the very long run AND/OR across the average of many possible future outcomes.

This means that for relatively short periods of time (not the long run, as in the long run we're all dead), and/or for conservative points of the expectation of the future, we'd have more of a preference for portfolios that are safer and have less 'sparse' portfolio weights. An example of a sparse portfolio weight is if you have say 48 assets available but decide to only allocate to say three of them because they have a higher expected SR/return. So sparse is just another word for concentrated.

Note: This obviously ties into proper robust portfolio optimisation for backtesting purposes

Sparseness is penalised in the short run because there is more chance of eithier very good or very bad outcomes than with non sparse weights, and thus larger variance between outcomes. For example, suppose you pick only three stocks. There is a risk that you happen to pick three crap stocks, and also a possibility that you pick three amazing ones. That is a much larger dispersion of outcomes than if you have a 48 stock portfolio.

Let's look at a picture:

Each of the grey lines shows the cumulative mean returns over a year for 1000 different bootstraps. The setup here is an average correlation of 0.5, 3 assets, risk free rate 3.75%, per asset SR 0.2735 and per asset standard deviation 30% with no leverage. This would equate to an average arithmetic return of 12.0% (the expected CAGR remember comes out to 8.95%). The black line shows the average across bootstraps, and that indeed comes out roughly where we would expect it (equivalent to a CAGR of 8.6%). But there is huge dispersion. If we convert each of the bootstrap outcomes into a CAGR figure we can get a histogram:

The MEAN of this distribution is 8.65%, which is the same as if we had took the daily mean return across all the distributions and worked out it's CAGR. The median however is much lower, and comes in around 5.35%. The 25% quantile is a loss: -10.8%. 

What if we did the same thing but with 48 assets, whose per instrument SR is just 0.25. We already know that the lower per asset SR exactly balances out the extra diversification, with an expected CAGR of 8.95% the same as before. Here is the histogram:

 The upside is capped, but the downside is also lower. As a result the 25% quantile is better: -9%. 

If I produce much longer time series, then things don't deviate as much. For 20 years of data rather than one, the 25% quantile with three assets is a CAGR of 1.56%. For 48 assets, with the lower per asset SR, it's still higher: 3%.  

Note: you have probably seen this sort of exercise before in bonds vs equities comparisons. If you care about say the 25% quantile then you'd probably prefer a portfolio with more bonds or cash for shorter time periods to avoid the potential bear market in stocks that would happen in any given year; whereas for longer time periods we expect the returns of stocks to be sufficiently high that even at the 25% quantile you'd prefer more stocks.

The TLDR on this section is that a more concentrated portfolio will have more dispersion of results, especially over short periods. So even if your stockpicking talent is enough to overcome the loss of diversification on average; you will need an even bigger advantage to be confident that you'll beat a more diversified portfolio say 3/4 of the time.

 

Diversification is for the rich, take two

Notice that in the histograms above we are capping the upside more with a diversified portfolio. If the two different sized portfolios above had equal per asset SR then the 99% percentile of the two over one year comes in at CAGR of 86% for three assets, and 73.9% for 48. That's versus medians of 5.0% and 5.8% respectively (with equal per asset SR diversification gets an edge). For me that tiny chance of a higher upside isn't worth the loss I get on average.

Of course over longer time periods, and considering even tinier top percentiles, you would get starker differences. But if your utility function is "I need to make more than X in time T, and if I don't make that number I place zero value on lower wealth" then sure, you would ultimately want to take buy riskier assets, use more leverage if possible, and have a more concentrated portfolio. That is one weird utility function though! It's perhaps a very gen Z influencer view of the world; you need to be rich or risk losing almost all of your money trying, with no other outcome making any sense. It doesn't make sense to this Gen X'er for sure. If your utility function is "I want as much money as possible in expectation (the median), or in the worst 1%, 10%, 25%... percentile outcome" then you should absolutely be diversifying.

The role of forecast error

Now let's assume we don't know the SR. This is more realistic! The way I have dealt with this before is in the context of a backtest where we need to discover the hidden parameters of a distribution that we assume is stable. That doesn't make as much sense here. Instead let's suppose we have a universe of 48 assets, three of which have a higher SR than the others. We don't know for sure which three are special! How special is special? Let's say a SR of 0.5 rather than the ~0.25 we have been assuming. That might not sound much of an improvement but over 20 years that means compounding to a 15x gain rather than a 3.3x gain. 

We begin with the assumption of zero skill. That means we unlikely to pick all 3 good stocks with a portfolio of three, or even just one. Of course with a portfolio of 48 we will always get the 3 good stocks, but their greatness will be diluted by them keeping company with 45 other shitty stocks. Just how shitty? If you set the 45 stocks to have a SR of 0.23, and the three good stocks at 0.5, then the 48 stock portfolio will have an average per stock SR of 0.247. 

We can now generate bazillions of backtests just like before. In each one we pick 3 or 48 stocks from a pool of 48. The results for 48 stocks will be very similar to those from before, since we always end up with the same per stock average or around 0.247. The results for 3 stocks will vary though. The average per asset SR across multiple bootstraps will be 0.247, but it will vary a lot on an individual basis. We have a (3/48)*(2/47)*(1/46) = 0.006% chance of picking three amazing stocks and having a per asset SR of 0.5. There is a (45/48)*(44/47)*(43/46)=82% chance of picking no good stocks and having a per asset SR of 0.23. That isn't the worst of it of course, since we will also have less diversification which will also punish our expected CAGR.

Here are some statistics of the CAGR distribution, running a one year backtest:

                      3 assets                 48 assets   

Median                  4.5%                      5.2%

25% percentile        -11.6%                     -8.6%

99% percentile         85.6%                     72.1%

As you would expect the 48 asset portfolio is superior, except at the very extreme right tail.

Now what happens if we have some skill? We calculate the numbers again using a skill ratio: R. A skill ratio of one means no skill, and just the normal chance of getting a high SR asset on any draw. A skill ratio of two means on our first draw the chance of getting a high SR asset is 2*(45/48) not (45/48). Notice the skill ratio doesn't affect the 48 asset portfolio, since that will always include 45 duffers and 3 special ones.

What skill ratio R is required for the median of the 3 assets to be the same as that for 48 assets? There is no closed form for this, we just have to play around until we get the right number. A skill ratio of 3 works pretty well (these are bootstrapped numbers so the outcome will vary slightly):

                 3 assets with skill R=3        48 assets   

Median                  5.2%                      5.2%

25% percentile        -10.0%                     -8.6%

99% percentile         86.8%                     72.1%

With a skill ratio of 9 we can match the 25% percentile:

                 3 assets with skill R=9        48 assets   

Median                  7.6%                      5.2%

25% percentile         -8.6%                     -8.6%

99% percentile         90.3%                     72.1%

How likely is a skill ratio of 9 in practice? That means our probability of picking one good stock is 56%, when the odds of doing it just by luck are just over 6%. I don't know, that sounds like an astonishing amount of skill to me. Yes some people will pick all three good stocks just by luck -  0.006% of the time, or one in 16,667 people. One of them could even be called Bill. 

The above analysis is with a 0.50 correlation; what would it look like with 0.7?

                      3 assets                 48 assets   

Median                  3.9%                      4.3%

25% percentile        -13.5%                    -12.0%

99% percentile         94.9%                     85.4%

Then with skill, R>1:

                      3 assets R=1.7              3 assets R=4  

Median                   4.3%                        5.5%

25% percentile          -13.3%                     -12.0%

99% percentile           95.0%                      95.1%  

TLDR: For a three asset portfolio to be better 3/4 of the time than a more diversified portfolio, when we don't know for sure which assets have the best SR, we need to be very skilled in picking stocks: between four and nine times better than pure luck. If we are concerned only with the average median outcome, then we 'only' need a skill level around two to three times better than pure luck. 

The role of other uncertainty

At this point I have to stop, since there is no easy way to model the other sources of uncertainty: parameter changes and distributional uncertainty. Suffice to say that in the real world it will be even harder than it is for the toy models above to have sufficient stockpicking skill to overcome the benefits of diversification.

Summary

My conclusions are unchanged - diversification always pays in expectation and at the lower tail of distributions unless you are very, very skilled. But hopefully reading this post has made the reasons why clearer.

Wordle (TM) and the one simple hack you need to pass funded trader challenges

An unusual (but quick) mid month post, as this is a live issue I thought I'd publish this whilst it's relevant.

There has been some controversy on X/Twitter about 'pay to play' prop shops (see this thread and this one) and in particular Raen Trading. It's fair to say the industry has a bad name, and perhaps this is unfairly tarnishing what may pass for good actors in this space. It's also perhaps fair to say that many of those criticising these firms, including myself, aren't as familiar with that part of the trading industry and our ignorance could be problematic. 

But putting all that aside, a question I thought I would try and answer is this - How hard is it to actually pass one of these challenges? As a side effect, it will also tell us what the optimal vol target is to use if we're taking part in one of these challenges. Hence the clickbait article heading. I know from experience this will open me up to having to filter out 500 spam comments a day, but f*** it. 

As well as modelling Raen, I also model a much dodgier challenge later in the post, from another company which I will name only as prop firm #2. Finally I close with some generic and unquantified thoughts on the subject. 

Standalone Python code here. You can play with this to model another firms challenges.

TLDR: 

• Raen you have reasonable chance of passing their first round challenge and you should use a vol target of [scroll down to find out!] to maximise your chances.
• Prop firm #2 and most of the 'industry' use a very long bargepole, I can lend you mine
• I remain skeptical of pay to play

As to what any of this has to do with the word game Wordle (TM), read on to find out.

IMPORTANT: This is not an endorsement of Raen. I have no association with them and I remain skeptical of this entire industry. Their CEO reached out to me after this blogpost was initially published, confirmed my understanding of the challenge parameters was correct, and gave me permission to use the firms name. I made one small correction to the post as a result of that contact.

The (relatively) good guys 

The rules of the Raen challenge are this:

• You must make 20%
• You can't lose more than 2% in a single day. There is no maximum trailing drawdown. So if you lose 1.99% every day forever, you're still in the game.
• You must trade for at least 30 trading days before passing the challenge
• It costs $300 a month to do the challenge. This isn't exactly the same Raen which charges a little more, but as a rounder number it makes it easier to directly see how many months we expect to take by backing out from the cost per month. I assume this is paid at the start of the month.

Note: this is just the 1st stage of the challenge. The rules for the 2nd stage are much more nebolous, but to be fair there are no charges for those. Like I said, this prop firm appears to be amongst the relatively good guys. 

 

I've also got these parameters:

• 256 business days a year, 22 business days a month (it's actually more like 21, but again this higher figure will make the prop firm look good)
• Random gaussian returns generated with no autocorrelation. This is extremely kind as it ignores the chance of fat tails that are somewhat common in finance.
• If we get stopped out we try again, which means restarting the challenge from scratch. There are no reset fees. I assume that this reset doesn't affect the timing of monthly fees (I can't find the answer to this question on the website, but this must be the case as otherwise the cost of resetting would be free and your best strategy would be to keep making massive bets every day and you would pass eventually and only ever have to pay the first month).
• We give up if we can't pass after trying for a year (there are no time limits in the challenge, but this speeds up the computation and seems like reasonable behaviour).
• I assume there are no other limits which make it hard to hit a given risk target. This is unlikely to be a constraint except for suboptimally high vol targets.

There are two clear variables we are missing: the expected Sharpe Ratio, and the vol, both required to generate the gaussian returns. The former is assumed to be exogenous (a question to answer is how hard are these challenges to pass - if you need a SR of 4 to pass them that suggests they are probably too hard), whilst the latter we can optimise for. Note that due to the drawdown and self imposed time limit the optimal vol target won't be equal to the usual Kelly optimal. In fact this subject is intellectually interesting as well as topical since it's the first time I've looked at optimisation with a drawdown/time constraint. 

I run this as a bootstrap exercise. We try and optimise: (a) minimise the median cost, (b) maximise the probability of being funded before we give up. 

OK so two simple graphs then. Each has a different line for each SR, and the x-axis is the vol target we are running at. The y-axis on graph one is the cost, with a minus sign so we have the natural thing of a high y-axis being good. On graph two the y-axis is the probability of passing before we give up, again obviously high y-axis is good.

Median cost of getting to stage two, lines are SR, x axis is annual vol target, y axis is cost (bigger minus numbers are higher costs)

Note that for SR/vol combinations where we have a less than 50% median chance of succeeding the median cost will be equal to the monthly cost * 12. This is the case for SR<1.5

Probability of getting to stage two, lines are SR, x axis is annual vol target, y axis is probability of success

What conclusions can we draw from this?

• The optimal vol target depends on your SR and whether you are focusing on costs or probability*
• To get a greater than 50% chance of passing we need an expected SR of 1.5 or higher. 
• The expected median cost with optimal vol is going to be be $2000 for a SR of 1.5, which you can get down to $1500 if you are the next RenTech (SR of 3). 
• The expected median time to pass is going to be about 7 months for a SR of 1.5 or about 5 months if you are the next RenTech

* Experts will recognise the vol target choice as the Wordle (TM) starting word problem (yes we finally got there). The best starting word for Wordle will depend on whether you are maximising your probability of winning, or trying to minimise the number of guesses you make. Similarly, are we trying to maximise our chance of passing the challenge, or minimising our likely cost? They are not quite the same thing.

The optimal vol looking at costs is around 15 - 20%. Looking at probability of passing, it's around 12% for very high SR traders, and more like 22% for low SR traders. Basically if you're crap you have to take a bit more risk to have a chance. If you're good you can chill. Given we're assuming Gaussian returns I'd be tempted to mark these figures down a bit, although note that for high SR traders using less than optimal vol is quite harmful (very steep lines) whilst using more than optimal is less painful (this is completely at odds with Kelly of course).

Since nobody knows what their SR is, I'd suggest using 15% as a vol target. If you are incredible that is slightly more than optimal, but you still have an 80% chance of passing. If you are less incredible it may be slightly less than optimal, but then you have no business passing this challenge anyway.

The not so good guys firm #2

Here is an example of another firm's level 1 challenge, I won't name them but they are currently on the 1st page of google results for the search term "trading prop challenge" so that narrows it down. This firm has several challenge tiers in the futures space, I've chosen the lowest; but all the conditions are the same just different notional capital and $ costs. 

The rules of the challenge are this:

• You must make 6%
• The maximum drawdown is 4%; trailing based on daily balances.
• If you lose more than 2% in a day, well basically you're stopped out at 2% but the challenge doesn't end. So your max loss in a day is 2%. In practice would be slightly more because of slippage but let's be generous here.
• There is a one time activation of $130 (not exact figures again but ballpark).
• You have to do the challenge in 30 days. It costs $100 to start each challenge. If you want to extend the challenge by 30 days it costs another $100. This equates to a monthly fee of $100, so we'll model it like that.
• If you need to reset (start again because you've gone boom) it's $80. This is on top of the monthly cost since it doesn't reset the number of days to zero before you have to pay a monthly fee again.
• There are optional data fees we will ignore, because there are enough fees here already.

Now, it's worth saying that there are many other terms and conditions that make firm #2 much dodgier and less likely to fund you or give you your profit share once funded (of course we're assuming firm #1 sticks to their word as well); but we're purely here to model the challenge itself.

Here are the graphs:

Median cost of getting to stage two, lines are SR, x axis is annual vol target, y axis is cost (bigger minus numbers are higher costs)

Probability of getting to stage two, lines are SR, x axis is annual vol target, y axis is probability of success

This is not what I had expected. I had expected the challenge to be much harder, so the firm could keep collecting the fees. But this challenge is easy to pass, just use vol more than 25%. Basically you get to flip a coin a couple of times and sooner or later it will turn up heads. This strategy will work even if you are a losing trader (SR -0.5) as shown. The only benefit of being a better trader is you will pass quicker and thus pay less. 

This is an incredibly badly designed challenge. It rewards higher volatility. It doesn't discriminate at all between good and bad traders. 

So eithier (i) there are other conditions in the (very hard to find) small print that in practice make the challenge hard to pass or (ii) it's a deliberate strategy to allow almost anyone to get to the next stage. The biggest red flag is that trading with this particular firm is sim only even after you have passed the challenge. They don't want to make the initial challenge too hard; they want you as a paying customer ASAP. And people who use too much vol are ideal customers for bucket shops.

The prop firms view

Of course what we're not doing here is looking at things from the prop firm's point of view. The challenge is designed to answer the question: "is this potential trader any good or just lucky?". At least that is if you are assuming they are genuinely looking for good traders. Which prop firm #2 definitely isn't, so let's focus on Raen.

The main shortcoming of these challenges is that 30 days or even a year is a wholly insufficient time to determine if anyone has any skill, unless they are very highly skilled indeed. And again, to be fair, the initial challenge of Raen is purely a screening exercise that will essentially tell you (a) if someone has a vague idea of how to manage risk and avoid a 2% daily drawdown and (b) is eithier very good (SR somewhere over 1) or just very lucky.

Someone who shoots for a vol target that is too high will almost certainly fail. However there is still a chance of a crap trader being lucky. But hopefully the second stage will weed them out. So we aren't too worried about type 1 errors.

However even relatively highly skilled traders (say SR 1 to 1.5) will only have a coinflip chance of passing. So there is still quite a big chance of a type 2 error and missing out on the next Nav Sarao*. Perhaps that's okay. They're probably only interested in people with a SR of over 2 anyway, where the passing percentage for a year will be over 60% if they use optimal vol. Of course I'm assuming all these people have several thousand dollars to stump up to a years worth of monthly fees. There will be many who don't, and therefore also miss out on potentially being funded even if they are good traders. So I would say the possibility of a type 2 error is quite high.

* this famous gentlemen who for all his faults was an incredibly succesful futures prop trader even when he wasn't breaking the law.

 

I'd say on balance that Raen's challenge is relatively well designed given all the caveats. It's simple, it's difficulty rating feels about right, and the 2% daily drawdown acts as a simple anti muppet filter. The fact there is an optimal vol is satisfying. It would be interesting to see their internal numbers on how many people pass the first and then the second challenge; and then go on to become good traders. That will tell us what their type 1 error actually is. 

But is this all really a good idea? Some unquantified and unqualified opinions

Putting aside the statistical debate, is this all really a good thing? For the traders trying out, or for the firms themselves (assuming they again are genuine). There are many red flags in this industry, having to pay to be considered for a 'job' is always bad (although Raen's CEO clarified to me that they also accept applicants who haven't passed the challenge, presumably with some kind of filter on experience); the fact that many places are purely bucket shops where you trade against the broker is awful (again not Raen), frankly the whole thing makes my stomach churn but I'm trying to be as fair as possible here and put emotions aside.

The world of trading has changed an awful lot. In this post the founder of Raen says their shop is for people who would never the opportunity to get into Jane Street (JS). But JS is looking for people with a very particular set of skills to do a certain kind of trading which you can't do unless you have the sort of resources JS has. 

Yes we can argue that the Jane Street filter is too strict (though they hired SBF, a man who did not understand how to size trading positions, so maybe not strict enough), but it's pretty silly to pretend that Jane Street would be interested in hiring the sort of people who have the ability to be point and click futures traders. It's really not the same at all.

Raen apparently has ex JS people working there and they are 'very succesful'. I am sure they are. I'm also sure that they're almost certainly not point and click traders eithier. But is it really realistic to replicate JS by hiring a completely different set of people, without any of the filters JS uses to get specific sets of skills, using a totally different process from what JS uses, and without most of JS resources; and then sit them next to ex JS traders from whom they will presumably absorb brilliance by osmosis?

Basically if for some reason you are trying to be the next JS why are you using a hiring process which is clearly for point and click traders? There are no references to APIs that I can see on any of these challenge websites, so I assume it's point and click they are looking for.

So, is the world of point and click prop traders too inaccessible? It's probably more accessible than it was 20 years ago from an IT and cost perspective. But admittedly if you're not trading costly and dodgy retail assets,  and want to trade futures, then no you can't really do this with the $3000 or so you'd need to pass even a good trading challenge. The $100k of (notional, real?) money you get from Raen is the bare minimum I suggest in my book. You would need less in equities though, but to be eg US PDT you need $25k (for now). 

But from my perspective, the whole point and click futures industry seems very... niche. The vast majority of professional traders now are basically quants, or heavily supported by quants, and/or using data other than the charts and order books fancied by the dozen monitor setups of the cliched point and click trader. It's an area of the market that really is very efficient and where the vast majority of point and click humans can't compete even if supported by execution algos, which is why I deliberately trade much... more... slowly. 

In fact I'd say there are now significantly more people employed by the likes of JS than by genuine and profitable point and click firms. 

So we're talking about getting access to a relatively tiny industry that is frankly a bit quaint and probably still shrinking. I can understand why many people want to get into it though. Who wouldn't want to gamble for a living, make millions of dollars a year, in a job which requires no qualifications (no Phd in astrophysics needed here!), which so many films and YouTube videos have glamorised, and which eithier requires almost no work or where hard work and effort will be rewarded (depending on which video you watch). 

I can believe that there are a very small number of people who have pointed and clicked for so long, that they really do have an ability to 'feel' a particular market very well, they can glance at an L2 order book and see patterns, they know which news and statistics to focus on, they know what other markets to look at, they know how to manage risk and size positions, they have built execution algos that enable them to compete not with HFT but certainly in the sub one day area... and they are certainly better traders than me or my systems. 

As to how you would select such people, I do not know. They are not my people. Personally I am very skeptical as to whether there really are people who can sit at a computer having never traded futures before except maybe in a simulator, with no training or market experience, and have some innate trading ability that enables them to have a high probability of passing a trading challenge with the sort of SR that would make most hedge funds weep with envy, and also that those challenges are the best way of being able to tell that someone has that innate ability. 

But once again, I'm not in this industry so what do I know. 

Summary

Raen: not a bad intial screening and a more than 50% chance of a pass with a SR above 1.5. But it will cost you more than $300. Budget for several thousand bucks and use a vol target of around 15% to optimise your chances.

Unamed prop firm #2 I googled and most of this industry: stay away for gods sakes.

Pay to play: morally dubious IMHO

Random futures traders having some sort of innate talent that can be found in this way: I doubt it

Taking an income from your trading account - probabilistic Kelly with regular withdrawals

Programming note: This post has been in draft since ... 2016!

One question you will see me asked a lot is 'how much money do I need to become a full time trader?'. And I usually have a handwaving answer along the lines of 'Well if you think your strategy will earn you 10% a year, then you probably want to be able to cover 5 years of expenses with no income from your trading strategy, so you need 15x your annual living expenses as an absolute minimum'. Which curiously often isn't the answer people want to hear, since objectively at that point they would already be rich and they want to trade purely to become rich (a terrible idea! people should only trade for fun with money they can afford to lose); and also because they want to start trading right now with the $1,000 they have saved up which wouldn't be enough to cover next months rent. 

But behind that slightly trite question there is a more deep and meaningful one. It is a variation of this question, which I've talked about a lot on this blog and in my various books:

"Given an expected distribution of trading strategy returns, what is the appropriate standard deviation or leverage target to run your strategy at?"

And the variation we address here is:

How does the answer to the above question change if you are regularly taking a specific % withdrawal from your account?

This has obvious applications to retail traders like me (although I don't currently take a regular withdrawal from my trading account which is only a proportion of my total investments, rather I sporadically take profits). But it could also have applications to institutional investors creating some kind of structured product with a fixed coupon (do people still do that?).

There is generic python code here (no need to install any of my libraries first, except optionally to use the progressBar function) to follow along with.

A brief reminder of prior art(icles)

For new readers and those with poor memories, here's a quick run through what I mean by 'probabilistic Kelly'. If you are completely new to this and find I'm going too quickly, you might want to read some prior articles:

https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html

https://qoppac.blogspot.com/2022/11/if-youre-so-smart-how-come-youre-not.html

https://qoppac.blogspot.com/2023/12/portfolio-optimisation-uncertainty.html

If you know this stuff backwards, then you can skim through very quickly just to make sure you haven't remembered it wrong.

Here goes then: The best measure of performance is the following - having the most money at the end of your horizon (which for this blogpost I will assume is 10 years, eg around 2,560 working days). We maximise this by maximising final wealth, or log(final wealth). This is known as the Kelly criterion. The amount of money you will have at the end of time is equal to your starting capital C, multiplied by the product of (1+r0)(1+r1)...(1+rT) where rt is the return in a given time period. The t'th root of all of that lot, minus one, is equal to the geometric mean. So to get the most money, we maximise the annual geometric mean of returns which is also known in noob retail trading circles as the CAGR.

If we can use any amount of leverage then for Gaussian returns the optimal standard deviation will be equal to the Sharpe Ratio (i.e. average arithmetic excess return / standard deviation). For example, if we have a strategy with a return of 15%, with risk free rate of 5%, and standard deviation of 20%; then the Sharpe ratio will be (15-5)/20 = 0.50; the optimal standard deviation is 0.50 = 50%; and the leverage required to get that will be 50%/20% = 2.5.

Note: For the rest of the post I'm going to assume Gaussian normal returns since we're interested in the relative effects of what happens when we introduce cash withdrawal, rather than the precise numbers involved. As a general rule if returns are negatively skewed, then this will reduce the optimal leverage and standard deviation target, and hence the safe cash withdrawal rate. 

Enough maths: it's probably easier to look at some pictures. For the arbitrary strategy with the figures above, let's see what happens to return characteristics as we crank up leverage (x-axis; leverage 1 means no leverage and fully invested, >1 means we are applying leverage, <1 means we keep some cash in reserve):

x-axis: leverage, y-axis: various statistics

The raw mean in blue shows the raw effect of applying leverage; doubling leverage doubles the annual mean from 15% to 30%. Similarly doubling leverage doubles the standard deviation in green from 20% to 40%. However when we use leverage we have to borrow money; so the orange line showing the adjusted mean return is lower than the blue line (for leverage >1) as we have to pay interest..

The geometric mean is shown in red. This initially increases, and is highest at 2.5 times leverage - the figure calculated above, before falling. Note that the geometric mean is always less than the mean; and the gap between them gets larger the riskier the strategy gets. They will only be equal if the standard deviation is zero. Note also that using half the optimal leverage doesn't halve the geometric return; it falls to around 14.4% a year down from just over 17.5% a year with the optimal leverage. But doubling the leverage to 5.0 times results in the geometric mean falling to zero (this is a general result). Something to bear in mind then is that using less than the optimal leverage doesn't hurt much, using more hurts a lot.

Here is another plot showing the geometric mean (Left axis, blue) and final account value where initial capital C=1 (right axis, orange); just to confirm the maximum occurs at the same leverage point:

x-axis: leverage, y-axis LHS: geometric mean (blue), y-axis RHS: final account value (orange)

Remember the assumption we're making here is that we can use as much leverage as possible. That means that if we have a typical relative value (stat arb, equity long short, LTCM...) hedge fund with low standard deviation but high Sharpe ratio, then we would need a lot of leverage to hit the optimal point. 

If we label our original asset A, then now consider another asset B with excess mean 10%, standard deviation 10%, and thus Sharpe Ratio of 1.0. For this second asset, assuming it is Gaussian (and assets like this are normally left skewed in reality) the optimal standard deviation will be equal to the SR, 100%; and the leverage required to get that will be 100/10 = 10x. Which is a lot. Here is what happens if we plot the geometric mean against leverage for both assets.

Optimal leverage (x axis) occurs at maximum geometric mean (y axis) which is at leverage 2.5 for A, and at leverage 10 for B (which as you would expect has a much higher geometric mean at that point). 

But if we plot the geometric mean (y axis) against standard deviation (x axis) we can see the optimium risk target is 50% (A) and 100% (B) respectively:

x-axis: leverage, y-axis geometric mean

 

Bringing in uncertainty

Now this would be wonderful except for one small issue; we don't actually know with certainty what our distribution of futures returns will be. If we assume (heroically!) that there is no upward bias in our returns eg because they are from a backtest, and we also assume that the 'data generating process (DGP)' for our returns will not change, and that our statistical model (Gaussian) is appropriate for future returns; then we are still left with the problem that the parameters we are estimating for our returns are subject to sampling estimation error or what I called in my second book 'Smart Portfolios', the "uncertainty of the past".

There are at least three ways to calculate estimation error for something like a Sharpe Ratio, and they are:

• With a distributional assumption, using a closed form formula eg the variance of the estimate will be (1+.5SR^2)/N where N is the number of observations, if returns are Gaussian. For our 2560 daily returns and an annual SR of 0.5 that will come out to a standard deviation of estimate for the SR of 0.32; eg that would give a 95% confidence interval for annual SR (approx +/- 2 s.d.) of approximately -0.1 to 1.1
• With non parametric bootstrapping where we sample with replacement from the original time series of returns
• With parametric monte carlo where we fix some distribution, estimate the distributional parameters from the return series and resample from those distributions

Calculation: annual SR = 0.5, daily SR = 0.5/sqrt(256) = 0.03125. Variance of estimate = (1+.5*.03125^2)/2560 = 0.000391, standard deviation of estimate = 0.0197, annualised = 0.0197*sqrt(256) = 0.32

For simplicity and since I 'know' the parameters of the distribution I'm going to use the third method in this post. 

(it would be equally valid to use the other methods, and I've done so in the past...)

So what we do is generate a number of new return series from the same distribution of returns as in the original strategy, and the same length (10 years). For each of these we calculate the final account value given various leverage levels. We then get a distribution of account values for different leverage levels. 

The full Kelly optimal would just find the leverage level at which the average account value was maximised, i.e. the median 50% percentile point of this distribution. Instead however we're going to take some more conservative distributional point which is something less than 50%, like for example 20%. In plain english, we want the leverage level that maximises the account value that we expect to get say two out of ten times in a future 10 year period (assuming all our assumptions about the distribution are true). 

Note this is a more sophisticated way of doing the crude 'half Kelly' targeting used by certain people, as I've discussed in previous blog posts. It also gives us some comfort in the case of our returns not being normally distributed, but where we've been unable to accurately estimate the likely left hand tail from the existing historic data ('peso problem').

Let's return to asset A and show the final value at different points of the monte carlo distribution, for different leverage levels:

x-axis leverage level, y-axis final value of capital, lines: different percentile points of distribution

Each line is a different point on the distribution, eg 0.5 is the median, 0.2 is the 20% percentile and so on. As we get more pessimistic (lower values of percentile), the final value curve slips down for a given level of leverage; but the optimal leverage which maximizes final value also reduces. If you are super optimistic (75% percentile) you would use 3.5x leverage; but if you were really conservative (10% percentile) you would use about 0.5x leverage (eg keep half your money in cash). 

As I said in my previous post your choice of line is down to your tolerance for uncertainty. This is not quite the same as a risk tolerance, since here we are assuming that you are happy to maximise geometric mean and therefore you are happy to take as much standard deviation risk as that involves. I personally feel that the choice of uncertainty tolerance is much more intuitive to most people than choosing a standard deviation risk limit / target, or god forbid a risk tolerance penalty variable.

Introducing withdrawals

Now we are all caught up with the past, let's have a look at what happens if we withdraw money from our portfolio over time. First decision to make is what our utility function is. Do we still want to maximise final value? Or are we happy to end up with some non positive value of money at the end of time? For some of us, the answer will depend on how much we love our children :-) To keep things simple, I'm initially going to assume that we want to maximise final value, subject to that being at least equal to our starting capital. As my compounding calculations assume an initial wealth of 1.0, that means a final account value of at least 1.0.

Inititally then I'm going to look at what happens in the non probabilistic case. In the following graph, the x-axis is leverage as before, and the y-axis this time is final value. Each of the lines shows what will happen at a different withdrawal rate. 0 is no withdrawal, 0.005 is 0.5% a year, and so on up to 0.2; 20% a year.

x-axis leverage, y-axis final value. Each line is a different annual withdrawal rate

At higher withdrawal rates we make less money - duh! - but the optimal leverage remains unchanged. That makes sense. Regardless of how much money we are withdrawing, we're going to want to run at the same optimal amount of leverage. 

And for all withdrawal rates of 17% or less, we end up with at least 1.0 of our final account value, so we can use the optimal leverage without any worries. For higher withdrawal rates, eg 20%, we can never safely withdraw all that amount, regardless of how much leverage we use. We'll always end up with less than our final account value even at the optimal leverage ratio.

For this Sharpe Ratio level then, to end up with at least 1.0 of our account value, it looks like our safe withdrawal rate is around 17% (In fact, I calculate it later to be more like 18%).

Safe withdrawals versus Sharpe Ratio

OK that's for a Sharpe of 0.5, but what if we have a strategy which is much better or worse? What is the relationship between a safe withdrawal rate, and the Sharpe Ratio of the underlying strategy?  Let's assume that we want to end up with at least 1.0x our starting capital after 10 years, and we push our withdrawal rate up to that point. 

X-axis Sharpe Ratio, y-axis safe withdrawal rate leaving capital unchanged at starting level

That looks a bit exponential-esque, which kind of makes sense since we know that returns gross of funding costs scale with the square of SR: If our returns double with the same standard deviation we double our SR, then we can double our risk target, which means we can use twice as much leverage, so we end up with four times the return. It isn't exactly exponential, because we have to fund borrowing. 

The above result is indifferent to the standard deviation of the underlying asset as we'd expect (I did check!), but how does it vary when we change the other key values in our calculation: the years to run the strategy over and the proportion of our starting capital we want to end up with?

x-axis amount of starting capital to end up with, y-axis withdrawal rate, lines different time periods in years

Each of these plots has the same format. The Sharpe Ratio of the underlying strategy is fixed, and is in the title. The y-axis shows the safe withdrawal rate, for a given amount of remaining starting capital on the x-axis (where 1.0 means we want to end up with all our remaining capital). Each line shows the results for a different number of years.

The first thing to notice is that if we want to maintain our starting capital, the withdrawal rate will be unchanged regardless of the number of years we are trading for. That makes sense - this is a 'steady state' where we are withdrawing exactly what we make each year. If we are happy to end up with less of our capital, then with shorter horizons we can afford to take a lot more out of our account each year. Again, this makes sense. However if we want to end up with more money than we started with, and our horizon is short, then we have to take less out to let everything compound up. In fact for a short enough time horizon we can't end up with twice our capital as there just isn't enough time to compound up (at what here is quite a poor Sharpe Ratio). 

x-axis amount of starting capital to end up with, y-axis withdrawal rate, lines different time periods in years

With a higher Sharpe, the pattern is similar but the withdrawal rates that are possible are much larger. 

x-axis amount of starting capital to end up with, y-axis withdrawal rate, lines different time periods in years

Withdrawals probabilistically

Notice that if you really are going to consistently hit a SR of exactly 1, and you're prepared to run at full Kelly, then a very high withdrawal rate of 50% is apparently possible. But hitting a SR of exactly 1 is unlikely because of parameter uncertainty. 

So let's see what happens if we introduce the idea of distributional monte carlo into withdrawals. To keep things simple, I'm going to stick my original goal of saying that we want to end up with exactly 100% of our capital remaining when we finish. That means we can solve the problem for an arbitrary number of years (I'm going to use 30, which seems reasonable for someone in the withdrawal phase of their investment career post retirement). 

What I'm going to do then is generate a large number of random 30 year daily return series drawn for a return distribution appropriate for a given Sharpe Ratio, and for each of those calculate what the optimal leverage would be (which remember from earlier is invariant to withdrawal rate), and then find the maximum annual withdrawal rate that means I still have my starting capital at the end of the investment period. This will give me a distribution of withdrawal rates. 

From that distribution I then take a different quantile point, depending on whether I am being optimistic or pessimistic versus the median.

X-axis: Sharpe Ratio. Y-axis: withdrawal rate (where 0.5 is 50% a year). Line colours: different percentiles of the monte carlo withdrawal rate distribution, eg 0.5 is the median, 0.1 is the very conservative 10% percentile.

Here is the same data in a table:

                   Percentile
SR     0.10   0.20   0.30   0.50    0.75
0.10    4.7    4.8    4.9    5.5    7.40
0.25    4.9    5.3    6.0    7.9   11.00
0.50    9.0   11.0   13.0   18.0   24.25
0.75   18.0   23.0   26.0   33.0   43.00
1.00   33.0   39.0   45.0   55.0   66.00
1.50   83.9   96.0  102.0  117.0  135.00
2.00  162.0  176.0  185.7  205.0  228.25

We can see our old friend 18% in the median 0.50 percentile column, for the 0.50 SR row. As before we can withdraw more with higher Sharpe Ratios.
Now though as you would expect, as we get more optimistic about the quanti, we would use a higher withdrawal rate. For example, for a SR of 1.0 the withdrawal rates vary from 33% a year at the very conservative 10% percentile, right up to 66% at the highly optimistic 75% percentile.
As I've discussed before nobody should ever use more than the median 50% (penultimate column) which means you're basically indifferent to uncertainty, and I'd be vary wary of the bottom few rows with very high  Sharpe Ratios, unless you're actually running an HFT shop or Jane Street in which case good luck.
Footnote: All of the above numbers were calculated with a 5% risk free rate. Here are the same figures with a 0% risk free rate. They are roughly, but not exactly, the above minus 5%. This means that for low enough SR values and percentile points we can't safely withdraw anything and expect to end up with our starting capital intact.

                    Percentile
SR     0.10   0.20   0.30   0.50   0.75
0.10    0.0    0.0    0.0    0.4    2.8
0.25    0.0    0.6    1.4    3.6    7.5
0.50    3.6    5.9    8.1   12.0   19.0
0.75   13.0   17.0   21.0   28.0   38.0
1.00   29.0   34.0   39.0   48.0   62.0
1.50   79.0   90.0   96.0  110.0  131.0
2.00  150.9  167.0  178.0  198.5  223.0

Conclusion

I find him strangely compelling and also very annoying. He is always sniggering and has a permanent smug look on his face. The videos are mostly set in exotic places where we are presumably supposed to envy Anton's lifestyle which seems to involve spending a lot of time flying around the world - not something I'd personally aspire to.

t

I can't comment on the quality of his education but at least he has the pedigree. He also has some interesting opinions about non trading subjects but then so do most trading "gurus". Mostly on trading, and on the financial industry generally, from what I've seen he talks mostly sense.

Anyway, one interesting thing he said is that you shouldn't use trading for income but only to grow capital. Something I mostly agree with. Mostly people who w

http://www.elitetrader.com/et/index.php?threads/how-much-did-you-save-up-before-you-decided-to-trade-full-time.298251/

As a procrastination technique (I'm supposed to be writing my second book) I've been watching the videos of Anton Kriel on youtube. For those of you who don't know him he's an english ex goldman sachs guy who retired at the age of 27, "starred" in the post modern turtle traders based reality trading show "million dollar traders", and now runs something called the institute of trading that offers very high priced training and mentoring courses.

I find him strangely compelling and also very annoying. He is always sniggering and has a permanent smug look on his face. The videos are mostly set in exotic places where we are presumably supposed to envy Anton's lifestyle which seems to involve spending a lot of time flying around the world - not something I'd personally aspire to.

I can't comment on the quality of his education but at least he has the pedigree. He also has some interesting opinions about non trading subjects but then so do most trading "gurus". Mostly on trading, and on the financial industry generally, from what I've seen he talks mostly sense.

Anyway, one interesting thing he said is that you shouldn't use trading for income but only to grow capital. Something I mostly agree with. Mostly people who w

http://www.elitetrader.com/et/index.php?threads/how-much-did-you-save-up-before-you-decided-to-trade-full-time.298251/

I'd quite a conservative person, so I'd probably conservatively assume my SR was around 0.50 (in backtest it's much higher than that, and even in live trading it's been a little bit higher), and use the most conservative 10% percentile. That implies that with a withdrawal rate a shade under 4% plus the risk free rate I'd still have my starting capital intact after any given period of time. 

Since I've used a risk free rate of 5%, that implies withdrawing the risk free rate plus another 4% on top, for a total of 9%.

If you're more aggressive, and have good reason to expect a higher Sharpe Ratio, then you could consider a withdrawal rate up to perhaps 30%. But this should only be done by someone with a track record of achieving those kinds of returns over several years, and who is comfortable with the fact that their chances of maintaining their capital are only a coin flip.

Note that one reason this is quite low is that in a conservative 10% quantile scenario I'd rarely be using the full Kelly (remember 0.50 SR implies a 50% risk target); this is consistent with what I actually do which is use a 25% risk target. With a 25% risk target, and SR 0.5 in theory I will make the risk free rate plus 12.5%. So I'm withdrawing around a third of my expected profits, which sounds like a good rule of thumb for someone who is relatively risk averse. 

Obviously my conservative 9% is higher than the 4% suggested by most retirement planners (which is a bit arbitrary as it doesn't seem to change when the risk free rate changes), but that is for long only portfolios where the Sharpe probably won't be even as good as 0.50; and more importantly where leverage isn't possible. Getting even to the half Kelly risk target of 25% isn't going to be possible without leverage with a portfolio that doesn't just contain small cap stocks or crypto.... it will be impossible with 60:40 for sure! But also bear in mind that my starting capital won't be worth what it's currently worth in real terms in the future, so I might want to reduce that figure further. 

Introducing max-GM (median@p), a new(?) performance statistic

Do you remember this post? https://qoppac.blogspot.com/2022/06/vol-targeting-cagr-race.html

Here I introduced a performance metric, the best annualised compounding return at the optimal leverage level for that strategy. This is equivalent to finding the highest geometric return once a strategy is run at it's Kelly optimal leverage.

I've since played with that idea a bit, for example in this more recent post amongst others I considered the implications of that if we have different tolerances for uncertainty and used bootstrapping of returns when optimising a stock and bond portfolio with leverage, whilst this post from last month does the same exercise in a bitcoin/equity portfolio without leverage.

In this post I return to the idea of focusing on this performance metric - the maximised geometric mean at optimal leverage, but now I'm going to take a more abstract view to try and get a feel for in general what sort of strategies are likely to be favoured when we use this performance metric. In particular I'd like to return to the theme of the original post, which is the effect that skew and other return moments have on the maximum achievable geometric mean. 

Obvious implications here are in comparing different types of strategies, such as trend following versus convergent strategies like relative value or option selling; or even 'classic' trend following versus other kinds.

Some high school maths

(Note I use 'high school' in honor of my US readers, and 'maths' for my British fans)

To kick off let's make the very heroric assumption of Gaussian returns, and assume we're working at the 'maximum Kelly' point which means we want to maximise the median of our final wealth distribution - same as aiming for maximum geometric return - and are indifferent to how much risk such a portfolio might generate.

Let's start with the easy case where we assume the risk free rate is zero; which also implies we pay no interest for borrowing (apologies I've just copied and pasted in great gobs of LaTeX output since it's easier than inserting each formula manually):

Now that is a very nice intuitive result!

Trivially then if we can use as much leverage as we like, and we are happy to run at full Kelly, and our returns are Gaussian, then we should choose the strategy with the highest Sharpe Ratio. What's more if we can double our Sharpe Ratio, we will quadruple our Geometric mean!

Truely the Sharpe Ratio is the one ratio to rule them all!

Now let's throw in the risk free rate:

We have the original term from before (although the SR now deducts the risk free rate), and we add on the risk free rate reflecting the fact that the SR deducts it, so we add it back on again to get our total return. Note that the higher SR = higher geometric mean at optimal leverage relationship is still true. Even with a positive risk free rate we still want to choose the strategy with the highest Sharpe Ratio!

Consider for example a classic CTA type strategy with 10% annualised return and 20% standard deviation, a miserly SR of 0.5 with no risk free rate; and contrast with a relative value fixed income strategy that earns 6.5% annualised return with 6% standard deviation, a SR of 1.0833

Now if the risk free rate is zero we would prefer the second strategy as it has a higher SR, and indeed should return a much higher geometric mean (since the SR is more than double, it should be over four times higher). Let's check. The optimal leverages are 2.5 times and 18.1 (!) times respectively. At those leverages the arithmetic means are 25% and 117% respectively, and the geometric means using the approximation are 12.5% for the CTA and 58.7% for the RV strategy.

But what if the risk free rate was 5%? Our Sharpe ratios are now equal: both are 0.25. The optimal leverages are also lower, 1.25 and 4.17. The arithmetic means come in at 12.5% and 27.1%, with geometric means of 9.4% and 24%. However we have to include the cost of interest; which is just 1.25% for the CTA strategy (borrowing just a quarter of it's capital at a cost of 5% remember) but a massive 15.8% for the RV. Factoring those in the net geometric means drop to 8.125% for both strategies - we should be indifferent between them, which makes sense as they have equal SR.

The horror of higher moments

Now there is a lot wrong with this analysis. We'll put aside the uncertainty around being able to measure exactly what the Sharpe Ratio of a strategy is likely to be (which I can deal with by drawing off more conservative points of the return distribution, as I have done in several previous posts), and the assumption that returns will be the same in the future. But that still leaves us with the big problem that returns are not Gaussian! In particular a CTA strategy is likely to have positive skew, whilst an RV variant is more likely to be a negatively skewed beast, both with fat tails in excess of what a Gaussian model would deliver. In truth in the stylised example above I'd much prefer to run the CTA strategy rather than a quadruple leveraged RV strategy with horrible left tail properties.

Big negative skewed strategies tend to have worse one day losses; or crap VAR if you prefer that particular measure. The downside of using high leverage is that we will be saddled with a large loss on a day when we have high leverage, which will significantly reduce our geometric mean.

There are known ways to deal with modifying the geometric mean calculation to deal with higher moments like skew and kurtosis. But my aim in this post is to keep things simple; and I'd also rather not use the actual estimate of kurtosis from historic data since it has large sampling error and may underestimate the true horror of a bad day that can happen in the future (the so called 'peso problem'); I also don't find the figures for kurtosis particuarly intuitive. 

(Note that I did consider briefly using maximum drawdown as my idiots tail effect here. However maximum drawdown is only relevant if we can't reduce our leverage into the drawdown. And perhaps counter intuitively, negative skewed strategies actually have lower and shorter drawdown)

Instead let's turn to the tail ratio, which I defined in my latest book AFTS. A lower tail ratio of 1.0 means that that the left tail is Gaussian in size, whilst a higher ratio implies a fatter left tail.

I'm going to struggle to rewrite the relatively simple 0.5SR^2 formulae to include a skew and left tail term, which in case will require me to make some distributional assumptions. Instead I'm going to use some bootstrapping to generate some distributions, measure the tail properties, find the optimal leverage, and then work out the geometric return at the optimal leverage point. We can then plot maximal geometric means against empirical tail ratios to get a feel for what sort of effect these have.

Setup

To generate distributions with different tail properties I will use a mixture of two Gaussian distributions; including one tail distribution with a different mean and standard deviation* which we draw from with some probability<0.5. It will then be straightforward to adjust the first two moments of the sample distribution of returns to equalise Sharpe Ratio so we are comparing like with like.

* you will recognise this as the 'normal/bliss' regime approach used in the paper I discussed in my prior post around optimal crypto allocation, although of course it will only be bliss if the tail is nicer which won't be the case half the time.

As a starting point then my main return distribution will have daily standard deviation 1% and mean 0.04% which gives me an annualised SR of 0.64, and will be 2500 observations (about 10 years) in length - running with different numbers won't affect things very much. For each sample I will draw the probability of a tail distribution from a uniform distribution between 1% and 20%, and the tail distribution daily mean from uniform [-5%, 5%], and for the tail standard deviation I will use 3% (three times the normal). 

All this is just to give me a series of different return distributions with varying skew and tail properties. I can then ex-post adjust the first two moments so I'm hitting them dead on, so the mean, standard deviation and SR are identical for all my sample runs. The target standard deviation is 16% a year, and the target SR is 0.64, all of which means that if the returns are Gaussian we'd get a maximum leverage of 4.0 times.

As always with these things it's probably easier to look at code, which is here (just vanilla python only requirements are pandas/numpy).

Results

Let's start with looking at optimal leverage. Firstly, how does this vary with skew?

Ignore the 'smearing' effect this is just because each of the dots in a given x-plane will have the same skew and first two moments, but slightly different distribution otherwise. As we'd expect given the setup of the problem the optimal leverage with zero skew is 4.0

For a pretty nasty skew of -3 the leverage should be about 10% lower - 3.6; whilst for seriously positive skew of +3 you could go up to 4.5. These aren't big changes! Especially when you consider that few real world trading strategies have absolute skew values over 1 with the possible exception of some naked option buying/selling madness. The most extreme skew value I could find in my latest book was just over 2.0, and that was for single instruments trading carry in just one asset class (metals).

What about the lower tail? I've truncated the plot on the x-axis for reasons I will explain in a second.

Again for the base case with a tail ratio of 1.0 (Gaussian) the optimal leverage is 4.0; for very thin lower tails again it goes up to 4.4, but for very fat lower tails it doesn't really get below about 3.6. Once again, tail ratios of over 4.0 are pretty rare in real strategies (though not individual instruments), although my sampling does sometimes generate very high tail ratios the optimal leverages never go below 3.6 even for a ratio of nearly 100.

Upper tail, again truncated:

Again a tail ratio of 1.0 corresponds to optimal leverage of roughly 4.0; and once again even for very fat upper tail ratios (of the sort that 'classical' trend followers like to boast about, the optimal leverage really isn't very much higher.

Now let's turn to Geometric mean, first as affected by skew:

As we'd expect we can generate more geometric mean with more skew... but not a lot! Even a 3 unit improvement in skew barely moves the return needle from 22.5% to 24.5%. To repeat myself it's rare to find trading strategies above or below 1.0, never mind 3.0. For example the much vaunted Mulvaney capital has a return skew on monthly returns of about 0.45. The graph above shows that will only add about 0.25% to expected geometric return at optimal leverage versus a Gaussian normal return distribution (this particular fund has extremely large standard deviation as they are one of the few funds in the world that actually runs at optimal leverage levels).

For completeness, here are the tail ratios. First the lower tail ratio:

Fat left tails do indeed reduce maximum optinmal geometric means, but not a lot.

Now for the upper tail:

Summary

I do like neat formulae, and the results for Gaussian normal distributions do have the property of being nice. Of course they are based on unreal expectations, and anyway who runs full Kelly on a real strategy expecting the returns to be normal and the SR to be equal to the backtested SR is an idiot. A very optimistic idiot, no doubt with a sunny disposition, but an idiot nonetheless.

For the results later in the post it does seem surprising that even if you have something with a very ugly distribution that you'd not really adjust your optimal leverage much, and hence see barely on impact on geometric mean. But remember here that I'm fixing the first two moments of the distribution; which means I'm effectively assuming that I can measure these with certainty and also that the future will be exactly like the past. These are not realistic expectations! 

And that is why in the past rather than choose the leverage that maximised geometric mean, I've chosen to maximise some more conservative point on the distribution of terminal wealth (where geometric mean would be the median of that distribution). Doing that would cause some damage to the negative skewed, fat left tail distributions resulting in lower optimal leverage and thus lower geometric means.

I have thought of a way of doing this analysis with distributional points, but it's quite computationally intensive and this post is already on the long side, so let's stop here.