Monday, 14 April 2025

Annual performance update returneth - year 11

Mad out there isn't it? Tarrifs on/off/on/partially off/on... USD/SP500/Gold/US10/Bitcoin all yoyoing like crazy. Seems a good moment to be slightly reflective.

I skipped my annual performance update last year, a little sad given it was my tenth anniversary. Mainly this is because it had become a lot of work, covering my entire portfolio. The long only stuff is especially hard, as all my analysis is driven by spreadsheets rather than nicely outputted from python. If I'm going to restart the annual performance, I'm going to do it just on my systematic futures portfolio. That's probably what most people are interested in anyway. 

This then is my 2025 performance update. As in previous years I track the UK tax year, which finishes on 5th April. That precise dating is especially important this year!! 

You can track my performance in (almost) real time, along with other information here,

TLDR: On a relative basis my performance was good, 3% long only vs -0.1% vs benchmark; -16.3% in futures vs ~-18% benchmarks. On an absolute basis, it's a mediocre year long only and my worst ever in futures.


Headline

My total portfolio performance was -1.9% last year; It would have been +3% without futures. That's a sign that this hasn't been a great year in the systematic world of futures trading. Spoiler alert - it's my worst ever. Whilst I'm not doing the full gamut of benchmarking this year, my benchmark of Vanguard 80:20 was down 0.1% (thanks to Trump), so I would have been well ahead of that if it wasn't for futures.

Everything else in this post just relates to futures.

Headline numbers, as a % of the starting capital:

MTM:        -17.8%
Interest:     2.4%
Fees:        -0.05%
Commissions: -0.29%
Slippage:    -0.56%

Net :       -16.3%

'Interest' includes dividends on 'cash like' short bond ETFs I hold to make a slightly more efficient use of my cash. Actually I ended up paying interest as I was a bit sloppy and didn't liquidate the ETFs when I lost money so ended up with negative cash balances. Given the interest rate was probably higher on the borrowing than the yield on the money market cash like funds; and given the tax disadvantage (I pay tax on dividends but can't offset losses from interest payments), this was dumb. I've now liquidated a chunk of my ETFs.

MTM - mark to market - also includes gains or losses on FX positions held to meet margin, and on the cash like ETFs. Broken down:

Pure futures:   -14.5%

Cash like ETFs:  -0.64%

FX:              -2.7%

So perhaps a better way of doing this would be to lump together all the interest, margin and ETF related flows, and call those 'cost of margin':

Futures MTM: -14.5%
Commissions:  -0.29%
Slippage:     -0.56%
Fees:         -0.05%

SUBTOTAL - PURE FUTURES: -15.3%

Cost of margin: -1.0%

Net :         -16.3%

Time series


You can see that after the peak in late April (which was also my all time HWM), we have two legs down; the first is almost recovered from; but the drawdown from mid July to the end of September is quite vicous at ~17% is probably the worst I have seen. 

Just about visible at the end of the chart is 'tariff liberation day' which has liberated me from a chunk of my money. Ouch, let's make ourselves feel better with a longer term plot

Benchmarks

My two benchmarks are the SG CTA index, and an AHL fund which like me is denominated in GBP.

Here is the raw performance:



It's not super fair as I run at a much higher vol target than the other two products, but here are some numbers:

           AHL       SG           ROB

Mean       4.5%      4.0%         12.9%

Stdev     10.7%      8.9%         16.8%

SR (rf=0)  0.42      0.45          0.76        

Those Sharpes would be lower, especially in the last couple of years, if a risk free rate was deducted.

My correlation is 0.66 with SG CTA, and 0.56 with AHL. Their joint correlation is 0.78. Perhaps because I'm not just trend following and carry. Both have been bad this year. More on that story later. To make things fairer then, let's do an insample vol adjustment so everything has the same vol:



I'm still winning, but it's a bit closer. Here are the annual figures:

          AHL SG Me

30/03/15 66.0% 51.4% 59.5%

30/03/16 -6.2% -3.7% 28.1%

30/03/17 -3.7% -12.6% 2.4%

30/03/18 8.7% -1.7% 2.0%

30/03/19 5.4% -2.7% 4.5%

30/03/20 22.0% 6.5% 33.8%

30/03/21 0.9% 11.9% -1.7%

30/03/22 -12.0% 33.1% 25.8%

30/03/23 8.3% 0.6% -7.6%

30/03/24 14.5% 24.3% 20.6%

30/04/25 -18.2% -17.9% -14.7%

Note: The reason I'm showing -14.7% here and 16.3% earlier, is that these figures are to the end of the relevant month (i.e. March 31st to March 31st), rather than the UK tax year; as I can only get monthly figures for the AHL fund.

I've highlighted in green the best performer in each year, red is the worst. You can see that, at least with my vol adjustment, it was a shocking year for the industry generally, and although this was my worst year on record, I did actually outperform (the unadjusted numbers are -12% for AHL and -10% for the CTA index so I was worse than those, but obviously would have looked even better earlier in the period).

The one stat I haven't included here is my favourite, geometric return or CAGR; mine was 12.0% vs 5.9% AHL and 6.4% SG (based on the leveraged up, vol adjusted numbers in the second graph; the figures would be worse for AHL and SG without this adjustment).

Is this fair? Well no, there are fees embedded in the AHL and SG numbers. My fees aren't in these figures, and are much higher - I charge no management fee, but my performance fee is 100% :-)


Market by market

Here are the numbers by asset class:

Equity  -6.0%
OilGas  -4.2%
FX      -2.7%
Metals  -1.5%
Bond    -1.3%
Vol     -0.5%
Ags     +3.2%

When you only make money in one sector, that's a bad year! My worst markets were: Gas-pen, MIB, China A construction, Gasoline, MSCI Taiwan, Platinum, MXP; with losses between 1.7% and 1%. My best markets were Coffee, MSCI singapore, Cotton#2 and S&P 500; with gains between 1.7% and 0.7%. So no concentration issues at least.

Because of my dynamic optimisation, the p&l explanations can be ... interesting. Coffee for example, I made 1.7%, but I only held Coffee for less than two weeks; from 31st January to 11th February 2025 (a period in which it went up rather sharply as it happens, but I missed out on the long rally that preceeded it). 

S&P 500 on the other hand, I actually traded:

That's price, position, and p&l. Nice trading. Now look at MIB, italian equities:
For some reason we only played the long side, and boy did we get chopped up.


Trading rules

Presented initially without comment, a bunch of plots showing p&l for each trading rule group (these are backtested numbers, and they are before dynamic optimisation).

















What a rogues gallery! It's quicker to list the rules that did make money or at least broke even:

- relative value skew
- relative carry
- fast relative momentum

This also explains, to an extent, why my performance isn't quite as bad as the industry; I would imagine that I have a little more allocation to this weird stuff than the typical CTA. The pure univariate momentum rules (breakout, momentum and normalised momentum) were especially bad. Even fast momentum has been crucified especially badly in the recent carnage, for obvious reasons.


Costs and slippage

I've already noted that my total slippage (diff between mid market price when I come to place an order, and where I get filled) was 0.56% of my starting capital; and commissions were 0.29%. That's an all in cost of 0.85% which is a little lower than what I usually pay; but as a % of my final capital it's 0.98%, the real number is somewhere around the 89bp - 93bp mark. Two years ago, the last time I checked I was paying 90bp of which 69bp was slippage and 21bp comissions. So slippage is down, commissions up slightly but net-net roughly similar.

Without my execution algo, if I had just traded at the market, I would have paid 1.53% in slippage; my simple algo earned 97bp and cut my slippage bill by two thirds. So that is a good thing.


Coming up

I'm not planning to do much with my futures system this year; I'm going to be busy writing a new book so let's see how it does going forward. Certainly feels like a bad year for trend following; unless or until the US economy gets tipped into recession and we get clear risk off markets. I think it unlikey we're going to get risk on until November 2028 or January 2029 :-) At least for the time being, it's not going great for me (-4% YTD), and it's going even worse for the industry generally.

Tuesday, 4 March 2025

Very.... slow... mean reversion .... and some thoughts on trading at different speeds

 Bit of a mixed bag post today. The golden thread connecting them is the idea that markets trend and mean revert at different frequencies.

- A review of the discussion around timeframes for momentum and mean reversion in 'Advanced Futures Trading Strategies', in light of this excellent recent paper (which I also discussed on the TTU podcast, here from 1:02:12 onwards).

- A mea culpa on the mean reversion strategies in 'Advanced Futures Trading Strategies'. TLDR - there is an error in the backtest and they don't work at least in the specified form.

- A new slow.... absolute mean reversion strategy inspired by a question from Paul Calluzzo on the aforementioned podcast episode.

Note: in this article I use the terms momentum and trend following interchangeably to both mean absolute momentum - not relative.


When do markets trend and mean revert?

When do markets trend? When do they not trend... perhaps even mean reverting? This is a very important question! 

You might think it would depend on the market, but actually there seem to be some fairly common patterns across many different instruments. Here is how I summarised by thoughts in my most recent book, Advanced Futures Trading Strategies (AFTS):

Multi-year horizons: Mean reversion sort of works (although the results are not statistically significant, as the value strategy in part three attests). Note: this value strategy is a relative value strategy that looks for mean reversion within asset classes. Such strategies are common in academic equity research.

Several months to one year horizon: Trend following works, but is not at it’s best (consider the slightly poorer results we get for EWMAC64 versus faster trend variations).

Several weeks to several month horizon: Trend works extremely well (consider the excellent performance of EWMAC8, EWMAC16 and EWMAC32).

Several days to one week: Trend is starting to work less well (EWMAC4 and especially EWMAC2 perform somewhat worse than slower variations, even before costs are deducted).

A few days: We might expect mean reversion to work?

Less than a day: We might expect trend to work?

Less than a second: Mean reversion works well (high frequency trading - HFT - is very profitable).      

Note that 'momentum works for months or years' and 'mean reversion / value works for years' is a very well known stylised fact which has been established in the literature for many decades; see for example this seminal paper. And given the existence of profitable CTAs with holding periods in the weekly to monthly range, it's hardly surprising that momentum works for shorter holding periods. Nor is the fact that HFT firms make a ton of money a secret.

However, in the region between high frequency trading and a horizon of a week or so I wasn't sure exactly what to expect, but I speculated that there would be a region where mean reversion would start to work (more on that later!), and I also thought trend following with holding periods in the 'few hours' range (mainly because there has been some sell side research on that). Note that since my own data is hourly at best, I couldn't really test anything with a horizon of less than about one day.

Fortunately someone came along to fill in this gap in our understanding, with this excellent paper:

"Trends and Reversion in Financial Markets on Time Scales from Minutes to Decades" by Sara A. Safari and Christof Schmidhuber

I won't summarise the paper in much detail (for example it has some interesting results around the relationship between trend strength and reversal), but they have the following pattern of results (from figure 10 in the paper):


Horizon over two years: Mean reversion works, becoming more effective at longer horizons. They used literally centuries of data to check this result. 

One week to two year horizon: Trend following works, but it's effectiveness peaks at around one year

One hour to one week horizon: Trend following works, getting gradually less effective as the time horizon shortens.

Two minute to 30 minute horizon: Mean reversion works, and is most effective at the 4-8 minute horizon


The key differences between my results and theirs only occur in my 'zone of speculation', where I was only guessing and they had actual evidence so let's go with them :-) In particular they have two 'crossing points' from when mean reversion stops working and momentum starts working (at just over two years, and somewhere between 30 minutes and one hour), giving the following broad ranges:

Horizon over two years: Mean reversion works.

One hour to two year horizon: Trend following works.

Two minute to 30 minute horizon: Mean reversion works

Whilst I had speculated that there was something more complicated going on. Even without evidence, Occams razor would suggest you should prefer their results to mine.

Another difference is that when I looked 'optimal points' for eg momentum I was concerned with Sharpe Ratio, but they are instead fitting a response function and seeing when it has the best statistical fitness. Because of the Law Of Active Management, Sharpe Ratio (loosely) scales inversely with the square root of time for a given level of prediction accuracy. So you if you are equally good at predicting one year trends, and 3 month trends, the latter will have twice the Sharpe Ratio of the former. Hence there are good reasons why my optimal SR point is different from their optimal response point; all other things being equal the optimal SR is going to be at a shorter horizon.

Combining the two pieces of research together, and thinking about what sorts of strategies we could be trading, we get this:

Horizon over two years: Mean reversion works. The optimal SR is probably quite flat for anything between three and ten years. Equity value, relative value within asset classes, and absolute mean reversion (of which more in a moment) are all nice strategies. But given their holding period you shouldn't expect high Sharpe Ratios unless you are Warren Buffett (hi Warren!). 

One to two years: Momentum will work but will be getting steadily worse as the timescale gets longer, both from a predictability perspective and a Sharpe Ratio viewpoint. Avoid.

Three months to one year horizon: Trend following works with high predictability, but is not at it’s highest Sharpe ratio due to the slow turnover. However, the advantage here is that this is a playing field that even retail punters with expensive trading costs can play in. Slower momentum strategies are all good.

Three weeks to three months horizon: Trend following probably has it's optimal Sharpe Ratio somewhere in this region, depending on the asset class. Any medium speed momentum strategies are good, and nearly all futures traders can play in this area if they avoid a few very expensive instruments.

Several days to three weeks: Trend is starting to work less well (because the improvement from trading faster is being overwhelmed by the deficit in response) and trading costs will start to bite except for the very cheapest futures (see calculation below), traded with exemplary execution. On the upside, trend following models at this speed will have the highest positive skew. Trade selectively.

A few hours to several days: Trend still just about works but but there are probably only a small number of futures where  you can overcome the bid/offer costs (although I hear costs are very low in Crypto, and there might be US traders who get zero commission able to trade highly liquid ETFs like SPY); I'd doubt though it would be worth doing. As the authors note, strong trends also tend to reverse strongly in this region (see AFTS for my own confirmation of this effect). Against that there have been the sell side papers on this subject, but they seem to rely on gamma hedging effects which may not persist. Avoid.

1 hour to a few hours: The authors in the paper note that the very weak trend effect here can't overcome the tick size effect. Avoid.

Two minute to 30 minute horizon: Mean reversion works, and is most effective at the 4-8 minute horizon from a predictive perspective; although from a Sharpe Ratio angle it's likely the benefits of speeding up to a two minute trade window would overcome the slight loss in predictability. There is no possibility that you would be able to overcome trading costs unless you were passively filled, with all that implies (see below). Automating trading strategies at this latency - as you would inevitably want to do - requires some careful handling (although I guess consistently profitable manual scalpers do exist that aren't just roleplaying instagram content creators, I've never met one). Fast mean reversion is also of course a negatively skewed strategy so you will need deep pockets to cope with sharp drawdowns. Trade mean reversion but proceed with great care.

Less than a second to two minutes: Not covered in the paper, but I would speculate that mean reversion continues to work, and the pre-cost Sharpe Ratio would also continue to improve as the horizon falls. Proceed with even more care.

Less than a second: High frequency trading works, and clearly has a very high Sharpe Ratio, but this is not for the amateur.


Notes on costs: 

The very cheapest equity index future I trade has a cost of around 0.2bp assuming we execute market orders; and vol of around 20% a year, for a SR cost of  about 1bp. Median single instrument SR on the optimal trend strategy (holding period around 3 weeks) is around 0.30. Predictability, as a regression coefficient, from the linked paper is around 6.5% at 3 weeks; and around 1.8% at 2 days (a reduction of 3.6x). Time scaling would improve the SR by 2.7x so the net effect is a 25% fall in SR to around 0.22 for a two day forecast horizon. 

If we take a third of that (my 'speed limit') or 0.22 SR for costs, then our annual cost budget is 0.07 SR or 700bp; implying we could perhaps safely trade a couple of times a day implying a two day forecast horizon (which means trading once a day) is possible. 

But the median future I get data for has a cost of around twenty times that, meaning a holding period of around two weeks is required to meet the 'speed limit'.

Do we have to pay the spread? Broadly speaking, if you are trading slowly, then you can afford to be more patient in your execution, using passive fills where possible (as I do myself). But as a fast trend follower who thinks the price is going to move away from you in the near future, it's probably harder to sit on your hands and wait. 

Alternatively if we are fast mean reverting traders then we can use passive fills by setting limits around where we think the equilibrium is. That of course runs the risk of adverse selection, but without doing this we are never going to make enough money to overcome the bid/offer if we're trading dozens of times a day. You may also be still liable to commissions unless you received exchange rebates from providing liquidity. Note since we earn the bid/offer spread from passive fills, it might be that the best instruments for this strategy are those with wider rather than narrower bid/offers.



Forgive my father, for I have sinned against the gods of backtesting...

Now in AFTS I introduce two strategies which trade mean reversion, with horizons of around a week (since I'd speculated that would work). It included a very elegant way of including limit orders to passively execute, and the second strategy introduced a very nice trend following overlay. And it looked great! But that obviously isn't consistent with the findings above.

Well gentle reader, I screwed up. As I said in my book:

"But what jumps out from this table is the Sharpe ratio. It is impressively large, and the first we have seen in this book that is over two. In my career as a quantitative trader I have always had a long standing policy: I do not trust a back tested Sharpe ratio over two. There are certainly plenty of reasons not to trust this one. 

Firstly, the historical back test period, just over ten years, is shorter than I would like. There are good reasons to suppose that the last ten years included unusual market conditions that might just have favored this strategy. Secondly, it is hard to back test a strategy deploying limit orders that effectively trades continuously using hourly data. There may well be assumptions or errors in my code that make the results look better than they really would have been."

The underlined section (not underlined in the book!) is key here; basically there was an implicit forward fill in my backtest as I calculated the equilibrium price including todays closing price (which of course I wouldn't have known in the morning). The real backtest shows basically no statistically significant return at all.

The good news is that the basic technology of this strategy should work well, at least pre-costs, with a much shorter time horizon; although for all the reasons above I haven't tried it myself (though I know others that have).




A new slow absolute mean reversion strategy

Since I'm taking away one strategy, let me replace it with another. In AFTS strategy twenty two is a 'value' strategy, which bets on mean reversion over five year periods in relative terms against an asset class index. It has crappy SR (basically zero), but positive alpha and improved overall SR when added to trend and carry strategies. 

But on a recent TTU podcast, herePaul Calluzzo asked me if I'd ever tested absolute mean reversion. Certainly I haven't on this blog. So let's do that.

I'll use a three year return for my forecast, which is slow enough to avoid the two year point where we know momentum probably still works; whilst being quick enough to avoid the death by sqrt(T) that will reduce my SR. We go long if the return is negative so:

Forecast = Price_t-3yrs - Price_t

To avoid the turnover being excessive (this is a slow forecast!), and because we should always vol scale:

Smoothed vol scaled Forecast = EWM_64(Forecast/ EW_std_dev(returns))

Drumroll...

It's not..... great (SR -0.48), apart from perhaps a recent pickup. You could argue that as a lot of my data starts in 2013, and the first five year return occurs in 2018, that it's actually profitable for many instruments and we've just been unlucky in the instruments we've traded before. The median SR is -0.06 though which doesn't completely support that argument. 

But really it would appear that at least with this construction absolute mean reversion isn't as good as the relative mean reversion I tested in AFTS.

OK so we've dropped a strategy with an unfeasibly high backtested SR, and I've replaced it with one that has a very poor backtested SR. Unfair? Well, life isn't fair.


Summary

Good things to trade:

Horizon over two years:  Cross sectional mean reversion, but possibly not absolute mean reversion. And similar type things like equity relative value.

One to two years: Nothing*

Three months to one year horizon: Trend following of pretty much anything

Three weeks to three months horizon: Trend following; avoiding very expensive instruments.

Several days to three weeks: Trend following; only the very cheapest instruments.

One hour to several days: Nothing*

Less than a 30 minute horizon: Mean reversion - the faster the better, but only with limit orders and with great care (the faster you are, the more care needs to be taken).

* or at least not outright momentum or mean reversion

Thursday, 6 February 2025

How much should we get paid for skew risk? Not as much as you think!

 A bit of a theme in my posts a few years ago was my 'battle' with the 'classic' trend followers, which can perhaps be summarised as:

Me: Better Sharpe!

Them: Yeah, but Skew!!

My final post on the subject (when I realised it as a futile battle, as we were playing on different fields - me on the field of empirical evidence, them on .... a different field) was this one, in which the key takeaway was this:

The backtest evidence shows that you can achieve a higher maximum CAGR with vol targeting, because it has a large Sharpe Ratio advantage that is only partly offset by it's small skew disadvantage. For lower levels of relative leverage, at more sensible risk targets, vol targeting still has a substantially higher CAGR. The slightly worse skew of vol targeting does not become problematic enough to overcome the SR advantage, except at extremely high levels of risk; well beyond what any sensible person would run.

And another more recent post was on Bitcoin, and why your allocation to it would depend on your appetite for skew. 

With those in mind I recently came to the insight that I could use my framework of 'maximising expected geometric mean / final wealth at different quantile points of the expectation distribution given you can use leverage or not'* to give an intuitive answer an intruiging question - probably one of the core questions in finance:

"What should the price of risk be?"

* or MEGMFWADQPOTED for short - looking actively for a better acronym - which I used in the Bitcoin post linked to above, but explain better in the first half of this post and also this one from a year ago

The whole academic risk factor literature assumes the price of risk often without much reasoning. We can work out the size of the exposure, and the risk of the factor, but that doesn't really justify it's price. After all, academics spent a long time justifying the equity risk premium

I think it would be fun to think about the price of different kinds of risk. Given the background above, I thought only about skew (3rd moment) risk but I will also briefly discuss standard deviation (2nd moment) risk. Generally speaking the idea is to answer the question "What additional Sharpe Ratio should an investor require for each unit of additional risk in the form of X?" Whilst this has certainly been covered by academics at some length, I think the approach of wrapping up into expressing risk preference as optimising for different distributional points is novel and means pretty graphs.

I'm going to assume you're familiar with the idea of maximising geometric return / CAGR / log(final wealth) at some distributional point (50% median or more conservative points like 10, 25%), to find some optimal level of leverage. If not enjoy reading the prior work.


The "price" of standard deviation risk - with and without leverage

To an investor who can use leverage, for Gaussian normal returns, this is trivial. We want the higest Sharpe Ratio asset, irrespective of what it's standard deviation is. Therefore the 'price' of standard deviation is zero. We don't mind getting additional standard deviation risk as long as it doesn't affect our Sharpe Ratio - we don't need a higher SR to compensate. Indeed in practice, we might prefer higher standard deviations since it will require less potential leverage that could be problematic if we are wrong about our SR estimates or assumptions about return distributions.

In classical Markowitz finance to an investor who cannot use leverage, the price of standard deviation is negative. We will happily pay for higher risk in the form of a lower Sharpe Ratio. We want higher returns at all costs; that may come at the cost of higher standard deviation so we aren't fully compensated for the additional risk, but we don't care. This is the 'betting against beta' explanation from the classic Pedersen paper. Consider for example an investment with a mean of 5% and a standard deviation of 10% for a Sharpe Ratio of 0.5 (I set the risk free rate to zero without loss of generality) . If the standard deviation doubles to 20%, but the mean only rises to 6%, well we'd happily take that higher mean. We'd even take it if the mean only increased by 0.00001%. That means the 'price' of higher standard deviation is not only negative, but a very big negative number.

But we are not maximising arithmetic mean. Instead we're maximising geometric mean, which is penalised by higher standard deviation. That means there will be some point at which the higher standard deviation penalty for greater mean is just too high. For the median point on the quantile distribution, which is a full Kelly investor, that will be once the standard deviation has gone above the Kelly optimal level. Until that point the price of risk will be negative; above it will turn positive.

Consider again an arbitrary investment with a mean of 5% and a standard deviation of 10%; SR =0.5. If returns are Gaussian then the geometric mean will be 4.5%. The Kelly optimal risk is much higher 50%, which means it's likely the local price of risk is still negative. So for example, if the standard deviation goes up to 20%, with the mean rising to say 6.5%, for a new (lower) SR of 0.325; we'd still end up with the same geometric mean of 4.5%. In this simple case the price of 10% units of risk is a SR penalty of 0.175; we are willing to pay 0.0175 units of SR for each 1% unit of standard deviation. 

If however the standard deviation goes up another 10%, then the maximum SR penalty for equal geometric mean we would accept is 0.025 units (getting us to a SR of 0.3 or returns of 6.5% a year on 30% standard deviation equating again to a geometric mean of 4.5%); and for any further increase in standard deviation we will have to be payed SR units. This is because the standard deviation is now 30% and so is the SR; we are at the Kelly optimal point. We wouldn't want to take on any additional standard deviation risk unless it is at a higher SR, which will then push the Kelly optimal point upwards.

So we'd need to get paid SR units to push the standard deviation up to say 40%. With 40% standard deviation we'd only be interested in taking the additional risk if we could get a SR of 0.3125 to maintain the geometric mean at 4.5%. Something weird happens here however, since 40% is higher than the new Kelly optimal we can actually get a higher geometric mean if we used less risk (basically by splitting our investment between cash and the new asset). To actually want to use that 40% of risk the SR would trivially have to be 40%. For someone who is remaining fully invested the price of standard deviation risk once you hit the Kelly optimal is going to be 1:1 (1% of standard deviation risk requiring 0.01 of SR benefit).

That is all for a Kelly optimal investor, but how would using my probabilistic methodology with a lower quantile point than the median change this? Well clearly, that would penalise higher standard deviations more, reducing the point at which standard deviation risk was negative.

Because the interaction of leverage and Kelly optimal is complex and will depend on exactly how close the initial asset is to the cutoff point, I'm not going to do more detailed analysis on this as it would be timeconsuming to write, and to read, and not add more intuition thatn the above. Suffice to say there is a reason why I usually assume we can get as much leverage as required!


The "price" of skew - with leverage

Now let's turn to skew (and let's also drop the annoying lack of leverage which makes our life so complicated). The question we now want to answer is "What is the price of skew: how many additional points of SR do we need to compensate us for a unit change in skew, assuming we can freely use leverage? And how does this change at different distributional points?". Returning to the debate that heads this post; is an extra 0.50 units of skew worth a 0.30 drop in SR when we go from continous to 'classical' trend following? We know that would only be the case if we were allowed to use a lot of leverage; which implies we were unlikely to be anything but a full Kelly optimising median distributional point investor. But at what distributional point does that sort of tradeoff become worth it?

To answer this, I'm going to recycle some code from this post and adapt it. That code uses a brute force technique to by mixing Gaussian returns to produce returns with different levels of skewness and fat tailed-ness, but with the same given Sharpe Ratio. We then bootstrap those returns at different leverage levels. That gives us a distribution of returns for each leverage level. We can then choose the optimal leverage that produces the maximum geometric return at a given distributional point (eg median for full Kelly, 10% to be conservative and so on). I then have an expected CAGR level at a given SR, for a given level of skew and fat tailness. By modifying the SR, skew and fat tailness I can see how the geometric return varies, and construct planes where the CAGR is constant. From that I can derive the price of skew (and fat tailness, but I will look at that in a momen) in SR units at different distributional points. Phew!

(Be prepared to set aside many hours of compute time for this exercise if you want to replicate...)


The "price" of skew: Kelly investor

Let's begin by looking at the results for the Kelly maximiser who focuses on the median point of the distribution when calculating their optimal leverage. 

The plots show 'indifference curves' at which the geometric mean is approximately equal. Each coloured line is for a different level of geometric mean. The plots are 'cross plots' that show statistical significance and the median of a cloud of points, as due to the brute force approach there is a cloud of points underneath.

Even then, there is still some non monotonic behaviour. But hopefully the broad message is clear; for this sort of person skew is not worth paying much for! At most we might be willing to give up 4 SR basis points to go from a skew of -3 to +3, which is a pretty massive range.



The "price" of skew: very conservative investor

Now let's consider someone who is working at the 10% quantile point.

If anything these curves are slightly flatter; at most the price of skew might be a couple of basis points. The intuition for this is that these people are working at much lower levels of leverage. They are much less likely to see a penalty from high negative skew, or much of a benefit from a high positive skew.


The "price" of lower tail risk: Kelly investor

Now let's consider the lower tail risk. Remember, a ratio of 1 means we have a Gaussian distribution, and a value above 1 means the left tail is fatter.


This may seem surprising; with a more extreme left tail it looks like you can have a higher SR. But the improvement is modest again, perhaps 5bp of SR at most.


The "price" of lower tail risk: 10% percentile investor

Once again, investors at a lower point on the quantile spectrum are less affected by changes in tail risk, requiring perhaps 3bp of SR in compensation.


How does the optimal leverage / skew relationship change at different percentiles?

As we have the data we can update the plots done earlier and consider how optimal leverage changes with skew. First for the Kelly investor:




Here each coloured line is for a different SR. We can see that for the lowest SR the optimal leverage goes from around 2.7 to 3.7 between the largest negative and positive skews; and for the higest from around 4.2 to 5.6. This is the same result as the last post: leverage can be higher if skew is positive, but not that much higher (from skew of -2 to +2 we can leverage up by around a third).

Here is the 10% investor:




The optimal leverage is lower as you would expect, since we are scaredy cats. It looks like the leverage range is higher though; for the highest SR strategies we go from around 1.7 to 2.8; a two thirds increase. And for the lower SR the rise in optimal leverage is even more dramatic. 


 

One final cut of the data cake

Finally another way to slice the cake is to draw different coloured lines for each level of skew and then see how the geometric mean varies as we change Sharpe Ratio. First the Kelly guy:


This is really reinforcing the point that skew is second order compared to Sharpe Ratio. Each of the bunches of coloured lines is very close to each other. At the very lowest SR at around 0.52 we only get a modest improvement in CAGR going from skew of -2.4 (purple) to +2.4 (red). We get a bigger improvement in CAGR when we add around 3bp of SR and move along the x-axis. Hence 5 units of skew are worth less than 3bp in SR. It's only at relatively high levels of SR that skew becomes more valuable; perhaps 5bp of SR for each 5 units of skew.


Here is the 10% person:


As we noted before there is almost no benefit from skew for the conservative investor (coloured lines close together at each SR point), except until SR ramps up. At the end 5 units of skew are worth the same as around 6bp of SR. 


Conclusion: Skew isn't as valuable as you might think

I started this post harking back to this question: is an extra 0.50 units of skew from 'traditional' trend following worth a 0.30 drop in SR? And the answer is, almost certainly not. The best price we get for skew is around 6bp for 5 units of skew. At that price, 0.5 units of skew should cost us less than 1bp in SR penalty. We're being charged about 50 times the correct price!!!

And this is for Kelly investors. For those with a lower risk tolerance, much of the time there is basically no significant benefit from skew.

That doesn't mean that you shouldn't know what your skew is, as it will affect your optimal leverage, particularly as we saw above if you are a conservative utility person (being such a person will also protect you if you think your skew or Sharpe ratio is better than it actually is, and that's no bad thing). And negatively skewed strategies at la LTCM with very low natural vol that have to be run at insane leverage will always be dangerous, particularly if you don't realise they are negatively skewed. 

But part of the problem with the original debate is a false argument by taking a true statement 'highly negatively skewed strategies are very dangerous with leverage' and extending it to 'you should be happy to suffer significantly lower Sharpe Ratio to get a marginally more positive skew' (which I have demonstrated is false). 

Anyway outside of that argument I think I have shown that to an extent the obsession with getting positive skew is a bit of an unhealthy one. Sure, get it if it's free, but don't pay much for it otherwise.