Wednesday, 7 November 2018

Is maths in portfolio construction bad?

First an apology. It's been quite a few months since my last blog post. I've been in book writing mode and trying to minimise outside distractions. Though looking at my media page since my last blog post I've done two conferences, a webinar, a book review, a guest lecture, a TV panel discussion and written six articles for So maybe I haven't done a great job as far as filtering out the noise goes.

Anyway the first draft of the book is now complete, and I though I'd write a blog post before I start smashing my next project (updating my course material for next semester, since you ask).

This post is about portfolio construction and is a response to the following article:

It's long but absolutely worth reading. To sum up the salient points, as I see them (and I apologise to the authors if I am misrepresenting them):

  • The "Diversification industry" is a con, including but not limited to portfolios labelled: risk efficient, maximum diversification, minimum variance, equal-risk contribution, inverse volatility-weighted, diversity-weighted, and so on.
  • Equal weighting is as good or better than any of this garbage
  • The 'rebalancing premium' is junk.
  • Maximum diversification type portfolios are often highly concentrated
  • By seeking to minimise market beta maximum diversification portfolios end up with the weirdest stocks
  • Low correlation is no protection against a crisis
  • These strategies naturally tilt towards small cap and value. And the authors have a particular problem with the small cap part of that.
  • Summary of the summary: 
    • "Maximum diversification" - bad. "Market cap weighted" - good.
I agree with a fair bit of the content of this article, and I'm a massive fan of people smashing their sticks into the piƱata of received industry wisdom. And it's true to say that many people are charging unreasonably high fees for something that can be done mechanically using publicly available methodologies.
But just because people use certain methodologies badly doesn't mean they are bad methodologies. I guess if I was to sum up what I'm now going to say it in a pithy way it would be: Maths in portfolio construction is fine if you don't use it naively. 

Before I start, two minor irritants:
  • At one point the authors conflate the idea of equal Sharpe Ratios, and CAPM. Not the same thing. Both assume risk adjusted returns are identical, but they use different measures of risk: absolute standard deviation and covariance with the market respectively.
  • It's Maths. With an 's'. Bloody Americans :-)

A quick primer on uncertainty and portfolio construction

Before we begin in earnest let's have a recap (for those that haven't read my second book, attended one of my talks or webcasts on the subject, or been my unfortunate victims students in the Queen Mary lecture theatre).

  • Classic portfolio optimisation is very sensitive to small changes in inputs; in particular it's very sensitive to small differences in Sharpe Ratio, and correlations - when those correlations are high. It's relatively insensitive to small changes in standard deviation, or to correlations when correlations are low.
  • It's extremely difficult to predict Sharpe Ratios, and their historic uncertainty (sampling variance) is high. It's relatively easy to predict standard deviations, and their historic uncertainty is low. Correlations fall somewhere in the middle.
  • If we assume we can't predict Sharpe Ratios, then some kind of minimum variance (if we have a low risk target or can use leverage) or maximum diversification portfolio will make theoretical sense
  • If we assume we can't predict Sharpe Ratios or correlations, then an inverse volatility portfolio makes the most sense
  • If we assume we can't predict anything, then an equal weight portfolio makes the most sense.

Yes, Equal weights and Market cap weighting are as good as anything... in the right circumstances

Like I said above these portfolios will make the most sense in theory if you can predict vol and correlation, but not Sharpe Ratios.

[Let's take it as a given for now that we can't predict Sharpe Ratios; in other words that all assets have equal expected Sharpe - I'll relax that assumption later in the post] 

If you can't predict volatility or correlation either, then equal weights makes more sense. Equal weights will also make sense if volatility and correlations are pretty much the same, or if vols are the same and the correlation structure is such that the portfolio can be decomposed into blocks, like sectors, of broadly equal size with homogeneous correlations. Something like this:

                    BankA         BankB        TechA        TechB
BankA                1.0           0.8           0.4         0.4
BankB                0.8           1.0           0.4         0.4
TechA                0.4           0.4           1.0         0.8
TechB                0.4           0.4           0.8         1.0

Assuming equal vol and Sharpe the correct portfolio to hold here (lowest risk, highest Sharpe Ratio, highest geometric mean) would be equal weights. This would also be the minimum variance / maximum diversification / equal risk portfolio  ... blah blah blah.

The question then is how realistic is to assume equal variances, equal correlations or an 'equal block' correlation structure. Actually for something like the S&P 500 it's not a bad assumption to make. 

It's also worth noting that for a relatively well diversified index like the S&P 500 there isn't going to be a huge amount of difference between market cap weighted, maximum diversification, and equal weighted. In my second book I ran the numbers for the Canadian TSX 60 index; a relatively extreme index by the standards of developed markets:

  • Equal weighted. This portfolio is quite concentrated: 58% is in just three sectors.
  • Market cap. This portfolio is also quite concentrated by firm: 9% in one firm, and even more so in sectors: 62% in just three sectors.
  • Equal risk across sectors (a sort of maximum diversification portfolio). This portfolio is quite concentrated by firm: 10% in one firm
Although these portfolios are quite different the expected geometric mean, assuming equal vol and Sharpe Ratio, come in very similar: 
  • 2.17% market cap weighted
  • 2.20% equal weighted
  • 2.21% equal sector risk 

(All excess returns based on my central assumptions about future equity growth - but it's the relative values that's important). 

4 lousy basis points. Let me remind you that's for the quirky Canadian index, which judging by the sector concentration in their stock index is a country where everyone is busy digging stuff up, drilling for other stuff, or flogging financial products to the diggers and drillers. For the naturally more diversified S&P 500 index there will be a negligible difference.

If your universe of assets is a large cap index of developed markets in the same country.... then there is almost no value theoretically in moving away from market cap or equally weighted towards some sort of funky 'diversified' weighting. 

It's no coincidence that the seminal work on equally weighted portfolios was done on US equities

But not every universe of assets has those characteristics. A cross asset, cross country portfolio is likely to have a much messier correlation matrix, and is extremely unlikely to have equal volatility. An emerging markets index or a small concentrated index like the DAX or OBX is going to produce equal weight or market cap portfolios that have serious concentration issues. Because volatility, and to a lesser degree correlation, are reasonably predictable it would be silly to throw away that kind of information if you're working in that context.

We should all hate sparse weights 

There is a world of difference between the theoretical results above, obtained by setting everything to equality, and what happens when you push real data into an optimiser. Even slight differences in Sharpe Ratios, correlations or volatility can produce extreme weights (also named sparse weights, depending on whether it's the zeros or the higher values that bother you).

Again from my second book: if I consider sector weighted S&P 500 portfolios then the difference between holding 11 stocks (one per sector) and holding all 500 stocks is a geometric mean of 2.18% versus 2.23% (again assuming equal volatility and Sharpe Ratio). Just 5 basis points. Statistically completely insignificant- surely any idiot could get those 5 basis points back and more by smart stock picking.  Here is the same point made by Adam Butler at the end of my recent webcasts

“… while mean variance optimisation is unstable in terms of portfolio weights it’s actually quite stable in terms of portfolio qualities… with quite different portfolio weights the means and variances are very similar” Adam Butler, ReSolve asset management 2018

My answer was that by having extreme allocations, or sparse weights, we're exposing ourselves to idiosyncratic risk. In theory this is being mostly diversified away leaving us only with systematic market risk, but this is one theoretical result I am very unhappy about. The joint Gaussian model of risk is a pretty good workhorse but we all know it's flawed; and the consequences of those flaws will become very evident if you're holding a sparse portfolio in a crisis situation when co-skewness type behaviour becomes apparent, or if certain asset prices to go to zero (firms do, occasionally, go bust).

So I'm certainly no friend to sparse weights; I think the theoretically small advantage of 'fully populated' portfolios is in reality much bigger. Of course there is a limit; I'd probably only be happy if I held all 30 DAX stocks, but I don't see the need to hold all five hundred S&P 500 stocks. Assuming the portfolio is reasonably well diversified I don't see the harm in holding only 100 out of 500 stocks. 

Solving this problem is straightforward - you can do it in an ugly way with constraints, or you use any of the well known techniques that do optimisation more robustly.

As a professional fund manager of course there is another issue, which is that if you don't have any exposure to a high profile, high performing stock then you're going to look pretty silly. Perhaps then you should add something so that you always have some exposure to say the top 10% of stocks by market cap (I'm only half joking here).

The dangerous world of low or negative correlations, and weird factor risk

I like to think of risk management as a waterbed. If you try and reduce your risk too much by in one area, then it will pop up in an unexpected place. This is most notable in long short portfolios of highly correlated assets, or of leveraged long only portfolios of negatively correlated assets. 

It's possible to construct portfolios with very low risk - low risk at least if you assume that a joint Gaussian risk model is correct, and that volatility and correlations are perfectly stable and predictable. But they're not. By pushing down the risk on the Gaussian part of the waterbed we're forcing the risk to pop up somewhere else. We're exposed to correlation risk, and probably liquidity risk (One word, four letters: LTCM).

These problems still exist in the land of unleveraged long only portfolio construction- but they aren't as serious. Any long only portfolio can be decomposed into some other long only portfolio, plus a bunch of long/short bets. The long/short stuff can indeed be dangerously toxic. But it's only part of the portfolio.

Also: this happens because we allow the mean variance optimiser to use the correlation matrix naively. Just because correlations are relatively predictable it doesn't mean we should trust the optimiser to use them sensibly. We don't have to do that; there are many techniques for more robust optimisation. 

The rebalancing premium

There are two rebalancing premiums; a theoretical one which is small, but definitely exists; and an empirical one. I believe the theoretical premium was first outlined by Fernholz and Shay. It definitely exists, but may not survive the impact of costs. 

An additional (and probably much larger) empirical rebalancing premium will exist if asset prices were mean reverting in some relative sense. Then you could sell high on overweight assets, and buy underweight assets cheaply. 

In case you haven't noticed this is the point in the post when I relax the assumption that Sharpe Ratios are inherently unpredictable, and hence that all Sharpe Ratios are equal in expectation; now we have some conditioning information which can predict Sharpes.

Historically most assets have exhibited the following pattern:
  • Short horizon, mean reverting
  • Medium horizon, trending
  • Long horizon, mean reverting
The horizons vary depending on the asset class (and have also done over time), but if you're operating in a time frame between about a week and a year you are probably in the trending zone, where rebalancing doesn't work. Unfortunately that's also a pretty neat fit for the typical frequencies when most people rebalance: monthly, quarterly, annually. So you are likely to do worse by rebalancing unless you speed up (which will end up costing more, unless you do it smartly with the use of no-trade buffers and limit orders) or slow down a lot (in which case your information ratio will take a nose dive).

To labour the point, that doesn't mean rebalancing is a bad idea, in the same way that maximum diversification portfolios aren't always a bad idea. It's just bad if you do it in a dumb way - the way many parts of the fund industry do it.

The arguments against small cap and value

'Diversified' portfolios of all flavours start from the premise that all stocks are equal until proven otherwise (due to information about correlation or volatility), whereas market cap weighting thinks that larger cap stocks are better (of course equal weighting thinks that all stocks are always equal). So yes, anything that isn't market cap weighting will have a tilt towards small cap relative to the market cap index. 

Of course another way of putting this is that market cap weighting has a tilt towards large cap stocks relative to any other index. It depends on your perspective - there is no 'true' benchmark.

The choice of market cap weighting as a starting point is historical, and it's justification is that it's the portfolio that all investors have to hold in aggregate. Of course that's true... but then I look at the FTSE 100 weights and I have to ask myself if I really expect HSBC to outperform RBS (both UK banks) to the extent that it deserves a weight that is 4 times bigger?

[Weirdly I own shares in HSBC but not RBS. Go figure]

Frankly if your universe is the S&P 500 or the FTSE 100, then you're not really not tilting towards small caps. You're tilting towards 'not quite so large' large caps. So the well known reasons why one might expect genuine small caps to outperform (fewer analysts covering, higher trading costs, less liquidity) are unlikely to be present. 

Some benefit of market cap weighting is that it tilts towards stocks that have done well recently, so benefiting from trend effects that occur in line with quarterly rebalancing, but this is quite a small effect which mainly works on the margins (when stocks are promoted into the index that have recently done well), and is certainly outweighed by the slower effects of mean reversion (in the long run stocks which have done well - and are more likely to be in the top end of a large cap index - will do worse versus the rest).

Bottom line - I don't think there is any evidence one way or the other that a massive megacap stock should outperform a relatively small large cap or vice versa. So I see no reason why the massive megacap stock should automatically get a higher portfolio weight, or vice versa. Inverse volatility weighting - as practiced by all the 'esorteric' weighting schemes apart from equal weighting - will probably underweight the smaller large caps in cash terms since they're normally riskier. Since volatility is the most predictable characteristic of asset returns I'm a big fan of using it in portfolio construction. On this point then equal weighting falls over compared to weighting schemes that use volatility as an input.

Arguably using volatility is a very crude way of measuring risk, and it might be that it understates the tail risks. Evidence suggests that small cap stocks have more positive skewness, but worse kurtosis. I'm not sure this is a significant issue.

Genuine small caps will probably outperform for good reason, but that's not really what we're talking about here.

I'm slightly more confused about where value comes in, as the original authors weren't clear. Of course there is an overlap between value and small cap; you're unlikely to find much value in stocks that are well covered by analysts, and which may also have gone up in price a lot recently (although not always; the #10 stock by market cap could well be the #1 stock that has fallen on hard times). 

[And there is no reason why megacaps can't be good value in their own right. According to my value screens HSBC - currently #1 in the FTSE 100 by market cap - is much better value than RBS. I knew there was a good reason why I owned HSBC]

But to make the same point again; if higher cap stocks are poorer value then a market cap weighted index will be tilted towards poorer value relative to any other form of weighting. You might not buy the value premium, but do you really believe an anti-value premium exists? There would need to be such a premium for market cap weighting to make sense compared to pretty much any other type of weighting. Once again higher value stocks are probably going to be more volatile, but using information about volatility will deal with that.

Yes it is disingenuous to smuggle value and small cap bias into an index that is ostensibly something else, but I really don't think that is happening here. Market cap indices are the biased ones - with tilts towards large cap, and perhaps value, versus any other kind of weighting.


Yes, the naive use of portfolio construction methods is dumb. Yes, equal and market cap weights will often do as good a job. Yes, letting your optimiser set sparse / extreme weights is stupid. But:

  • Outside of the universe of large cap stock indices it makes a lot of sense to use the relatively predictable components of asset returns - volatility and correlation.
  • Using volatility as an input makes a lot of sense - it's highly predictable, and will help reduce your exposure to potentially problematic assets.
  • Plenty of well known techniques exist to do portfolio optimisation in robust sensible ways.
  • Sensible rebalancing can be fun and profitable
  • It's market cap weighted indices that are biased, and not in a direction that's likely to be profitable.
I also can't help feeling that now would be a good opportunity to plug next weeks talk I'm doing on portfolio construction with uncertainty, at The Thalesians.

Monday, 9 July 2018

Vol Targeting and Trend Following

  • We are long.
  • The price jumps up. Good.
  • But this means the risk goes up
  • So cut our position, just as we're finally making serious money. 

How can this make sense?

This is a post about volatility targeting - dynamically adjusting your positions according to your estimate of market volatility - in the context of trend following systems. I blindly do this when building all my trading strategies without thinking about it - but is this a good thing to do?

It's also a post about how you generally have to balance different criteria when judging backtests - there are no free lunches in finance.

What do we mean by volatility targeting?

It's not obvious whether "Volatility Targeting" is referring to the practice of scaling positions by volatility for a given level of conviction or targeting a constant portfolio or position level volatility regardless of conviction. 

To be clear for the unfamiliar the way I run a CTA style strategy is:

1- Decide on a level of conviction
2- From that infer the volatility target for a given position
3- Scale the position to a given volatility target

There is then another stage that some people do which I don't agree with:

4-  Rescale the leverage in the entire portfolio to some fixed target.

Essentially doing the last thing will throw away the absolute average level of conviction that you have across your system. I don't think you should do this, although it's very popular in eg equity neutral portfolios (mainly for historical reasons, take a bow Fama and French). To make things confusing this is sometimes called "Vol targeting". But it's not what I'm discussing in this post (maybe another one).

Another thing some people do is run a binary system, in which the level of conviction is essentially fixed. Doing this will throw away all the information you have about conviction, both absolute and relative. Again I think this is sub optimal.

To be clear then what I am defending here is stage 2: the scaling of positions to a given volatility target, irrespective of whether conviction was involved. It's implicit here that the volatility target is dynamic, otherwise what you're doing is just some kind of long run risk budgeting exercise.

Why do people like trend following

Trend following is considered a nice thing, because it's return profile is:

  • a majority of time periods when we have small lossess
  • a minority of time periods when we have big gains. Generally these come when other asset classes are suffering

This sort of return distribution will contain both positive skew (at least when measured at an appropriate time interval - at least monthly if not annually) and high kurtosis. Skew is an asymmetric measure of return 'non-Gausianness' (if that's a word), whereas kurtosis is a symmetric measure - it just means we have 'fat tails', without specifying which tail we're talking about.

Positive skew is generally agreed to be a good thing (to own a negatively skewed asset I'd want paying, in the form of higher expected Sharpe Ratio), but high kurtosis is generally agreed to be a bad thing, because it means we're going to get surprisingly large returns on both the up and down side. It makes no sense to talk about kurtosis that only existed on the right hand side of the distribution.

But a combination of positive skew and more kurtosis will give you more mass on the right hand side.

A possible case for not vol targeting

Some people don't like vol targeting because they think it degrades the nice property of trend following: that extra mass on the right hand tail.

Essentially not vol targeting will make sense if there is an asymmetric effect in the markets: where we tend to cut our positions on vol spikes in winning positions more than we do on losing positions. This indeed would lower your skew, and this would indeed be a bad thing. It would be better to stop vol targeting, and be rewarded with higher returns on winning positions, even after taking into account the higher losses on losers.

To be clear if you could get higher positive skew for free this would be a good thing. However if you have to pay for your higher positive skew with higher kurtosis then that wouldn't be so good. But intuitively removing vol targeting will mean worse kurtosis - vol targeting will tend to trim the tails of both sides of the distribution. This also ignores the first two moments of the distribution: if higher positive skew means a worse Sharpe Ratio would I be happy?

In general terms then it's unlikely that you can get positive skew for free without giving up something else: kurtosis or Sharpe Ratio. There are plenty of situations when this sort of trade off is present - for example you can boost your Sharpe Ratio by consistently selling option vol, but that give you rather unpleasant kurtosis and make your skew more negative.

All this is a theoretical discussion - let's see what actually happens to the moments of the return distribution when we remove vol targeting.

Empirical evidence

It's relatively easy to test this sort of thing with pysystemtrade. Here's an account curve for 37 futures markets using the system in chapter fifteen of my first book (with carry removed, since the original article was about trend following), and also the monthly distribution of returns:

Account curve with vol targeting
Distribution of monthly returns with vol targeting

This is a system which vol targets using the last month or so of returns. Vol targeting also increases costs, and all the analysis in this post are done after costs.

Now for the counterfactual. It's actually quite hard to 'turn off' vol targeting as it's not obvious what you'd replace it with: would you for example give all markets the same cash position and ignore vol completely? That would lead to some very distorted results indeed! I decided to continue to use vol to scale positions, but a very long term vol which didn't move around for each market; so basically cross sectional vol budgeting, without the time series adjustment to vol. I went with this set of config changes:


In plain english we will:

  • Calculate the vol over the first 4 years of data (because I only have about 4 years of data for many instruments)
  • Backfill and use that vol for the first 4 years (so forward looking, but <shrugs>)
  • After that use a very slow moving average of vol (half life of 30 years)
This is as close to fixed vol as you can get. Here's the account curve and the distribution:

Account curve without vol targeting

Distribution of monthly returns without vol targeting

Well the account curve clearly isn't as good. The distribution is harder to read: it looks like there are some outliers that weren't there before on both the left and right tails.

Here are some statistics that reinforce this result (all based on monthly returns):

                    With vol targeting          Without
Skew                    +1.08                    +2.46
Sharpe                   0.92                    0.569
Sortino                  1.62                    0.867
Min return             -32.6%                   -55.6%
Max return             +47.7%                  +100.6%
Kurtosis                 5.28                    33.0
1% point                -16.5%                  -18.1%
99% point               +31.8%                  +30.2%

To summarise then removing vol targeting leads to:

  • Higher skew
  • Worse Kurtosis
  • Worse Sharpe Ratio

Now depending on your utility function you might argue this is a trade worth taking. If you cared about Skew above all else then maybe you'd accept this deal. Personally I wouldn't take this deal, but you might have a very strange utility function indeed.

But... and there is a big but here... I'm not sure how significant these results are. Skew and Kurtosis are like anything else statistical estimators, which means they are subject to uncertainty, and they're also subject to being affected by a couple of outliers.

(By the way a formal T-test on the Sharpe Ratio difference in the curves has a statistic of 3.59, so the difference is indeed significant to something like 99.97%)

If we use a more robust measure of left and right tail - the 1% and 99% points on the distribution of returns - we can see that removing vol targeting leads to slightly worse outcomes on the left tail (1%), and more surprisingly a slightly worse outcome on the right tail as well (99% point). We were sold no vol targeting as a product to improve our right tail, and we don't see it.

This strongly suggests that the skew and kurtosis numbers are being heavily driven by one or two outliers.

Formally if we bootstrap the distribution of skew for each curve we get this:
Distribution of monthly skew estimate with vol targeting
And this without vol targeting:

Distribution of monthly skew estimate without vol targeting
Notice the much wider range of uncertainty, and the weird bimodal distribution, characteristic of a statistic that is being driven by one or two outliers.

How do we get round this? Well both the largest positive and negative returns occur in 1979 - 1980; when there weren't many instruments trading in the data. Let's recheck the statistics, but this time ignoring everything before January 1981; this is still over 36 years of data:

                    With vol targeting            Without

Skew                    +0.45                    +0.64
Sharpe                   0.78                    0.52
Min return             -26.5%                   -25.4%
Max return             +33.4%                   +33.6%
Kurtosis                 3.12                    3.9
1% point                -16.3%                  -14.2%
99% point               +22.2%                  +22.0%

The improvement in Skew, and worsening Kurtosis, are both still there but nowhere near as dramatic. The minima and maxima, and 1% / 99% points, are almost identical. It looks like there might be a slight improvement in the left tail without vol targeting, and a slight worsening in the right tail - which is the opposite of what we'd expected - but the values are not significantly different. And, sadly, the drop in Sharpe Ratio is still present (and it is still very significant).


On the face of it vol targeting does indeed seem to remove some of the positive skew from trend following. But there are a few caveats:

  • The improvement in Skew can be heavily influenced by one or two outliers in the data
  • It looks like the improvement in Skew doesn't in fact lead to a better right tail
  • The kurtosis is definitely worse, although again this could be influenced by outliers; taking these out the degradation in Kurtosis is still there but not as dramatic
  • There is a substantial reduction in Sharpe Ratio, with or without outliers

So yes, maybe, there is something in the idea that vol targeting involves giving up some of the positive skew that trend following gives you, at least with monthly data. But the cost is terribly high: about a third of our Sharpe Ratio! This is the old 'no free lunch in finance' idea - we can improve one moment of our return distribution, but it usually involves giving something up. Another word for this is the 'waterbed' effect - when we push down on the skew part of our waterbed to ensure a better nights sleep, the water just moves somewhere else (the kurtosis and Sharpe Ratio parts of the bed).

I can't help thinking there are cheaper ways of getting positive skew; like maybe buying some out of the money straddles as an overlay on your trading system.

Finally, it's also worth reading this recent paper by my old shop, AHL, which goes into more detail on this subject.

Acknowledgements - I'd like to thank Mark Serafini who accidentally inspired this blog post with a LinkedIn post that turned out to be on an entirely different topic, and Helder Palaro who found that post for me.

Friday, 8 June 2018

Kelly versus Classical portfolio theory, and the two kinds of uncertainty premium

Since I was a young lad there has been an ongoing fight in Financial Academia 'n' Industry between two opposing camps:

  • In the red corner are the Utilitarians. The people of classical finance, of efficient frontiers, of optimising for maximum return at some level of maximum risk.

  • In the blue corner are the Kellyites. Worshipping at the feet of John Kelly and Ed Thorpe they have only one commandment in their holy book: Thou shalt maximise the expectation of log utility of wealth.

This post is sort of about that battle, but more generally it's about two different forms of uncertainty for which humans have varying degrees of stomach for, and how they should be accounted for when deciding how much volatility your trading or investment portfolio should have: "Risk" (which we can think of as known unknowns, or at least the amount of volatility expected from a risk & return model which is calibrated on past data) and "Uncertainty" (which we can think of as unknown unknowns, or to be more precise the unknowability of our risk & return model).

The Kellyites deny the existence of "risk appetite" (or at least they deny it's importance), whereas the Utilatarians embrace it. More seriously both camps seriously underestimate the importance of uncertainty; which will be more the focus of this post.

This might seem somewhat esoteric but in laymans term this post is about answering an extremely critical question, which can be phrased in several equivalent ways:

  • What risk target should I have?
  • How much leverage should I use?
  • How much of my capital should I bet on a given position?
This post was inspired by a twitter thread on this subject and I am very grateful to Rob Hillman of Neuron Advisors for pointing me towards this. I've blogged about this battle before, here (where I essentially address one interesting criticism of Kelly) and I've also talked about Kelly generally here

If you're unfamiliar with arithmetic and geometric returns it's probably worth rereading the first part of this post here, otherwise you can ignore these other posts (for now!).

Classic Utilitarian portfolio optimisation

To make live easier I'm going to consider portfolios of a single asset. The main difference I want to highlight here is the level of leverage / risk that comes out of the two alternatives, rather than the composition of the portfolio.

I'm sure the readers of this blog don't need reminding of this but basically Utilitarians tend to do portfolio optimisation like this: specifying the investors utility function as return minus some penalty for variance. Which for a single asset with leverage, if we assume the risk free rate is zero or that the return is specified as an excess return (it doesn't matter for the purposes of this post) becomes this:

Maximise f.E(r) - b*[f.E(s)]^2

Where f is the leverage factor (f=1 incidentally means fully invested, f=2 means 100% leverage, and so on), r is the expected return, E is the expectation operator, risk s is measured as the standard deviation of returns on the unleveraged asset, is the coefficient of risk aversion (you'll often see 1/2 here, but like, whatever). This is a quadratic utility function. In this model risk tolerance is an input, here defined as a coefficient of aversion.

Of course we could also use a different utility function, like one which cares about higher moments, but that will probably make the maths harder and definitely mean we have to somehow define further coefficients establishing an investors pain tolerance for skew and kurtosis.

This specification isn't so much in fashion in industry; it's hard enough getting your investors to tell you what their risk appetite is (few people intuitively understand what 150% standard deviation a year feels like, unless they're crypto investors or have money with Mr C. Odey). Imagine trying to get them to tell you what their coefficient of risk aversion is. Easier to say "our fund targets 15% a year volatility" which most people will at least pretend to understand. So we use this version instead:

Maximise portfolio returns = weights.E(asset returns)

Subject to: portfolio risk = function of weights and E(asset covariance)<= maximum risk

Which for one asset, with no risk free rate is:

Maximise f.E(r)
Subject to f.E(s)<=s_max

Where s_max is some exogenous maximum risk tolerance specified by the investor

Importantly (a) maximum risk depends on the individual investors utility function (which is assumed to be monotonically increasing in returns up to some maximum risk at which point it drops to zero - yeah I know, weird) and (b) the return here is arithmetic return (c) we only care about the first two moments since risk is measured using the standard deviation. Again risk is an input into this model (as a tolerance limit this time, rather than coefficient), and the optimal leverage comes out.


Under Kelly we choose to find the portfolio which maximises the expectation of the log of final wealth. For Gaussian returns (so again, not caring about the 3rd or higher moments which I won't do throughout this post) it can be shown that the optimal leverage factor f* is:

f* = (r - r_f) / s^2

(If you aren't in i.i.d. world then you can mess around with variations that account for higher moments, or just do what I do - bootstrap)

Where r is the expected arithmetic portfolio mean, r_f is the risk free rate and s is the standard deviation of portfolio returns without any leverage (and with expectations operators implicit - this is important!). f=1 incidentally means fully invested, f=2 means 100% leverage, and so on. Noting that the risk of a portfolio with leverage will be f*s this means we can solve for the target risk s*:

s* = f*. s = (r - r_f) / s

Notice that this thing on the right is now the Expected Sharpe Ratio. This is my favourite financial formula of all time: optimal Kelly risk target = Expected Sharpe Ratio. It has the purity of E=mc^2. But I digress. Let's take out the risk free rate for consistency:

f* = r  / s^2

s* = r  / s

Importantly for the battle in this world we don't specify any risk tolerance, or coefficient of risk aversion, or utility function. Assuming an investor wants to end up with the highest expected log utility of final wealth (or as I said here, the highest median expectation of final wealth) they should just use Kelly and be done with it.

The battle

Let's recap:

Kelly:       f* = r / s^2      s* = r / s
Utilitarian: f* = s_max / s    s* = s_max

With no risk free rate; f*= optimal leverage, s* = optimal risk, s_max is maximum risk (both standard deviations) and r = expected return.

Importantly the two formula don't usually give the same answers (unless r /s = s_max; i.e. the Sharpe Ratio is equal to the risk tolerance), and the relative answer depends on the Sharpe Ratio you're using versus typical risk appetite.

CASE ONE: Kelly leverage< Utilitarian leverage

If you're investing in a long only asset allocation portfolio then a conservative forward looking estimate of Sharpe Ratio (like those in my second book) would be about 0.20. If we assume expected return 2% and standard deviation 10% then the optimal Kelly risk target will be 20%, implying a leverage factor of 2. But that only gives 4% return! If the utilitarian investor is rather gung ho and has a risk appetite of 30% then the optimal leverage for them would be 3.

CASE TWO: Kelly leverage > Utilitarian leverage

If you're running a sophisticated quant fund with a lot of diversification and a relatively short holding period then a Sharpe Ratio of 1.0 may seem reasonable. For example if a stat-arb equity neutral portfolio has expected return 5% and standard deviation 5% (assuming risk free of zero) then the optimal Kelly risk target will be 100%, which implies a leverage factor of 20 (!). But a utilitarian investor may only have a risk appetite of 15%, in which case the optimal leverage will be 3. And indeed most equity neutral funds do run at leverage of about 3.

For case one I refer you back to my previous post, in which I said that nobody, no matter what their risk appetite should invest more than Kelly if you believe my logic that it is MEDIAN expected portfolio value that matters rather than mean. Essentially where the utility optimisation gives you a higher leverage than Kelly you should ignore it and go with Kelly.

Case two is a little more complicated, and the solution quoted in a thousand websites and papers is "Most investors find the risk of full Kelly to be too high - we recommend they use half Kelly instead". Frankly this is a bit of a cop-out by the Kelly people, which admits to the existence of risk appetite.

The compromise

I personally believe in risk appetite. I believe that people don't like lumpy returns, and some are more scared of them than others. Nobody has the self discipline to invest for 40 years and completely ignore their portfolio value changes in the interim.

But I also believe that using more than full Kelly is dangerous, insane, and wrong.

So this means the solution is easy. Your risk target should be:

s* = Minimum(r / s , s_max)

Where the first term is of course the Kelly optimum without the risk free, and the second is the risk tolerance beloved of utilitarian investors. And your leverage should be:

f* = Minimum(r / s^2, s_max / s)

What we know, and what we don't know

In case you haven't noticed I find this battle a little tiresome (hence my pretty superficial attempt at 'solving' it), and mainly because it completely ignores something incredibly important. We have two guys in the corner of a room arguing about whether Margin Call or The Big Short is the best film about the 2008 financial crisis (a pointless argument, because both v. good films), whilst a giant elephant is in the corner of the room. Running towards them, about to flatten them. Because they haven't seen it. They're too busy arguing. Have I made the point sufficiently, do you think?

What is the elephant in this particular metaphorical room. It's this. We don't know r. Or s. Of the Sharpe Ratio, r/s (ignoring risk free of course). And without knowing these figures, we don't have a hope in hell of finding the right leverage factor.

We have to come up with a model for them, based on historical data, because that's what quant finance people do. Which means there is the risk that:

  • It's the wrong model (non Gaussian returns, jumps, autocorrelation...)
  • The parameters of the model aren't stable
  • The parameters of the model can't be accurately measured
This triumvirate of problems should be recognisable to people familiar with my work, and you already know that I feel it is most productive to focus on the third problem for which we have relatively straightforward ways of quantifying our difficulties (using the classical statistical workhorse of sampling distribution).

Parameter uncertainty (and the other issues) isn't such an issue for standard deviation; we are relatively good in finance at predicting risk using past data (R^2 of regressions of monthly standard deviation on the previous month is around 0.6, compared to about 0.01 for means and Sharpe Ratios). So let's pretend that we know the standard deviation.

However the Sharpe Ratio is the key factor in working out the optimal leverage and risk target for Kelly (which even for Utilitarians should act as a ceiling on your aspirations). The sampling distribution of Sharpe Ratio is highly uncertain.

The effect of parameter uncertainty on Sharpe Ratio estimates

There is an easy closed form formula for the variance of the Sharpe Ratio estimate under i.i.d returns given N returns:

w = (1+ 0.5SR^2)/N

(If you aren't in i.i.d. world then you can mess around with formulas that account for higher moments, or just do what I do - bootstrap)

We need an example. Let's just pick an annual Sharpe Ratio out of the air: 0.5. And assume the standard deviation is 10%. And 10 years of monthly data. But if you don't like these figures feel free to play with the example here in google docs land (don't ask for edit access - make your own copy).

Here is the distribution of our Sharpe Ratio estimate:

The concept of "Uncertainty appetite"

Now let's take the distribution of Sharpe Ratio estimate, and map it to the appropriate Kelly risk target:

Yeah, of course it's the same plot, since s* = r/s = SR. You should however mentally block off the negative part of the x-axis, since we wouldn't bother running the strategy here and negative standard deviation is meaningless. And here is the plot for optimal leverage (r/s^2):

So to summarise the mean (and median, as these things are Gaussian regardless of the underlying return series) optimal risk target is 50% and the mean optimal leverage is 5. Negative leverage sort of makes sense in this plot, since if an asset was expected to lose money we'd short it.

At this point the Kellyites would say "So use leverage of 5 or if you're some kind of wuss use half Kelly which is 2.5" and the Utilitarians might sniff and say "But my maximum risk appetite is 15% so I'm going to use leverage of 1.5". Since the optimal standard deviation of 50% is relatively high it's very likely that we'd get a conflict between the two approaches here.

But we're going to go beyond that, and note that there is actually a lot of uncertainty about what the optimal leverage and risk target should be. To address this let's introduce the concept of uncertainty appetite. This is how comfortable investors are with not knowing exactly what their optimal leverage should be. It is analogous to the more well known risk appetite, which is how comfortable investors are with lumpy returns.

Someone who is uncertainty blind would happily use the median points from the above distributions- they'd use full Kelly, assuming of course that their risk appetite wasn't constraining them to a lower figure. And someone weird who is uncertainty loving might gamble and assume that the true SR lies somewhere to the right of the median, and use a higher leverage and risk target than full Kelly.

But most people will have a coefficient of uncertainty aversion (see what I did there?). They'll be uncomfortable with full Kelly, knowing that there is a 50% chance that they will actually be over gearing. We have to specify a confidence interval that we'd use to derive the optimal leverage, with uncertainty aversion factored in.

So for example suppose you want to be 75% sure that you're not over-geared. Then you'd take the 25th percentile point off the above distributions: which gives you an expected Sharpe Ratio of about 0.29, a risk target of 29% and optimal leverage of 2.9.

Here are a few more figures for varying degrees of uncertainty:

Confidence interval         Optimal risk       Optimal leverage 

       <5.8%                     Don't invest anything
       10.0%                   9.2%                 0.93
       15.0%                  17.1%                 1.71
       20.0%                  23.2%                 2.32
       30.0%                  33.3%                 3.33
       40.0%                  41.9%                 4.19
       50.0%                  50.0%                 5.00 

Incidentally the famous half Kelly (a leverage of 2.5) corresponds to a confidence interval of about 22%. However this isn't a universal truth, and the result will be different for other Sharpe Ratios.

What this means in practice is that if you're particularly averse to uncertainty then you'll end up with a pretty low optimal Kelly risk target. How does this now interact with risk appetite, and the Utilitarian idea of maximum risk tolerance? Well the higher someones aversion to uncertainty, the lower their optimal risk target will be, and the less likely that an exogenous maximum risk appetite will come into play.

Now someone who is averse to uncertainty will probably also be averse to the classical risk of lumpy returns. You can imagine people who are uncertainty averse but not risk averse (indeed I am such a person), and others who are risk averse but not uncertainty averse, but generally the two probably go together. Which also raises an interesting philosophical point about the difference between them, as we'd rarely be able to distinguish between the two kinds of uncertainty except in specific experiments or unusual corner cases.


Both the Kellyites and the Utilitarians have good points to make - you should never bet more than full Kelly no matter how gung ho you are, and risk appetite is actually a thing even if few investors really know have quadratic utility functions.

But both are missing the real point, which is that there is a lot of uncertainty about what the Sharpe Ratio and hence optimal leverage really is. Assuming some conservatism and a degree of uncertainty appetite this produces a Kelly optimal revised for uncertainty which will be lower than the uncertainty blind full Kelly. This then makes risk appetite less relevant as a constraint, and the whole battle becomes a moot point.

Monday, 16 April 2018

Trading performance - year four

Time flies, and it's time for another annual update on the performance of my own investment and trading. Previous updates can be found here, here and here. These updates follow the UK tax year; from 6th April to 5th April, as I have to do my taxes anyway it makes sense to analyse everything at the same time.

Following the mind numbing detail of the performance analysis there are some concluding thoughts on life, the universe, and everything pertaining to systematically investing / trading for a living.

Investments and benchmarking

My investments fall into the following categories:

  1. In my investment accounts:
    1. UK stocks
    2. Various ETFs, covering stocks, bonds, and gold
  2. In my trading account:
    1. Various ETFs, covering stocks and bonds
    2. A futures contract hedge against those long only ETFs in 2.1, so that the net Beta is around zero
    3. Futures contracts traded by my fully automated trading system
    4. Cash needed for futures margin, and to cover potential trading losses (there is also some cash in my investment accounts, but it's pretty much a rounding error)

I'm excluding from this analysis the value of our house (and any debt secured against it), defined benefit pensions, and my 'cash float' - roughly 3 months of household expenditure that I keep segregated away from my brokerage accounts. Anyone who is living wholly or partly off investment income would do well to keep a similar float, as dividends do not arrive in smooth lumps throughout the year.

For the purposes of benchmarking it's then convenient to aggregate my investments in the following way:

  • A: UK single stocks, benchmarked against any dirt cheap FTSE 100 ETF (FTSE 350 is probably a better benchmark but these ETFs tend to be more expensive).
  • B: Long only investments: All ETFs (in both investment and trading accounts) and UK stocks, benchmarked against a cheap 60:40 fund. This is the type of top down asset allocation portfolio I deal with in my second book.
  • C: Equity neutral: The ETFs in my trading account, plus the equity hedge. Benchmark is zero.
  • D: Futures trading: Return from the futures contracts traded by my fully automated system. This is the type of portfolio I deal with in chapter 15 of my first book. Benchmarks are a similar fund run by my ex employers, or any CTA index of your choosing. The denominator of performance here is the notional capital at risk in my account.
  • E: Trading account value: This is essentially everything in my trading account, and consists of equity neutral + futures trading. No relevant benchmark.
  • F: Everything: Long only investments, plus futures hedge, plus futures trading. For the benchmark here again I use a cheap 60:40 fund, but I include the value of any cash included in my trading account, since if I wasn't trading I could invest this. 
If you prefer maths, then the relationship to the first set of categories is:

A = 1.1
B = 1.1 + 1.2 + 2.1 
C = 2.1 + 2.2
D = 2.3
E = 2.1 + 2.2 + 2.3 = C + D
F = 1.1 + 1.2 + 2.1 + 2.2 + 2.3 + 2.4 = B + 2.2 + D + 2.4

I include this to point out that in many cases you can't just 'add-up the figures included here across categories.

UK Stocks portfolio

My UK stock investments have been in a period of transition for the last few years. The goal is to run these fully systematically (though not in an automated fashion), with all assets held in tax free accounts so that capital gains tax does not eat up returns. The system I use is described in this post I wrote here; with the twist that I now enforce industry diversification. It is in fact very similar to the list of filters I describe in chapter 11 on equity investing in my second book; with the addition of a stop-loss / momentum selling rule.

However there are some legacy issues, in particular I have a couple of large positions which I've been tactically reducing my exposure to (to maximise the use of capital gains allowances).

Anyway enough of a preamble, here are the numbers as a % of initial capital value:

Dividends: 5.8%
Mark to market: 12.0%
Total return: 17.8%

As with previous years the total return figure is misleading as I was a net seller of UK stocks; calculating the IRR I get 18.3%. This compares extremely well with the benchmark which came in at 2.2% (don't get excited - this is probably the high point of this post!).

I've actually outperformed the FTSE with my UK stock picking over each of the last 4 years; this is nowhere near a statistically significant record (and I'm pretty sure that ) but it does give you pause for thought. I'll come back to that thought later.

I owned 13 UK stocks at some point during the year (starting and ending with 10 stocks, three of which were replaced according to the systems rules (KIE, MARS and PFC). The other trades I did were further top slicing of the large position in STOB. Stellar performers were ICP, RMG, BKG and STOB (all of which earned over 25% measured with simple total return); whilst all the stocks I sold ended badly down (partly reflecting the stop loss which meant they were sold on a loss, but also reflecting the fact they didn't sharply recover by year end making me look like an idiot).

Current holdings then are:

ICP 18.8%
STOB 17.1%
BKG 10.4%
VSVS 9.5%
RMG 8.9%
LGEN 7.6%
GOG 7.5%
HSBA 7.4%
IBST 6.9%
BP 6.0%

The relatively large positions in ICP and STOB are historic rather than deliberate; further tactical top slicing should reduce these when tax allowances allow.

Long only investment portfolio

My long only investment portfolio as a whole (which includes the UK shares above, plus ETFs regardless of which account they are in or whether they are hedged) is constructed according to the principals in "Smart Portfolios".

The results here aren't quite as impressive:

Dividends: 4.4%
Mark to market: -3.1%
Total return: 1.3%
IRR: 1.33%
Benchmark: 1.31%

I know for a fact many people are thinking I would have been better off in Bitcoin. Clearly the ETF part of my portfolio dragged down the equity performance (UK equities are roughly 20% of my overall long only investment portfolio).

I was a small net seller of UK stocks, but a net buyer of ETFs (with some net buying overall as I was able to reinvest some additional capital). Generally I was a seller of bonds and a buyer of equities, as discussed last year the asset allocation model I use looks at 12 month momentum to tilt between bonds and equities (this is in part three of my second book). I noted in the previous update that equities were outperforming bonds, so a tilt towards equities away from my strategic allocation is warranted.

Right now MSCI world equities are up around 16% over one year, versus global bonds down around 1.8% so this pattern is unchanged.

I won't look at the current make up or risk exposure of my ETF portfolio just yet, since it only makes sense holistically including the equity hedge.

Trading account

Although the make up of my trading account is complex I only have nice graphs that show the value of everything in it, so here they are:

Since inception

Last 12 months

The good news is I reached a new HWM in February; the bad news is that like the rest of the CTA universe I then got hammered and ended up flat for the year.

And here is the breakdown (all values normalised by my notional capital at risk):

Mark to market: 0.1%
Dividends 3.7%
Commissions: -0.00%

Subtotal: 3.8%

Mark to market: 0.25%
Commissions: -0.00%

Subtotal: 0.25%

Total for stocks and hedge: 4.1%

Gross profit: 0.48%
Commissions: -0.77%
Slippage: -0.47% (Bid ask spread cost -0.91%, less execution algo profit 0.43%)
Interest and fees: -0.09%
FX adjustments: -2.8%

Total for futures: -3.7%

Grand total: 0.4%

A sea of flatness then; basically I made no money trading futures, and then earned some dividends which paid for FX losses. These FX losses aren't unusually large (in context the same numbers for the last few years are -0.8%, +3.2% and +1.7%); but in a year with such flat performance elsewhere they stand out more than they ought to.

These FX losses are essentially the MTM of non GBP cash held in my futures account. Some of this cash is required for initial margin; if I didn't have this then I'd pay interest to borrow foreign currency which seems nuts. However there is also some excess cash, not required for margin. I took the decision not to 'sweep' this cash regularly back into GBP, as a proper hedge fund would do, which would minimise account volatility. Ultimately I'd rather have diversified currency holdings, although there is no right answer to this argument. In the long run I essentially view these FX gains and losses as noise with an expected zero mean.

Commissions and slippage are in line with backtest and previous years.

Some return statistics:

Standard deviation of returns (based on weekly, annualised): 23.8% versus long term target 25%
Average drawdown: 6.3%
Max drawdown: -17.2%
Worst day: -5.7%

Best day: +6.7%

Some trade statistics:

Profit factor: 0.98
Percent wins: 41.4%
Win/loss ratio: 1.5
Average holding period, winning trades: 32 days
......................................, losing trades: 21 days

It's probably instructive to review this performance in the context of the last few years, including some benchmark figures. 'Bench1' is this AHL fund, using monthly returns from April to March in each year, and a new benchmark 'Bench2' is the SG CTA index. Both have returns scaled up to match my volatility. Remember the benchmark should only be compared against futures trading, not the equity neutral component of the portfolio. Also note that the 'Bench1' fund has an explicit GBP hedge; so won't be as careless with cash exposure as I have been.

Year:    14/15   15/16   16/17   17/18

Hedge:   -1.1%,  16.3%,  14.4%,  4.1%
Futures: 58.2%,  23.2%, -14.0%, -3.7%
Net:     57.2%,  39.6%,   0.3%,  0.4%

Bench1: 106.9%, -10.6%,  -6.2%, 16.4%
Bench2:          -6.7%*,-21.9%, -3.8%

* From 13th April 2015

So for this year at least I'm roughly in line with the CTA index, but behind the better performing AHL fund. This is a similar pattern to last year.

From another point of view my Sharpe Ratio since inception is still running at around 0.98; whilst for the AHL benchmark over the same period it's 0.86 (without risk free rate). None of these figures are statistically significant; and I personally couldn't care less whether I outperform or not, but it's still interesting to look at these figures occasionally (though annually is probably enough - I don't miss the days when institutional pressure meant I had to check in on competitor performance on a monthly basis or even more frequently!).

Digging more deeply it looks like the winning sectors were Volatility and bonds; with losses in FX and Energy. On an individual instrument level gainers were: Palladium, BTP, VSTOXX and Nasdaq and losers: Soybeans, Gas, Korean 3 year bonds and JPYUSD.

Let's look at the good news first; here is Palladium:

Classic picture of a long up trend which we ride until it finishes. After that the signal isn't strong enough to warrant a position. Interestingly Gold was not a profitable market this year, showing the advantages of intra sector diversification. Now for Italian BTP bonds:

A more nuanced example here; the system basically benefits from two clear uptrends each lasting a matter of weeks, and then bides its time in between. VSTOXX is particularly interesting:

The price shows a gradual downward trend; partly due to the rolldown effect that is particularly pronounced in vol markets; and also because vol levels did decline to very low levels (as I discussed at the time). Then in February there was a pronounced spike in vol that took a lot of people by surprise. Because I prefer to stay at least two months out in the contract space I didn't see such a sharp rise in price levels as in 'spot' implied vol, but it's also clear that I didn't have any position on between October and March, and thus avoided the spike entirely.

What gives? Well basically I ran out of margin head room, and because VIX and V2X were very margin hungry I closed my positions in them. So a bit of luck there.

Now the bad news. Soybeans:

Classic stuff where a choppy market results in gradual losses as we get whipsawed like crazy. The other losing markets show similar pictures so I won't bore you.

Holistic view of overall performance

Looking at my entire portfolio the raw numbers come in like this (dividing by a total for assets that includes cash held in my futures and other investment accounts):

Dividends: 4.1%
Mark to market: -3.5%
Total return: 0.56%
.... of which UK stocks: 3.3%
.... of which ETFs: -2.0%
.... of which futures + hedge p&l: -0.65%

IRR: 0.6%

This is a similar picture to last year: a slight under-performance against the benchmark.

Risk exposures

Here are my current cash weights across the entire portfolio:

Bonds:    25.1%
Equities: 65.3%
Other:     3.3% (property & gold)
Cash:      6.4%

Unlike last year the figures here already show the rebalancing I did at year end; as I already mentioned this included a further reallocation away from underperforming bonds to equities.

I prefer to look at risk allocations, which are (with last year in brackets) and [my strategic target allocations in square brackets]:

Bonds:    13.1%  (17%) [25%]
Equities: 59.4%  (54%) [50%]
Other:     2.9%  (3%)  [3%]
Futures*: 24.5%  (26%) [22%]
* Trading, futures hedge offsets equities exposure

Again note the tilt towards equities given their strong relative momentum. Regionally my exposures are (each row adding up to 100% of each asset class):

Asia EM Euro UK US Other
Bonds 0.0% 25.7% 27.8% 4.4% 33.7% 8.4%
Equity 13.5% 27.4% 20.5% 28.8% 9.3% 0.5%

These don't exactly match to the figures in the model portfolios in my second book, partly for historic reasons (the UK equity exposure is still quite high), partly because of availability (eg of Asian bond ETFs), and partly as I tilt towards higher yielding funds.

Some thoughts

It would be nice to make more money, so an interesting question is how? Without digging too deep it looks like my systematic UK equity trading is doing relatively okay, and my futures trading could be better. This leads one to ask a number of questions.

Does the apparent out performance of my UK equities warrant a higher allocation than a Smart Portfolios investor would give it? To put it another way what is the benefit of a long only systematic portfolio exposed to multiple risk factors, versus vanilla market cap weighted? There is some benefit, but in my book I recommend not adjusting portfolio weights too much in the expectation of higher relative Sharpe Ratio. Indeed I'm currently at around 17% of total portfolio risk in UK equities, versus the 4% or so I recommend for a UK investor in my book. If I continue to top slice my outsized positions in ICP and STOB I will still only get down to 15%. In conclusion I think it is worth continuing to tactically reduce my UK equities exposure.

Improving my futures trading is something on the 'to-do' list. Arguably it would make more sense to introduce another asset class; perhaps cover more individual equities, start a long:short portfolio, or look into options. However this will involve far more work than I'm prepared to do so I'm sticking to futures. At some point when I get pysystemtrade to the point where it can replace my current trading system I will be in a position to start looking at some improvements here.