Portfolio optimisation, uncertainty, bootstrapping, and some pretty plots. Ho, ho, ho.

Optional Christmas themed introduction

Twas the night before Christmas, and all through the house.... OK I can't be bothered. It was quiet, ok? Not a creature was stirring... literally nothing was moving basically. And then a fat guy in a red suit squeezed through the chimney, which is basically breaking and entering, and found a small child waiting for him (I know it sounds dodgy, but let's assume that Santa has been DBS checked*, you would hope so given that he spends the rest of December in close proximity to kids in shopping centres)

* Non british people reading this blog, I could explain this joke to you, but if you care that much you'd probably care enough to google it.

"Ho ho" said the fat dude "Have you been a good boy / girl?"

"Indeed I have" said the child, somewhat precociously if you ask me.

"And what do you want for Christmas? A new bike? A doll? I haven't got any Barbies left, but I do have a Robert Oppenheimer action figure; look if you pull this string in his stomach he says 'Now I am become Death destroyer of worlds', and I'll even throw in a Richard Feynman lego mini-figure complete with his own bongo drums if you want."

"Not for me, thank you. But it has been quite a long time since Rob Carver posted something on his blog. I was hoping you could persuade him to write a new post."

"Er... I've got a copy of his latest book if that helps" said Santa, rummaging around in his sack "Quite a few copies actually. Clearly the publisher was slightly optimistic with the first print run."

"Already got it for my birthday when it came out in April" said the child, rolling their eyes.

"Right OK. Well I will see what I can do. Any particular topic you want him to write about in this blog post?"

"Maybe something about portfolio optimisation and uncertainty? Perhaps some more of that bootstrapping stuff he was big on a while ago. And the Kelly criterion, that would be nice too."

"You don't ask for much, do you" sighed Santa ironically as he wrote down the list of demands.

"There need to be really pretty plots as well." added the child. 

"Pretty... plots. Got it. Right I'll be off then. Er.... I don't suppose your parents told you to leave out some nice whisky and a mince pie?"

"No they didn't. But you can have this carrot for Rudolf and a protein shake for yourself. Frankly you're overweight and you shouldn't be drunk if you're piloting a flying sled."

He spoke not a word, but went straight to his work,And filled all the stockings, then turned with a jerk. And laying his finger aside of his nose, And giving a nod, up the chimney he rose! He sprang to his sleigh, to his team gave a whistle, And away they all flew like the down of a thistle. But I heard him exclaim, ‘ere he drove out of sight,

"Not another flipping protein shake..."

https://pixlr.com/image-generator/ prompt: "Father Christmas as a quant trader"

Brief note on whether it is worth reading this

I've talked about these topics before, but there are some new insights, and I feel it's useful to combine the question of portfolio weights and optimal leverage into a single post / methodology. Basically there is some familar stuff here but now in a coherent story, plus some new stuff.

And there are some very nice plots.

Somewhat messy python code is available here (with some data here or use your own), and it has no dependency on my open source trading system pysystemtrade so everyone can enjoy it.

Bootstrapping

I am a big fan of bootstrapping. Some definitional stuff before I explain why. Let's consider a couple of different ways to estimate something given some data. Firstly we can use a closed form. If for example we want the average monthly arithmetic return for a portfolio, we can use the very simple formula of adding up the returns and dividing by the number of periods. We get a single number. Although the arithmetic mean doesn't need any assumptions, closed form formula often require some assumptions to be correct - like a Gaussian distribution. And the use of a single point estimate ignores the fact that any statistical estimate is uncertain. 

Secondly, we can bootstrap. To do this we sample the data repeatedly to create multiple new sets of data. Assuming we are interested in replicating the original data series, the new set of data would be the same length as the original, and we'd be sampling with replacement (or we'd just get the new data in a different order). So for example, with ten years of daily data (about 2500 observations), we'd choose some random day and get the returns data from that. Then we'd keep doing that, not being bothered about choosing the same day (sampling with replacement), until we had done this 2500 times. 

Then from this new set of data we estimate our mean, or do whatever it is we need to do. We then repeat this process, many times. Now instead of a single point estimate of the mean, we have a distribution of possible means, each drawn from a slightly different data series. This requires no assumptions to be made, and automatically tells us what the uncertainty of the parameter estimate is. We can also get a feel for how sensitive our estimate is to different variations on the same history. As we will see, this will also lead us to produce estimates that are more robust to the future being not exactly like the past.

Note: daily sampling destroys any autocorrelation properties in the data, so it wouldn't be appropriate for example for creating new price series when testing momentum strategies. To do this, we'd have to sample larger chunks of time period to retain the autocorrelation properties. For example we might restrict ourselves to sampling entire years of data. For the purposes of this post we don't mind about autocorrelation, so we can sample daily data.

Bootstrapping is particularly potent in the field of financial data because we only have one set of data: history. We can't run experiments to get more data. Bootstrapping allows us to create 'alternative histories' that have the same basic character as our actual history, but aren't quite the same. Apart from generating completely random data (which itself will still require some assumptions - see the following note), there isn't really much else we can do.

Bootstrapping helps us with the quant finance dilemma: we want the future to be like the past so that we can use models calibrated on the past in the future, but the future will never be exactly like the past. 

Note: that bootstrapping isn't quite the same as monte carlo. With that we estimate some parameters for the data, making an assumption about it's distribution. Then we randomly sample from that distribution. I'm not a fan of this. We have all the problems of making assumptions about distribution, and of uncertainty about the parameter estimates we use for that distribution. 

Portfolio optimisation

With all that in mind, let's turn to the problem of portfolio opimisation. We can think as this as making two decisions:

• Allocating weights to each asset, where the weights sum to one
• Deciding on the total leverage for the portfolio

Under certain assumptions we can seperate out these two decisions, and indeed this is the insight of the standard mean variance framework and the 'security market line'. The assumption is that enough leverage is available that we can get to the risk target for the investor. If the investor has a very low risk tolerance, we might not even need leverage, as the optimal portfolio will consist of cash + securities.

So basically we choose the combination of asset weights that maximises our Sharpe Ratio, and then we apply leverage to hit the optimal risk target (since SR is invariant to leverage, that will remain optimal). 

To begin with I will assume we can proceed in this two phase approach; but later in the post I will relax this and look at the effect of jointly allocating weights and leverage.

I'm going to use data for S&P 500 and 10 year Bond futures from 1982 onwards, but which I've tweaked slightly to produce more realistic forward looking estimates for means and standard deviations (in fact I've used figures from this report- their figures are actually for global equities and bonds, but this is all just an illustration). 

My assumptions are:

• Zero correlation (about what it has been in practice since 1982)
• 2.5% risk free rate (which as in standard finance I assume I can borrow at)
• 3.5% bond returns @ 5% vol
• 5.75% equity returns @ 17% vol

This is quite a nice technique, since it basically allows us to use forward looking estimates for the first two moments (and first co-moment - correlation) of the distribution, whilst using actual data for the higher moments (skew, kurtosis and so on) and co-moments (co-skew, co-kurtosis etc). In a sense it's sort of a blend of a parameterised monte-carlo and a non parameterised bootstrap.

Optimal leverage and Kelly

I'm going to start with the question of optimal leverage. This may seem backwards, but optimal leverage is the simpler of the two questions. Just for illustrative purposes, I'm going to assume that the allocation in this section is fixed at the classic 60% (equity), 40% (bonds). This gives us vol of around 10.4% a year, a mean of 4.85%, and a Sharpe Ratio of 0.226

The closed form solution for optimal leverage which I've written about at some length, is the Kelly Criterion. Kelly will maximise E(log(final wealth)) or median(final wealth), or importantly here it will maximise the geometric mean of your returns.

Under the assumption of i.i.d. Gaussian returns optimal Kelly leverage is achieved by setting your risk target as an annual standard deviation equal to your Sharpe Ratio. With a SR of 0.226 we want to get risk of 22.6% a year, which implies running at leverage of 22.6 / 10.4 = 2.173

That of course is a closed form solution, and it assumes that:

• Return parameters are Guassian i.i.d. (which financial data famously is not!)
• The return parameters are fixed
• That we have no sampling uncertainty of the return parameters
• We are fine running at fully Kelly, which is a notoriously aggressive amount of leverage

Basically that single figure - 2.173 - tells us nothing about how sensitive we would be to the future being similar to, but not exactly like, the past. For that we need - yes - bootstrapping. 

Bootstrapping optimal leverage 

Here is the bootstrap of my underlying 60/40 portfolio with leverage of 1.

Each point on this histogram represents a single bootstrapped set of data, the same length as the original. The x-axis shows the geometric mean, which is what we are trying to maximise. You can see that the mean of this distribution is about 4.1%. Remember the arithmetic mean of the original data was 4.85%, and if we use an approximation for geometric mean that assumes Gaussian returns then we'd get 4.31%. The difference between 4.1% and 4.31% is because this isn't Guassian. In fact, mainly thanks to the contribution of equities, it's left tailed and also has fat tails. Left fat tails result in lower Geometric returns - and hence also a lower optimal leverage, but we'll get to that in a second.

Notice also that there is a fair bit of distributional range here of the geometric mean. 10% of the returns are below 2%, and 1% are below 0.4%.

Now of course I can do this for any leverage level, here it is for leverage 2:

The mean here is higher, as we'd probably expect since we know the optimal leverage would be just over 2.0 if this was Gaussian. It comes in at 4.8%; versus the 7.2% we'd expect if this was the arithmetic mean, or the 5.04% that we would have for Gaussian returns.

Now we can do something fun. Repeating this exercise for many different levels of leverage, we can take each of the histograms that are producing and pull various distributional points off them. We can take the median of each distribution (50% percentile, which in fact is usually very close to the mean), but also more optimistic points such as the 75% and 90% percentile which would apply if you were a very optimistic person (like SBF, as I discussed in a post about a year ago), and perhaps more usefully the 25% and 10% points. We can then plot these:

How can we use this? Well, first of all we need to decide what our tolerance for uncertainty is. What point on the distribution are you optimising for? Are you the sort of person who worries about the bad thing that will happen 1 in 10 times, or would you instead be happy to go with the outcome that happens half the time (the median)?

This is not the same as your risk tolerance! In fact, I'm assuming that your tolerance for risk is sufficient to take on the optimal amount of leverage implied by this figure. Of course it's likely that someone with a low tolerance for risk in the form of high expected standard deviation would also have a low tolerance for uncertainty. And as we shall see, the lower your tolerance for uncertainty, the lower the standard deviation will be on your portfolio.

(One of the reasons I like this framing of tolerance is that most people cannot articulate what they would consider to be an appropriate standard deviation, but most people can probably articulate what their tolerance for uncertainty is, once you have explained to them what it means)

Next you should focus on the relevant coloured line, and mentally remove the odd bumps that are due to the random nature of bootstrapping (we could smooth them out by using really large bootstrap runs - note they will be worse with higher leverage since we get more dispersion of outcomes based on one or two bad days eithier being absent or repeated in the sample), and then find the optimium leverage.

For the median this is indeed at roughly the 2.1 level that theory predicts (in fact we'd expect it to be a little lower because of the negative skew), but this is not true of all the lines. For inveterate gamblers at 90% it looks like the optimum is over 3, whilst for those who are more averse to bad outcomes at 10% and 25% it's less than 2; in fact at 10% it looks like the optimium could easily be 1 - no leverage. These translate to standard deviations targets of somewhere around 10% for the person with a 10% risk tolerance . 

Technical note: I can of course use corrections to the closed form Kelly criterion for non Gaussian returns, but this doesn't solve the problem of parameter estimation uncertainty - if anything it makes it worse.

The final step, and this is something you cannot do with a closed form solution, is to see how sensitive the shape of the line is to different levels of leverage, thus encouraging us to go for a more robust solution that is less likely to be problematic if the future isn't exactly like the past. Take a slightly conservative 25% quantile person on the red line in the figure. Their optimium could plausibly be at around 1.75 leverage if we had a smoother plot, but you can see that there is almost no loss in geometric mean from using less leverage than this. On the other hand there is a steep fall off in geometric mean once we get above 1.75 (this assymetry is a property of the geometric mean and leverage). This implies that the most robust and conservative solution would be to choose an optimal leverage which is a bit below 1.75. You don't get this kind of intuition with closed form solutions.

Optimal allocation - mean variance

Let's now take a step backwards to the first phase of solving this problem - coming up with the optimal set of weights summing to one. Because we assume we can use any amount of leverage, we want to optimise the Sharpe Ratio. This can be done in the vanilla mean-variance framework. The closed form solution for the data set we have, which assumes Gaussian returns and linear correlation, is a 22% weight in equities and 78% in bonds. That might seem imbalanced, but remember the different levels of risk. Accounting for this, the resulting risk weights are pretty much bang on 50% in each asset. 

As well as the problems we had with Kelly, we know that mean variance has a tendency to produce extreme and not robust outcomes, especially when correlations are high. If for example the correlation between bonds and equities was 0.65 rather than zero, then the optimal allocation would be 100% in bonds and nothing in equities.

(I actually use an optimiser rather than a single equation to calculate the result here, but in principal I could use an equation which would be trivial for two assets - see for example my ex colleague Tom's paper here - and not that hard for multiple assets eg see here).

So let's do the following; boostrap a set of return series with different allocations to equities (bond allocation just 100% - equity allocation), then measure the Sharpe Ratio of each allocation/bootstrapped return series, and then measure the distribution of those Sharpe Ratios for different distributional points.

Again, each of these coloured lines represents a different point on the distribution of Sharpe Ratios. The y-axis is the Sharpe Ratio, and the x-axis is the allocation to equities; zero in equities on the far left, and 100% on the far right. 
Same procedures as before: first work out your tolerance for uncertainty and hence which line you should be on. Secondly, find the allocation point which maximises Sharpe Ratio. Thirdly, examine the consequences of having a lower or higher allocation - basically how robust is your solution.
For example, for the median tolerance (green line) the best allocation comes in somewhere around 18%. That's a little less than the closed form solution; again this is because we haven't got normally distributed assets here. And there is a reasonably symettric shape to the gradient around this point, although that isn't true for lower risk tolerances.
You may be surprised to see that the maximum allocation is fairly invarient to uncertainty tolerance; if anything there seems to be a slightly lower allocation to equities the more optimistic one becomes (although we'd have to run a much more granular backtest plot to confirm this). Of course this wouldn't be the case if we were measuring arithmetic or even geometric return. But on the assumption of a seperable portfolio weighting problem, the most appropriate statistic is the Sharpe Ratio. 
This is good news for Old Skool CAPM enthusiasts! It really doesn't matter what your tolerance for uncertainty is, you should put about 18% of your cash weight - about 43% of your risk weight in equities; at least with the assumption that future returns have the forward looking expectations for means, standard deviations, and correlations I've specified above; and the historic higher moments and co-moments that we've seen for the last 40 years.

 

Joint allocation

Let's abandon the assumption that we can seperate our the problem, and instead jointly optimise the allocation and leverage. Once again the appropriate statistic will be the geometric return. We can't plot these on a single line graph, since we're optimising over two parameters (allocation to equities, and overall leverage), but what we can do is draw heatmaps; one for each point on the return distribution.
Here is the median:

The x-axis is the leverage; lowest on the left, highest on the right. The y-axis is the allocation to equities; 0% on the top, 100% on the bottom. And the heat colour on the z-axis shows the geometric return. Dark blue is very good. Dark red is very bad. The red circle shows the highest dark blue optimum point. It's 30% in equities with 4.5 times leverage: 5.8% geometric return.
But the next question we should be asking is about robustness. An awful lot of this plot is dark blue, so let's start by removing everything below 3% so we can see the optimal region more clearly:

You can now see that there is still quite a big area with a geometric return over 5%. It's also clear from the fact there is variation of colour within adjacent points that the bootstrapped samples are still producing enough randomness to make it unclear exactly where the optimium is; and this also means if we were to do some statistical testing we'd be unable to distinguish between the points that are whiteish or dark blue. 
In any case when we are unsure of the exact set of parameters to use, we should use a blend of them. There is a nice visual way of doing this. First of all, select the region you think the optimal parameters come from. In this case it would be the banana shaped region, with the bottom left tip of the banana somewhere around 2.5x leverage, 50% allocation to equities; and the top right tip around 6.5x leverage, 15% allocation. And then you want to choose a point which is safely within this shape, but further from steep 'drops' to much lower geometric returns which means in this case you'd be drawn to the top edge of the banana. This is analogous to avoiding the steep drop when you apply too much leverage in the 'optimal leverage' problem. 
I would argue that something around the 20% point in equities, leverage 3.0 is probably pretty good. This is pretty close to a 50% risk weight in equities, and the resulting expected standard deviation of 15.75% is a little under equities. In practice if you're going to use leverage you really should adjust your position size according to current risk, or you'd get badly burned if (when) bond vol or equity vol rises.
Let's look at another point on the distribution, just to get some intuition. Here is the 25% percentile point, again with lower returns taken out to better intuition:

The optimal here stands out quite clearly, and in fact it's the point I just chose as the one I'd use with the median! But clearly you can see that the centre of gravity of the 'banana' has moved up and left towards lower leverage and lower equity allocations, as you would expect. Following the process above we'd probably use something like a 20% equity allocation again, but probably with a lower leverage limit - perhaps 2.

Conclusion

Of course the point here isn't to advocate a specific blend of bonds and equities; the results here depend to some extent on the forward looking assumptions that I've made. But I do hope it has given you some insight into how bootstrapping can give us much more robust outcomes plus some great intuition about how uncertainty tolerance can be used as a replacement for the more abstract risk tolerance. 
Now go back to bed before your parents wake up!

The State Of Vol

 I'm sometimes asked where I get my ideas for new trading strategies from. The boring truth is I rarely test new trading strategies, and I mostly steal ideas when I feel in the mood. Today for example I saw this tweet post on twitter X:

Think high market volatility leads to high returns? No. Periods with high market volatility are followed by high volatility, but not necessarily better returns. A volatility effect through time. Check out the graph below, based on the US equity market from 1927 to 2023. The… pic.twitter.com/23rrTwIf9O

— Pim van Vliet (@paradoxinvestor) October 2, 2023

The original paper is here  (requires subscription or academic institution membership)

Now I've written in the past about how volatility levels affect the profits of trading strategies, in particular momentum where there is a pretty striking effect (see my new book Advanced Futures Trading Strategies [AFTS] for more info), and I've also written about how the level of past vol affects future vol (and indeed, this incredible predictability of vol is a cornerstone of the inverse volatility sizing formula that lies at the heart of my trading strategies), but I don't think I've ever written about the level of volatility's effect on future price movements. 

There is quite a simple story here which if you can see the paper is in figure 1: absolute returns are pretty flat across different vol regimes, and because vol is pretty autocorrelated from month to month, we'd expect return/vol to be higher if recent vol was lower. To put it another way, there is a link here with my previous post about CAPM; this story is sort of about the time series version of CAPM in which higher vol should be rewarded with higher returns (and hence Sharpe Ratios should not show any pattern conditional on relatively volatility levels), and the opposite happens in single stocks in an analogy to the fact that CAPM doesn't work in cross section for single stocks, in fact we should 'bet against beta' and buy low beta stocks.

But the paper is written for single stocks; a risky asset, and one with some idiosyncratic distributional features. Will the results hold for the wider universe of futures markets? 

Data and definitions

I am not interested in reproducing the results in the paper since I'm more concerned with whether this is a profitable trading strategy sitting within my usual framework, so I will do things a bit differently.

I use my usual set of futures markets, after removing duplicates (eg micro and mini versions, or cross listings) it comes in at 211 instruments. Data goes back to 1970, and I use my standard vol estimate calculated in % terms, which I divide by an exponential moving average of the same with a 10 year vol halflife to get a relative vol measure. If this is 1, then the vol we're seeing is going to be typical of the level over the last couple of decades or so, if it's higher than 1 then the vol is higher, and so on. Again readers of AFTS will recognise this measure. The paper doesn't quite do this; it just looks at the level of vol versus the in sample historic distribution. This means we can't use that forecast as a trading signal.

For the y-axis of response, I will use the returns in the following month normalised by the volatility estimate (in price differences this time) at the beginning of the month for reasons I have discussed before. As I'm trading futures, this is also the Sharpe Ratio. The original paper uses a straight one month moving window of returns for vol estimation, and the pure ex-post Sharpe Ratio using the realised vol in the following month, so some small difference there. However because of the autocorrelation of vol, it shouldn't affect things too much; if anything my results probably should be a bit better because I'm dividing an unknown future estimate of mean by a known current estimate of standard deviation to get Sharpe Ratio, rather than having unknown mean/unknown sigma.

The original papers uses single stocks, I use a massive variety of futures allowing me to see if this effect persists at the stock index level and for other asset classes. 

Some comparable graphs

Let's start with producing the same kind of graphs as in the paper. So each month I look at the current level of vol versus it's historic average, and then rank it on an in sample basis (yes I know, but for the time being...), and then bucket the ranking into quintiles. I measure the risk adjusted return in the following month and take a median of those Sharpe Ratios; medians being somewhat more robust than mean.

Here is the result for the S&P 500:

I use the same ordering; 1 on the left is the lowest quantile of vol, 5 on the right is the highest. That picture is not quite as compelling as the original paper; but ignoring the 2nd bar it has the right pattern: higher recent vol means lower risk adjusted returns in the following month. Nevertheless we are only using 492 data points compared to the thousands we can get with single stocks; plus I am suspicous of anyone that uses a single instrument to prove anything, so let's expand out to all equity futures (with months stacked, which will give a higher weight to instruments with more history):

That is .... not what we'd hoped for.

The name is Bond:

Not great. What about Vol (VSTOXX, VIX)?

OK only two markets, but there is an apparent reverse effect here. You don't want to be long vol on average, but perhaps the worst time to be is when vol of vol is very low.

FX:

At best noise, at worst the reverse of what we expect.

Metals

That is the strongest, wrong way round, effect we have seen. Agricultural:

Just noise really. And finally energies:

The wrong way round, and inconsistent.

It's probably futile, but what happens if we pool our results across all markets?

There could be a story there- higher vol means higher risk adjusted returns, unless vol is very high; or we could be looking at the aggregated result of jamming together instruments that behave very differently*, plus a bunch of instruments that are just noise. 

[* it looks like markets fall into two camps, one where the effect is the 'right way' round (high vol, lower returns; perhaps the S&P 500 is here), others where it is the wrong way (most strikingly, vol), and the third of the two camps doesn't really have a strong effect]

Just to give a point of comparison, here's the bucket plot if I replace the ratio of vol with the level of the 32/128 day momentum forecast level (something we know is very successful as a forecast at a portfolio level, although it is weak in say equities):

Clearly this vol effect is not as strong as momentum.

A trading rule

Although the results aren't what we expect, this could be profitable as a 'wrong way round' rule, which buys when vol is higher, and sells when vol is lower. This won't work for very high vol (since quintile 5 shows a worse return for quintiles 2,3 and 4, but then we would hit forecast capping up there anyway. That isn't much different to what happens with faster momentum rules anyway; if you plot response versus forecast level it's not as good for extreme forecasts (see AFTS for more).

In my first book, Systematic Trading, I said that implementing a rule as a wrong way rule after discovering it is a form of data mining so bear in mind that the backtest results here will effectively be overstated [in reality, you couldn't have implemented this rule until enough historical evidence that the effect was the 'wrong' way round existed; until that moment of statistically significant realisation we would eithier be trading the money losing 'right way round' rule, or a combination of the two, eithier of which is inferior to being all in on 'wrong way round' from the start of the backtest].

I'd add a couple of tweaks as well; firstly rather than quantile buckets which are too coarse and in sample; I will use the percentile of the ratio versus historic levels as the forecast (the same approach I use for the volatility overlay rule in AFTS) , and also chuck in my standard ewm(span=10) smooth to reduce noise on what should be a monthly holding forecast (this reduces turnover from 25 times a year to 10 times without affecting profitability).

Before continuing, it might be worth thinking about how this interacts with the volatility overlay on a trend following rule (which cuts exposure long or short when vol is high).

• If vol is low, we'd be SHORT from a vol rule
• ... and we'd have a stronger trend following signal
• if markets are trending up we'd be net flat
• If markets are trending down we'd have a strong sell
• If vol sits in the middle, we'd be FLAT from a vol rule
• .... and we've have an unadulterated trend signal
• If markets are trending up we'd be long
• If markets are trending down we'd be short
• If vol is high, we'd be LONG from a vol rule
• .... and have a weaker trend following signal
• If markets are trending up we'd be modestly long
• If markets are trending down we'd be flat

Clearly this will work differently for risk on/risk off markets; risky markets like equities will most probably go up with low vol and down with high vol, thus occupying the two lines shown with italics. In these situations we'd have no position on (assuming these are the only two trading rules we're using). Since trend following doesn't work super well in equity indices, this might improve our lives somewhat.

Risk off markets, the most extreme of which is VIX/VSTOXX, will occupy the lines in bold as they have high vol when they are going up, and low vol vol the way down. Notice that this will introduce a short bias (modest long on the way up, strong short on the way down); since these markets tend to lose money on the long side again we might expect a benefit here.

Other markets - which form the majority of futures instruments - will be less clear, so let's see how it goes.

Here's the code for the purely backward looking vol quantile calculation, and the actual trading rule:

def get_vol_quantile_points(self, instrument_code):
    self.log.debug("Calculating vol quantile for %s" % instrument_code)
    daily_vol = self.parent.rawdata.get_daily_percentage_volatility(instrument_code)
    ten_year_vol = daily_vol.rolling(2500, min_periods=10).mean()
    normalised_vol = daily_vol / ten_year_vol

    normalised_vol_q = quantile_of_points_in_data_series(normalised_vol)

    return normalised_vol_q

def quantile_of_points_in_data_series(data_series: pd.Series) -> pd.Series:
    ## With thanks to https://github.com/PurpleHazeIan for this implementation
    numpy_series = np.array(data_series)
    results = []

    for irow in range(len(data_series)):
        current_value = numpy_series[irow]
        count_less_than = (numpy_series < current_value)[:irow].sum()
        results.append(count_less_than / (irow + 1))

    results_series = pd.Series(results, index=data_series.index)
    return results_series

def vol_rule(vol_quantile_points: pd.Series, smooth: int = 10):
    # vol quantile points sits in space 0 to 1.0
    raw_forecast = (vol_quantile_points - 0.5) * 40  ## sits in space -20 to +20
    smoothed_forecast = raw_forecast.ewm(span=smooth).mean()
    return smoothed_forecast

System test

I'm testing this using my current trading system with 147 instruments and relevant instrument weights. I'll be using the 'static' rather than 'dynamically optimised' flavour of the system, to get a feel for pure performance before the noise added by optimisation. This also means I'm going to put in an unrealistically large slug of capital, and remove a few markets that are too expensive to trade, bringing me down to 138 markets. I estimate instrument weights and the IDM (which peaks at 2.15) using my usual optimisation defaults.

To begin with, let's look at the performance of the vol strategy by itself. You're all dying to see it, so here is the money shot with the full account curve before and after costs:

Well it's not terrible but it's not amazing eithier. The Sharpe is a mere 0.10, which is not exactly knocking it out of the park... at best we've tapped the ball and it's dribbled a few feet away. 

Can this thing add value when combined with momentum (particularly given the discussion above)? Let's keep it simple and just use a single ewmac rule, 16/64. Correlation between the rules is actually a little negative, so let's do some god-awful in sample fitting and allocate 10% of our portfolio to the new vol rule.

It's not really worth plotting as these two systems will both look pretty similar, but this relatively small allocation to our new putative signal does indeed push up the Sharpe Ratio from 1.10 to 1.15, with the Sortino rising by a similar amount. Costs are slightly reduced, skew falls slightly (from 0.55 to 0.48), but the more robust lower tail ratio is unchanged (see AFTS for a definition). 

Summary

It does seem like the predictive effect of vol in single equities isn't replicated across the futures universe; if anything the effect is reversed although it is not as strong as in the original paper. Rather than buying instruments with lower vol to get higher risk adjusted returns, we should do the opposite. Rather than 'time series CAPM' failing, and a 'bet against time series standard deviation levels', we in fact see an even stronger 'time series CAPM' where risk adjusted returns aren't constant with relative vol levels, but actually improve when volatility is relatively high.

If we use this idea to construct a simple trading rule the result is not the world's greatest standalone signal. But there does seem to be some promise in adding it to a trend following strategy due to it's negative correlation.

Does CAPM work across and within asset classes - done correctly

 I haven't posted much recently because I've been busy with other stuff, and I only post when I feel like I have something to say (the advantages of not having a paid for subscription service!). But I was compelled to post by this tweet:

CAPM doesn't work within asset classes, but it does do a decent job of explaining cross-asset class returns.https://t.co/y5beyDIh0j? pic.twitter.com/T7SDBkgoor

— Dan Rasmussen (@verdadcap) September 25, 2023

Which links to this article: https://mailchi.mp/verdadcap/asset-class-capm

... which in turn generated a fair amount of heat and light, since there are two key mistakes in the article and tweet. In truth these are a manifestation of a single mistake, which is a mis-definition of CAPM. CAPM remember says that there is only one risk factor, market risk, and excess security returns are equal to the covariance of security/market returns (Beta) multiplied by market returns. And excess returns are equal to the risk free rate. 

But in the article they plot standard deviation versus mean, minus inflation. So they are confusing both inflation and the risk free rate, but also getting covariance and standard deviation mixed up. The latter error pointed out by several posters on twitter, although there is a mini argument suggesting that in the uber CAPM model with freely available leverage all securities should lie on the capital markets line, and hence have the same Sharpe Ratio, and hence all you need to do is plot excess return vs standard deviation (although again, excess return is versus the risk free rate NOT inflation!). 

Anyway I thought it would be interesting to redo this plot, but correctly. After all it's an interesting topic that speaks to the benefits of diversification. TLDR: the original authors conclusion is correct (CAPM works across asset classes, but not within) even if their methods are badly flawed.

Data

I use monthly returns data pulled from my dataset of over 200 futures instruments, from which I annualise mean, standard deviation and Sharpe Ratios. As futures markets these are automatically excess returns. I have history back to 1970 for some markets, and the original plot only goes back to 1973, but for reasons that I will explain in a minute I will start my analysis in 1983. To define an asset class 'market' I start with a simple equal weighted index of all the futures that had returns in that month. Arguably I should use market cap weightings, but I don't have these to hand and in any case the results probably won't change much (since there are, eg, more US futures equity indices and the US is a big part of the global equity market). Note that this means due to diversification in theory the standard deviation of each market index will fall over time. I could correct for this, but it is not significant.

Another slightly weird thing about the original plot is that it actually splits out certain asset classes; which seems to rather undermine the argument; for example small and big equity markets, short and long term bonds (ST, LT), and different credit quality bonds.I don't have enough futures with enough history to do a split between small and big equity, nor do I have enough HY/IG bond futures to be confident the results would be meaningful, but I able to include a lot more markets and asset classes. So I have:

• Bonds (and at this stage I won't seperate these into ST/LT) - these are mostly government bonds (39 markets)
• Equities of all types (58)
• Metals (rather than just Gold in the original piece, 21)
• Energies (rather than just Oil, 20)
• Agricultural (38)

I don't include FX, since you can argue if it's really an asset class, and because it includes a mishmash of things that are bets for and against the dollar, emerging markets, and so on. And I don't include volatility, since this usually only has two markets in it (VIX and VSTOXX). 

Equity indices are late to the futures trading party, and my data for these doesn't start until late 1982. So for strict comparability I remove everything before January 1983. Again, this doesn't affect the final results all that much.

Plotting Sharpe Ratio

Let's first drop the incorrect definition of excess return, and plot excess mean versus standard deviation plots (to reiterate, as these are future the returns are automatically excess of the risk free rate). Note that means that everything on a straight line will have the same Sharpe Ratio.

Looks pretty good! And indeed if we look at the statistics including the Sharpe Ratios, we can see there is not that much difference between the SR, certainly nothing statistically significant:

        mean   std    sr
Ags     0.02  0.12  0.20
Bond    0.02  0.04  0.55
Equity  0.08  0.16  0.48
Metals  0.04  0.18  0.21
OilGas  0.09  0.28  0.34

Although we only have five data points, it does seem that there is a roughly positive relationship between excess mean over risk free and standard deviation.

Bringing in Beta

Having verified the original results after substituting the risk free rate for inflation, let's now bring in Beta. Under CAPM we'd expect that if we plotted excess mean against covariance rather than standard deviation, we'd again find a positive relationship. That should make assets with lower correlations look more attractive; that reminds me here's the correlation matrix:

        Ags  Bond  Equity  Metals  OilGas
Ags     1.00 -0.11    0.21    0.37    0.27
Bond   -0.11  1.00    0.09   -0.06   -0.14
Equity  0.21  0.09    1.00    0.26    0.12
Metals  0.37 -0.06    0.26    1.00    0.28
OilGas  0.27 -0.14    0.12    0.28    1.00

The problem of course is how to measure Beta, i.e. what is the 'market' that we are regressing our returns on. That's a hard enough problem when considering equities, but here we should really include every investable asset in the world, weighted by market cap back to 1983. I don't have those figures to hand!!

Instead I'm going to opt for another quick and dirty solution, namely to create a market index in the following proportions:

• Ags 10%
• Bonds 40%
• Equities 30%
• Metals 10%
• Oil and Gas 10%

This is based on some roughly true things; bonds and equities form most of the investable universe and there are more bonds issued than equities. And since most people are probably starting with a bonds/equities based portfolio, considering the diversification available versus something that is mostly that is probably a reasonable thing to do.

If you prefer you can do something else like risk parity (which would be about 50% bonds, with the other asset classes roughly splitting the rest), but it probably won't make that much difference. 

This market index has a standard deviation of 7.4% a year, and a mean of 4.8%; it's SR of 0.64 as you would expect is superior to it's constituents.

Let's have a look at the betas and alphas, also correlation with the market (corr), standard deviations and Sharpe Ratios:

          std   corr   beta     sr  alpha
Ags     0.119  0.471  0.765  0.199 -0.011
Bond    0.039  0.183  0.098  0.553  0.017
Equity  0.162  0.826  1.822  0.484 -0.008
Metals  0.176  0.566  1.353  0.215 -0.026
OilGas  0.277  0.541  2.027  0.338 -0.006

We can see that to an extent higher standard deviation also means higher beta, but not always; equities and metals have virtually the same standard deviation but equities have a higher beta because they are more correlated. There is also a weak relationship between alpha and SR.

Let's now redo the scatter plot but this time with Beta on the x-axis and adding the market portfolio:

The obvious outperformance of Bonds aside, this again does like a clear case of supporting the CAPM for the case of across asset classes; if anything it's clearer than before.

Intra market

Now let us address the point in the post which is mentioned but briefly; the fact that CAPM doesn't work within asset classes. This is not a new finding. Indeed there is the mysterious result of Beta making an excellent counter signal ('Betting against Beta' Pedersen and Frazzini JFE 2014) at least in individual equities. It seems that lower Beta stocks have excess Alpha compared to higher Beta stocks; one story that explains that is that if Beta is synomonous with standard deviation (which as discussed, it ain't exactly), then we'd need higher leverage to hold low Beta stocks and not everyone can or wants to leverage to the hilt.

This is perhaps a more interesting study to do, since we could potentially use any positive result here as a trading signal; buying instruments within an asset class that have low Beta (or low standard deviation), and shorting those that are high Beta. Once again we run up against the definition of 'the market' in each asset class, but I will stick with the simple equal weighted across time version I have been using so far.

Here follows a blizzard (correct collective noun?) of plots. Firstly, here's excess mean against standard deviation (the original Sharpe Ratio plot):

A big caveat here is that different instruments may have wildly different data histories. With that said, there is mostly no evidence here of a similar Sharpe Ratio. The exception is bonds. There does seem to be a relationship between duration (which is highly correlated to standard deviation) and excess return; and we also see that High Yield which is riskier than most of the goverment bonds has a higher return. In other worse, the bastardised version of the CAPM using vol rather than Beta does work within one asset class, which is perhaps why the authors of the original post decided to treat bonds as several different asset classes :-)

Now let's do things 'properly' and look at excess mean versus Beta:

Interestingly the positive result in Bonds is slightly different here; it mostly holds true that we get higher excess return for more Beta with the exception of high yield bonds. These are negatively correlated to the rest of the universe, and as a result have negative Beta. My returns for the high yield bond future go back to 2000, so this isn't a fluke down to a limited number of returns. However for government bonds, again it seems that CAPM holds true.

For a giggle let's reproduce the plots from the 'Betting Against Beta' paper, and plot Alpha vs Beta. CAPM predicts a horizontal line, whilst the original paper found a downward sloping line.

With the possible exception of oil and gas, there isn't much to write home about here. It doesn't look like CAPM or Betting against Beta is particularly compelling within asset classes that contain futures. 

(Note that in any case this isn't a proper test of Betting against Beta as a trading signal, since everything is in sample and not time varying)

Summary

Sloppy execution aside, the key findings of the original paper are correct; CAPM doesn't really work within asset classes, unless you lump all bonds into a single asset class in which case it works just fine, but it does work across asset classes. 

Clustering trading rule p&l

 I recently upgraded my live production system to include all the extra instruments I've added on recently. I also did a little consolidation of trading rules, simplifying things slightly by removing some rules that didn't really have much allocation, and adding a couple from my new book. As usual I set the instrument weights and forecast weights using my handcrafting methodology, which is basically a top down method that involves clustering things into groups in a hierarchical fasion.

In my backtests I do this clustering using the correlation matrix as a guide, but for production weights I use heurestics. So for instruments I say things like 'bonds are probably more correlated with each other than with other assets' and form the clusters initially as asset classes. And for forecast weights, which allocate across trading rules, I say things like 'momentum type trading rules are probably more correlated with each other', so I end up using a hierarchy like this:

• Convergent (eg carry and mean reversion), Divergent (eg momentum)
• Generic trading rule (eg EWMAC)
• Specific trading rule variation (eg EWMAC2,8)

Now I recently tested this clustering method for instruments in this blog post. OK it was 17 months ago, but it felt recent to me. Basically I used a clustering methodology and threw in the actual correlation matrix to see how the grouping turned out. It was quite interesting. So I thought it would be quite interesting to do a similar thing with forecast weights. Effectively I am sense checking my heuristic guidelines to see if they are completely nuts, or vaguely okay.

Some code.

Getting the correlation matrix

Well you might think this is easy, but it's not. The correlation matrix here is the correlation of returns for a given set of trading rules and variations. But returns of what? A single instrument, like the S&P 500? That obviously may be unrepresentative of the sample generally, and we're not going to do this exercise for the 200+ instruments I have in my dataset now. What about the correlation of average returns taken across a bunch of instruments, or perhaps the average of the correlations taken across the same bunch - note these aren't quite the same thing. For example an average of correlations will give every instrument the same weight, wheras an average of returns will give a higher weighting to instruments with more data history.

And if we are doing averaging, do we just do a simple average - which will be biased since 37% of my futures are equities? Or do we use the instrument weights?

The good news is it probably won't make too much difference. Given enough history, the correlation of trading rule forecast returns is pretty similar across instruments. But we probably want to avoid overweighting certain asset classes, or equally weighting instruments without much history. So I'm going to go for taking the return correlation of portfolios for each trading rule. Each portfolio consists of the same trading rule being traded in all the instruments I trade, weighted by my actual instrument weights. 

Now I don't actually trade all rules in all instruments, because of trading costs, and sometimes because the instrument has certain flaws, but what we are trying to get here is as much information as possible to build a robust correlation matrix. I will also use pre-cost returns; not that it will make any difference, but the point here is to discover how similar rules are to each other, which doesn't depend on costs.

Finally note that I have 135 instruments with instrument weights, because some of my 208 are duplicates (eg micro and mini S&P 500), or I can't legally trade them, or for some other reason.

Results: N=2

Let's kick things off then:

Cluster 1 'convergent'
['mrinasset160', 'carry10', 'carry30', 'carry60', 'carry125', 'relcarry', 

'skewabs365', 'skewabs180', 'skewrv365', 'skewrv180']
Cluster 2 'divergent'
['breakout10', 'breakout20', 'breakout40', 'breakout80', 'breakout160', 'breakout320', 

'relmomentum10', 'relmomentum20', 'relmomentum40', 'relmomentum80', 

'assettrend2', 'assettrend4', 'assettrend8', 'assettrend16', 

'assettrend32', 'assettrend64', 

'normmom2', 'normmom4', 'normmom8', 'normmom16', 'normmom32', 'normmom64', 

'momentum4', 'momentum8', 'momentum16', 'momentum32', 'momentum64', 

'accel16', 'accel32', 'accel64']

An absolutely perfect convergent vs divergent split. The labels by the way are added by me, not the code.

Results: N=3

Cluster 1 'convergent' (Unchanged)
['mrinasset160', .... ]

Cluster 2 'fast divergent'
['breakout10', 'breakout20', 

'relmomentum10', 'relmomentum20', 'relmomentum40', 'relmomentum80', 

'assettrend2', 'assettrend4', 

'normmom2', 'normmom4', 

'momentum4', 'accel16']

Cluster 3 'medium and slow divergent'
['breakout40', 'breakout80', 'breakout160', 'breakout320', 

'assettrend8', 'assettrend16', 'assettrend32', 'assettrend64', 

'normmom8', 'normmom16', 'normmom32', 'normmom64', 

'momentum8', 'momentum16', 'momentum32', 'momentum64', 

'accel32', 'accel64']

This is why we are doing this exercise - we've just discovered something interesting: fast momentum like trading rules have more in common with other fast momentum trading rules, than they do with slow variations of themselves.

Results: N=4

Cluster 1 'convergent mean reversion'
['mrinasset160']
Cluster 2 'convergent skew and carry'
['carry10', 'carry30', 'carry60', 'carry125', 'relcarry', 'skewabs365', 'skewabs180', 'skewrv365', 'skewrv180']
Cluster 3 'fast divergent - unchanged'
['breakout10', 'breakout20', ....]
Cluster 4 'medium and slow divergent - unchanged'
['breakout40', 'breakout80', ....]

Results: N=5

Now it's the turn of the (relatively) slow divergent to be split up:

Cluster 1 'convergent mean reversion (unchanged)'
['mrinasset160', 'mrwrings4']
Cluster 2 'convergent skew and carry (unchanged)'
['carry10', 'carry30', 'carry60', ....]
Cluster 3 'fast divergent - unchanged'
['breakout10', 'breakout20', ....]

Cluster 4 'slow divergent'
['breakout160', 'breakout320', 
'assettrend32', 'assettrend64', 
'normmom32', 'normmom64', 
'momentum32', 'momentum64']
Cluster 5 'medium speed divergent'
['breakout40', 'breakout80', 
'assettrend8', 'assettrend16', 
'normmom8', 'normmom16', 
'momentum8', 'momentum16', 
'accel32', 'accel64']

Again it's the speed of trading that is the differentiator here, not the trading rule.

Results: N=6

We break relative momentum off from it's counterparts in what was previously cluster 3:

Cluster 1 'convergent mean reversion (unchanged)
['mrinasset160']
Cluster 2 'convergent skew and carry' (unchanged)
['carry10', 'carry30', 'carry60', ...]
Cluster 3 'fast divergent - unchanged'
['breakout10', 'breakout20', ....]

Cluster 3
['relmomentum10', 'relmomentum20', 'relmomentum40', 'relmomentum80']
Cluster 4
['breakout10', 'breakout20', 
'assettrend2', 'assettrend4', 
'normmom2', 'normmom4', 
'momentum4', 'accel16']

Cluster 5 'slow divergent' (unchanged - was cluster 4)
['breakout160', 'breakout320',...]
Cluster 6 'medium speed divergent' (unchanged - was cluster 5)
['breakout40', 'breakout80',...]

Results: N=7

And now acceleration comes away from the other slow rules:

Cluster 1 'convergent mean reversion (unchanged)
['mrinasset160']
Cluster 2 'convergent skew and carry' (unchanged)
['carry10', 'carry30', 'carry60', ...]

Cluster 3 'relative momentum' (unchanged)
['relmomentum10', 'relmomentum20', 'relmomentum40', 'relmomentum80']
Cluster 4 'fast divergent' (unchanged)
['breakout10', 'breakout20'...]

Cluster 5 'slow divergent ex. accel'
['breakout160', 'breakout320', 'assettrend32', 'assettrend64', 'normmom32', 'normmom64', 'momentum32', 'momentum64']
Cluster 6 'slow acceleration'
['accel32', 'accel64']
Cluster 7 'medium speed divergent' (unchanged - was cluster 6)
['breakout40', 'breakout80',...]

Results: N=8

Skew and carry seperate:

Cluster 1 'convergent mean reversion (unchanged)
['mrinasset160']
Cluster 2 ('skew)
['skewabs365', 'skewabs180', 'skewrv365', 'skewrv180']
Cluster 3 ('carry')
['carry10', 'carry30', 'carry60', 'carry125', 'relcarry']

Cluster 4 'relative momentum' (unchanged)
['relmomentum10', 'relmomentum20', 'relmomentum40', 'relmomentum80']
Cluster 5 'fast divergent' (unchanged)
['breakout10', 'breakout20'...]

Cluster 6 'slow divergent ex. accel'
['breakout160', 'breakout320',...]
Cluster 7 'slow acceleration' (unchanged)
['accel32', 'accel64']
Cluster 8 'medium speed divergent' (unchanged)
['breakout40', 'breakout80',...]

Results: N=11

Let's skip ahead a bit, and also show all the instruments in each group for this final iteration:

Cluster 1 'slow asset mean reversion'
['mrinasset160']
Cluster 2 'skew'
['skewabs365', 'skewabs180', 'skewrv365', 'skewrv180']
Cluster 3 'carry'
['carry10', 'carry30', 'carry60', 'carry125', 'relcarry']
Cluster 4 'slow relative momentum'
['relmomentum10', 'relmomentum20']
Cluster 5 'fast relative momentum'
['relmomentum40', 'relmomentum80']
Cluster 6 'divergent speed 2'
['breakout20', 'assettrend4', 'normmom4', 'momentum4']
Cluster 7 'divergent speed 1 (fastest)'
['breakout10', 'assettrend2', 'normmom2', 'accel16']
Cluster 8 'divergent speed 5 (slowest)'
['breakout160', 'breakout320', 'assettrend32', 'assettrend64', 'normmom32', 'normmom64', 'momentum32', 'momentum64']
Cluster 9 'slow acceleration'
['accel32', 'accel64']
Cluster 10 'divergent speed 4'
['breakout80', 'assettrend16', 'normmom16', 'momentum16']
Cluster 11 'divergent speed 3'
['breakout40', 'assettrend8', 'normmom8', 'momentum8']

A new heirarchy for handcrafting trading rules

With that all in mind, a better heirarchy would be something a bit like this:

• Convergent
• Mean reversion
• Skew
• Equal split between 4 skew rules
• Carry
• Outright carry
• Relative carry
• Divergent
• Speed 1 (fastest: turnover > 45)
• acceleration - nothing fast enough
• relmomentum10
• other trend
• breakout10
• assettrend2
• normmom2
• momentum4
• Speed 2 (22 < turnover <45)
• acceleration16
• relmomentum20
• other trend
• breakout20
• assettrend4
• normmom4
• momentum8
• Speed 3 (12 < turnover < 22)
• acceleration32
• relmomentum40
• other trend
• breakout40
• assettrend8
• normmom8
• momentum16
• Speed 4 (7 < turnover < 12)
• acceleration64
• relmomentum80
• other trend
• breakout80
• assettrend16
• normmom16
• momentum32
• Speed 5 (4 < turnover < 7)
• other trend
• breakout160
• assettrend32
• normmom32
• momentum64
• Speed 6 (turnover > 4)
• other trend
• breakout320
• assettrend64
• normmom64

As you can see I (roughly) used turnovers to group the divergent rules, although these groupings aren't quite right I thought it better to go for some nice neat sequences. And this also doesn't exactly follow how the clustering above works eithier. But this would certainly be a better way of doing things than the grouping entirely by trading rule, which as we've seen doesn't make sense for divergent rules where speed is more important.

Now of course there are a lot of caveats with this; first of all it's entirely in sample, but given how stable and persistent correlation of trading rules returns are over time, we'd probably get very similar results with a purely backward looking approach. Secondly we're ignoring things like costs, and the possibility that some rules may do better than others, but we can deal with that when we actually use the above structure to set instrument weights.