CTA allocations, QE, meta-prediction, and conditional return distributions
The US QE2 programme began in November 2010, and it finished in overdue 2014. We knew that QE turned into retaining interest fees fairly low and solid. We knew sooner or later it might quit, and interest fees could upward push.
(I'm penning this as a trader rather than an economist, and the excellent distinction among the outcomes of Fed Fund rate rises and the slowing or reversing of asset purchases do now not difficulty us right here)
Our number one difficulty became how our models might react while this event befell. Would our techniques cope, or might they be badly underwater? Given that the QE unwind is still underway that is nevertheless very a whole lot a stay dialogue.
The problem here is one of meta-prediction. We aren't trying to predict interest rates, or the returns of assets like US bond futures; we already have our underlying trading strategies for that: momentum, carry, and perhaps a few other bits and pieces. Instead we're trying to make predictions about the returns of the trading strategies. Does momentum do better in particular regimes? Can we identify when carry is likely to under perform? If so we can reallocate our risk capital into the most favourable place. Essentially this is a problem of factor timing. Like Cliff says, factor timing is hard. But let's try anyway.
Some messy python code the use of pysystemtrade is right here
An idiots manual to meta-prediction
Meta-prediction requires essential components:
- A strategy, or strategies, whose returns we can back test historically.
- A conditioning variable to identify a particular regime (for simplicity I'm assuming that regimes are discrete rather than continuous here)
Assuming we've those ingredients our job is straightforward: partition our historic strategy returns by way of regime, and check whether or not there may be any difference in method overall performance. We then discover the regime this is closest to what we assume to occur subsequent, and determine if our strategy will do better or worse than average.
Choice of strategies
To keep matters especially simple I'm going to use the four core buying and selling rule variations from chapter 15 of my first ebook: Carry, and 3 versions of momentum - speedy, medium, slow (moving average crossovers with speeds sixteen/64, 32/128, 64/256).
The desire of gadgets is more thrilling. Obviously we might see clearer effects if we checked out Eurodollar futures and US bonds; even clearer if we aligned the device with the conditioning variable (eg the use of a 2 yr interest price to see what occurs with 2 year bond futures). Of course there are in all likelihood to be spillover consequences into different bonds, and probably into different asset classes (the worry of growing interest fees appears to have been one of the reasons of the latest sharp promote off in equities). To preserve matters simple in this submit I'm broadly speaking going to look only at the subsequent US hobby fee associated futures:
- Eurodollar (traded approximately 3 years out on the curve)
- US 5 year
- US 10 year
- US 20 year (I don't actually trade this but I have the data so why not)
(The US 2 yr bond destiny has inadequate facts, so I'm ignoring that. Also I'm focusing basically on US QE in this publish. Finally I'm ignoring the possibility of extrapolating from the sooner Japanese QE test, and seeing what lessons this would have for america)
Conditioning variable
To be beneficial conditioning variables need to have some key homes:
- Quantifiable
- Present in history
- Meaningful historic variation
- Reasonably distributed
- Ex-Ante
'Quantifiable' is, I wish, self explanatory to the average reader of this weblog. By 'found in records' I imply that we have a protracted historic file of the variable. QE fails badly in this degree - we've got by no means had QE in the US earlier than.
The OIS - LIBOR spread is an example of a range that did not have meaningful version previous to 2007. It also is not 'moderately dispensed' - the distribution prior to 2007 is absolutely one of a kind from what followed.
A key part of making meta predictions is to use an ex-ante rather than an ex-post variable. Official US recessions are an example of an ex-post variable - the official announcement is made around a year after each change in regime. Ex-post analysis is interesting, but useless when it comes to making predictions.
Given that QE itself is flawed, what is a variable that is a good QE proxy and satisfies all of the above conditions? A naive description of the effect of QE reversing would be something like this “Interest rates are low, but have started rising”. The level and change in interest rates would seem to be appropriate variables. To keep things ex-ante we'd need to ensure we measured the level at the start of the period (if we're using daily returns this just means lagging the rate by a day). Similarly the change needs to be up to the start of the period. Let's use the change in the previous 12 months up to the start of the return period.
Which hobby rate have to we use? The Fed Funds fee is an apparent one. But given we are concerned with QE, which was designed to make bond yields lower, I suppose we should focus on government bond yields. The average maturity of US debt is notably quick, around four-6 years. So I'm going to use this 5 yr regular adulthood bond price.
Note: the consequences are comparable with the Fed Funds Rate
Here is the hobby price stage:
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| US five year constant adulthood charge (https://www.Quandl.Com/information/FRED/DGS5-5-Year-Treasury-Constant-Maturity-Rate) |
I'm no longer top notch glad with this variable. If we situation on fee degree we're going to in all likelihood turn out to be partitioning on time: pre 1995 and put up 1995. I'm not sure how significant the effects from a good way to be. Here is the 12 month change:
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| Rolling 12 month exchange in five year US interest rates |
The amplitude of modifications is honestly higher when hobby prices are better; but I'd argue that a 100bp boom is a long way more huge now than it was lower back within the early Eighties. However the hassle isn't as critical as for the extent: this is mostly a reasonably desk bound collection.
To cope with this I'm going to apply a normalisation to the level: I will divide by the rolling 20 year average of the interest rate. Since interest rate cycles normally last about 10 years this will give us an indication of where we are in the rate cycle; a far more useful conditioning variable.
Here is the normalised level:
Much nicer. After adjustment you may see that hobby prices are genuinely in a better a part of the cycle than the unadjusted charge would suggest. Here is the normalised alternate:
- Normalised interest rate level (divided by 10 year average)
- Normalised interest rate change over last 12 months
Some naive consequences
Interest rate tiers (normalised)
To kick things off here is the account curve for Eurodollar carry, coloured to show the two regimes for normalised interest rate levels.
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| Carry account curve for Eurodollar futures, conditional on normalised interest rate regime |
Notice how we are presently in a 'high' interest charge surroundings, and that apart from the early 1980's returns look to had been higher while charges are low. An open query right here is whether or not we must use return or Sharpe Ratio. It looks as if the volatility is probably one-of-a-kind within the high interest fee surroundings. I'm going to use Sharpe Ratio, but you have to endure this in thoughts.
Here are the results for Eurodollar futures throughout unique buying and selling regulations:
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| Sharpe Ratio for Eurodollar futures throughout trading rule variations, conditioned on normalised five yr fees |
The 'low' charge surroundings here is a normalised range of zero.15 to 0.Seventy one; while for 'excessive' it is from zero.Seventy one to one.Fifty seven (those buckets are divided at the median fee of the conditioning variable). The rate is presently over 1.0; so at least on a normalised foundation we're really in a 'excessive' rate environment (in evaluation in 2013 just before the 'taper tantrum' the adjusted charge changed into at an all time low of zero.15). |
There seems to be a few susceptible evidence that 'excessive' is better than low, in particular for momentum.
Here are the results for the bond futures:
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| Sharpe Ratio for 5 year futures across trading rule variations, conditioned on normalised 5 year rates |
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| Sharpe Ratio for 10 year futures across trading rule variations, conditioned on normalised 5 year rates |
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| Sharpe Ratio for 20 year futures across trading rule variations, conditioned on normalised 5 year rates |
These are very blended effects. Because of this, and due to the fact the normalisation makes things slightly elaborate, I do not suppose there is anything well worth pursuing right here.
Interest charge adjustments (normalised)
Now let's flow directly to looking at hobby charge adjustments. We'll begin with Eurodollar, and then circulate up through the tenors.
Here is the account curve for Eurodollar and the slowest momentum rule, hacked to show the one of a kind regimes:
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| Carry account curve for Eurodollar futures, conditional on normalised hobby rate trade regime |
Notice how we're presently in a growing regime (orange), and how precise performance is sort of absolutely restricted to falling charge regimes (blue). Notice also that the trading rule reduces it's volatility whilst we lose money; this is a conventional pattern for fashion following. This additionally means that the use of conditional Sharpe Ratio is perhaps a touch unfair; absolutely the losses may be smaller when quotes are rising even supposing the SR looks certainly bad.
Anyway, do those results maintain throughout different trading policies?
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| Eurodollar futures, performance of trading regulations conditioned on normalised price adjustments |
'Fall' manner the normalised interest fee alternate was inside the variety -zero.Forty eight to -0.04 over the preceding one year before the applicable day. 'Rise' manner the fee changed was inside the variety -0.04 to zero.40. The purpose for this skew in buckets is of route that interest rates have in the main fallen inside the period we're the usage of, and the normalisation does not quite correct for this. Changing the buckets so that they cover a strictly bad and tremendous variety won't have an effect on the results.
The current 12 month change in rates is +0.26, so we're solidly in the rising interest rate environment here.
That is a completely constant sample but let's have a look at if this is repeated across different devices:
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| US 5 year bond futures, performance of trading rules conditioned on normalised rate changes |
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| US 10 year bond futures, performance of trading rules conditioned on normalised rate changes |
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| US 20 year bond futures, performance of trading rules conditioned on normalised rate changes |
There is a very consistent pattern here: recent rises in interest rates are bad news for carry, and really bad news for momentum (especially the slowest kind). Here is a nice summary chart that shows what happens if we lump all the different futures into a single portfolio (equally weighted):
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| Portfolio of US bond & rate futures, performance of trading rules conditioned on normalised rate changes |
This sort of makes sense: if rates are rising then the net effect of still positive carry plus negative price movements can lead to a 'choppy' total return series; choppy prices are seriously bad news for any kind of trend following; if carry stays long it will benefit from positive total return even if it's much much smaller than what we see in falling rate environments.
What about across portfolios? Here is the performance of each instrument, after applying some sensible forecast weights: system.config.forecast_weights = dict(ewmac16_64 = 0.2, ewmac32_128 = 0.2, ewmac64_256 = 0.2, carry=0.4)
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| Portfolio of trading rules, Sharpe Ratio across instruments, conditioned on normalised rate changes |
Focusing on the pattern of Sharpe Ratios it looks like you might want to up your allocation to US 5 years, but reduce fixed income generally.
Now we may have done a little data mining to get to this point, but blimey! That is one strong result! Consistently falling Sharpe Ratio as we move from a regime of recently falling rates to one of recently rising. It appears to be a very convincingnull points for fixed income momentum and carry in the current regime.
Some less naive results
The results above look compelling, and no doubt have many people working in CTAs rushing to deallocate from fixed income momentum as we speak. If we were working for the sell side, where our job was to generate flow rather than do proper research, we'd probably stop there.
But they miss out on an important point: we're only seeing the average conditional return, not thedistribution of conditional returns. This is important because the average doesn't tell us how significant the difference is between the returns we're seeing. More specifically what we want is the sampling distribution of the Sharpe Ratio estimate.
We know from Andrew Lo that for i.i.d. returns this has a standard deviation of root(1+.5SR^2).
(For non-normal distributions check out Opdyke)
Rather than mucking about with fancy formula that aren't quite accurate anyway let's bootstrap the relevant distributions. To avoid plot overload I'm going to do these for each trading rule variation individually, for a portfolio of instruments.
Here's the plot for carry
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| Histogram of sampling estimate for SR, across instruments, for carry rule, conditioned on normalised yield change |
There is clearly a serious difference between the performance here. The p-value for a t-test of the performance across these two regimes comes in at just over 1%.
Legend key: Range of conditioning variable, Mean Sharpe Ratio estimate, Independent t-Test p-value for comparing with the worst Sharpe Ratio
The results for other trading rules are equally strong or stronger, so to cut to the chase here is a plot for the entire portfolio of fixed income trading rules and instruments:
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| Histogram of sampling estimate for SR, across instruments, for all rules, conditioned on normalised yield change |
The t-test p-value is pretty significant here, again well under 5%. It turns out that there is a pretty substantial difference in fixed income strategy performance across different interest rate regimes.
By the way although the result is significant, it isn't as clear cut as the p-value above might lead you to believe. If we cut our data into finer regimes then we get the following Sharpe Ratios:
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| Average Sharpe Ratio for portfolio of all fixed income instruments and trading rules conditional on normalised interest rate regime |
Due to the way the regimes are created each bucket as an identical length of return history, but the rate regimes may be of different widths.
A few more experiments
Lest we be accused of p-hacking, here is the result for the unadjusted yield change:
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| Histogram of sampling estimate for SR, across fixed income instruments, for all rules, conditioned on raw yield change |
If anything the results here are more significant than for the normalised yield change. For reference the current 12 month trailing interest rate rise is +0.64%; so we do fall in the orange distribution, but only just. The results here aren't quite as relevant for current meta-predictions than those for the normalised change.
Fixed income is buried, but what about the rest of our CTA portfolio? Should we try a different asset class? Here are the results for S&P 500:
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| Histogram of sampling estimate for SR, S&P 500 futures, for all rules, conditioned on raw yield change |
It looks like rising rate environments are slightly better for momentum and carry in stocks but this is a long way from being significant. Generally we don't see such stark effects for momentum and carry in non fixed income instruments.
Finally here are the results for a couple of long only portfolios (i.e. we're now making predictions not meta-predictions). First long only equally weighted for all four of our fixed income instruments (actually these are inverse vol weighted, not equal weighted):
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| Histogram of sampling estimate for SR, across fixed income instruments, for long only portfolio, conditioned on adjusted yield change |
The results here make sense on one level, but not on another. This plot shows us that if interest rates have recently been rising then we should expect to do relatively badly from fixed income (although the p-value isn't that significant). Another way of putting that is that momentum works: if interest rates have been rising then total return from long only bond portfolios won't be that great (although this measure of momentum is the 12 month change in a single yield, rather than a moving average crossover on the adjusted price of whatever). However we already know that slow momentum is best avoided in fixed income when yields have been rising.
And here is long only S&P 500:
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| Histogram of sampling estimate for SR, S&P 500 futures as long only portfolio, conditioned on adjusted yield change |
It looks like rising rate environments are slightly better for stocks (due to higher economic confidence?) but again this is a long way from being significant. Interesting though and possibly the source of a trading rule idea for stock index prediction.
Summary: what should we do?
Here is a quick summary of the Sharpe Ratios that we've seen so far for each normalised interest rate change regime, plus the p-values when comparing the two regimes.
Falling Rising P-value
SP500 long 0.04 0.45 28.00%
FI long 0.9 0.44 13.50%
SP 500 CTA 0.68 0.88 69.00%
FI CTA 0.91 0.18 2.00%
EDOLLAR CTA 0.96 0.18 2.20%
US5 CTA 0.84 0.36 18.20%
US10 CTA 0.86 0.25 6.50%
US20 CTA 0.58 0.12 14.00%
FI carry 0.92 0.14 1.20%
FI slow mom 0.7 -0.25 0.20%
FI med mom 0.64 -0.02 2.50%
FI fast mom 0.66 0.10 7.30%
It looks like in a rising interest rate environment you should make the following portfolio adjustments:
- Long only: Shift to stocks and out of bonds (although bonds still do about as well as stocks in a rising rate environment - it's just that they're not doing as well as when rates are falling and stock performance is flat)
- CTA vs long only: Perhaps slightly reduce your overall CTA exposure, but not by much (stock CTA strategies actually do a little better, and CTA returns won't be much lower if managers deallocate from fixed income)
- CTA asset allocation: Shift out of fixed income and into other asset classes.
- CTA fixed income instrument weighting: You might want to slightly overweight 5 years at the expense of other tenors
- CTA fixed income forecast allocation: You might want to slightly overweight faster momentum at the expense of slower momentum (the slow momentum loses out most when we don't get a tailwind of general reductions in yield). On a relative basis carry holds up reasonably well versus momentum.
By the way 'shift' doesn't imply a complete reallocation. I'd be wary of changing my weights by more than a factor of 0.5 / 1.5, even with p-values of 2% or less. So for example if your CTA portfolio is 30% in fixed income; then the largest reduction I'd countenance would be to shift it to 15% in fixed income.
Concluding thoughts
I will be honest - I was surprised by these results - and I didn't set out to find them (always a risk with any piece of research). It's unusual to find meta-predictions that work. Out of loyalty to CTAs generally, and to fixed income specifically, I was rather hoping to find no significant effects.
Have I discovered a holy grail of factor timing? It depends what you mean by factor timing. I haven't shown that we can predict when momentum or carry can do well relative to buy and hold for a given asset class. All I've confirmed is that rising interest rates are not ideal for fixed income, and the results show that is true not only for long only, but also for most strategies that you might care to mention.
But if 'bond momentum' and 'bond carry' are factors, then sure I've found a pretty good predictor of when it does or doesn't work: recent rises in interest rates. Although remember from before that momentum will partially turn itself off when it is losing money, due to weaker signals as you'd expect to get when rates are rising.
I probably won't personally do anything with these results because it's an extra layer of complexity (and I haven't formally back-tested using interest rate changes to alter weightings, which means I haven't accounted for switching costs; plus there is the question as to how we approach say other countries like Germany where I don't have enough data to prove whether this works).
But they are still interesting food for thought.
