Wednesday, 27 March 2019

Risk targeting and dynamic asset allocation: absolute or relative momentum?

Quite a few of my recent blog pieces have been picked up by the lovely folk at allocate smartly. So I thought I'd write an asset allocation piece, as the readers of my second book "Smart Portfolios" probably feel neglected with the lack of articles on investment rather than trading.


Absolute or relative momentum?


The motivation for this comes from a table in my second book, which exposes an interesting problem. Here is the table (actually a slightly modified version of it, so you won't recognise the precise numbers), and I'll explain what it means and what the problem is:


                   Arithmetic mean  Geometric mean   Std. Deviation   Sharpe Ratio

Fixed weight            8.37%         8.04%             8.14%             1.03
Relative momentum       9.26%         8.89%             8.62%             1.07
Absolute momentum       8.93%         8.61%             7.96%             1.12


Fixed weight: This is a portfolio with 75:25 risk weightings in US equities and US bonds (using the last 12 months of monthly returns to calculate the appropriate volatility for risk weighting; this works out to roughly 60:40 cash weightings on average).

Relative momentum: This portfolio tactically rebalances the strategic fixed weights using the relative 12 month total risk adjusted return of equities and bonds. The rebalancing is a 'tilt' to account for forecasting uncertainty; the maximum tilt is to 148% of the original portfolio weight, and the minimum is 60% of the original. The relative momentum portfolio is always fully invested.

Absolute momentum: This portfolio tactically rebalances the strategic fixed weights according to the absolute 12 month total risk adjusted return of equities and bonds, again using a 'tilt'. The absolute momentum portfolio may not be fully invested if momentum is relatively weak in one or both assets. The minimum investment is 60% (which is not unusual), and the average is 93%.

All portfolios are rebalanced monthly, using data from January 1954 to March 2016 (I could update this, but I wanted to use the same data as in the book, and it wouldn't affect the results much). Returns shown are excess returns, net of the risk free rate.

Let's do some basic analysis of these results. Using absolute momentum results in a slightly higher risk than for fixed weights, because equities have spent more time going up in a risk adjusted sense. I call this the historical volatility boost. That more than compensates for the fact we aren't always fully invested, which drags down risk. The average cash weight to equities is 67% versus the 61% under fixed weights. But the extra risk is well rewarded with a higher arithmetic and geometric return, and a higher Sharpe Ratio.

Absolute momentum is a super popular asset allocation methodology, because people like the 'downside protection' of being partly in cash when markets are selling off.

Relative momentum has even higher risk; again it has a systematic bias towards equities and a historical volatility boost, but because we are always fully invested that all hits the 'bottom line' in the form of higher risk. The average cash weight to equities is 70%. The extra risk is rewarded with a higher arithmetic and geometric mean return, but the Sharpe Ratio is actually lower than for absolute momentum (though still better than fixed weights).

Relative momentum is less popular amongst the general public, as it seems hard to justify a big allocation to bonds just because they aren't falling quite as fast as equities. It also has a worse Sharpe Ratio, so 'theoretically' it's inferior (if you're an investor who can use leverage).

In my book I rather blandly concluded that relative momentum was better due to the higher geometric mean.

However, we're not comparing like with like. Strictly speaking we should probably compare relative momentum with an absolute version that has a higher strategic allocation to equities, so that their risk levels are comparable. To put it another way: is it better* to use relative momentum, or to use absolute momentum and crank up your strategic risk target to compensate for the reduction in risk?

Already we can see that this is a variation of the classic dilemma that investors without access to leverage and high risk tolerance have: should I opt for the highest Sharpe Ratio, or for something with higher risk (and return) but a lower Sharpe Ratio? However this story is more complicated, because we have two moving parts: the original risk weights, and the choice of rebalancing strategy (fixed weights, absolute, or relative). The interaction of these will produce portfolios with different return and risk profiles.

* The dilemma would be the same** for any type of forecast, but momentum is a popular and well understood rule to establish conditional returns.

** Strictly speaking the idea of an 'absolute' forecast requires some kind of equilibrium value at which we have a zero position. So dividend yield as a forecast wouldn't be helpful for absolute weighting, but something like (divided yield - interest rates)*** would make sense. 

*** the 'Fed model'


The experiment


The general question we want to answer is: For a given risk tolerance, what is the best choice of strategic risk weights and rebalancing strategy?

My criteria will be to judge a particular outcome by looking at the geometric mean (my reasons for choosing that are documented here), and the standard deviation of returns.

The range of strategic risk weights I will consider are from 10% equities 90% bonds, up to 90% equities 10% bonds. All strategic risk weight portfolios will be fully invested. Note that people with really low risk appetites will be best served by the maximum Sharpe Ratio portfolio plus a cash allocation; however I won't consider that option here. After all the problem we are exploring is most acute for investors with higher risk appetites.

To make the results starker, I'm going to allow the two tactical portfolios to 'tilt' all the way from 10% to 200% of the original strategic weight. Obviously this won't affect the fixed weights. For less aggressive tilts the relative results will be the same, but the numbers will be closer together.

First let's look at the Sharpe Ratios:



The black line is what you'd expect; the maximum SR portfolio is roughly 50:50 in equities and bonds. Absolute momentum is mostly inferior to the other options except for relatively high allocations to equities. Relative momentum shows declining performance as we increase the risk weight.

However these differences in SR might not be significant (I'll discuss this later in the post), but more importantly 'we can't eat Sharpe Ratios' if we're not leveraged investors, so let's instead focus on the geometric means and standard deviations.



Each line shows a classic 'efficient frontier', with one line for fixed weights, one for relative weights, and one for absolute weights. Each cross is a different strategic allocation, in 10% steps. So the first black cross on the bottom end of the fixed weights line is 10% risk weight in equities, the next cross is 20% in equities, and so on up to 90% on the top right end of the line.

We can safely ignore all the portfolios with lower risk than 30% equities; for these we'd be better off mixing the maximum Sharpe Ratio portfolio with cash.

It's clear from this graph that the out performance of relative momentum is pretty consistent. For a given risk target relative momentum is better than fixed weights or absolute momentum. It also looks like there is no benefit from using a risk target of greater than 80% in equities.


Which strategic portfolio weights should we use?


There is an important question that is not easily answered by the graphs above: how much should I adjust my strategic weights to compensate for the effect of applying a relative or absolute momentum tactical weight?



So, for example, if you want a standard deviation of 8% you could use:


  • a fixed risk weight of ~75% to equities
  • tactical absolute weighting with a strategic risk weight of ~68% to equities 
  • tactical relative weighting with a strategic risk weight of ~40% to equities


That is some substantial difference!

How robust are these results?


First let's consider the differences in geometric means. I'm extremely confident that 12 month momentum is a robust effect that has existed in the past, though we can argue about whether it will continue in the future. So I'd expect both types of momentum to beat fixed weights.

What about the out performance of relative momentum? Cross sectional momentum across asset classes is a less popular idea (though super popular within asset classes e.g. across stocks), but it would be surprising if there was a substantial difference between the two types of forecast.

However in a long only portfolio absolute momentum is operating with one hand tied behind it's back, as it cannot go short. This might explain the relatively poor performance of absolute momentum. Even when it has a slightly higher Sharpe Ratio (for relatively high equity weightings), the reduction in volatility means that absolute momentum can't compete on a geometric mean basis.

What about the differences in standard deviations? By construction the standard deviation for absolute momentum will always be lower than that for relative momentum.

The reasons for the increase in standard deviation when using relative momentum is less robust (risk also rises for absolute momentum, except for very low or very high equity allocations). In theory, if both equities and bonds had the same average forecast going forward, then the standard deviation would be the same for relative momentum as it is for fixed weights.

Radically reducing your strategic weight to equities to compensate for the expected 'volatility boost' from your tactical overlay might not be wise. The existence of a risk boost is probably the least robust finding here - I wouldn't be 100% sure it will exist in the future.


Conclusion


I'm reasonably happy that my superficial analysis in "Smart Portfolios" was correct when put through a more thorough test: relative momentum gives a higher geometric mean than absolute momentum, except for investors with low tolerance to risk. Therefore for most investors it's preferable.

In terms of more specific advice, the graphs above suggest the optimal portfolios are:


  • If you can use leverage, the highest Sharpe Ratio comes from using relative momentum tactical weighting with a risk weight to equities of somewhere between 30% (15:85 equity/bonds in cash weights based on current vols) and 50% (30:70 equity/bonds in cash weights). Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past is absent. Recommend: Strategic risk weights 50% equity 50% bond, cash weights 30% equity 70% bonds, relative momentum tactical weighting.

  • If you can't use leverage and have a high risk tolerance, the highest geometric mean comes from using  relative momentum tactical weighting with a risk weight to equities of somewhere between 60% (40:60 in cash weights based on current vols) and 90% (80:20 in cash weights).  Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past is absent. Recommend: Strategic risk weights 90% equity 10% bond, cash weights 80% equity 20% bonds, relative momentum tactical weighting.

  • If you can't use leverage and have a modest risk tolerance (but higher than 8% standard deviation a year): I'd use relative momentum but with a lower risk weight. If you don't buy the 'risk boosting' story then you will need between 60% and 90% risk weighting in equities; if you do buy the story and believe history will repeat itself, between 30% and 60% risk in equities. Recommend: Strategic risk weights 60% equity 40% bond, cash weights 40% equity 60% bonds, relative momentum tactical weighting.

  • If you can't use leverage and have a low risk tolerance (lower than 8% standard deviation a year): I'd invest in the maximum Sharpe Ratio portfolio (see above), and blend it with cash.



28 comments:

  1. great article as always. And I fully bought your risk weighting methodology and actually manage my own portfolio like that.

    I was just wondering. Do you think the whole momentum weighting is actually worth it? like on a real world implementation stand-point? ok, sharpe is around 0.1 worse but this is a backtest.. and keeping track of all the 12m performance and weighting it makes the handling slightly complicated.
    You think I leave returns on the table for not implementing it?

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    1. I find this question fascinating :-) Let me answer it first and then explain why it's such an interesting question.

      Let's say the out performance is just 0.2% a year - much less than the figures show but better to be pessimistic. If your portfolio was $100K then you'd earn an extra $200 a year, or $16 a month. With spreadsheets and automatic data feeds it is probably going to take you about half an hour a month to do this. So if your hourly pay is more than $32 it probably isn't worth doing this. This simple analysis ignores compounding effects. That all suggests that only people with quite large portfolios and quite low hourly pay should bother. Basically retired people.

      In a real world situation, once costs are accounted for, then you still get most of the benefit from rebalancing annually. The breakeven with a $100K portfolio will go up to $400 an hour. That is a wider set of people!

      (Incidentally, and I say this in my book, if your portfolio is too small then the costs of rebalancing even annually are too high, and you should stick with fixed weights. I can't remember the exact figure but I think it's about $15K. Final note is that tax may also have an effect but that is hard to model)

      Why is this question interesting? Because it gives us an interesting angle on the whole efficient market hypothesis. Arguably the market is efficient if 'Smart Beta', such as momentum, is only available to those who are willing to put the time into capturing it. We can view 'Smart Beta' as a reward for time spent capturing it. For retail investors that reward is too small to make it worth their while. But if you have a large institutional portfolio then amortising the cost of a couple of guys with a spreadsheet and a Bloomberg terminal over hundreds of millions of pounds makes it all worthwhile.

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  2. Hi Rob,

    Just finished your second great book, and there are a few points I'm not clear about (this post touches one of them):

    The effect of rebalancing weights for historical volatility. From your book, it seems like you don't think it matters much if you use the historical estimates of standard deviation for each asset class, or use a 12 month trailing volatility estimate (either with or without exponential MA). Since the first option (constant volatility estimate) means you ignore changes in volatility when rebalancing, there should be a difference between them. What about giving recent volatility even more weight, like using a 60 day trailing volatility? I would expect that shortening the historical volatility window will result in lowering the overall volatility of the portfolio.
    Using a momentum model for adjusting weights (either relative or absolute) means recent volatility does effect your weights, but by how much? How would the fixed weight option from this post compete against momentum if you would have shortened the volatility window from 12 months to 60 days? (maybe with a 'no trade zone' to reduce costs)

    It is also not clear to me why you conclude that relative momentum is the better choice. I understand why it is superior when looking at geometric mean, but risk wise? How do they compare when looking at draw-downs during fat-tail events? What about the possibility that future returns of either asset class will exhibit longer periods of negative returns? wouldn't that make absolute momentum a winner in risk adjusted returns?

    I have a relatively large portfolio, with a moderate risk tolerance. I chose to go with the compromise risk weights you suggested in your book - 30% risk weight for bonds (I don't fill comfortable allocating higher risk weight to bonds). I'm leaning towards using the relative momentum model, but would like to have some of that 'down-side' protection that the absolute momentum model offers. It seems like combining relative momentum with a shorter historical volatility window might do the trick. Does that make sense to you? If so, I need to know how much weight I'm already giving recent volatility just by using the momentum model.

    Thanks a lot for all the valuable and practical knowledge in your books + the great tools (pysystemtrade & spread sheets).

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    1. I will do a multipart answer here.

      "What about giving recent volatility even more weight, like using a 60 day trailing volatility? I would expect that shortening the historical volatility window will result in lowering the overall volatility of the portfolio."

      The sweet spot for predicting future volatility is something like a 30 to 60 day window. However predicting future volatility better doesn't necessarily improve returns, and it certainly wouldn't reduce realised volatility. In my fully automated system I use something like a 30 day lookback, but in a slower manual system the gains from doing this are minimal vs the extra work involved.

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    2. "Using a momentum model for adjusting weights (either relative or absolute) means recent volatility does effect your weights, but by how much? How would the fixed weight option from this post compete against momentum if you would have shortened the volatility window from 12 months to 60 days? (maybe with a 'no trade zone' to reduce costs)"

      As I already said reducing the vol lookback would hardly make any difference to performance.

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    3. "It is also not clear to me why you conclude that relative momentum is the better choice. I understand why it is superior when looking at geometric mean, but risk wise?"

      As I said in the post absolute momentum always has lower risk than relative momentum for a given strategic risk allocation. So to compare them fairly we need to jointly look at both risk allocation and choice of filter. Once that is done relative momentum generally wins.

      "How do they compare when looking at draw-downs during fat-tail events? "

      I haven't looked at this. I'm not super keen on using draw down as a risk measure, since there is a lot more parameter uncertainty. Clearly in a tail event where both equities and bonds fell then absolute momentum would do better. In a tail event where one went up and one went down (2008) relative momentum would do better. Depending on how you define 'tail event' there may only have been a few events in each category so we are unlikely to get results that are statistically significant.

      "What about the possibility that future returns of either asset class will exhibit longer periods of negative returns? wouldn't that make absolute momentum a winner in risk adjusted returns?"

      Obviously it would. For that to happen however the future would have to be significantly different from a very long period of the past (1954 to 2006). As someone who builds trading models based on backtests I always assume that I can't predict the future, but that the future will be sufficiently like the past that a simple trading model will continue to perform as it did historically.

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    4. "I have a relatively large portfolio, with a moderate risk tolerance. I chose to go with the compromise risk weights you suggested in your book - 30% risk weight for bonds (I don't fill comfortable allocating higher risk weight to bonds). I'm leaning towards using the relative momentum model, but would like to have some of that 'down-side' protection that the absolute momentum model offers. It seems like combining relative momentum with a shorter historical volatility window might do the trick. Does that make sense to you? "

      Using a shorter historical window is fine if you are happy with the extra work involved, but like I said it won't reduce your volatility or improve your returns, just give you more accurate risk targeting.

      If you prefer absolute momentum, then by all means use it. Just be aware of the caveats in this post:

      - your risk will be reduced, so you may want to increase your strategic weight to equities. Generally speaking it's going to be harder to get your risk right with absolute momentum as there is more uncertainty about what it will be (ironic if you're going to reduce your volatility lookback, since the only benefit of that is to get better risk targeting).
      - based on my analysis it will underperform relative momentum

      "If so, I need to know how much weight I'm already giving recent volatility just by using the momentum model."

      I don't really understand this statement. Using the momentum model doesn't 'give more weight to recent volatility'. It just changes your strategic weights.

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    5. Thanks for clearing the whole volatility issue to me. I just ran a few simple tests, and the difference between shorter and longer volatility windows is indeed very small.

      If I may, one more question: Is there a difference between relative and absolute momentum for someone who pays a higher tax rate for short term profits (US)?

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    6. "Is there a difference between relative and absolute momentum for someone who pays a higher tax rate for short term profits (US)?"

      I wouldn't have thought so. Generally momentum is probably more tax efficient than fixed weights, but I am not sure the different forms of momentum would be very different.

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    7. "I don't really understand this statement. Using the momentum model doesn't 'give more weight to recent volatility'. It just changes your strategic weights."

      The question was about 'punishing' volatility a bit more. For example, what effect would using a 1.25 F factor for the momentum model will have on the results? would it be the same as just allocating more risk to bonds in the first place?

      "Generally momentum is probably more tax efficient than fixed weights"

      I'm assuming this is because of the 'trend following' effect: letting your winners run longer and cutting your losses faster.

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    8. "The question was about 'punishing' volatility a bit more. For example, what effect would using a 1.25 F factor for the momentum model will have on the results? would it be the same as just allocating more risk to bonds in the first place?"

      Sorry I don't understand what 'using a 1.25F factor' means. Nor do I understand what you mean by 'punishing volatility'...

      "Generally momentum is probably more tax efficient than fixed weights" I'm assuming this is because of the 'trend following' effect: letting your winners run longer and cutting your losses faster."

      Yes

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    9. "Sorry I don't understand what 'using a 1.25F factor' means. Nor do I understand what you mean by 'punishing volatility'..."

      It just means that when you calculate your assets' sharpe ratio, you multiply the standard deviation with a constant which is larger than 1.

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    10. "It just means that when you calculate your assets' sharpe ratio, you multiply the standard deviation with a constant which is larger than 1."

      Ok, frankly that's weird. I can't think of any reason why that would make sense.

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  3. thanks for the answer. fully agree. probably I'm just lazy and somewhat sceptical to buy more of the things which went up more so the fixed weighting is appealing to me.

    on another note. do you do consulting to some extent? I will now get my ass up and implement the momentum weighting in a proper sheet. is there any way you could cross check if I did the implementation correctly in my spreadsheet? also I would have some other questions to you. I would be absolutely happy to pay you for that (unless you charge Goldman like fees :-))



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    1. I'm afraid I don't have time to do any consulting right now, and to be honest I do charge quite high fees when I do!

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  4. Hi Rob,
    A question regarding the choice of 12 month returns as a parameter for the momentum model.
    Do you think averaging a couple (or more) return periods would decrease parameter sensitivity? Kind of what you do in your trend-following model with the different variations for each trading rule.

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    1. Yes this is a good thing to but more work. So up to you.

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    2. You could use an average of 3...18 month averages. I probably wouldn't go shorter or longer, or mean reversion will kick in.

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    3. Hi Rob,
      In order to average multiple return/std periods, do I need to annualize both the total returns and the STD for each time period?

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  5. This comment has been removed by the author.

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    1. I am reproducing your comment here as I think there was a mixup, you deleted a duplicate comment and I deleted the other duplicate comment:

      "Hi Rob, your "Smart Portfolios" book is possibly the most practical book on investing I have ever read (out of well over 100+ books).

      I imagine some of the readers of your book may be near retirees (late 50's, retirement horizon within 5-10 years).

      From this perspective I have a few questions. Hopefully you will be kind enough to answer:

      1) When one is making monthly portfolio withdrawals in retirement, a major drawdown is significantly more impacting on "risk of ruin" than for someone who is still in the accumulation phase and can wait it out. So for a retiree, might absolute momentum be a better approach?

      2) In your book, you treated relative and absolute momentum as mutually exclusive options. Had you looked into using both relative and absolute momentum together (dual momentum) and what the geometric mean of that would be?

      3) There are significant asset classes besides equities and bonds (gold, managed futures, market neutral, etc.). Would it be smart to include them in a portfolio to increase diversification and thus geometric mean, and if so, why did you choose to exclude them from your book "Smart Portfolios" and focus exclusively on equities and bonds?

      4) Do you have any specific recommendations for retiree readers of your book in the withdrawal phase of their retirement?

      Thanks very much for all the incredible effort I can only imagine you must have put into your book to make it so detailed and useful."

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    2. And now the answer (excellent questions by the way)

      (1) & (4) Curiously I have been thinking about giving a more formal treatment to people in the withdrawal phase of their investing lives, and it will probably be covered in my next book. Generally if you are making withdrawals then lower risk is more important. So yes absolute momentum would make more sense; but on the downside it will reduce returns, and assuming you are relying on dividend income to fund your retirement you will have periods without any income at all when you will have to sell assets - psychologically problematic.

      2) I wasn't sure what you meant by using absolute and relative momentum together (since they are indeed mutually exclusive), then I googled 'dual momentum' and now I understand. The standard dual momentum model seems to switch entirely between asset classes, but let's leave that difference aside. As far as I can tell, dual momentum isn't 'absolute' in the way I understand it since it's always invested in something. The fallback is bonds, though weirdly you do that if the momentum of your assets is less than t-bill returns, for absolute momentum you would be investing in t-bills if that was the case). If we replace t-bill returns with bond returns then you've basically got an all or nothing relative momentum.

      3) Literally the whole of chapter 9 is devoted to other asset classes...

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  6. Hi Rob, your "Smart Portfolios" book is possibly the most practical book on investing I have ever read (out of well over 100+ books).

    I imagine some of the readers of your book may be near retirees (late 50's, retirement horizon within 5-10 years).

    From this perspective I have a few questions. Hopefully you will be kind enough to answer:

    1) When one is making monthly portfolio withdrawals in retirement, a major drawdown is significantly more impacting on "risk of ruin" than for someone who is still in the accumulation phase and can wait it out. So for a retiree, might absolute momentum be a better approach?

    2) In your book, you treated relative and absolute momentum as mutually exclusive options. Had you looked into using both relative and absolute momentum together (dual momentum) and what the geometric mean of that would be?

    3) Do you have any specific recommendations for retiree readers of your book in the withdrawal phase of their retirement?

    Thanks very much for all the incredible effort I can only imagine you must have put into your book to make it so detailed and useful.

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    1. I finally found your original comment! Please read my answers above if you haven't already done so.

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  7. Hi Rob,

    In your answer above you reference your "next book".

    "(1) & (4) Curiously I have been thinking about giving a more formal treatment to people in the withdrawal phase of their investing lives, and it will probably be covered in my next book."

    Which book are you referring to... would that be "Leveraged Trading" or an as yet to be released book?

    Thanks very much.

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