Tuesday 9 January 2024

Skew preferences for crypto degens

An old friend asking for help... how can I resist? Here is the perplexing paper:


And here is the (not that senstional) abstract:

Bitcoin (BTC) returns exhibit pronounced positive skewness with a third central moment of approximately 150% per year. They are well characterized by a mixture of Normals distribution with one “normal” regime and a small probability of a “bliss” regime where the price appreciation is more than 100 times at the annual horizon. The large right-tail skew induces investors with preferences for positive skewness to add significant BTC holdings to equity-bond portfolios. Even when BTC is forecast to lose half of its value in the normal regime, investors with power utility optimally add 3% allocations to BTC when the probability of the bliss regime is around 1%. Cumulative Prospect Theory investors are even more sensitive to positive skewness and hold BTC allocations of around 3% when the probability of the bliss regime is 0.0006 and the mean of BTC in the normal regime corresponds to a loss of 90%.

3% in BTC doesn't sound too crazy to me, but what has been really setting the internet on fire is this out of context quote from later in the paper:

Starting with a 60-40 equity-bond portfolio, which is produced with a risk aversion of 𝛾 = 1.50, the optimal BTC allocation is a large 84.9%! The remainder of the portfolio, 15.1% is split 60-40 between equities and bonds. Although BTC has an extremely large volatility of 1.322 (see Exhibit 1), the pronounced positive skewness leads to large allocations and dominates in the utility function (see equation (9)). The certainty equivalent compensation required to not invest in BTC is close to 200%. [my emphasis]

Here's my English translation of this:

- Bitcoin has pronounced positive skew 

- Some people really like positive skew (people  with 'power utility' and 'cumulative prospect theory' preferences)

- This justifies a higher allocation to Bitcoin than they would otherwise have, since it has lots of positive skew (both on an outright basis, and as part of a 60:40 portfolio).

- There is a 'Bliss' regime when Bitcoin does really well ('goes to the moon') but which isn't very likely

- Even if there is a tiny probaility of this happening, and if things are generally terrible in the non bliss regime, then people who like positive skew should have more Bitcoin. Some of them should have a lot!

Now, I could just as easily write this:

- Lottery tickets have (very!) pronounced positive skew 

- Some people really like positive skew 

- This justifies a higher allocation to lottery tickets than they would otherwise have

- There is a 'Bliss' regime when lottery tickets do very well ('winning the jackpot') but which isn't very likely

- Even if there is a tiny probaility of this happening, and if things are generally terrible in the non bliss regime, then people who like positive skew should have more lottery tickets. Some of them should have a lot!

I see nothing here that I can argue with (sorry Ben)! And it certainly doesn't require an academic to make the argument that people who like lottery ticket type payoffs, and think that there is a chance that Bitcoin will go up a lot, should buy more Bitcoin. But I think there is a blogpost to be written about the interaction of skew prefences and allocations; and hopefully one that is perhaps easier to interpret. Two key questions for me are:

- to what extent does the expectation of return distributions affect allocations?

- just how far from 'skew neutral' does ones prefence have to be before we allocate significant amounts to Bitcoin

Luckily, I already have an intuitive framework for analysing these problems, which I used in a fairly complete way in my previous post - bootstrapping the return distribution. 


The goal then, is to understand the asset allocation that comes out of (a) a set of return distributions and (b) a preference for skew.

For the return distributions we have two broad approaches we can use. Firstly, we can use actual data. Secondly, we can use made up return distributions fitted to the actual data. This is what the paper does, mostly "We use monthly frequency data at the annual horizon from July 2010 to December 2021 for BTC and from January 1973 to December 2021 for stocks and bonds. The univariate moments for each asset are computed using the longest available sample, and the correlation estimates are computed with the common sample across the assets."

The paper also uses a third approach, which is to see what happens if they mess with the return distributions once fitted by changing the probability of 'Bliss'.

I'm going to use the first approach, which is to use real return data at least initially. Other slight differences, I will use returns from July 2010 to November 2023 for all three assets, I will use excess rather than total returns (which given the low interest rates in the period makes almost no difference) with futures prices for S&P 500 (equity proxy) and US 10 year bonds (bond proxy), with Bitcoin total return deflated by US 3 month treasury yields, and I'm going to use daily rather than monthly data to improve my sample size.

The next consideration is the utility preference of the investor. I am going to assume that the investor wants to maximise the Nth percentile point of the distribution of geometric returns. This is the approach I have used before which requires no assumptions about utility function and allows an intuitive measure of risk preference to be used by modifying N. 

As I have noted at length, someone with N=50 is a Kelly optimiser. That is the absolute maximum you should bet, irrespective of your appetite for skew or risk. Thus the Kelly bettor must have the maximum possible appetite for skew. Someone with N<50 would be very nervous about the downside and much more worried about small losses than the potential for large gains; and hence they would have less of a preference for positive skewed assets.

I personally think this is a much more intuitive way to proceed than randomly choosing utility functions and risk aversion parameters, and choosing from a menu of theoretical distributions. The downside is that isn't possible to decompose skew and risk preferences, since both have been replaced with a different measure - the 'appetite for uncertainty'.

An important point is that maximising CAGR will naturally lead to a higher allocation to crypto than you would get from the more classical method of maximising mean subject to some standard deviation constraint or risk aversion penalty. 

The method I will use then is:

- sample the returns data repeatedly to create multiple new sets of data.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). 

- from this new set of data and a given set of possible portfolio allocations, estimate the geometric return

- for a given set of allocations, take the Nth percentile of the distribution of geometric means

- plot the Nth percentile for each allocation to work out roughly where the optimal might be

I say 'roughly', because as readers of previous posts on this subject are aware, we never know exactly where the optimal is when bootstrapping, which is a much better reflection of reality than the precise analytical calculations done by the original authors. Still, we can get a feel for how the optimal changes as we vary N (skew preference).

Note: As a fan of Red Dwarf, the use of the term 'Bliss' in this context is very confusing!

The data

Since we're pretending to be proper academics, here are the summary statistics of the real data:

Annualised mean:
equity 0.256
bonds 0.000
bitcoin 1.280

Annualised standard deviation:
equity 0.160
bonds 0.064
bitcoin 0.816

equity bonds bitcoin
equity 1.000 -0.245 0.069
bonds -0.245 1.000 -0.014
bitcoin 0.069 -0.014 1.000
Sharpe ratio:
equity 0.783
bonds 0.166
bitcoin 1.612

equity -0.431
bonds 0.294
bitcoin 0.970

Note that if anything the statistics here are more favourable to Bitcoin than in the original paper. Importantly, we are assuming that as in the past Bitcoin will more than double every year on average (the figure in bold), and that it will have a Sharpe Ratio well north of 1.0. Given these raw statistics, it isn't then very suprising regardless of skew preferences that we would potentially dump a large part of our portfolio into Bitcoin. And indeed, if I run these numbers through my optimisation the optimal position is 100% in Bitcoin for a Kelly maximiser.

To add another line to my 'dumb' bullet point translation of the paper earlier:

- if you think Bitcoin will go up a lot like it did in the past, you should only own Bitcoin

To make things more realistic and interesting, I'm doing to massage the data to reflect what I think is a fairly conservative forward looking position: All assets will have the same Sharpe Ratio (which I will set arbitrarily at an annualised 0.5). I achieve this by shifting the mean returns up or down respectively, which means all the other return characteristics remain the same - only Sharpe Ratio and means are affected. Note that this also means that bonds will look better relative to equities.

This still implies that Bitcoin will, on average, go up by 40% a year, which means it will double every two years. Personally I still think this is extremely optimistic, but I'm going to put my own views to one side for this exercise.

Note: even if you are not a Bitcoin skeptic, it seems unlikely that Bitcoin will behave in the same way going forward as it did when it was worth less than $1,000 and had the market cap of a penny stock rather than a decent sized country; both the mean, skewness, and the standard deviation have reduced in the last few years since Bitcoin has become a bigger market.


Right, let's see some pictures. 

The following heatmap shows what happens to the median of the distribution of bootstrapped geometric returns (Kelly maximiser, with maximum appetite for skew) as we allocate to equities (y-axis) and Bitcoin (x-axis). The allocation to bonds will be whatever is leftover. The white area is where we can't allocate, since we are putting more than 100% into the portfolio, and my working assumption is here is that leverage isn't allowed (if it was, we'd have much more bonds, much less equities and Bitcoin, and use leverage to maximse CAGR).

The optimal allocation to Bitcoin is somewhere around 50% with equities taking most of the rest. So even the most gung-ho optimistic skew loving nutjob shouldn't put more than half their wealth in BTC. For context, a 50% equity and BTC portfolio would have a standard deviation of around 42%, nearly 3 times the risk of equities. To be Kelly optimal, that implies the Sharpe Ratio would need to be at least 0.42. This is a much higher risk target than pretty much every hedge fund uses.

Now let's see what happens if we reduce our N to the 25% percentile point. Importantly: this is roughly the N that produces a 60:40 portfolio considering a portfolio with only equities and bonds. So we can think of this as the 'base case' for risk and skew preference. Again with CAGR below 4% washed out to produce a more granular z axis:

You can see the optimal allocation to Bitcoin is lower here, around 30%, with perhaps 50% in equities and the rest in bonds.

What about N=10%?

Again, we are looking at a bit less again in Bitcoin; with something around 20% with perhaps 70% in equities and the rest in bonds. This would give you something with a standard deviation not much higher than equities, at least in theory.

Summary- ignore everything I have said

The original paper has been toted around the internet to say that you should have 85% of your portfolio in Bitcoin ('this is optimal'). But:

- this is a single figure taken out of context from a much more nuanced paper; note again that the abstract does not include such an extreme figure
- it assumes that historic Bitcoin performance is matched going forward, including performance from 2010 back when BTC cost less than $1 and the total 'market cap' was less than $200,000. 
- it assumes particular risk aversion, preferences for skew and utility functions; such that you would hold quite a lot of Bitcoin even if you thought it's performance would generally be bad except in rare 'Bliss' regimes. Basically it says 'if you like lottery tickets, you are going to love Bitcoin!'.

In this post I take a different approach which hopefully is more intuitive for the non economist, and gives a bit more insight into the interplay between return skew and skew preference, which is also useful beyond the narrow problem of allocating to crypto currency. But what you couldn't or shouldn't do is take anything I or anyone else has written, and claim it 'proves' that the 'optimal' allocation to Bitcoin is x%. All it can do is say based on these assumptions and assuming this set of preferences what your allocation should be. That can quite easily come out to 85%, or 100%. It can also quite easily come out to less than 1%, or even zero.  

What's my own personal allocation to Bitcoin, I hear you ask? On a long only basis it is zero, and nothing I have written here will change that. Partly this is because of my long standing and well known aversion to this 'asset class', both in principle* and in practice**.

* to summarize it's a ponzi that wastes energy with the ownership structure of a pyramid scheme, and which will never be useful for anything except the current use cases: 1% illegal money transfer, 99% gambling
** it's a real pain and very expensive to buy Bitcoin 'properly' i.e. owning your own coins and putting them into cold wallet storage 

But it's also because unlike in this example, there are more than three assets in the world! Concretely, I trade well over 100 futures; of which just a couple are crypto coins. Accordingly it also makes no sense to me to put more than a few % of my trading account into crypto - an account where I can go long and short and hence my personal biases are irrelevant.

My allocation to Bitcoin and Ether in my futures trading strategy is a touch under 5%. And those are risk weights; the equivalent cash weight would be lower: as I write this my position in Bitcoin is long 3 micro futures with a notional value of perhaps £12K or around 3% of my trading capital. Of course it could just as easily be zero, or a short position...

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