@investingidiocy looking for my favorite statistics wonk to critique this. https://t.co/xWQMEl5LUU

— bengraham (@bengraham) January 5, 2024

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

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4042239

And here is the (not that senstional) abstract:

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.

## Setup

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

Annualised mean:

equity 0.256

bonds 0.000

bitcoin1.280

Annualised standard deviation:

equity 0.160

bonds 0.064

bitcoin 0.816

Correlation:

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

Skew:

equity -0.431

bonds 0.294

bitcoin 0.970

**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.

**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.

*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.*

## Results

## Summary- ignore everything I have said

**But:**

**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.

**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*