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In general, the more optimistic we are on the prospects of an investment, the more of it we’ll want to own. However, at extreme levels of bullishness, the normal relationship can be turned on its head and it can make sense to own less of an asset the more we like it. It’s hard to think of everyday examples that work like this, so this poses something of a puzzle. It’s a problem which gets little attention in mainstream finance, because we rarely witness high enough forecasts of investment quality to observe this effect in the wild.^{2} For example, we can thank the digital-asset revolution and its band of optimistic crypto-asset enthusiasts for bringing this problem into focus, which also gives us some valuable insights that apply in other domains of financial decision-making.

**Risk aversion **

We’ll assume a representative Investor with a relatively moderate level of risk-aversion.^{3} For those familiar with the Kelly Criterion, we’re assuming an investor with twice the risk-aversion of a Kelly Bettor.^{4} Many individuals find that while “full” Kelly may be appropriate for cash management in blackjack or poker, it calls for too much risk when applied to total wealth. In gambling circles, our Base-Case investor would be called a “half-Kelly bettor”, and would find himself in the company of quite a few famous investors who also size their risks in line with this level of risk-aversion. More concretely, such an investor would be *indifferent* to a 50/50 coin flip which could increase his wealth by 50% or decrease it by 25%.^{5}

**Crypto risk and return**

It is hard to know how to accurately model the price behavior of digital assets, but most would agree that they do not follow the continuous Geometric Brownian motion of finance textbooks, and their returns are far from being well-described by the standard Normal distribution. Financial assets in general (and digital assets in particular) experience price jumps and fat tails, and their variability and expected returns are difficult to estimate and can change dramatically over time. Additionally in the case of digital assets, many investors recognize some risk of losing their investment through hacking, hard forks, loss of private keys, etc – more like seeing one’s house burn down rather than experiencing a bad run in the market. To cut through all the uncertainty of crypto return distributions, we’re going to radically simplify and assume that to some chosen horizon there’s a 50% chance the asset goes to 0, and a 50% chance it goes up by a factor of *P*, the “payoff ratio”.