From Model Thinking, November 5, 2024:
The end times portfolio is long
In general, it's a very good sign to have skin in the game: to reflect your claims in your deeds. For instance, the AI safety movement worries about AI causing an enormous catastrophe in the near future. What would be skin-in-the-game for this position?
Tyler Cowen argues that the way for them to show their sincerity would be to short the market, since they expect huge damages from AI eventually. (He further implies that their not shorting the market makes light of the x-risk view.)
Others argue that those worried about xrisk should go long on AI-related stocks – since actually, if AI is to be powerful enough to cause catastrophe, related stocks will have incredible gains for some period before that. It also buys insurance.
Another way of betting your beliefs would be to increase your current consumption (since you don't expect to be able to realise investment gains).
And another is to zero out your retirement contributions, since you think you won’t get to draw down.
So, does a lack of “endurance to collect on [your] insurance” imply shorting the market? Note that this is not investment advice.
Model
To answer which approach is the best, we need to model a portfolio that allows us to test each option. In our portfolio, there is three assets: first, a risk-free bond, which gives us the return r; second, a risky AI stock, with a mean return of μ₁ and a standard deviation σ₁₁; and third, a risky non-AI stock, with a mean return of μ₂ and standard deviation σ₂₂.
We want to maximise our return, while minimising our risk, and so we can represent returns as the annual returns for every year in the future, discounted back into the present, and our risk aversion with respect to income1
***
Finally, provided that the two stocks are not perfectly correlated, we should reduce our portfolio risk by owning some of both of them. The more uncorrelated the two stocks are (correlation is measured by σ₂₁), the more we would like to spread between them (holding returns constant). This means that the equation to calculate our optimal holdings and consumption looks like this:
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Several things immediately appear. Firstly, the optimal portfolio is independent of discounting - so high estimates of doom do not suppress stock holdings. This is because as the mean return and standard deviation of stocks doesn’t change across time, the risk profile of any given allocation is time-invariant - so you don’t want to change your desired portfolio if you start valuing the future more or less vs the present.2
Secondly, if covariance is negative and both risky assets return more than bonds, optimal holdings of non-AI stock are always positive. This is because negative covariance means risk can be reduced by holding at least some of each stock, and the second condition means doing so is a better deal than bonds.
Thirdly, although optimal consumption is increasing in both discounting and expected returns, η determines the relative degree of each - under low levels of risk aversion expected returns have no effect (η = 1), while under high levels mortality is greatly suppressed in importance.
Parameter estimation
High estimates of doom are theoretically compatible with high estimates of covariance though - so we need to establish upper bounds of individual’s beliefs on this to check whether any set of beliefs could justify shorting the market....
....MUCH MORE
