One extension to bring this closer to people's model of AI risk is to have a time T where there's a chance that utility goes to zero (AI takeover) or a person gets a one-time utility boost from the capital they have left (i.e. their money buys them some amount of expected utility in a post-singularity future).
I think you would end up with a Merton-like portfolio for the time leading up to T with some cash saved for after the singularity.
If you are certain about the time when singularity/doom occurs, this becomes the finite-horizon version of the Merton model. However, given that there is presumably some uncertainty as to when/if a singularity with occur, you'll need to discount at some rate to represent your uncertainty - which as you note leads to the same portfolio as in the post but higher consumption.
The model includes consumption - it just doesn't rise in response to the changes in expected income growth here because almost all of the rise in income growth is used to buy safety rather than higher returns (the average annual rate of return on the portfolios just goes from 11% to 13%), as the new assets while higher return are also much higher variance. In general, the new AI stocks introduced are too high variance relative to the market as a whole for an individual to want to short the market and use the funds to buy either AI stocks or bonds.
The optimal portfolio would depend on discounting if either optimal holdings varied by wealth level (so how much you cared about now vs the richer future mattered) or if you'd have preferred the past invested differently.
The first of these isn't the case because the utility function has constant relative risk aversion - which means that, for any level of wealth, an individual will always pick the same choice between options:
1) Probability p wealth rises by x%, probability 1-p wealth changes by y%
2) Wealth changes by guaranteed z%
For given x,y,z,p regardless of their wealth. Given normal distribution's of returns, their optimal risk choice stays constant regardless of their expected level of wealth.
The second of these isn't the case either, because consumption and investment decisions are separable in the model. Although you would today prefer your past self to have consumed less, you wouldn't like them to have invested differently (because optimal investments don't depend on wealth or any other characteristic that would have changed) - so with neither of these the case, your optimal portfolio is independent of your discount rate.
There is definitely AI risk, just not in the direction people expect. There’s a reason why Warren Buffett is currently sitting on his largest cash pile yet.
Cool thanks for writing this up!
One extension to bring this closer to people's model of AI risk is to have a time T where there's a chance that utility goes to zero (AI takeover) or a person gets a one-time utility boost from the capital they have left (i.e. their money buys them some amount of expected utility in a post-singularity future).
I think you would end up with a Merton-like portfolio for the time leading up to T with some cash saved for after the singularity.
If you are certain about the time when singularity/doom occurs, this becomes the finite-horizon version of the Merton model. However, given that there is presumably some uncertainty as to when/if a singularity with occur, you'll need to discount at some rate to represent your uncertainty - which as you note leads to the same portfolio as in the post but higher consumption.
This... assumes there is no consumption? Or that you can't spend money now to reduce risk later??
Overall I don't understand what's going on here.
> the optimal portfolio is independent of discounting - so high estimates of doom do not suppress stock holdings
I don't understand why. Could you say a bit more about the intuition here?
The model includes consumption - it just doesn't rise in response to the changes in expected income growth here because almost all of the rise in income growth is used to buy safety rather than higher returns (the average annual rate of return on the portfolios just goes from 11% to 13%), as the new assets while higher return are also much higher variance. In general, the new AI stocks introduced are too high variance relative to the market as a whole for an individual to want to short the market and use the funds to buy either AI stocks or bonds.
The optimal portfolio would depend on discounting if either optimal holdings varied by wealth level (so how much you cared about now vs the richer future mattered) or if you'd have preferred the past invested differently.
The first of these isn't the case because the utility function has constant relative risk aversion - which means that, for any level of wealth, an individual will always pick the same choice between options:
1) Probability p wealth rises by x%, probability 1-p wealth changes by y%
2) Wealth changes by guaranteed z%
For given x,y,z,p regardless of their wealth. Given normal distribution's of returns, their optimal risk choice stays constant regardless of their expected level of wealth.
The second of these isn't the case either, because consumption and investment decisions are separable in the model. Although you would today prefer your past self to have consumed less, you wouldn't like them to have invested differently (because optimal investments don't depend on wealth or any other characteristic that would have changed) - so with neither of these the case, your optimal portfolio is independent of your discount rate.
There is definitely AI risk, just not in the direction people expect. There’s a reason why Warren Buffett is currently sitting on his largest cash pile yet.
https://backseatpolicycritic.substack.com/p/ai-is-a-scam