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Levered ETFs exhibit path dependency, or "volatility drag", because they reset their leverage daily, which means you can't calculate the return without knowing what the interest rate does in between the 3% rise

The entire section is based on a first-order approximation, as explicitly noted in the post (which is also why we set aside e.g. the important issue of convexity). This point is of course correct!

A related point: The US stock market has averaged 10% annual returns over a century. If your style of reasoning worked, we should instead buy a 3x levered S&P 500 ETF, get 30% return per year, compounding to 1278% return over a decade, handily beating out 162%.

This calculation, like that of many other commenters, estimates the *total* return. What matters is *risk-adjusted* return (a la Sharpe ratio). If you think the market is literally wrong with certainty, then the bet could be literally risk-free ("infinite Sharpe", speaking loosely). If you aren't 100% certain, then you have a finite risk-adjusted return, but still high -- how high depends on your confidence level (etc).

Equities, on the other hand, have risk!

We welcome other criticisms to discuss, but comments like your first line are not helpful!

Thanks for these comments!

- The short answer here is: yes agreed, the level of real interest rates certainly seems
*consistent*with "market has some probability on TAI and some [possibly smaller] probability on a second dark age". - Whether that's a possibility
*worth putting weight on*-- speaking for myself, I'm happy to leave that up to readers.- (ie: seems unlikely to me! What would the story there be? Extremely rapid diminishing returns to innovation from the current margin, or faster-than-expected fertility declines?)

- As you say, perhaps the possibility of the stagnation/degrowth scenario would have
*other*implications for other asset prices, which could be informative for assessing likelihood.

Yes, to emphasize, the post is meant to define the situation under consideration as: "something close to a 10x increase in growth; or death". We're interested in this scenario only because it's the modal scenario in the particular world of LW/EA/AI safety.

The logic of the argument does not apply as forcefully to "smaller" changes (which could potentially still be quite large), and would not apply at all if AI did not increase growth (ie did not decrease marginal utility of consumption)!

To summarise, the effect on equities seems ambiguous to you, but it's clearly negative on bonds, so investors would likely tilt towards equities.

"Negative for bonds" does not imply "shift investment from bonds to stocks", though. It could mean "shift toward short bonds" or "shift investment from bonds, to just invest less overall".

In addition, the sharpe ratio of the optimal portfolio is decreased (since one of the main asset classes is worse)

I would push back on this too, for a related reason -- the optimal portfolio can include "go short bonds", which might now have a higher expected return.

I think the standard asset pricing logic would be: there is one optimal portfolio, and you want to lever that up or down depending on your risk tolerance and how risky that portfolio is. So, whether you 'take less total exposure to risky assets' depends on whether the argument here updates your view on how 'risky' the future is (Tyler Cowen has argued this, I'm not sure it's super clear cut though).

Here's another way of putting things, that I'll post here for reference:

Suppose I think Google is undervalued, because it is going to have a $1T dividend in 2030, and the market doesn't realize this.

1. I buy Google today at some cheap price.

2. Possibility 1: *before *2030, the market "corrects" and realizes that it was undervaluing Google. The stock price rises, and I receive capital gains.

3. Possibility 2: the market does not "correct" before 2030. I still get the big dividend in 2030, and was able to get it for a cheap price in 2023.

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The above seems exactly analogous to the case with existential risk.

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Suppose I think bonds are overvalued, because in 2030 the world is going to blow up.

1. I short real rates today.

2. Possibility 1: *before *2030, the market "corrects" and realizes that it was overvaluing bonds. Rates rise, and I receive capital gains.

3. Possibility 2: the market does not "correct" before 2030. I still was able to take out a cheap loan in 2023 (i.e. by selling short bonds), and don't have to pay it off in 2030 when the world ends.

1. Very interesting, thanks, I think this is the first or second most interesting comment we've gotten.

2. I see that you are suggesting this as a *possibility*, rather than a *likelihood*, but I'll note at least for other readers that -- I would bet against this occurring, given central banks' somewhat successful record at maintaining stable inflation and desire to avoid deflation. But it's possible!

3. Also, I don't know if inflation-linked bonds in the other countries we sample -- UK/Canada/Australia -- have the deflation floor. Maybe they avoid this issue.

4. Long-term inflation swaps (or better yet, options) could test this hypothesis! i.e. by showing the market's expectation of future inflation (or the full [risk-neutral] distribution, with options).

(duplicating from LW)

Thanks for these comments. In short, to all of your questions, the answer is "yes". Some specific comments:

1. This is perhaps already clear, but it might be worth emphasizing that the economic logic is: real rates are particularly use for *forecasting*, since the sign of the effect is rather unambiguous for the TAI scenario; but it's possible the *expected returns* could be higher for trading on other bets, if you're willing to make stronger assumptions (e.g. "compute will be important").

2. Re: equities, the appendix post (especially #4 there) summarizes how we're thinking about this. To spell out a bit more:

An approximation for stock pricing is the Gordon growth formula, , where

- P is stock price (i.e. market cap)
- D is some initial level of dividends
- r is the real rate
- g is the growth rate of dividends over time

For the equity market *as a whole*, a natural approximation is that the growth rate of dividends equals the growth rate of the economy. And as we pointed out in section I, a first-order approximation for the Euler equation under certainty ("the Ramsey rule") is

Combining the Ramsey rule and the Gordon growth formula, we have

How to interpret this? As a benchmark, suppose theta=1. That's log utility (which I think is the benchmark used in a lot of EA, e.g. at OpenPhil, and has some support in the literature). Then you have P=D/rho. That is, price is future profits discounted by your rate of time preference -- raising or lower the growth rate doesn't affect the stock price at all, because it 'cancels out' in a specific way.

So, that denominator is picking up the 'Merton optimality' that you mention. And I guess the reason I wrote all of this out was to reply to this:

Perhaps with equities you might expect both returns and the interest rate to rise by 3%, which would cancel out

Yes! But also they might not cancel out. It could go either way depending on theta ¯\_(ツ)_/¯. To my knowledge it's an active area of debate ('financial economists think theta < 1, macroeconomists think > 1').

If you really want to nerd out, Cochrane has extended wordy discussion here and Steinsson has long slides here (theta is the inverse of the elasticity of intertemporal substitution).

This is perhaps more than you asked for, and yet I'm not sure if this answered exactly what you were asking. Let me know if not!

If you haven't seen, a long analysis of this: