The Stone City

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Friday, July 01, 2005

Singularity Discounting

The Singularity (a.k.a. "the Rapture for nerds") is widely expected, among technophiles (who are best positioned to know), within 20 to 40 years. On the other hand, the U. S. Dollar (and other developed currencies) trades bonds and swaps implying a liquid lending market out to 30 or 40 years. In addition, that yield curve (of implied forward rates) is at present essentially flat, with annual rates ranging from 4.85 to 5.05% over the interval in question. Can we deduce anything from this? For the sake of argument, assume that the Singularity has small temporal extent, so we can treat it as instantaneous.

One feature of a singularity is that today's national currencies will become worthless; in addition, any prior investment, either in possessions or in the means of production, will become worthless. This complicates our analysis by introducing a "0 divided by 0" problem. However, we reason as follows: some expenditures (personal upgrades and/or transient pleasures) will be justified up to the moment of the Singularity. Thus a discounting curve built by a rational actor with foreknowledge of the date of the Singularity ("the Date") would show discount factors (the present value of a unit future payment) declining smoothly to zero as the Date approached.

A substantial population of investors with such a belief would distort the yield curve; thus the flatness of the yield curve is an indicator of their absence. We can quantify this a little, but only if we extrapolate a yield curve behavior in the absence of any Singularity.

For example, if we imagine that annual yields after 20 years would, in the absence of a singularity, approach a steady state of 3.5% (a rough historic low) with a decay rate of 0.1/year (i.e., a half-life of about 7 years), the relative change in the 40-year discount factor (i.e., the implied probability of a Singularity) is only 4%. The exact number is dependent on our extrapolation, but the implication is clear: the financial markets do not believe in any Singularity.