This was pretty surprising to me, because I’ve always assumed discount rates should be timeless. Why should it matter if I can trade $1 today for $2 tomorrow, or $1 a week from now for $2 a week and a day from now? Because the money-making mechanism survived. The longer it survives, the more evidence we have it will continue to survive. Loosely, if the hazard rate H is proportional to the survival (and thus discount) probability U, we get
U=kH,ddtU=−UH⟹U(t)=11+tk.
More rigorously, suppose there is some distribution of hazards in the environment. Maybe the opportunity could be snatched by someone else, maybe you could die and lose your chance at the opportunity, or maybe the Earth could get hit by a meteor. If we want to maximize the entropy of our prior for the hazard distribution, or we want it to be memoryless—so taking into account some hazards gives the same probability distribution for the rest of the hazards—the hazard rate should follow an exponential distribution
Pr[H(0)=h]∝e−kh.
By Bayes’ rule, the posterior after some time t is
Pr[H(t)=h]∝e−(k+t)h
and the expected hazard rate is
E[H(t)]=1k+t.
By linearity of expectation, we recover the discount factor
U(t)=11+tk.
I’m now a little partial to hyperbolic discounting, and surely the market takes this into account for company valuations or national bonds, right? But that is for another day (or hopefully a more knowledgeable commenter) to find out.
The Utility Engineering paper found hyperbolic discounting.
Eyeballing it, this is about
U(t)=11+t6 months.This was pretty surprising to me, because I’ve always assumed discount rates should be timeless. Why should it matter if I can trade $1 today for $2 tomorrow, or $1 a week from now for $2 a week and a day from now? Because the money-making mechanism survived. The longer it survives, the more evidence we have it will continue to survive. Loosely, if the hazard rate H is proportional to the survival (and thus discount) probability U, we get
U=kH,ddtU=−UH⟹U(t)=11+tk.More rigorously, suppose there is some distribution of hazards in the environment. Maybe the opportunity could be snatched by someone else, maybe you could die and lose your chance at the opportunity, or maybe the Earth could get hit by a meteor. If we want to maximize the entropy of our prior for the hazard distribution, or we want it to be memoryless—so taking into account some hazards gives the same probability distribution for the rest of the hazards—the hazard rate should follow an exponential distribution
Pr[H(0)=h]∝e−kh.By Bayes’ rule, the posterior after some time t is
Pr[H(t)=h]∝e−(k+t)hand the expected hazard rate is
E[H(t)]=1k+t.By linearity of expectation, we recover the discount factor
U(t)=11+tk.I’m now a little partial to hyperbolic discounting, and surely the market takes this into account for company valuations or national bonds, right? But that is for another day (or hopefully a more knowledgeable commenter) to find out.