dPV(x)= F (ze *° dz j (2.2)
where q is the appropriate time discount rate.
Note the implication
F (z)dz= dPV(x)et?™), (2.3)
showing that q is the growth rate that raises the value of dPV(x) to F (z)dz over
period z—x. Since this differential component of asset value defers all positive cash flow until moment z, and cannot in itself be affected by later transfers in, q simplifies by (A1.6a) to rate of return. This was Boehm Bawerk’s insight, although he was not mathematical, in equating time preference rate to rate of return r. Thus
(2.2) and (2.3) give
dPV(x)dx=F, (zje"’ dz and F (z)dz= dPV(xJe™’™, (A2.4)
where r is the appropriate rate of return and time discount rate equivalently.
But what determines appropriate r in these equations? Rate of return varies with risk among different assets at the same time, and varies over time with economic circumstances. Most sources I have seen treat r in (A2.4) as a variable to be
integrated over (x, z). | myself long believed the same.
My view now looks to the context. The asset as a whole will typically have received many differential investments before time x, and may receive many after. Each at inception will have been priced by the owner’s time preference rate then. But my
theme in risk theory is that assets can be traded or modified to the current owner's
APPENDIX A: The Argument in Notation 3/7/16 5
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