The Ex Ante Equation™
The Ex Ante Equation calculates asset-holding income, i.e., a value that can be consumed with the expectation that the residual value will appreciate to the whole current value over the course of the next period. Specifically, the equation is
Asset-Holding Ex Ante Income = [(r/(1+r)] × Asset Value . (1)
This equation can be derived from the present-value formula. Given an infinite length stream of constant periodic payments beginning in the current instant, the present value is
present value = p + p /(1+r)1 + p /(1+r)2 + p /(1+r)3 …, (2)
where p = constant periodic payment.
Solve for p to obtain asset-holding ex ante income by initially dividing by p:
present value/p = 1 + 1 / (1+r)1 + 1 / (1+r)2 + 1 / (1+r)3 … (3)
Multiply both sides by (1+r):
(1+r) × present value/p = (1+r) + 1 + 1/(1+r)1 + 1/(1+r)2 … (4)
The right-hand side of equation 4 equals the right-hand side of equation 3, so,
(1+r) × present value/p = (1+r) + present value/p . (5)
Subtract present value/p from both sides:
r × present value/p = (1+r) . (6)
Multiply both sides by p/(1+r):
Asset-Holding Ex Ante Income = (r/(1+r)) × present value = p . (7)
The term “ex ante” was possibly coined by Nobel laureate economist J.R. Hicks. The Ex Ante Equation can also be derived from Hicks’ definition of income, assuming that all consumption occurs at the start of the period:
"The purpose of income calculations in practical affairs is to give people an indication of the amount which they can consume without impoverishing themselves. Following out this idea, it would seem that we ought to define a man's income as the maximum value which he can consume during a week, and still expect to be as well off at the end of the week as he was at the beginning." (Hicks, J. R. (1946). Value and Capital, 2nd ed, p. 172)
In terms of applying the Ex Ante Equation, the asset value is ideally the current fair-market value. Depending upon the context, the value of r can be the cost of capital, the discount rate used for present-value calculation, a risk-adjusted determination, or the risk-free interest rate, because of the No Arbitrage Theorem of option pricing theory and practice.
See the graph for a demonstration of how asset value recovers after being reduced by asset-holding ex ante income.
