RDP 2015-10: The Life of Australian Banknotes Appendix A: Deriving the Feige Steady-state Equation

Deriving the Feige equation begins by specifying the cumulative number of transactions, T*(t), performed by all banknotes by time t:

where G is the total number of lifetime transactions a banknote can perform before becoming unfit (which is assumed to be constant over time), D(s) is the number of banknotes returned to the central bank for destruction by time s, I(s) is the number of new banknotes issued by the central bank by time s, and γ (s) equals the average proportion of lifetime transactions performed by banknotes on issue at time s.

The number of banknote transactions per year, T(t), is given by:

since Feige (1989) assumes that the average banknote in circulation has completed half of its lifetime transactions (i.e. γ(s) = 0.5). It follows that the average annual number of transactions per banknote, Z(t), is given by:

where C(t) equals banknotes on issue at time t.

So given the average life of a banknote can be calculated as the number of lifetime transactions relative to the number of transactions it undergoes annually, substituting in the expressions above, we arrive at the final formula:

Rather than using the number of banknotes on issue at a point in time, in practice, currency issuers often use the average number of banknotes on issue over the year to eliminate variations in the life of banknotes that would arise due to seasonal fluctuations in demand.