Exchange Rates and the Matrix-Tree Theorem
Classical theories of exchange rates, such as Mint Par and Purchasing Power Parity (PPP), have the desirable property that they define a network of exchange rates in which no profit by 'compound arbitration' or 'cyclic arbitrage' is possible. Conversely, it will be shown that any network of exchange rates, satifying the condition that no profit by cyclic arbitrage is possible, can be defined by a 'potential quotient' analogous to the one used in PPP theory. Given a network with specified cash flows between countries, the balance of payments determines a unique system of exchange rates for which no cyclic arbitrage is possible. An explicit formula for the potential of a currency in this situation is obtained by using the matrix-tree theorem. This provides us with a 'universal' form of PPP theory, in which the potential of a currency is defined in terms of the totality of cash flows in the network.