RDP 8610: Equilibrium Exchange Rates and a Popular Model of International Asset Demands: An Inconsistency 5. Example: Many Assets

The above results will now be generalised to show that the equilibrium solution for (the stochastic part of) the exchange rate process in these models continues to be a function only of the stochastic parts of the processes for asset prices and therefore that the model is internally inconsistent. Assume that in addition to a real riskless asset in each country, there are (S−2) other assets of which (s−1) are supplied by the home country and are supplied by the foreign country. Depending on his nationality, an investor will face the investment opportunity sets,

where In terms of the general model of Section 2, these equations define,

for the home country and,

for the foreign country. These definitions may then be used to obtain the optimal asset shares a and from equation (17) and the processes for wealth,

where ∑ denotes the sum over j=2,…s and denotes the sum over

Equilibrium in the asset markets is defined by equation (8) above. In this version of the model these market clearing conditions become,

The balance of payments equilibrium condition, equation (9), may be rewritten as,

As previously, for this equation to hold, both the drift (or dt) terms and the noise (or dz) terms on each side of the equation must be equal. Consider the noise terms,^[29]

which can only be true if one of the σdz's is a linear combination of the others, or if the coefficients on each σdz term sum to zero. The second possibility requires that,

for the coefficients of −σ_idz_i,

for the coefficients of σ_edz_e and,

for the coefficients of

These constraints must hold for all tε[0,∞). Hence Ito's Lemma may be applied to differentiate both sides of each of these relationships. Since the a_j's and 's are constant, this again means that the coefficients on each of the σdz in the resulting weighted sums of and dw must be zero. Differentiating equation (26a) gives,

Similarly, equation (26b) requires that

and differentiating equation (26c) gives,

This process of differentiation and equating coefficients on the different σdz terms may again be repeated ad infinitum. Notice that the term is the foreign country's total holdings (in terms of budget shares) of home country assets and the term is the home country's total holdings of foreign assets. There are thus only three possibilities. Either the price processes are such that each country holds none of the others' assets, ever (that is, the “own” assets completely dominate the “other” assets), or each country holds all of its wealth in a single asset. Neither of these is an admissable solution to a model that is intended to explain international portfolio holdings. The third possibility is that at least one of the following equations must hold,

However, given the asset market equilibrium conditions in equation (23), the first of these conditions will hold if and only if the total supply of home country assets is zero, and the second requires that total supply of foreign country assets to be zero. Clearly, neither of these can apply. Hence, neither do the constraints in equation (26).

Therefore, the only solution to the stochastic part of the balance of payments constraint (equation (25)) is that one of the σdz terms is a linear combination of the others – i.e., that the real exchange rate dz may be expressed as a function of the asset price dz's. This implies that the investment opportunity set, as it is perceived by either agent, is spanned by (S−1) of the S assets and that the covariance matrix of excess returns, Ω in equation (7), is singular. Hence the S assets are not distinct and the original maximisation problem is misspecified.

Footnote

The coefficients on the σ_Qdz_Q terms in equation (24) cancel out. [29]