RDP 8610: Equilibrium Exchange Rates and a Popular Model of International Asset Demands: An Inconsistency 4. Example: Two Risky Assets

Consider now a version of the model that allows for risky “own” assets. Assume that each country supplies an asset which is risky in real terms for a domestic investor and also carries exchange rate risk for a foreign investor. There are no other assets and S = 2 again. The investment opportunity sets are,

These equations define,

for the home country and,

for the foreign country. These definitions may again be used to obtain the optimal asset shares a and from the model presented in Section 2 and,

Equilibrium in the asset markets is again defined by equation (12), and the balance of payments equilibrium condition by equation (13). The constraint on the noise terms is now that either,

or

The first possibility is again ruled out by the argument at the end of the previous section. The second implies that there is a riskless return on a zero wealth portfolio that is long in the foreign asset and short in the home country asset. The absence of opportunities for riskless arbitrage in equilibrium implies that these two assets generate the same expected return. They are thus perfect substitutes. Hence, the covariance matrix of returns (Ω in equation (7)) as it is perceived by either agent is singular, the original maximisation problem is misspecified, and the equilibrium real exchange rate cannot be represented by a geometric Brownian motion process with an idiosyncratic risk component.

These two examples cover all the possibilities in a world where each country supplies only one asset and S = 2. In particular, the second contains the case of a nominally riskless asset in each country, discussed by Branson and Henderson (1985). It may be obtained by interpreting σ₁dz₁ as −σQdz_Q and Equation (20) then implies that the nominal exchange rate has zero variance, i.e., σ_Edz_E = 0. The two assets are both perceived by all agents to be nominally riskless in the agent's own domestic currency, and are thus perfect substitutes (in equilibrium). Clearly, imposing purchasing power parity (zero real exchange rate risk) in such a situation will only restrict the parameterisation of price level risk. It can not constrain nominal exchange rate risk.