Description of Simple Global
Equilibrium Example

(from Chapter 6 of Modern Investment
Management an Equilibrium Approach by Robert Litterman)

The problem is to compute the equilibrium excess returns and weights for a USD investor and a JPY investor, where each can invest in the equities of their country, the equities of the other country, and either of the currencies. Given that we're computing excess returns the excess return to a USD investor of holding USD is 0, and similarly for a JPY denominated investor holding JPY.

We can solve this simple problem, 2 countries, each country having an equity asset and each country with a unique currency, using standard linear techniques.

We start with the reverse optimization equation from Mean-Variance optimization.

- d,j Fraction of investors wealth (d = US investor, j = JP investor) invested in each asset
- Σ Covariance matrix for assets
- μ Vector of asset excess returns

We have the following linear equations for the asset weights and returns for the investor with the US dollar as the base currency.

We have the following linear equations for the asset weights and returns for the investor with the JP yen as the base currency.

We also have the results from Siegel's paradox

And for the equities

We introduce several equilibrium relations, first the sum of all investments in US equities must equal the market capitalization of US equities., and second the sum of all investments in JP equities must equal the market capitalization of jp equities. We assume that each country, the total wealth of the country is equal to the total market capitalization for the country.

and

Finally, we constrain the system to have total lending of each currency match total investment in each currency at equilibrium.

Finally we also constrain each investor to be 100% invested through the following constraints.

This yields 15 equations in 14 unknowns, so we can drop off one of the fully invested constraints and still have a proper linear system. We can thus build a matrix A and vector b from the various formulas above and solve a system of the form Ax = b for the unknown x vector.