Assume that the bank makes a swap agreement with a "counterparty"-a global, non-financial firm that is headquartered in a foreign country and operates in both countries. In \(t=0\), the firm invests in projects in both countries, and in \(t=1\) all projects return. The firm hence has the demand to convert foreign currency to home currency in \(t=0\) before it can invest in home country, and to convert home currency to foreign currency in \(t=1\) after home projects return. The firm's demand for currency exchange in both periods is exactly complementary to the bank's demand, so that it is feasible to serve as the bank's counterparty in the swap agreement: in \(t=0\), the firm and the bank exchange currencies at the spot exchange rate \(S_{0}\), and at the same time, agree to exchange currencies again in \(t=1\) at the forward exchange rate \(F_{0}\).

The firm is totally funded by its own wealth, \(W\) (in foreign currency), and as a non-financial firm, it does not hold any reserves.

In \(t=0\), the profit-maximizing firm chooses to invest \(\hat{H}\) in home country, and \(W-\hat{H}\) in foreign country; it thus swaps \(\hat{H}\) with the bank. In \(t=1\), the firm's investment in home/foreign country returns, with the return function being the logarithm of investment, i.e., \(\mathcal{H}(\cdot)=\ln (\cdot) / \mathcal{F}(\cdot)=\ln (\cdot)\).

(a) Swap market equilibrium 

i. Specify the firm's optimization problem. Using the firstorder condition, derive the firm's swap supply \(\hat{H}\).
ii. In equilibrium, the firm's swap supply is entirely taken by the bank. Express the equilibrium volume of swap as a function of \(r^{h}, r^{f}, W\), and \(S_{0}\).

(b) Comparative statics 

i.Define \(\Delta r\) as the interest rate differential between home and foreign countries. How does the equilibrium volume of swap vary with \(\Delta r\) ?
ii. Using your result, combining with the bank's budget constraint on swap (9.19), show how a bank's foreign reserves vary with \(\Delta r\). Interpret your result.
iii. Furthermore, show how a bank's home reserves vary with \(\Delta r\). Interpret your result.