Optimal Exchange Rates, Savings, Reserves, and Loans: A Malliavin Calculus Approach

We present novel formulae for the determination of the optimal values of the exchange rate, reserves, savings, and external (foreign currency) debt. Based on current account formulae, a stochastic differential equation is developed that relates the net worth of an economy to its exchange rate, savings, reserves, and the level of external debt. Using the martingale optimality principle, the optimal values for the exchange rate, interest rate, reserves, and the external debt are derived. Applied to Egypt, we find that its actual nominal exchange rate has stayed above its optimal value since the early 1990s, except for two years, and that the actual external debt-to-net worth ratio is higher than the optimal one for most of the time.