Application of delay differential equations to foreign exchange reserves

Abstract

First, we apply the theory of the delay differential equation to model the policy decisions of the central banks to deplete and accumulate the reserves during the global financial crisis. In the process we obtain a general first order monotonic delayed differential equation which we solve by transforming it to a recurrence relation using the theory of differential transform to generate the transformed recurrence equation. We then solve this recurrence equation to get a system of four linear equations which we solve to get general solutions of the reserve accumulations. Finally, we express the target reserves (i.e. gold or US dollars) as a function of the obtained reserve accumulation general solutions in a polynomial of time t to the fourth order. To do the analysis, we then contract two realist economic scenarios of the obtained polynomial series solutions for the target reserves; 1) for the fast depletion and fast accumulation of the reserves during the time of the global financial crisis, and 2) for the slow depletion and slow accumulation of the reserves during the time of the global financial crisis. For the two cases, we were able to model the policy decisions that can be made by the central banks of countries for the depletion and accumulation of the reserves with a fixed time delay of the response of the economy to the impact of these policy decisions. In our findings, we found that the effects of the delay factor, if it is closer to 0 it prolongs the accumulation process of the reserves, and if is lager, it shorten the accumulation process. In addition, we have also observed that the same effects can be obtained by varying the other model variables while the delay factor is kept constant. Finally, the inclusion of foreign exchange rate regime in our model will certainly require us to involve stochastic differential equations analysis. By doing this, we will then get a full understanding of the movement of the reserves in and out of a country in the form of the accumulation and depletion of the reserves