Estimating the probability of rare and extreme events is a crucial issue in the risk estimation of exchange rate returns. Extreme Value Theory (EVT) is a well developed theory in the field of probability that studies the distribution of extreme realizations of a given distribution function, or of a stochastic process, satisfying certain assumptions. This work has fitted the Generalized Pareto Distribution (GPD) proposed by EVT to the excesses returns over the threshold to estimate quantiles in the tails of independent and identically distributed residuals and asymptotic properties of the estimators were given. The results were applied to estimate extreme quantiles in the Rwanda exchange rate process.
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