Estimation of Tail Index and Value-at-Risk for the TA25 and the USD-ILS Exchange Rate Under Assumption of Pareto Distribution

Abstract

While it is clear that returns of financial assets are not well described by the normal distribution, it is unclear how best to describe them. One distribution suggested is the Pareto distribution. I apply an extreme value theory framework to estimate the tails of the distributions of returns of the TA25, the Tel-Aviv Stock Exchange's leading stock index and the USD-ILS exchange rate, under the assumption returns follow the Pareto distribution. I find that the left tail of the TA25 is lighter than that of the S&P500, suggesting less extreme events in the TA25 and the right tail of the USD-ILS exchange rate is heavier than the left tail, indicating that there are more extreme events when the ILS weakens against the USD than vice versa. This may be due to central bank intervention. I use the estimated tail indexes to assess the value-at-risk of the TA25 and USD-ILS and find the estimations fit historical values well for relatively high percentiles, which are most problematic to estimate.