The selection of upper order statistics in tail estimation is notoriously difficult. Most methods are based on asymptotic arguments, like minimizing the asymptotic mse, that do not perform well in finite samples. Here we advance a data driven method that minimizes the maximum distance between the fitted Pareto type tail and the observed quantile. To analyse the finite sample properties of the metric we organize a horse race between the other methods. In most cases the finite sample based methods perform best. To demonstrate the economic relevance of choosing the proper methodology we use daily equity return data from the CRSP database and find economic relevant variation between the tail index estimates. This paper is published as part of the Systemic Risk Centre's Discussion Paper Series. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means without the prior permission in writing of the publisher nor be issued to the public or circulated in any form other than that in which it is published.Requests for permission to reproduce any article or part of the Working Paper should be sent to the editor at the above address. Working Paper AbstractThe selection of upper order statistics in tail estimation is notoriously difficult. Most methods are based on asymptotic arguments, like minimizing the asymptotic mse, that do not perform well in finite samples.Here we advance a data driven method that minimizes the maximum distance between the fitted Pareto type tail and the observed quantile. To analyse the finite sample properties of the metric we organize a horse race between the other methods. In most cases the finite sample based methods perform best. To demonstrate the economic relevance of choosing the proper methodology we use daily equity return data from the CRSP database and find economic relevant variation between the tail index estimates.

Worst-case analysis is used among financial regulators in the wake of the recent financial crisis to gauge the tail risk. We provide insight into worst-case analysis and provide guidance on how to estimate it. We derive the bias for the non-parametric heavy-tailed order statistics and contrast it with the semi-parametric extreme value theory (EVT) approach. We find that if the return distribution has a heavy tail, the non-parametric worstcase analysis, i.e. the minimum of the sample, is always downwards biased and hence is overly conservative. Relying on semi-parametric EVT reduces the bias considerably in the case of relatively heavy tails. But for the less-heavy tails this relationship is reversed. Estimates for a large sample of US stock returns indicate that this pattern in the bias is indeed present in financial data. With respect to risk management, this induces an overly conservative capital allocation if the worst case is estimated incorrectly.

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