Three Essays on Financial Risk Management and Fat Tails.

Summary In this thesis, we study the various impacts of model misspecification and examine how to manage model uncertainty. We analyze the impact of ignoring fat tails on a forecast comparison test result in the first chapter, and then study the effects of ignoring the dynamics of the risk premium of returns on the amount of bank capital requirements in the second chapter. The third chapter provides a robust way to determine capital requirements in the face of model uncertainty, i.e., a lack of knowledge of the true data generation process. In the first chapter, we analyze forecast comparison tests under fat tails. Forecast comparison tests are widely used to compare the performance of two or more competing forecasts. The critical value is often obtained by the classical central limit theorem (CLT) or by the stationary bootstrap (Politis and Romano, 1994) with regularity conditions, including the one where the second moment of the loss difference is bounded. We show that if the moment condition is violated, the size of the test using classical normal asymptotics can be strongly distorted. As an alternative approach, we propose to use a "subsampling" method (Politis, Romano and Wolf, 1999) robust to heavy tails. In the empirical study, we analyze several variance prediction tests. By examining several estimators of the tail index, we show that the second moment of the loss difference is likely to be unbounded, especially when the popular squared error function is used as the loss function. We also find that the test result may change if "subsampling" is used. The second chapter explores the effect of misspecification in the conditional mean dynamics on the determination of bank capital requirements. In Basel II, capital requirements for market risk are determined based on a risk measure called Value-at-Risk (VaR). When VaR is calculated, it is often assumed that the conditional average return on an asset is constant over time. However, it is well documented that the predictability of returns increases as the forecast horizon lengthens. The contribution of this chapter is to demonstrate the problems associated with ignoring the conditional mean dynamics when we calculate VaR. We find that although models with constant and time-varying conditional mean may be statistically indistinguishable, the implied VaR may differ. This result then raises another question about how to produce VaR when one recognizes the time variability of the conditional mean but there is uncertainty about its current value. The third chapter proposes a solution to the question raised in the second chapter by examining a robust way to determine capital requirements. We propose to determine capital reserves on a worst-case basis. That is, we choose the maximum value in a set of ES forecasts mapped from the forecaster's pre-selected set of models. Assuming that the risk premium is taken to be non-negative, we show that robust ES can in fact be achieved with a model in which the conditional mean is constant and the risk premium is always zero. This finding serves as an answer to the question raised in Chapter 2 and justifies assuming a constant conditional mean.