Value-at-risk

Value-at-risk: Techniques to Account for Leptokurtosis and Asymmetric Behavior in Returns Distributions
Abstract
The last few years in the financial markets have shown great instability and high volatility. In order to capture the amount of risk a financial firm takes on in a single trading day, risk managers use a technology known as value-at-risk (VaR). In recent years, VaR analysis has been the prominent risk measure because it allows the manager to take the risk of an entire portfolio and reduce it to a single number that can be compared to other portfolios as well as other trading days. There are many methodologies available to calculate VaR, and each has its limitations. Many past methods have included a normality assumption, which can often produce misleading figures as most financial returns are characterized by skewness (asymmetry) and leptokurtosis (fat-tails). This paper compares the Student-t, autoregressive conditional heteroskedastic (ARCH) family of models, and extreme value theory (EVT) as a means of capturing the fat-tailed nature of a returns distribution. The paper concludes that one cannot utilize a single model as the best approach, due to the tradeoffs between accuracy and computational time, the confidence interval, and the type of asset.
1. Introduction History has shown that billions of dollars can be lost in a short time due to failure in controlling financial risks. With such extreme events, the development of reliable methods to monitor financial risk has become increasingly important (Bormetti 2006, Gilli and Kellezi 2006). Risk measures are critical for characterizing financial investment decisions (Hwang and
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Corresponding author:
146 Multi. Res. Bldg., Notre Dame, IN, USA. +1-574-631-9371 tovaert@
Purpose The last few years in the financial markets have shown great instability and high volatility. In order to capture the amount of risk a financial firm takes on in a single trading day, risk managers use a technology known as value-at-risk (VaR). There are many methodologies available to calculate VaR, and each has its limitations. Many past methods have included a normality assumption, which can often produce misleading figures as most financial returns are characterized by skewness (asymmetry) and leptokurtosis (fat-tails). Design/methodology/approach This paper compares the Student-t, autoregressive conditional heteroskedastic (ARCH) family of models, and extreme value theory (EVT) as a means of capturing the fat-tailed nature of a returns distribution. Findings Recent research has utilized the third and fourth moments to estimate the shape index parameter of the tail. Other approaches, such as extreme value theory, focus on the extreme values to calculate the tail ends of a distribution. By highlighting benefits and limitations of the Student-t, autoregressive conditional heteroskedastic (ARCH) family of models, and the extreme value theory, one can see that there is no one particular model that is best for computing VaR (although all of the models have proven to capture the fat-tailed nature better than a normal distribution). Originality/value This paper details the basic advantages, disadvantages, and mathematics of current parametric methodologies used to assess value-at-risk (VaR), since accurate VaR measures reduce a firm’s capital requirement and reassure creditors and investors of the firm’s risk level.
Value-at-risk: Techniques to Account for Leptokurtosis and Asymmetributions
Lindsay A. Lechner and Timothy C. Ovaert*
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University of Notre Dame, Notre Dame, IN, USA
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Pederson 2004); as each institution measures the amount of risk it takes on over a period of a day, week, month, or year. As imposed by the Basel Committee on Banking Supervision, a financial institution is obligated to meet capital requirements to cover potential losses due to sources of risk during normal operations: namely, credit risk, operational risk, and market risk (Bormetti 2006). As noted by Hopper (1996), a financial firm may want to understand potential losses to its portfolios in order to better allocate its funds and plan for payments to investors. In recent years, value-at-risk (VaR) has become the most popular among risk managers as the best and simplest method to predict losses of an asset, portfolio, or even an entire firm. The VaR of a portfolio measures the maximum loss suffered during a specific time horizon within a given confidence level (i.e. 99%), conditioned such that the composition remains unchanged. For example, if VaR is $2M, then the average loss of the portfolio will not exceed $2M over the oneday horizon on 99 out of 100 trading days with a 99% confidence level. VaR has become a standard component for risk management because of its conceptual simplicity and accuracy of estimating risk at a reasonable computational cost. Of course, this utility does not always imply reliability. There are many different methods to calculated VaR, ranging from a simple historical simulation to a complex semi-parametric approach. Most VaR estimates are evaluated from historically-estimated probability density functions (PDFs), which limit the predictive power of future risk measures (see Figure 1 for return series PDF). The major source of error in predicting future VaRs from historical data is the actual shape of the PDF, which can differ significantly from the one used in the past. For this reason, economists are now focusing their efforts on creating a robust parametric VaR model that