Academic journal article Federal Reserve Bank of New York Economic Policy Review

Methods for Evaluating Value-at-Risk Estimates

Academic journal article Federal Reserve Bank of New York Economic Policy Review

Methods for Evaluating Value-at-Risk Estimates

Article excerpt


In August 1996, the U.S. bank regulatory agencies adopted the market risk amendment (MRA) to the 1988 Basle Capital Accord. The MRA, which became effective in January 1998, requires that commercial banks with significant trading activities set aside capital to cover the market risk exposure in their trading accounts. (For further details on the market risk amendment, see Federal Register [1996].) The market risk capital requirements are to be based on the value-at-risk (VaR) estimates generated by the banks' own risk management models.

In general, such risk management, or VaR, models forecast the distributions of future portfolio returns. To fix notation, let [y.sub.t] denote the log of portfolio value at time t. The k-period-ahead portfolio return is [[Epsilon].sub.t+k] = [y.sub.t+k] - [y.sub.t] Conditional on the information available at time t, [[Epsilon].sub.t+k] is a random variable with distribution [f.sub.t+k]. Thus, VaR model m is characterized by [], its forecast of the distribution of the k-period-ahead portfolio return.

VaR estimates are the most common type of forecast generated by VaR models. A VaR estimate is simply a specified quantile (or critical value) of the forecasted [] The VaR estimate at time t derived from model m for a k-period-ahead return, denoted [] (k,[Alpha]), is the critical value that corresponds to the lower [Alpha] percent tail of [] In other words, VaR estimates are forecasts of the maximum portfolio loss that could occur over a given holding period with a specified confidence level.

Under the "internal models" approach embodied in the MRA, regulatory capital against market risk exposure is based on VaR estimates generated by banks' own VaR models using the standardizing parameters of a ten-day holding period (k = 10) and 99 percent coverage ([Alpha] = 1). A bank's market risk capital charge is thus based on its own estimate of the potential loss that would not be exceeded with 1 percent certainty over the subsequent two-week period. The market risk capital that bank m must hold for time t + 1, denoted [], is set as the larger of [] (10,1) or a multiple of the average of the previous sixty [] (10,1) estimates, that is,


where [] is a multiplication factor and [] is an additional capital charge for the portfolio's idiosyncratic credit risk. Note that under the current framework [] [is greater than or equal to] 3.

The [] multiplier explicitly links the accuracy of a bank's VaR model to its capital charge by varying over time. [] is set according to the accuracy of model m's VaR estimates for a one-day holding period (k = 1) and 99 percent coverage, denoted [](1,1) or simply [] [multiplied by] [] is a step function that depends on the number of exceptions (that is, occasions when the portfolio return [[Epsilon].sub.t+1] is less than []) observed over the last 250 trading days. The possible number of exceptions is divided into three zones. Within the green zone of four or fewer exceptions, a VaR model is deemed "acceptably accurate," and [] remains at its minimum value of three. Within the yellow zone of five to nine exceptions, [] increases incrementally with the number of exceptions. Within the red zone of tell or more exceptions, the VaR model is deemed to be "inaccurate," and [] increases to its maximum value of four.


Given the obvious importance of VaR estimates to banks and now their regulators, evaluating the accuracy of the models underlying them is a necessary exercise. To date, two hypothesis-testing methods for evaluating VaR estimates have been proposed: the binomial method, currently the quantitative standard embodied in the MRA, and the interval forecast method proposed by Christoffersen (forthcoming). …

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