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Another name for the normal distribution method is the variance-covariance approach. Unlike the normal distribution method, the historical simulation HS is a nonparametric method. It does not assume a particular distribution of the asset returns. Historical simulation forecasts risk by assuming that past profits and losses can be used as the distribution of profits and losses for the next period of returns. The VaR "today" is computed as the p th-quantile of the last N returns prior to "today.

The preceding figure shows that the historical simulation curve has a piecewise constant profile. The reason for this is that quantiles do not change for several days until extreme events occur. Thus, the historical simulation method is slow to react to changes in volatility. The first two VaR methods assume that all past returns carry the same weight.

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The exponential weighted moving average EWMA method assigns nonequal weights, particularly exponentially decreasing weights. The most recent returns have higher weights because they influence "today's" return more heavily than returns further in the past. The formula for the EWMA variance over an estimation window of size is:. A value of the decay factor frequently used in practice is 0.

This is the value used in this example. For more information, see References. In the first part of this example, VaR was estimated over the test window with three different methods and at two different VaR confidence levels. The goal of VaR backtesting is to evaluate the performance of VaR models. Clustering of VaR failures indicates the lack of independence across time because the VaR models are slow to react to changing market conditions.

Modified Value at Risk

A common first step in VaR backtesting analysis is to plot the returns and the VaR estimates together. To highlight how the different approaches react differently to changing market conditions, you can zoom in on the time series where there is a large and sudden change in the value of returns. For example, around August A VaR failure or violation happens when the returns have a negative VaR. A closer look around August 27 to August 31 shows a significant dip in the returns.

On the dates starting from August 27 onward, the EWMA follows the trend of the returns closely and more accurately. Consequently, EWMA has fewer VaR violations two 2 violations, yellow diamonds compared to the Normal Distribution approach seven 7 violations, blue stars or the Historical Simulation method eight 8 violations, red squares. Besides visual tools, you can use statistical tests for VaR backtesting. The summary report shows that the observed level is close enough to the defined VaR level. The failure ratio shows that the Normal95 VaR level is within range, whereas the Normal99 VaR Level is imprecise and under-forecasts the risk.

To run all tests supported in varbacktest , use runtests. Both confidence levels got rejected in the conditional coverage independence, and time between failures independence cci and tbfi columns. This result suggests that the VaR violations are not independent, and there are probably periods with multiple failures in a short span. Also, one failure may make it more likely that other failures will follow in subsequent days.

For more information on the tests methodologies and the interpretation of results, see varbacktest and the individual tests. Using a varbacktest object, run the same tests on the portfolio for the three approaches at both VaR confidence levels. Regarding independence, most tests pass the conditional coverage independence test cci , which tests for independence on consecutive days. Notice that all tests fail the time between failures independence test tbfi , which takes into account the times between all failures.

This result suggests that all methods have issues with the independence assumption. For the year , all three methods pass all the tests.

However, for the year , the test results are mostly rejections for all methods. The EWMA method seems to perform better in , yet all methods fail the independence tests.

Value at risk - Wikipedia

To get more insight into the independence tests, look into the conditional coverage independence cci and the time between failures independence tbfi test details for the year To access the test details for all tests, run the individual test functions. Tools for Fundamental Analysis.

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Value-at-Risk Calculation - Historical Simulation

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