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Methods for determining value at risk

Methods for determining value at risk description/claims

This application claims priority to U.S. Provisional Application No. 60/200,742, filed May 1, 2000; U.S. Provisional Application No. 60/200,743, filed May 1, 2000; U.S. Provisional Application No. 60/200,744, filed May 1, 2000; and U.S. Provisional Application No. 60/274,174, filed Mar. 8, 2001. The contents of the above applications are incorporated herein in their entirety by reference.

For banks and other financial institutions, risk measurement plays a central role. Risk levels must conform to the capital adequacy rule. An error in the computed risk level may thus affect a bank's investment strategy. The state of the art is measuring risk by analyzing daily data: using one market price per working day and per financial instrument. In this description, the stochastic error of such a risk measure is demonstrated in a new way, concluding that using only daily data is insufficient.

The challenge for statisticians is to analyze the limitations of risk measures based on daily data and to develop better methods based on high-frequency data. This description meets this challenge by introducing the time series operator method, applying it to risk measurement and showing its superiority when compared to a traditional method based on daily data.

Intra-day, high frequency data is available from many financial markets nowadays. Many time series can be obtained at tick-by-tick frequency, including every quote or transaction price of the market. These time series are inhomogeneous because market ticks arrive at random times. Irregularly spaced series are called inhomogeneous, regularly spaced series are homogeneous. An example of a homogeneous time series is a series of daily data, where the data points are separated by one day (on a business time scale which omits the weekends and holidays).

Inhomogeneous time series by themselves are conceptually simple; the difficulty lies in efficiently extracting and computing information from them. In most standard books on time series analysis, the field of time series is restricted to homogeneous time series already in the introduction (see, e.g., Granger C. W. J. and Newbold P., 1977, Forecasting economic time series, Academic Press, London; Priestley M. B., 1989, Non-linear and non-stationary time series analysis, Academic Press, London; Hamilton J. D., 1994, Time Series Analysis, Princeton University Press, Princeton, N.J.) (hereinafter, respectively, Granger and Newbold, 1977; Priestley, 1989; Hamilton, 1994). This restriction induces numerous simplifications, both conceptually and computationally, and was almost inevitable before cheap computers and high-frequency time series were available.

U.S. Provisional Application No. 60/200,743, filed May 1, 2000, discloses a new time series operator technique, together with a computationally efficient toolbox, to directly analyze and model inhomogeneous as well as homogeneous time series. This method has many applications, among them volatility or Value-at-Risk (VaR) computations tick by tick.

A comparison is made herein between VaR results based on daily data, sampled at a certain daytime, and results based on tick-by-tick data and the new time series operator technique. If using daily data, a surprising and (for practitioners) alarming sensitivity against the choice of the sampling daytime is observed. The stochastic noise seems higher than acceptable to risk managers. An alternative VaR computation based on tick-by-tick data and a new time series operator technique is shown to have similar properties, except for two advantages: distinctly reduced noise and availability of up-to-date results at each tick.

The time series operators can also be used in the formulation of old and new generating processes of time series. This opens new ways to develop process equations with new properties, also for inhomogeneous time series.

A preferred embodiment comprises a method for determining value-at-risk based on tick-by-tick financial data. Major steps of the method comprise the following: (1) financial market transaction data is electronically received by a computer; (2) the received financial market transaction data is electronically; (3) a time series z is constructed that models the received financial market transaction data; (4) an exponential moving average operator is constructed; (5) an operator is constructed that is based on the exponential moving average operator; (6) a causal operator Ω[z] is constructed that is based on the iterated exponential moving average operator; (7) values of predictive factors are calculated; (8) the values calculated by the computer are stored in a computer readable medium, and (9) value-at-risk is calculated from the values stored in step (8).

FIG. 1 illustrates the relationship between a time series Z and a time series operator Ω.

FIG. 2 depicts an example of a causal kernel ω(t) of a moving average.

FIG. 3 depicts a graph of a kernel of a simple EMA operator.

FIG. 4 depicts graphs of selected EMA operator kernels.

FIG. 5 depicts graphs of selected MA operator kernels.

• Provisional Patent
• Utility Patent

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