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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 10

Decay Factors

In this chapter we explore a class of estimators that has become very popular in finance and risk management for analyzing historical data. These models hint at the limitations of the type of analysis that we have explored in previous chapters.

MEAN

In previous chapters, we showed that the best linear unbiased estimator (BLUE) for the sample mean of a random variable was given by:

(10.1) Numbered Display Equation

For a practitioner, this formula immediately raises the question of what value to use for n. Because this chapter is concerned with historical data, what value to choose for n is equivalent to asking how far back in time we should look for data. Should we use 10 years of data? One year? Thirty days? A popular choice in many fields is simply to use all available data. If we have only 20 days of data, use 20 days; if we have 80 years, use 80 years. While this can be a sensible approach in some circumstances, it is much less common in modern finance. Using all available data has three potential drawbacks. First, the amount of available data for different variables may vary dramatically. If we are trying to calculate the mean return for two fixed-income portfolio managers, and we have 20 years of data for one and only two years of data for another, and the last two years have been particularly good years for fixed-income portfolio managers, a direct comparison of the means ...

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