Footnotes: Statistical Sampling as a Mangaement Tool in Banking
This result is derived by noting that , where Y is the total market value of the portfolio and X is the total book value, and using:
Both x and Sx are calculated directly from the population, that is, from the book values, so they are known. By inserting these two values into the formula above for the confidence interval, we now can solve for “n,” the sample size needed.
Note that the value of “a” in the linear equation is not determined. However, it is not necessary to use regression analysis to estimate it because it is reasonable to assume that it is zero if one is willing to assume that the market value of an asset is zero when the book value is zero.
This result is obtained by observing some basic results for gamma distributions, namely that for the population S2x= (p+q)/ a2, S2y = p/a2, E(X) = (p+q)/a, E(Y) = p/a and r=p/(p+q) in terms of the parameters of the joint conditional gamma, and then using method of moments techniques we can substitute the known values of and S2x to solve for functions of the parameter values.
Data about the book values of a portfolio should be readily obtainable from accounting records and anticipated recovery rates can be based on previous studies or an a priori assumption based on general knowledge of the portfolio
* Charles D. Cowan is Director, Management Consulting, Price Waterhouse.
This article was prepared when Mr. Cowan held the position of statistician in the FDIC's Division of Research and Statistics.