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Financial Applications for Weibull AnalysisReliability engineering may be described as the use of applied statistics for engineering evaluation purposes. Naturally, the statistical techniques incorporated in reliability engineering can be used for many other purposes. One field that employs many of the same statistical techniques is that of financial analysis. In this article, we will look at one of the ways Weibull analysis (i.e. statistical analysis with the Weibull distribution) can be used in a financial context by examining its utility in conducting intangible asset valuation. Potential applications for intangible asset valuation include credit card default rates, season ticket renewal rates for sports and entertainment venues, stock brokerage accounts and life insurance. This article uses the example of a newspaper's subscription accounts to explore the methods and results of this analysis type. Tangible and Intangible Assets Intangible asset valuation, as the name implies, is a method of assigning monetary worth to intangible assets such as business accounts or subscriptions. The concept is best illustrated with an example. Suppose a newspaper is being put up for sale. In order to determine the sale price, all of the assets of the newspaper must be assessed and assigned a dollar value. Material assets are relatively easy to assign a price to  the buildings, printing presses, office equipment, newspaper boxes and so forth can all have price tags associated with them. These are the tangible assets. Assigning a dollar value for intangible assets is a little trickier. The newspaper's subscriber base, for example, would be an intangible asset. These subscribers represent a potential revenue stream for the newspaper and they are definitely an asset. However, the behavior of the subscribers will dictate the amount of revenue that the subscriptions will generate. It would be possible for all of the subscribers to cancel their subscriptions as soon as the newspaper is sold, thus reducing their value to zero. Conversely, these subscribers could continue to renew their subscriptions over a long period of time, resulting in a large amount of revenue for the newspaper. Therefore, the trick with assigning value to the subscriber base involves being able to predict the rate at which the subscribers will cancel their subscriptions (i.e. the "failure rate" for the subscriptions). It should become apparent that the analysis involved in determining the subscriber cancellation rate is very similar to classical reliability analysis. Instead of analyzing data to estimate when a physical object will cease to operate, the analysis is performed to estimate when an account will be closed. When realistic estimates of account life have been determined, further financial analyses can be performed to determine the net present value of the intangible asset. Estimating the Life of Intangible Assets We will continue with our newspaper example and perform a sample calculation to estimate the longevity of the current subscriber base. Ideally, an analysis would be performed on all of the available data. In some instances, the analysis is performed on a representative sample of the entire population. For the sake of simplicity, we will look at a sample of just ten subscribers. Table 1 shows the open and closed dates for ten subscription accounts.
Table 1: Table of subscription account open and closed dates It is a relatively simple matter to convert the data from Table 1 to the familiar life data format, with timestofailure and timestosuspension. Five accounts in the sample have been closed and these will be considered "failures." The timetofailure for each account is simply the difference between the opening and closing date. The remaining five accounts are still active. These will be considered "suspensions" and the suspension time will be calculated as the difference between the opening date and the valuation date of 3/8/2002. Table 2 shows the data in the familiar format for life data analysis.
Table 2: Account life data described as timestofailure and timestosuspension This data set can now be analyzed with the usual life data analysis methodology. Using maximum likelihood estimation (MLE) in ReliaSoft's Weibull++ software, the Weibull parameters for this data set are β = 1.1815 and η = 586.2 days. Figure 1 displays a Weibull probability plot of the analysis.
Figure 1: Weibull probability plot for newspaper subscription data Calculating the Weibull parameters is just the first step, however. In order to derive information suitable for financial analysis, further calculations need to be performed. The concept of conditional probability is often used in these calculations. In traditional reliability analysis, the conditional probability calculation returns the probability that a unit will survive a mission of a certain length, given that it has already survived operation for a given amount of time. It can be looked at as the "reliability of used equipment." The equation for conditional reliability is given by:
The concept of conditional reliability can be used with the results of the Weibull analysis to create a survival table. In this example, the survival table lists the probabilities of the accounts remaining open for additional time periods based on the amount of time that they have already been active. Figure 2 shows a survival table for the data in the newspaper subscription example. The analysis has been grouped into time periods of 30 days for ease of use.
Figure 2: Survival table for newspaper subscription example This table indicates that brand new subscriptions stand an 89.6% chance of surviving 90 days without being cancelled. Similarly, a subscription that has existed for 600 days has an 83.1% chance of surviving an additional 90 days without being cancelled. This information can then be used to develop survival probabilities for the five surviving accounts in our data sample. There are two methods used to go about this: one method that incorporates all of the surviving account information and a simpler method that uses the average age of the surviving accounts. Table 3 shows the expected number of failures for the five surviving accounts. Note that the ages of the accounts have been rounded to the nearest 30 day period.
Table 3: Expected numbers of failures for the surviving accounts The fractional number of expected failures is due to the fact that we are dealing with a very small data set and each of the account age categories has only one account. Given the same parameters but with 100 active accounts in the 630 day category, the expected number of closed accounts for an additional 120 days would be approximately 20. This information can then be summed up to determine the total number of failures for each additional time period. With this information, it is an easy matter to calculate the percent surviving for the entire population of active subscription accounts (1 minus the ratio of expected failures to total survivors). Figure 3 shows the percent surviving for the five active accounts in our newspaper example.
Figure 3: Percent surviving over time for active accounts This process can be simplified by using the mean age of the surviving accounts to determine the percent surviving for the entire population. The mean age of the surviving accounts for the newspaper example is 395 days. This value can then be used to generate the same type of table. Figure 4 shows the percent surviving table based on the average age of the active subscriptions. As can be seen, the values are very close to those in Figure 3.
Figure 4: Percent surviving over time based on mean age of surviving accounts This information can then be used to attach a dollar value to the surviving accounts, based on the probability of account closure, the anticipated revenue stream and so forth, so that the net present value of the surviving accounts can be determined. These calculations are outside the scope of the current article. 

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