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Financial Applications for Weibull Analysis

Reliability 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 as well. 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 for this type of analysis.

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. It is relatively easy to assign a price to material assets. 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 to intangible assets is more complex. 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. However, 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 representative sample of just ten subscribers. Table 1 shows the opening and closing dates for ten subscription accounts.

 Account Number Date Opened Date Closed 1 1/2/00 2/8/01 2 2/4/00 OPEN 3 6/9/00 OPEN 4 7/14/00 11/24/00 5 11/5/00 6/15/01 6 2/22/01 1/3/02 7 3/26/01 OPEN 8 8/15/01 OPEN 9 12/29/01 3/1/02 10 2/14/02 OPEN

Table 1: Representative sample of subscription accounts as of3/8/02,
including open and close dates

It is a relatively simple matter to convert the data from Table 1 to the familiar life data format, with times-to-failure and times-to-suspension. Five accounts in the sample have been closed and these will be considered "failures." The time-to-failure 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 3/8/02. Table 2 shows the data in the familiar format for life data analysis.

 Account Number F or S (Inactive or Active) Time to F or S (days) 1 F 403 2 S 763 3 S 637 4 F 133 5 F 222 6 F 315 7 S 347 8 S 205 9 F 62 10 S 22

Table 2: Account life data described as times-to-failure (F)
and suspensions (S)

This data set can now be analyzed with the usual life data analysis methodology. Using maximum likelihood estimation (MLE) in ReliaSoft's Weibull++ 6 software, the Weibull parameters for this data set are determined to be 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 for the newspaper subscription data 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. Table 3 shows a survival table for the data in the newspaper subscription example.

 Account Age (Days) % Surviving 30 Days % Surviving 60 Days % Surviving 90 Days % Surviving 120 Days % Surviving 150 Days % Surviving 180 Days 0 97.0600 93.4559 89.6486 85.7696 81.8886 78.0486 30 96.2867 92.3641 88.3676 84.3690 80.4127 76.5285 60 95.9261 91.7755 87.6227 83.5138 79.4798 75.5421 90 95.6732 91.3439 87.0606 82.8552 78.7503 74.7612 120 95.4750 90.9979 86.6024 82.3118 78.7503 74.4054 150 95.3107 90.7069 86.2129 81.8458 77.6176 73.5363 180 95.1696 90.4546 85.8727 81.4364 77.1543 73.0316 210 95.0456 90.2311 85.5697 81.0702 76.7383 72.5770 240 94.9346 90.0302 85.2962 80.7384 76.3602 72.1627 270 94.8339 89.8475 85.0463 80.4345 76.0131 71.7817 300 94.7417 89.6792 84.8162 80.1539 75.6920 71.4287 330 94.6565 89.5236 84.6025 79.8930 75.3931 71.0995 360 94.5773 89.3785 84.4031 79.6491 75.1132 70.7910 390 94.5031 89.2425 84.2159 79.4199 74.8499 70.5005 420 94.4334 89.1144 84.0395 79.2036 74.6012 70.2259 450 94.3675 88.9934 83.8725 78.9988 74.3655 69.9654 480 94.3051 88.8785 83.7139 78.8041 74.1414 69.7177 510 94.2457 88.7693 83.5630 78.6187 73.9278 69.4813 540 94.1891 88.6650 83.4188 78.4415 73.7236 69.2554 570 94.1350 88.5652 83.2808 78.2719 73.5280 69.0389 600 94.0832 88.4696 83.1485 78.1091 73.3403 68.8310 630 94.0334 88.3777 83.0213 77.9526 73.1597 68.6311 660 93.9854 88.2892 82.8989 77.8019 72.9859 68.4384 690 93.9393 88.2040 82.7808 77.6566 72.8181 68.2526 720 93.8947 88.1216 82.6668 77.5162 72.6561 68.0731 750 93.8516 88.0420 82.5565 77.3804 72.4994 67.8994 780 93.8099 87.9650 82.4498 77.2490 72.3479 67.7311

Table 3: Survival table for newspaper subscription example

The analysis has been grouped into time periods of 30 days for ease of use. This table indicates, for example, 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 4 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.

 Account Age (Days) % Surviving 30 Days % Surviving 60 Days % Surviving 90 Days % Surviving 120 Days % Surviving 150 Days % Surviving 180 Days 0 97.0600 93.4559 89.6486 85.7696 81.8886 78.0486 30 96.2867 92.3641 88.3676 84.3690 80.4127 76.5285 60 95.9261 91.7755 87.6227 83.5138 79.4798 75.5421 90 95.6732 91.3439 87.0606 82.8552 78.7503 74.7612 120 95.4750 90.9979 86.6024 82.3118 78.7503 74.4054 150 95.3107 90.7069 86.2129 81.8458 77.6176 73.5363

Table 4: 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). Table 5 shows the percent surviving at each time interval for the five active accounts in our newspaper example.

 Additional Time (Days) Percent Surviving Additional Time (Days) Percent Surviving 30 94.7589 30 94.5003 60 89.6787 60 89.2373 90 84.7837 90 84.2087 120 80.0843 120 79.4111 150 75.5847 150 74.8398 180 71.2854 180 70.4893 210 67.1847 210 66.3536 240 63.2794 240 62.4263 270 59.5652 270 58.7003 300 56.0368 300 55.1686 330 52.6887 330 51.8240 360 49.5148 360 48.6591 390 46.5089 390 45.6665 420 43.6645 420 42.8390 450 40.9750 450 40.1692 480 38.4340 480 37.6500 510 36.0651 510 35.2746 540 33.7717 540 33.0359 570 31.6376 570 30.9274 600 29.6268 600 28.9427 630 27.7331 630 27.0754 660 25.9507 660 25.3196 690 24.2741 690 23.6694 720 22.6978 720 22.1192 Table 5: Survival table based on all surviving account information Table 6: Survival table based on mean age of surviving 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 percent surviving values. Table 6 shows the percent surviving at each time interval based on the average age of the active subscriptions. As can be seen, the values are very close to those in Table 5.

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 discussion.

Conclusion
As this article demonstrates, the Weibull distribution analysis techniques that are commonly used in reliability engineering can also be used for other applications. In addition to the example provided here for asset valuation of newspaper subscription accounts, these techniques can be applied to other intangible assets, such as season ticket renewal rates, life insurance accounts, credit card default rates, stock brokerage accounts, etc. In each case, existing account information can be converted to “failure” and “suspension” data for statistical analysis.

ReliaSoft’s Weibull++ software was used to perform this analysis. On the Web at http://www.ReliaSoft.com/Weibull/.

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