Background
A large corporation, with numerous divisions and affiliated companies, realized that their pension calculations for employees who retired or were terminated were incorrect. For a number of reasons, including ethical behavior by management coupled with the potential for a class-action lawsuit by the former employees, the company opted to be proactive and determined that pension benefits needed to be recalculated.
Although management recognized the need to recalculate these benefits, the hurdles they encountered were significant. These included but were not limited to:
- The absence of complete and comprehensive electronic data or hard copy files to support many of the earlier pension calculations;
- The changes in the pension plans over the years due to the corporate mergers and acquisitions that occurred and were continuing; and
- The multiple pension plans that provided benefits to more seasoned employees.
General Approach
Although these issues were troublesome, the absence of electronic files, in particular, required a creative approach to quantify the effect of the errors in the pension benefit calculations. Given that there were numerous pension plans in question and several thousand employees were eligible to receive benefits from these plans, it was determined that the efforts to recalculate the correct pension benefit amounts for each employee was, logistically, impossible. As such, an approach was put forth to firm management that required:
- A statistical sample of pension recipients to be selected;
- Correct pension benefits to be calculated for the statistical sample only;
- The results of the calculations for the statistical sample to be extrapolated to the relevant population of all pension recipients;
- The calculation of an average benefit to be paid to each eligible employee, adjusted for age at the time of retirement/termination; years of service with the company; pension plan membership; and payout option (lump sum or annuity); and
- Corporate approval of this methodology.
With corporate buy-in, it was determined that statistical sampling and estimation was an appropriate and credible technique to be used for these calculations. In particular, with the understanding from corporate management and the stakeholders themselves that corrected pension benefits would be based on an average adjusted calculation from a representative sample of the population, we applied statistical methodology in order to select a sample that satisfied the tenets of robust statistical theory. That is, the estimates derived from the sample would be unbiased and precise. Moreover, any one employee would likely receive an estimated benefit that may not be equivalent to what his or her benefit would have been if a new pension calculation were performed for all employees.
There are substantial benefits to statistical sampling. These include:
- Learning something about a population without examining each item in it;
- Reducing cost, labor and time associated with a review or test;
- Producing objective and defensible results;
- Providing rational predictions about the population; and
- Determining a plausible range (at a specified degree of confidence) around the estimates derived from the sample.
Implementation
In order to select a sample that was representative of the population, we requested a list of all retired or terminated employees who were eligible for retirement benefits during the period that the pension calculation was inaccurate. Representations from the company indicated that the pension algorithm calculation was known to be incorrect during only certain years and for only certain plans. (If these representations had not been made and even tested, it would have been necessary to increase the scope of our investigation and include eligible pension recipients over a longer time period and from additional pension plans.)
Employee-specific characteristics provided by the company included:
- Name and Social Security number;
- Age at time of retirement/termination;
- Hire date and years of service;
- Termination and pension end dates;
- Termination reason;
- Affiliated pension plan;
- Affiliated subsidiary or company; and
- Annuity or lump sum amount. (Annuity or lump sum amounts were potentially incorrect for many of the recipients. As was explained to us, the date of retirement or termination within a calendar year was correlated with the correctness or incorrectness of the pension calculation. Additionally, the pension amounts for a large number of employees were missing.)
This data was the starting point for our statistical sample implementation. More specifically, using these variables singly or in combination, we determined that a stratified sample was the most effective design to determine unbiased and precise estimates of the pension liabilities for the entire population. After examining numerous combinations of variables, we ultimately determined that stratifying by the pension amount - with the expectation that a large number of employees had incorrect pension amounts - was sufficient to satisfy the precision and confidence requirements set forth initially.
Under many circumstances, it is quite appropriate to randomly select a sample from a population. However, it is frequently the case that the population from which a sample is to be selected is heterogeneous - that is, there is a great deal of variability in the item that one is interested in measuring or testing. Thus, in order to obtain estimates that are accurate from a sample drawn from a highly variable population, a large number of pension records may need to be sampled and then evaluated. Clearly, this method is still advantageous to testing, measuring or evaluating all items from a population, but may be more onerous or costly than the client anticipated.
