Mathematics of retirement income

Summary: The development and allocation of retirement income can involve significant mathematical analysis.

Planning for retirement is one of the most important financial responsibilities a person faces. Ideally, after working for several decades, a person will be in a financial position to sustain a desired lifestyle during retirement. In the United States, the source of retirement income can be from a combination of one or more of the following: Social Security; an employer-sponsored pension plan; individual savings, including individual retirement accounts (IRAs); or other mechanisms. The U.S. government has provided military pensions to disabled veterans and widows since the Revolutionary War. This benefit expanded after the U.S. Civil War to include nearly any veteran who had served honorably for some minimum time. Southern states also paid Confederate veterans.

By the early twentieth century, state, municipal, and city governments were paying pensions to their employees, especially firemen and policemen. Teachers were the next large group to receive benefits. Private pensions started in the late nineteenth century with the American Express Company and several railroads. When the 1926 Revenue Act exempted pension trust income from taxes, companies had a new incentive to provide employee pensions, which became commonplace by the 1930s. Social Security was designed in 1935 to extend pension benefits to those not covered by a private pension plan. In the early twenty-first century, Social Security benefits are the main source of retirement income for most retirees, though this varies greatly depending on income from earnings, assets, and private pensions.

Each of these potential sources of retirement income can involve significant mathematical and financial analysis to estimate an individual’s retirement needs, determine necessary pre-retirement financial planning, and evaluate the potential uncertainty associated with personal and economic factors. Specialized mathematicians known as “actuaries” work for governments and industries to design financially sound insurance and pension programs that help meet people’s retirement needs.

At the same time, as the professionals at the American Pension Corporation (a major pension administrator) assert, good actuaries “are more than just mathematicians—[they] take great pride in [their] ability to dissect and communicate the intricacies of pension administration in layman’s language.”

Pensions

A pension provides a stream of income during retirement. It is typically sponsored by a person’s employer—either a corporation or governmental entity. The amount and timing of the retirement income stream provided by a pension are a function of several factors, such as the worker’s salary, the proportion of that salary invested into the pension plan, any matching funds or contributions to the pension fund provided by the employer, the length of the worker’s tenure with the employer, and the investment performance of the pension fund. There are two types of pension plans: defined-benefit (DB), and defined-contribution (DC). DB plans, which have to some extent been phased out in the private sector but are still common in the public sector, define the benefits that will be paid to the worker during retirement. Assuming the solvency of the pension plan—a significant issue in itself—a worker covered under a DB plan is guaranteed to receive the benefits defined by the plan.

Because of potential difficulties in adequately funding DB plans, many (particularly private sector) employers converted to DC plans during the last several decades of the twentieth century. With DC plans, the retirement benefits are not specified; rather, the plan defines the periodic contributions to be invested during the worker’s life, and then the retiree receives an income stream based on the actual accumulated amount of the investment fund. Relative to DB plans, this means that the employer’s risk of inadequate retirement benefit funding is reduced, and that some risk has been transferred to the employee, who faces an uncertain pension income stream.

Mathematics of Pensions

The mathematics associated with pensions involves both “future (or accumulated) value” and “present value” concepts. The general idea is that a worker (or the sponsoring employer) accumulates a retirement fund by setting aside and investing periodic amounts during the working years. Then, upon retirement, this accumulated amount ideally represents sufficient funds with which to provide the retiree an adequate stream of income until death. This retirement income stream may be obtained by leaving the funds invested and withdrawing a certain amount per year, or through the purchase of an annuity, which provides the payment stream. In most cases, these two approaches are mathematically equivalent.

While somewhat straightforward conceptually, achieving an adequately funded and effective retirement plan (especially DB plans, which generally involve more sophisticated and extensive mathematical and financial considerations than DC plans) is a challenging mathematical and actuarial problem. Some of the parameters involved in a pension analysis, and for which assumptions must be made, include the following:

• 1. Periodic contributions to the pension investment fund—usually expressed as a percentage of worker salary during employment.
• 2. Size of the retirement income stream needed or desired—generally estimated as a percentage of projected salary immediately prior to retirement.
• 3. Rate of return on the invested retirement funds, both before and after retirement.
• 4. Changes in worker salary throughout employment.
• 5. Impact of inflation on the worker’s buying power.
• 6. Taxation rules and regulations, both during employment and in retirement.
• 7. Longevity and mortality.

Along with these assumptions, actuaries use mathematics and computer modeling to determine potential answers to questions such as how much must a worker (or employer) invest every month (or year) into a retirement plan in order to successfully achieve that worker’s financial goals in retirement?

IRAs and Social Security

In addition to having an employer-sponsored pension plan, a worker can supplement retiree income with personal savings. One such mechanism is one or more types of IRA. While the rules surrounding IRAs are extensive, they can have potential advantages for some people, including certain tax-advantaged properties.

Social Security is a particularly contentious issue in the twenty-first century. Some have compared Social Security to a type of scam called a “Ponzi scheme” in which a growing pool of new investors’ money is used to pay the promised returns to previous investors. Despite superficial resemblances (for example, current taxpayer money is used to pay variable benefits to others), Social Security is not a savings plan or investment account, but rather a tax, which nullifies the comparison. However, there have been proposals to replace Social Security with an investment program, using a variety of calculations and probabilistic mathematical models to try to demonstrate its cost-effectiveness and the likelihood of the system’s impending failure.

Another major financial issue related to social security is the potential misuse of Social Security numbers. Initially issued to track workers for taxation and benefits, these nine-digit numbers are now assigned routinely at birth and have grown over time into the role of a unique identifier for creditors, schools, employers, and others who want to assign codes to individuals. Modern identity theft, which usually involves a person using a fake or stolen social security number to obtain credit or other benefits, has been on the rise as a result of Internet growth and the widespread collection of personal data. Mathematicians have calculated that a person making up a false social security number in 2010 has about a 50% chance of matching a real number. Faking multiple numbers results in an almost-guaranteed match very quickly.

These calculations have been used to counter thieves’ assertions that they did not know numbers they were using were real. Social Security numbers themselves are not random (for example, the first three digits are a numerical code for geographic location), and mathematical and computer methods have used publicly available data, like date and place of birth, to successfully predict most or all of a person’s social security number. There are also concerns that the government will run out of Social Security numbers, which are not reused after a person dies. Some calculations suggest that the supply will be exhausted early in first half of the twenty-second century. Alternative proposals include using alphanumeric or hexadecimal strings, which offer more permutations for a series of nine “digits.” Others suggest including a security checksum in the number to decrease fraudulent use.

Bibliography

Anderson, Arthur. Pension Mathematics for Actuaries. Winsted, CT: ACTEX Publications, 2006.

Harding, Ann, and Anil Gupta. Modeling our Future: Population Ageing, Social Security and Taxation. Amsterdam, The Netherlands: Elsevier Science, 2007.

Muksian, Robert. Mathematics of Interest Rates, Insurance, Social Security, and Pensions. Upper Saddle River, NJ: Prentice Hall, 2002.