Pension Mathematics (2) PDF

# Chapter 4 Pension Plan population Theory ## Basic Concepts This chapter deals with the population of pension plan members, which consists of several subpopulations. Active employees, of course, make up the primary group within the plan's membership, and the following discussion focuses on them....

# Chapter 4 Pension Plan population Theory ## Basic Concepts This chapter deals with the population of pension plan members, which consists of several subpopulations. Active employees, of course, make up the primary group within the plan's membership, and the following discussion focuses on them. Retired employees represent another subpopulation, and for present purposes, this group is assumed to consist only of members who retired directly from the service of the employer. This is in contrast to defining a retired member as any one of several types of benefit recipients . Vested terminated employees make up a third group, which can be further divided into those in the benefit deferral period and those receiving benefits. Disabled employees make up the fourth subpopulation for plans providing disability benefits, and beneficiaries, generally surviving spouses, make up a fifth subpopulation. ## Stationary Population The discussion of pension plan populations begins at the most elementary level, namely, with the concept of a stationary population. A population is considered to be stationary when its size and age distribution remain constant year after year. If the decrement rates associated with the population are constant, and if a constant number of new entrants flows into the population ## 4. Pension Plan Population Theory each year, a stationary condition will exist after n years, where n equals the oldest age in the population less the youngest age. Understanding the concept of a stationary population can be facilitated by considering a simplified example. Assume that a population has four ages (x, x+1, x+2, and x+3), that the rates of decrement for each age are 1/4, 1/3, 1/2, and 1, respectively, and that 100 new employees are hired each year, all at age x. The first few years experience for a population exposed to these conditions is given in Table 4-1. **Table 4-1 Development of a Stationary Population** | Year | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | New Entrants - 100- | | | | | 100 | | 2 | New Entrants - 100- | 75 | | | | 175 | | 3 | New Entrants - 100- | 75 | 50 | | | 225 | | 4 | New Entrants - 100 | 75 | 50 | 25 | | 250 | | 5 | New Entrants - 100 | 75 | 50 | 25 | 0 | 250 | | 6 | New Entrants - 100 | 75 | 50 | 25 | 0 | 250 | | Stationary Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | After the first year, the original group of 100 employees hired at age x are age x+1 and total 75 in number. Since 100 new employees are hired at age x each year, the total population at the beginning of year 2 is 175 members. Continuing this process for three years (i.e., until the beginning of year 4), produces a population with a constant size of 250 members and a stationary distribution, as shown in percentage form at the bottom of Table 4-1. Thus, the population becomes stationary after n years, If the population is assumed to be continuous over each age interval, rather than discrete as is assumed in this chapter for simplicity, then n would be equal to the first age at which no survivors exist less the youngest age. In other words, one year would be added to the value of n for the continuous case. where n in this case is equal to three (i.e., the oldest age in the population, x+3, less the youngest age, x). Since pension benefits are tied to service, it is also relevant to point out that the service distribution of a stationary population also becomes constant after n years. Table 4-1 shows that 40 percent of the stationary population has zero years of service, 30 percent has one year of service, and so forth. A pension plan population, unlike the example shown in Table 4-1, has multiple entry ages, and it is logical to inquire whether or not the concept of a stationary population applies in this case. To show that it does apply, one need only conceptualize a multiple entry age population as a series of single entry age populations, with each subpopulation representing a given entry age. Consequently, a multiple entry age population of active employees will become stationary after m years, where m equals the largest retirement-age/entry-age spread among the various subpopulations. ## Mature Population The concept of a mature population is only slightly different from, and somewhat more general than, a stationary population. In fact, a stationary population is a special case of a mature population. Both concepts involve a constant year-to-year age and service distribution, but whereas the stationary population maintains a constant size, this need not be the case for a mature population. If the increments to the population (newly hired employees) increase at a constant rate, the population will attain a constant percentage age and service distribution in precisely the same length of time as required for a population to become stationary. Moreover, the size of the mature population will grow at precisely the same rate as the growth in new entrants. These characteristics are illustrated in Table 4-2 where the number of new entrants is doubled each year (i.e., a 100 percent growth rate). The decrement assumptions are the same as those used in Table 4-1. The age distribution, which is shown at the bottom of the table, is constant year after year, but considerably This assumes, of course, that the entry age distribution of new entrants is fixed. different from the age distribution developed in Table 4-1. This is the case for the service distribution also. **Table 4-2 Development of a Mature Population** | Years | Ages: x | x + 1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 | | | | | 100 | | 2 | 200 | 75 | | | | 275 | | 3 | 400 | 150 | 50 | | | 600 | | 4 | 800 | 300 | 100 | 25 | | 1,225 | | 5 | 1,600 | 600 | 200 | 50 | 0 | 2,450 | | 6 | 3,200 | 1,200 | 400 | 100 | 0 | 4,900 | | 7 | 6,400 | 2,400 | 800 | 200 | 0 | 9,800 | | 8 | 12,800 | 4,800 | 1,600 | 400 | 0 | 19,600 | | Mature Age Distribution: | 65% | 25% | 8% | 2% | 0% | 100% | Throughout the remainder of this book, the term mature population will be used, even in those cases where a stationary population applies, since it is the more general of the two concepts. ## Undermature and Overmature Populations A population is considered to be undermature if its age and service distribution has a larger proportion of younger, short-service employees than that of a mature population that faces the same decremental factors, is of the same size, and experiences the same entry age distribution. An overmature population is one that has a disproportionately large number of employees at older ages and with longer periods of service than that of a mature population based on the same decrement and entry age assumptions. Generally, growing industries are characterized by firms having undermature populations, while declining industries have firms with overmature populations. An example of an undermature population is given in Table 4-3, where the number of new entrants increases by 100 employees each year, representing a continually decreasing rate of growth. The membership distribution in this example, parenthetically expressed in percentage , asymptotically approaches the same distribution as that for the stationary population discussed previously in Table 4-1. After 100 years the population's age and service distribution is nearly identical to that of the corresponding stationary population . **Table 4-3 Development of an Undermature Population** | Years | Ages: x | x+1 | x + 2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 (100%) | | | | | 100 | | 2 | 200 (73%) | 75 (27%) | | | | 275 | | 3 | 300 (60%) | 150 (30%) | 50 (10%) | | | 500 | | 4 | 400 (53%) | 225 (30%) | 100 (13%) | 25 (3%) | | 750 | | 5 | 500 (50%) | 300 (30%) | 150 (15%) | 50 (5%) | 0 | 1,000 | | 6 | 600 (48%) | 375 (30%) | 200 (16%) | 75 (6%) | 0 | 1,250 | | 7 | 700 (47%) | 450 (30%) | 250 (17%) | 100 (7%) | 0 | 1,500 | | 9 | 800 (46%) | 525 (30%) | 300 (17%) | 125 (7%) | 0 | 1,750 | | ... | ... | ... | ... | ... | ... | ... | | 100 | 10,000 (40.4%) | 7,425 (30.0%) | 4,900 (19.8%) | 2,425 (9.8%) | 0 | 24,750 | | Asymptotic Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Finally, Table 4-4 illustrates the development of an overmature population, created by hiring 1,000 employees in the first year and 100 fewer employees each year thereafter. The degree to which the population is overmature is determinable by comparing its age and service distribution to that of the stationary popu- lation based on the same decrement rates (given in the lower part of Table 4-4 for convenience). **Table 4-4 Development of an Overmature Population** | Years | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 1,000 (100%) | | | | | 1,000 | | 2 | 900 (55%) | 750 (45%) | | | | 1,650 | | 3 | 800 (41%) | 675 (34%) | 500 (25%) | | | 1,975 | | 4 | 700 (35%) | 600 (30%) | 450 (23%) | 250 (13%) | | 2,000 | | 5 | 600 (34%) | 525 (30%) | 400 (23%) | 225 (13%) | 0 | 1,750 | | 6 | 500 (33%) | 450 (30%) | 350 (23%) | 200 (13%) | 0 | 1,500 | | 7 | 400 (32%) | 375 (30%) | 300 (24%) | 175 (14%) | 0 | 1,250 | | 9 | 300 (30%) | 300 (30%) | 250 (25%) | 150 (15%) | 0 | 1,000 | | Stationary Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Throughout the remainder of this book, the term mature population will be used, even in those cases where a stationary population applies, since it is the more general of the two concepts. ## Undermature and Overmature Populations A population is considered to be undermature if its age and service distribution has a larger proportion of younger, short-service employees than that of a mature population that faces the same decremental factors, is of the same size, and experiences the same entry age distribution. An overmature population is one that has a disproportionately large number of employees at older ages and with longer periods of service than that of a mature population based on the same decrement and entry age assumptions. Generally, growing industries are characterized by firms having undermature populations, while declining industries have firms with overmature populations. An example of an undermature population is given in Table 4-3, where the number of new entrants increases by 100 employees each year, representing a continually decreasing rate of growth. The membership distribution in this example, parenthetically expressed in percentage , asymptotically approaches the same distribution as that for the stationary population discussed previously in Table 4-1. After 100 years the population's age and service distribution is nearly identical to that of the corresponding stationary population . **Table 4-3 Development of an Undermature Population** | Years | Ages: x | x+1 | x + 2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 (100%) | | | | | 100 | | 2 | 200 (73%) | 75 (27%) | | | | 275 | | 3 | 300 (60%) | 150 (30%) | 50 (10%) | | | 500 | | 4 | 400 (53%) | 225 (30%) | 100 (13%) | 25 (3%) | | 750 | | 5 | 500 (50%) | 300 (30%) | 150 (15%) | 50 (5%) | 0 | 1,000 | | 6 | 600 (48%) | 375 (30%) | 200 (16%) | 75 (6%) | 0 | 1,250 | | 7 | 700 (47%) | 450 (30%) | 250 (17%) | 100 (7%) | 0 | 1,500 | | 9 | 800 (46%) | 525 (30%) | 300 (17%) | 125 (7%) | 0 | 1,750 | | ... | ... | ... | ... | ... | ... | ... | | 100 | 10,000 (40.4%) | 7,425 (30.0%) | 4,900 (19.8%) | 2,425 (9.8%) | 0 | 24,750 | | Asymptotic Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Finally, Table 4-4 illustrates the development of an overmature population, created by hiring 1,000 employees in the first year and 100 fewer employees each year thereafter. The degree to which the population is overmature is determinable by comparing its age and service distribution to that of the stationary popu- lation based on the same decrement rates (given in the lower part of Table 4-4 for convenience). **Table 4-4 Development of an Overmature Population** | Years | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 1,000 (100%) | | | | | 1,000 | | 2 | 900 (55%) | 750 (45%) | | | | 1,650 | | 3 | 800 (41%) | 675 (34%) | 500 (25%) | | | 1,975 | | 4 | 700 (35%) | 600 (30%) | 450 (23%) | 250 (13%) | | 2,000 | | 5 | 600 (34%) | 525 (30%) | 400 (23%) | 225 (13%) | 0 | 1,750 | | 6 | 500 (33%) | 450 (30%) | 350 (23%) | 200 (13%) | 0 | 1,500 | | 7 | 400 (32%) | 375 (30%) | 300 (24%) | 175 (14%) | 0 | 1,250 | | 9 | 300 (30%) | 300 (30%) | 250 (25%) | 150 (15%) | 0 | 1,000 | | Stationary Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Throughout the remainder of this book, the term mature population will be used, even in those cases where a stationary population applies, since it is the more general of the two concepts. ## Undermature and Overmature Populations A population is considered to be undermature if its age and service distribution has a larger proportion of younger, short-service employees than that of a mature population that faces the same decremental factors, is of the same size, and experiences the same entry age distribution. An overmature population is one that has a disproportionately large number of employees at older ages and with longer periods of service than that of a mature population based on the same decrement and entry age assumptions. Generally, growing industries are characterized by firms having undermature populations, while declining industries have firms with overmature populations. An example of an undermature population is given in Table 4-3, where the number of new entrants increases by 100 employees each year, representing a continually decreasing rate of growth. The membership distribution in this example, parenthetically expressed in percentage , asymptotically approaches the same distribution as that for the stationary population discussed previously in Table 4-1. After 100 years the population's age and service distribution is nearly identical to that of the corresponding stationary population . **Table 4-3 Development of an Undermature Population** | Years | Ages: x | x+1 | x + 2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 (100%) | | | | | 100 | | 2 | 200 (73%) | 75 (27%) | | | | 275 | | 3 | 300 (60%) | 150 (30%) | 50 (10%) | | | 500 | | 4 | 400 (53%) | 225 (30%) | 100 (13%) | 25 (3%) | | 750 | | 5 | 500 (50%) | 300 (30%) | 150 (15%) | 50 (5%) | 0 | 1,000 | | 6 | 600 (48%) | 375 (30%) | 200 (16%) | 75 (6%) | 0 | 1,250 | | 7 | 700 (47%) | 450 (30%) | 250 (17%) | 100 (7%) | 0 | 1,500 | | 9 | 800 (46%) | 525 (30%) | 300 (17%) | 125 (7%) | 0 | 1,750 | | ... | ... | ... | ... | ... | ... | ... | | 100 | 10,000 (40.4%) | 7,425 (30.0%) | 4,900 (19.8%) | 2,425 (9.8%) | 0 | 24,750 | | Asymptotic Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Finally, Table 4-4 illustrates the development of an overmature population, created by hiring 1,000 employees in the first year and 100 fewer employees each year thereafter. The degree to which the population is overmature is determinable by comparing its age and service distribution to that of the stationary popu- lation based on the same decrement rates (given in the lower part of Table 4-4 for convenience). **Table 4-4 Development of an Overmature Population** | Years | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 1,000 (100%) | | | | | 1,000 | | 2 | 900 (55%) | 750 (45%) | | | | 1,650 | | 3 | 800 (41%) | 675 (34%) | 500 (25%) | | | 1,975 | | 4 | 700 (35%) | 600 (30%) | 450 (23%) | 250 (13%) | | 2,000 | | 5 | 600 (34%) | 525 (30%) | 400 (23%) | 225 (13%) | 0 | 1,750 | | 6 | 500 (33%) | 450 (30%) | 350 (23%) | 200 (13%) | 0 | 1,500 | | 7 | 400 (32%) | 375 (30%) | 300 (24%) | 175 (14%) | 0 | 1,250 | | 9 | 300 (30%) | 300 (30%) | 250 (25%) | 150 (15%) | 0 | 1,000 | | Stationary Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Throughout the remainder of this book, the term mature population will be used, even in those cases where a stationary population applies, since it is the more general of the two concepts. ## Undermature and Overmature Populations A population is considered to be undermature if its age and service distribution has a larger proportion of younger, short-service employees than that of a mature population that faces the same decremental factors, is of the same size, and experiences the same entry age distribution. An overmature population is one that has a disproportionately large number of employees at older ages and with longer periods of service than that of a mature population based on the same decrement and entry age assumptions. Generally, growing industries are characterized by firms having undermature populations, while declining industries have firms with overmature populations. An example of an undermature population is given in Table 4-3, where the number of new entrants increases by 100 employees each year, representing a continually decreasing rate of growth. The membership distribution in this example, parenthetically expressed in percentage , asymptotically approaches the same distribution as that for the stationary population discussed previously in Table 4-1. After 100 years the population's age and service distribution is nearly identical to that of the corresponding stationary population . **Table 4-3 Development of an Undermature Population** | Years | Ages: x | x+1 | x + 2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 (100%) | | | | | 100 | | 2 | 200 (73%) | 75 (27%) | | | | 275 | | 3 | 300 (60%) | 150 (30%) | 50 (10%) | | | 500 | | 4 | 400 (53%) | 225 (30%) | 100 (13%) | 25 (3%) | | 750 | | 5 | 500 (50%) | 300 (30%) | 150 (15%) | 50 (5%) | 0 | 1,000 | | 6 | 600 (48%) | 375 (30%) | 200 (16%) | 75 (6%) | 0 | 1,250 | | 7 | 700 (47%) | 450 (30%) | 250 (17%) | 100 (7%) | 0 | 1,500 | | 9 | 800 (46%) | 525 (30%) | 300 (17%) | 125 (7%) | 0 | 1,750 | | ... | ... | ... | ... | ... | ... | ... | | 100 | 10,000 (40.4%) | 7,425 (30.0%) | 4,900 (19.8%) | 2,425 (9.8%) | 0 | 24,750 | | Asymptotic Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Finally, Table 4-4 illustrates the development of an overmature population, created by hiring 1,000 employees in the first year and 100 fewer employees each year thereafter. The degree to which the population is overmature is determinable by comparing its age and service distribution to that of the stationary popu- lation based on the same decrement rates (given in the lower part of Table 4-4 for convenience). **Table 4-4 Development of an Overmature Population** | Years | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 1,000 (100%) | | | | | 1,000 | | 2 | 900 (55%) | 750 (45%) | | | | 1,650 | | 3 | 800 (41%) | 675 (34%) | 500 (25%) | | | 1,975 | | 4 | 700 (35%) | 600 (30%) | 450 (23%) | 250 (13%) | | 2,000 | | 5 | 600 (34%) | 525 (30%) | 400 (23%) | 225 (13%) | 0 | 1,750 | | 6 | 500 (33%) | 450 (30%) | 350 (23%) | 200 (13%) | 0 | 1,500 | | 7 | 400 (32%) | 375 (30%) | 300 (24%) | 175 (14%) | 0 | 1,250 | | 9 | 300 (30%) | 300 (30%) | 250 (25%) | 150 (15%) | 0 | 1,000 | | Stationary Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Throughout the remainder of this book, the term mature population will be used, even in those cases where a stationary population applies, since it is the more general of the two concepts. ## Undermature and Overmature Populations A population is considered to be undermature if its age and service distribution has a larger proportion of younger, short-service employees than that of a mature population that faces the same decremental factors, is of the same size, and experiences the same entry age distribution. An overmature population is one that has a disproportionately large number of employees at older ages and with longer periods of service than that of a mature population based on the same decrement and entry age assumptions. Generally, growing industries are characterized by firms having undermature populations, while declining industries have firms with overmature populations. An example of an undermature population is given in Table 4-3, where the number of new entrants increases by 100 employees each year, representing a continually decreasing rate of growth. The membership distribution in this example, parenthetically expressed in percentage , asymptotically approaches the same distribution as that for the stationary population discussed previously in Table 4-1. After 100 years the population's age and service distribution is nearly identical to that of the corresponding stationary population . **Table 4-3 Development of an Undermature Population** | Years | Ages: x | x+1 | x + 2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 100 (100%) | | | | | 100 | | 2 | 200 (73%) | 75 (27%) | | | | 275 | | 3 | 300 (60%) | 150 (30%) | 50 (10%) | | | 500 | | 4 | 400 (53%) | 225 (30%) | 100 (13%) | 25 (3%) | | 750 | | 5 | 500 (50%) | 300 (30%) | 150 (15%) | 50 (5%) | 0 | 1,000 | | 6 | 600 (48%) | 375 (30%) | 200 (16%) | 75 (6%) | 0 | 1,250 | | 7 | 700 (47%) | 450 (30%) | 250 (17%) | 100 (7%) | 0 | 1,500 | | 9 | 800 (46%) | 525 (30%) | 300 (17%) | 125 (7%) | 0 | 1,750 | | ... | ... | ... | ... | ... | ... | ... | | 100 | 10,000 (40.4%) | 7,425 (30.0%) | 4,900 (19.8%) | 2,425 (9.8%) | 0 | 24,750 | | Asymptotic Age Distribution: | 40% | 30% | 20% | 10% | 0% | 100% | Finally, Table 4-4 illustrates the development of an overmature population, created by hiring 1,000 employees in the first year and 100 fewer employees each year thereafter. The degree to which the population is overmature is determinable by comparing its age and service distribution to that of the stationary popu- lation based on the same decrement rates (given in the lower part of Table 4-4 for convenience). **Table 4-4 Development of an Overmature Population** | Years | Ages: x | x+1 | x+2 | x+3 | x+4 | Total Size | |---|---|---|---|---|---|---| | 1 | 1,000 (100%) | | | | | 1,000 | | 2 | 900 (55%) | 750 (45%) | | | | 1,650 | | 3 | 800 (41%) | 675 (34%) | 500 (25%) | | | 1,975 | | 4 | 700 (35%) | 600 (30%) | 450 (23%) | 250 (13%) | | 2,000 | | 5 | 600 (34%) | 525 (30%) | 400 (23%) | 225 (13%) | 0 | 1,750 | | 6 | 500 (33%) | 450 (30%) | 350 (23%) | 200 (13%) | 0 | 1,500 | | 7 | 400 (32%) | 375 (30%) | 300 (24%) | 175 (14