Three Pension Cost Methods under Varying Assumptions

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Three Pension Cost Methods under Varying Assumptions Linda S. Grizzle Brigham Young University - Provo Follow this and additional works at: Part of the Mathematics Commons BYU ScholarsArchive Citation Grizzle, Linda S., "Three Pension Cost Methods under Varying Assumptions" (2005). All Theses and Dissertations This Selected Project is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 THREE PENSION COST METHODS UNDER VARYING ASSUMPTIONS by Linda Grizzle A project submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Brigham Young University August 2005

3 Copyright c 2005 Linda Grizzle All Rights Reserved

4 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a project submitted by Linda Grizzle This project has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date David Clark, Chair Date Chris Grant Date Dennis Tolley

5 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the project of Linda Grizzle in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date David Clark Chair, Graduate Committee Accepted for the Department Tyler Jarvis Graduate Coordinator Accepted for the College Earl M. Woolley, Dean College of Physical and Mathematical Sciences

6 ABSTRACT THREE PENSION COST METHODS UNDER VARYING ASSUMPTIONS Linda Grizzle Department of Mathematics Master of Science A pension plan administrator promises certain benefits in the future in exchange for labor today. In order to budget for this expense and create more security for the participant, the administrator uses a pension cost method. Each cost method assigns a portion of the future liability to the current year. This is called the normal cost. We calculate the normal cost under three cost methods using different annuity, interest and inflation assumptions. Then we make comparisons between cost methods as well as between assumption changes. The cost methods considered in this paper are the unit credit cost method, projected unit credit cost method, and the entry age cost method. Both the constant dollar and the constant percent versions of the entry age cost method are considered.

7 ACKNOWLEDGMENTS I would like to thank Dr. Clark for creating a project that prepared me to work in pension administration and funding. He helped me to form my education around my professional goals and was always available to answer my questions in the process. I would also like to thank my parents for their endless love and support in all of my educational and professional aspirations.

8 Table of Contents 1 Introduction 1 2 Fundamental Concepts 2 3 Benefit Allocation Cost Methods Unit Credit Normal Cost Calculations Explicit Supplemental Cost Methods Unit Credit Strengths and Weaknesses Unit Credit With Projections Normal Cost Calculations Projected Unit Credit Strengths and Weaknesses Cost Allocation Cost Methods Entry Age Level Dollar Method Normal Cost Calculations Under the Constant Dollar Method Entry Age Constant Dollar Strengths and Weaknesses Constant Percent Method Normal Cost Calculations Under the Constant Percent Method Entry Age Constant Percent Strengths and Weaknesses Aggregate Overall Comparison 27 6 Normal Costs Under Varying Assumptions Life Annuity Interest Rate Change Unit Credit Method Projected Unit Credit Method Entry Age Constant Dollar Method Entry Age Constant Percent Method Inflation Rate Change Unit Credit Method Projected Unit Credit Method Entry Age Constant Dollar Method Entry Age Constant Percent Method Inflation and Interest Rate Change Unit Credit Method Projected Unit Credit Method Entry Age Constant Dollar Method Entry Age Constant Percent Method Assumption Comparison Conclusion 49 References 51 vii

9 1 Introduction Why is pension funding necessary? The easiest way to finance pension benefits is the current disbursement or pay-as-you-go method. There are two main disadvantages with this sort of financing method [2]. The first is a budgeting problem. The expense of future benefits is closely related to the current payroll. If current disbursement financing is used, the expense is deferred. Thus current disbursement financing charges too little in the early years and too much in the future. The second disadvantage is a lack of security for the employee. Current disbursement financing relies on the employer s continued ability and willingness to pay the owed benefits. Funding a pension is a partial solution to these two difficulties. There are other benefits to funding a pension plan. One added benefit is that a funded plan offers the employer or plan sponsor more flexibility. The employer has more options in the amount and timing of each contribution. Another benefit is that a funded plan avoids an undue drain on the guarantee fund of the Pension Benefit Guarantee Corporation, a government organization set up to offer more security to pensioners. Finally, the greatest advantage to funding is that the Employee Retirement Income Security Act of 1974 (ERISA) makes the pay-as-you-go financing method illegal for private pension plans. When choosing a method for funding a plan, various characteristics of a cost method are desirable [1]. Some pension providers want a cost method to produce an annual cost that is a level percentage of payroll. Others desire the method to generate an actuarial liability at least equal to the actuarial present value of all the accrued benefits payable in the event of plan termination. Conversely, some want the generated actuarial liability not to be grossly larger than the plan termination liability. Finally, most pension providers want the cost method to provide sufficient flexibility, thereby meeting the short term needs of the employer or plan sponsor. The four desirable characteristics are somewhat contradictory and therefore no cost method can satisfy each one. Thus, different cost methods are preferred by employers to 1

10 fund actuarial liabilities. We will consider three such methods: unit credit, projected unit credit, and entry age. We will also briefly mention a fourth method called the aggregate cost method. Attention will be paid to the extent to which each method satisfies the desirable characteristics. Then we will consider the effect of differing assumptions on each cost method. While the mathematics behind each cost method is standard throughout pension literature, notation and nomenclature are not. We will use notation consistent with Pension Mathematics with Numerical Illustrations [3]. 2 Fundamental Concepts For funding, accounting, and tax purposes, the expected cost of a pension plan for a group of participants is allocated to the years of service that give rise to that cost. The technique used to do this is called an actuarial cost method or funding method. The portion of the actuarial present value of future benefits for an individual participant or the plan as a whole assigned to a particular year is called the normal cost. Since each actuarial cost method allocates the present value of future benefits to each year according to different patterns, the normal cost will be different under each method. Actuarial cost methods generally divide the present value of future benefits into two parts, the portion attributable to the past and the portion attributable to the future. The part attributable to the past is called the actuarial liability while the part assigned to the future is called the present value of future normal costs. Retrospectively, the actuarial liability is the accumulated value of the normal costs of the plan decreased by the benefit and expense disbursements. Prospectively, the actuarial liability is the difference between the actuarial present value of future benefits and the actuarial present value of future normal costs. Theoretically, the actuarial accumulated value of normal costs from entry age to retirement age will be equal to the actuarial liability for the employee s pension benefit at 2

11 retirement. Similarly, the actuarial liability for a given year should be equal to the assets of the plan due to previous contributions. There are several reasons why these equalities do not hold [3]. First, the experience of the plan will be different from the original actuarial assumptions. This difference is called an actuarial gain or loss. Also, the actuarial assumptions may be changed periodically. Another cause of difference occurs because the benefit formula may be changed from time to time with the change being retroactive. The plan sponsor also may grant benefit credits to years prior to the establishment of the plan. A fifth reason is the sponsor might contribute more or less than the normal cost under the cost method used. The difference between the actuarial liability and the assets of a given year is called the unfunded liability of the plan. When the actual liability differs from the expected liability, the unfunded liability is liability-based. An asset-based unfunded liability occurs when the actual assets differ from the expected assets. The change from year to year in the unfunded liability is amortized by supplemental costs. Supplemental costs can be determined explicitly or implicitly, on an individual basis or on a group basis for the entire plan. Explicit supplemental cost methods amortize the unfunded liability in a way that is not related to the actuarial cost method. Explicit methods are the most widely used of all supplemental cost methods [3]. Implicit methods calculate supplemental costs according to the same principles used in calculating the normal costs. 3 Benefit Allocation Cost Methods Actuarial cost methods can be broadly classed into two categories [1]. A benefit allocation cost method is one that allocates the benefits of the plan to various plan years and then determines the actuarial present value of these benefits. A cost allocation cost method is one that allocates the actuarial present value of all the projected future benefits to specific periods without allocating the benefits themselves. 3

12 Two of the four cost methods examined here are classified as benefit allocation cost methods. They are the unit credit cost method and the projected unit credit cost method. 3.1 Unit Credit The accrued benefit for the unit credit cost method is defined by the plan and is usually used when the annual benefit accrual is a flat dollar amount or a constant percentage of the participant s current annual salary. The benefit assigned to each year is defined to be the expected increase in the participant s accumulated plan benefit during that year. The actuarial liability is the accrued benefit multiplied by a whole life annuity due starting at age r and discounted for interest and survival to age x and is given by r (AL) x = B x r x p (T x ) v r x ä r where r (AL) x = the actuarial liability for a plan participant of age x with retirement age r, B x = the accrued benefit at the beginning of age x, r xp (T ) x = the probability of surviving all causes of decrement to the age of retirement, v = the discount factor which is equal to i where i is the interest rate, ä r = a whole life annuity due that pays one at the beginning of the year starting at age r. The normal cost under the unit credit cost method is the increase in the participant s plan benefit during the year, multiplied by a whole life annuity due starting at retirement and discounted for interest and mortality. The formula is given by 4

13 r (NC) x = b x r x p (T x ) v r x ä r where b x is the benefit accrual during age x. For a plan containing ancillary benefits such as vested or disability benefits or those benefits given to a surviving spouse, the normal cost calculations become more complicated. The normal cost is then where T (NC) x = b x [ r 1 k=x k xp x (T ) v k x (q (t) k v F k + q (d) k d F k + q (m) k s F k )] + b r x F r T (NC) x = the normal cost for both ancillary benefits and retirement benefits, q (t) k = the probability that the participant will leave active status and be eligible for vested benefit status between ages k and k + 1, q (d) k = the probability that the participant will leave active status due to a disability between ages k and k+1, q (m) k = the probability that the participant will die between ages k and k+1. Each function F k represents the value of the the benefit payable at each decrement. v F k = g (v) k r k 1p (m) k+1 vr k ä r d F k = g (d) d k wp (m) k+1 vw+1 ä d k+w+1 s F k = M g (s) k v1 ä k+u+1 r F r = r x p (T x ) v r x ä r where 5

14 g (v) k = the function that gives the proportion of the vested accrued benefit at age k, g (d) k = the function that gives the proportion of the accrued benefit provided if disability occurs between ages k and k + 1, w = the waiting period before disability benefits start, d wp (m) k+1 = the probability that a disabled life age k + 1 lives w years, ä d k+w+1 = a life annuity based on the disabled life mortality, M = the probability that a deceased participant has a surviving spouse, g (s) k = the function that gives the proportion of the accrued benefit provided to a surviving spouse if the participant dies, u = the number of years, positive or negative, that when added to the participant s age, gives the assumed age of the surviving spouse. Normal costs become even more complicated if the plan provides for a number of retirement dates. If r is the earliest retirement age and r is the age by which all participants are assumed to be retired, the normal cost equation is r (NC) x = b x r 1 k=x g(r) k r k x p(t x ) q (r) k vk x ä k where g (r) k = the function that defines the proportion of the accrued benefit payable if the participant retires at age k, q (r) k = the probability of retiring at age k. 6

15 The actuarial liability corresponding to either ancillary benefits or multiple retirement ages is found by replacing b x with B x in each equation Normal Cost Calculations To calculate the normal costs under the unit credit cost method, a set of assumptions must be made. Some cost methods are not appropriate for some sets of assumptions, but for comparison purposes, we will use the same set of assumptions under each cost method. First we must make assumptions about the employee s attained age salary. An employee s salary will increase from year to year due to inflation and company productivity as well as his own progression through his career. This last increase is called a merit-based salary increase. For our calculations, we will assume a four percent inflation increase and a one percent productivity increase. Finally, the merit-based salary increases in a company can be estimated by comparing the salaries of participants at different ages and years of service. We will use the merit salary scale found in Pension Mathematics as table 2-10 [3]. Now assuming that an age thirty entrant has a salary of one, the salary at any age thereafter can be determined by the formula s x = (SS) x [(1 + I)(1 + P )] x y where (SS) x is the merit salary scale at age x, I is the inflation rate, and P is the rate of productivity increase. For example, to calculate the salary of an age thirty-two participant as a multiple of the entry age thirty salary, we find the merit salary scale at age thirty-two. This is Now the age thirty-two salary is given by s 32 = (1.0706)[(1.04)(1.01)] 2 = A table of calculated attained age salaries with a corresponding graph for an age thirty entrant are given below. 7

16 Attained Age Salaries as a Percent of Entry Age-30 Salary Age Salary Age Salary Age Salary Age Salary Salary Scale for an Age 30 Entrant Multiple of Entry Age Salary Age The benefit for the model pension plan is a final average benefit, meaning the accrued benefit is a fraction of the average salary of the last five service years. We will use 1.5 percent of the final five year average salary. Under the unit credit cost method, b x must be calculated for each age between entry and retirement. In order to calculate b x, we must first calculate B x. The formula for B x under the final average benefit plan is B x =.015(x y)(.2) r 1 t=x 5 s t where r is the retirement age, y is the entry age, and s t is the attained age salary of a participant age t. 8

17 Now b x is the difference between B x+1 and B x. The model plan has a single retirement age of sixty-five. Calculating the accrued benefit for an age thirty entrant that is now age thirty-two gives B 32 =.015(2)(.5)[ ] =.0313 where the attained age salary values come from the above table. Similarly, the accrued benefit at age thirty-three is given by B 33 =.015(3)(.33)[ ] =.049. Now we can calculate the benefit accrual at age thirty-two. b 32 = = The benefit accrual and accrued benefit for an age thirty entrant under the unit credit cost method is given in the table below. 9

18 Unit Credit Benefit Accrual and Accrued Benefit Functions As a Percent of Entry Age-30 Salary Age Benefit Accrual Accrued Benefit Now that we have calculated the benefit accrual, we have the first element to the formula for the normal cost of future retirement benefits. Next we need the probability that a participant will continue in employment until age sixty five. This value can be obtained from a service table. We use the ultimate table 3-2 from Pension Mathematics [3]. Because we are using an ultimate table, our calculations do not consider the length of service factor in the probability of a participant continuing employment. 10

19 The next element of the unit credit normal cost formula is the discount function. First, we need to define our interest rate assumptions. We assume a one percent risk free rate, a three percent risk premium, and four percent inflation. This sums to an eight percent interest rate. Now the discount function can be found by i v t = ( 1 + i )t. A table of discount function values is below where year one corresponds to v 35, year two to v 34, etc. Discount Function for t Years Year Discount Year Discount Year Discount Year Discount The last element is the present value of a life annuity due starting at age sixty-five. Assuming a continued 8% interest rate, the life annuity value is 8.60 [3]. Now we can calculate the normal cost of future retirement benefits at age thirty. r (NC) 30 = b p (T ) 30 v35 ä 65 = (.015)(.167)(.0676)(8.60) = A table of normal costs for an age thirty entrant under the unit credit cost method as a percent of the attained age salary is given below with a corresponding graph. 11

20 Unit Credit Normal Costs for Entry Age 30 as a Percent of Attained Age Salary Age Cost Age Cost Age Cost Age Cost Unit Credit Normal Cost as a Percent of Attained Age Salary Percent Age Notice that while costs as a percent of attained salary are low early on, they rise quickly in the later years. Since salary is also increasing due to merit, productivity, and inflation, the increase in the dollar amount of the normal costs is even greater Explicit Supplemental Cost Methods As stated previously, the unfunded liability is the difference between the actuarial liability and the assets of the plan. (UL) t = (AL) t (Assets) t. 12

21 The change in the unfunded liability during year n is the difference between the actual unfunded liability at the beginning of year n + 1 and the expected unfunded liability at n + 1. A change in the unfunded liability due to experience variations can be determined by ( n (UL)) = (UL) n+1 [(UL) n + (NC) n (Cont) n ](1 + i). where (Cont) n is the contribution made in year n. The change in the unfunded liability attributed to assumption or benefit changes can be determined by calculating the actuarial liability at time n with the previous set of assumptions and benefits and then noting the difference. The Financial Accounting Standards Board gives accounting requirements that say the change in the unfunded liability must be amortized. Supplemental costs do this. Under the unit credit cost method, the supplemental costs that occur because of the initial unfunded liability, plan changes, assumption changes, and actuarial gains and losses are all amortized using an explicit method. There are three methods used to explicitly amortize supplemental costs. They are the straight line method, the constant dollar amortization method, and the constant percent amortization method. The straight line method uses payments equal to the interest on the outstanding balance plus 1 m of the original debt for an m-year amortization period. The supplemental cost payment for the beginning of the jth year after the change in the unfunded liability occurs is given by the formula (SC n ) j = d[( n ULB) j 1 m ( nul)] + 1 m ( nul) 13

22 where (SC n ) j = the supplemental cost payment made at the beginning of the jth year associated with the nth unfunded liability change (1 j m), d = i the rate of discount equal to ( 1 + i ), ( n ULB) j = the unfunded liability balance at the beginning of the jth year for the nth unfunded liability change. If the change in the unfunded liability is $100, then the supplemental cost for the first year under the straight line method that amortizes the unfunded liability in fifteen years, assuming a 8% interest rate is (SC n ) 1 = [ (100)] (100) = The constant dollar amortization method is widely used. The calculation of the supplemental cost payment is given by (SC n ) j = n(ul) ä m. If the change in the unfunded benefit for year n is $100, then the supplemental cost payment under the constant dollar approach that amortizes the liability in fifteen years with a 8% interest rate is given by (SC n ) = 100 ä 15 = = The constant percent amortization method is argued as best for salary-based benefit formulas [3]. The series of supplemental cost payments under this method are increasing 14

23 due to the non-merit portion of the salary increase assumption. This method will be approximately equal to a constant percent of payroll. The supplemental cost payment is given by where (SC n ) j = n(ul) sä m [(1 + I)(1 + P )] j 1 sä m = salary based m-year annuity where the payments increase by inflation and productivity assumptions, I = the inflation component of the salary assumption, P = the productivity component of the salary assumption. To calculate the supplemental cost payments under a constant percent method where the unfunded liability is amortized in fifteen years, first we need to calculate the salary based temporary annuity certain. sä 15 = 1 + v(1.04(1.01) + v 2 (1.04) 2 (1.01) v 14 (1.04) 14 (1.01) 14 = 1 (v(1.04)(1.01))15 1 v(1.04)(1.01) = Now we can calculate the first supplementary payment under the constant percent method assuming there is $100 of unfunded liability. (SC n ) 1 = [(1.04)(1.01)]0 = 8.04 A table comparing the supplemental cost payments under each method for $100 of unfunded liability amortized over fifteen years is given below with a corresponding table. 15

24 Supplemental Costs per $100 of Unfunded Liability Under Alternative Methods Year Straight Line Constant Dollar Constant Percent Supplemental Costs per $100 of Unfunded Liability Dollars Straight Line Constant Dollar Constant Percent Years Unit Credit Strengths and Weaknesses The unit credit cost method is satisfactory in terms of the security of the plan once the unfunded liability is reduced to zero [2]. It is a simple and easy to explain method when the benefit formula lends itself to allocation of benefits to specific service years. Otherwise, the calculations become complicated and confusing. One of the weaknesses of the unit 16

25 credit cost method is that it generates a normal cost that is likely to rise from year to year. Also, the actuarial liability may fall short of the present value of accrued benefits payable in the event of plan termination. 3.2 Unit Credit With Projections The difference between the calculations for the unit credit method and the projected unit credit method under the same actuarial assumptions is in the allocation of benefits to each year. Otherwise, each formula is identical. The accrued benefit at the beginning of age x under the projected unit credit cost method is determined by the projected benefit at retirement. The benefit at retirement is pro-rated over the number of years of service. The accrued benefit is defined by the following formula. where B x = B r (x y) (r y) B r = the projected accrued benefit at retirement, r = the age of retirement predetermined by the plan, y = the age at which the participant entered the plan. The actuarial liability for the projected unit credit method is r (AL) x = x y r y B r r xp (T ) x v r x ä r. Notice that the actuarial liability is a fraction of the present value of the future benefits where the formula for the present value of future benefits is r (P V F B) x = B r r x p (T ) x v r x ä r. 17

26 The normal cost of retirement benefits for the unit credit with projections cost method is given by r (NC) x = B r r y r x p (T ) x v r x ä r. For a plan containing ancillary benefits the normal cost is given by T (NC) x = [ r 1 k=x B k k y k x p (T x ) v k x (q (t) k v F k + q (d) k d F k + q (m) k s F k )] + Br r y r F r. The formula for normal costs for a plan that allows multiple retirement ages is r (NC) x = r 1 k=x B k k y g(r) k r k x p(t x ) q (r) k vk x ä k. The actuarial liability under plans allowing for either ancillary benefits or multiple retirement ages is found by multiplying each of the benefit components by (x y) Normal Cost Calculations Under the model plan assumptions, the projected accrued benefit at retirement is as is seen in the unit credit benefit calculation table. Then the normal cost at age thirty of future retirement benefits is r (NC) 30 = B r p (T ) 30 v35 ä 65 = (.167)(.0676)(8.60) =.0130 A table of the normal costs for an age thirty entrant under the projected unit credit cost method as a percent of attained age salary is given below. Notice that the normal costs in the accompanying graph do not increase as sharply as those under the unit credit method. 18

27 Projected Unit Credit Normal Costs for Entry Age 30 as a Percent of Attained Age Salary Age Cost Age Cost Age Cost Age Cost Projected Unit Credit Normal Cost as a Percent of Attained Age Salary Percent Age The supplemental costs that arise in a plan using a projected unit credit cost method are amortized using any of the three explicit methods described under the unit credit cost method Projected Unit Credit Strengths and Weaknesses The projected unit credit cost method is more conservative than the standard unit credit method. The projected method generates a larger actuarial liability which therefore results in a greater build up of funds. Thus, the actuarial liability is more likely to be 19

28 equal to the plan termination liability. It is also considered more conservative because a vested employee s termination results in an actuarial gain for the plan since his funded benefit is usually larger than the vested benefit. Another strength of the projected unit credit cost method is that the normal costs are not as steeply rising as the normal costs of the unit credit method [1]. In fact the projected method may result in decreasing normal costs as a percentage of total payroll [2]. 4 Cost Allocation Cost Methods The second category of cost methods is the cost allocation cost method. A cost allocation method is one that assigns the actuarial present value of all the projected future benefits to specific periods without allocating the benefits themselves. The entry age cost method is an example of a cost allocation method. 4.1 Entry Age There are two methods for calculating the actuarial liability and normal cost under the entry age cost method. One is the level dollar method and the other is the constant percent method. In either case, by using salary projections, the participant s total prospective benefit at retirement is estimated. The actuarial present value of the benefit at retirement at the participant s entry age is calculated and then allocated to each year of the participant s total prospective service. The constant dollar method assigns the present value of benefits to each year in a constant dollar amount. The constant percent method assigns the value to each year using a constant percentage of the participant s estimated salary from year to year Level Dollar Method The Level Dollar Method allocates the actuarial present value of future benefits to each year in a constant dollar amount and is usually used for a plan that provides a flat dollar 20

29 benefit for each year of service [1]. The projected benefit at retirement is amortized with a temporary life annuity due that commences at the entry age of the participant and ends at retirement. The actuarial liability under this method is r (AL) x = ä T y:x y ä T y:r y B r r x p (T x ) v r x ä r where y = the entry age of the participant, ä T y:x y ä T y:r y = the actuarial present value of an (x y) year temporary life annuity due, = the actuarial present value of an (r y) year temporary life annuity due. Notice that, like the projected unit credit cost method, the actuarial liability for the level dollar method is a fraction of the present value of the future benefits. Therefore, the formula for the actuarial liability can be written as r (AL) x = ä T y:x y r (P V F B) x. ä T y:r y The normal cost under the constant dollar method is calculated using the formula r(nc) x = r (P V F B) y ä T y:r y where y is the entry age of the participant. by If ancillary benefits are included in the plan provisions, the normal cost can be found T (NC) x = v (P V F B) y + d (P V F B) y + s (P V F B) y + r (P V F B) y ä T y:r y where v (P V F B) y is the actuarial present value of future vested benefits. The other forms of the present value of future benefits are related to disability, a surviving spouse, and retirement respectively. Some form of ancillary benefits are required under ERISA and 21

30 therefore real-world normal cost calculations will involve some, if not all, of the elements in the above formula. When the plan provides for multiple retirement dates, normal costs under the constant dollar method are calculated using the formula r (NC) x = r (P V F B) y r ä T y:r y where r is the earliest retirement age and r ä T y:r y represents the present value of a temporary employment-based annuity including early retirement decrements from age r to r Normal Cost Calculations Under the Constant Dollar Method In order to calculate the normal cost of an age thirty entrant under the entry age constant dollar cost method, first we need to calculate the present value of future retirement benefits evaluated at age thirty. 65 (P V F B) 30 = B p (T ) 30 v35 ä 65 = (4.7011)(.167)(.0676)(8.60) =.456 Now, to find the normal cost at any age, we divide this value by the present value of a temporary annuity due. 65 (NC) 30 = 65 (P V F B) 30 = =.066 ä 30:35 where the value of ä 30:35 under the model assumptions is found by calculating 34 t=0 tp 30 v t. 22

31 The normal cost under the constant dollar method has a constant dollar value as expected from the name. However, the cost as a percentage of attained age salary varies. A table of the percents is given below as well as a corresponding graph. Entry Age Constant Dollar Normal Costs for Entry Age 30 as a Percent of Attained Age Salary Age Cost Age Cost Age Cost Age Cost Percent Entry Age Constant Dollar Normal Cost as a Percent of Attained Age Salary Age Notice that as a percentage of salary, the normal costs are decreasing. 23

32 4.1.3 Entry Age Constant Dollar Strengths and Weaknesses One weakness of this method is that the plan usually starts with a large unfunded liability. This occurs because both entry age methods calculate the normal costs from the age the participant started employment even if this date is before the commencement of the plan. The contribution of normal costs obviously do not start until the inception of the plan, possibly leaving a number of normal costs to be accounted for in the unfunded liability. However, one of the strengths of the constant dollar method is the security it offers for the employee. During the early years, the normal costs of the entry-age constant dollar method are larger than those under the benefit allocation plans and are therefore more stable. Also, the actuarial liability generated by the constant dollar method is larger than the plan termination liability. Some feel this method generates an actuarial liability that is excessively greater than the plan termination liability Constant Percent Method The constant percentage method differs from the constant dollar method in that the normal costs are a level percentage of salary. It is appropriate for plans that state the benefit as a percentage of salary[1]. The actuarial liability under an entry age constant percent method is given by r (AL) x = sä T y:x y sä T y:r y B r r x p (T x ) v r x ä r where s ä T y:r y denotes the present value of an employment-based annuity due with payments equal to the employee s attained age salary based on a unit salary at age y. The formula used to calculate this annuity is sä T y:r y = r 1 t=y s t t yp (T y ) v t y s y where s t is the current dollar salary for a participant age t. The normal cost of retirement benefits under the constant percentage method is found by first calculating the percentage of the present value of future salary that is equal to the 24

33 present value of future benefits. This percentage then becomes the constant percentage multiplied by the current dollar salary of a participant to find the associated normal cost. The formula for the normal cost is r (NC) x = r (P V F B) y s sä T s x. y y:r y The normal cost for future ancillary and retirement benefits is given by T (NC) x = v (P V F B) y + d (P V F B) y + s (P V F B) y + r (P V F B) y s sä T s x. y y:r y When multiple retirement ages are allowed, the equation for normal costs becomes r (NC) x = r (P V F B) y s r sä s T x. y y:r y Normal Cost Calculations Under the Constant Percent Method We have previously calculated all of the elements of the normal cost of an age thirty entrant under the entry age constant percent method except one. The ultimate values for the salary-based temporary annuity due under the model assumptions can be found by calculating Now the normal cost at age thirty is r 1 t=x s t s x t xp (T ) x v t x. 65 (NC) 30 = = 65 (P V F B) 30 s 30 sä (T ) 30: (1)(11.42) (1) =.0335 s 30 Notice that the fraction portion of the calculation is a constant for normal costs evaluated at all years for an age thirty entrant. The last attained age salary element is the only changing factor. As a result, the normal cost as a percent of the attained age salary is constant as expected. Therefore, as a percent of salary, the normal cost under 25

34 the entry age constant percent cost method is 3.45 for all ages with an age thirty entrance into the plan. The supplemental costs generated by the unfunded liability of a plan using either version of the entry age method are calculated using one of the three explicit methods previously described Entry Age Constant Percent Strengths and Weaknesses The entry-age constant percent cost method is less conservative than the constant dollar method but more conservative than the original unit credit method. The actuarial liability is greater than the plan termination liability. However, as with the constant dollar method, some believe the difference is excessive. The method is centered on normal costs that are a constant percentage of salary and therefore normal costs are stable. Also, the constant percent method generates a supplemental liability and therefore allows more flexibility. 4.2 Aggregate The aggregate is another cost prorate method worth mentioning here. However, we will not calculate the normal costs. There are also two methods used to calculate the normal costs under the aggregate cost method. As in the entry age cost method, these two variations are based on a level dollar and level percentage of salary calculation. However, while the entry-age method calculates normal costs by summing individual costs, the aggregate method calculates normal costs on a group basis. Under both the constant dollar and constant percent aggregate methods, no actuarial liability directly emerges. The Internal Revenue Service states that the actuarial liability is not directly calculable [1], but some say the actuarial liability is exactly equal to the present value of future benefits less the present value of future normal costs and is therefore equal to the assets of the plan plus any unfunded liability. The normal cost for the constant dollar version is found using the formula 26

35 r (NC) = ( lx,y l x,y )[ r (P V F B) y lx,y ä T ] y:r y where = the summation over all entry age combinations, l x,y = the number of age-y entrants that are currently age x. The normal cost under the constant percent method is calculated using r (NC) = ( lx,y l x,y s x,y )[ r (P V F B) y lx,y s sä T ] y y:r y where s x,y is the salary at age x for an age-y entrant. The aggregate cost methods produce no supplemental liability and therefore offer more security. The price for this security is paid in the flexibility of the method. The aggregate method is conservative when experience gains dominate and is not conservative when experience losses prevail. The contributions to the plan are simple and smooth and the computations are relatively easy. However, the initial contributions are very high. Some consider the assets built by aggregate methods to be unnecessarily high. 5 Overall Comparison From the graph below, we can see, once again that the greatest increase in normal costs is under the unit credit cost method. The projected unit credit method lessens the increase. The entry age constant dollar cost method normal costs actually decrease as a percentage of salary. The entry age constant percent method naturally is a constant. 27

36 Normal Costs Under Various Cost Methods as a Percent of Attained Age Salary Percent Unit Credit Projected Unit Credit Entry Age Constant Dollar Entry Age Constant Percent Age 6 Normal Costs Under Varying Assumptions In order to further explore and compare the various cost methods, it is beneficial to observe how the normal costs change under differing assumptions. We will consider the normal costs after changing the interest rate in the life annuity, the inflation rate, and both the inflation rate and the overall interest rate. 6.1 Life Annuity Interest Rate Change In our original calculations, we assumed we would be able to purchase an annuity with an 8% interest rate at the time of retirement. Now we will consider the change in normal costs under each method when we assume the annuity purchased to cover retirement benefits has a 6% interest rate. The present value of such an annuity with payments of one,is 9.73 [3]. Since our productivity, inflation, and pre-retirement interest rate remain the same as 28

37 our original assumptions, salary, benefits, and the discount factor calculations also remain the same Unit Credit Method The newly calculated normal costs under the unit credit method with a 6% interest rate in the life annuity are given in the table below with a corresponding graph. Unit Credit Normal Costs for Entry Age 30 With a Change in Life Annuity Assumptions Age Cost Age Cost Age Cost Age Cost

38 Unit Credit Normal Cost with Life Annuity Change Percent of Salary Age No Assumption Change Life Annuity Change The normal costs under both methods in the early years are close. In the later years, however, the normal costs differ by a little less than four percentage points with the 6% annuity assumption having the greater costs Projected Unit Credit Method The normal costs under the projected unit credit method with the new life annuity assumption are given in the table below with a corresponding graph. 30

39 Projected Unit Credit Normal Costs for Entry Age 30 With a Change in Life Annuity Assumptions Age Cost Age Cost Age Cost Age Cost Projected Unit Credit Normal Cost with Life Annuity Change Percent of Salary No Assumption Change Life Annuity Change Age In the later years, the normal cost under the new life annuity assumption is only about 1.3 percentage points higher than under the original assumptions. The projected unit credit method normal costs are less affected with a new life annuity assumption than the original unit credit method. 31

40 6.1.3 Entry Age Constant Dollar Method The normal cost values under the entry age constant dollar method with the 6% life annuity assumption are given below as a percent of attained age salary. Entry Age Constant Dollar Normal Costs for Entry Age 30 With a Change in Life Annuity Assumptions Age Cost Age Cost Age Cost Age Cost Entry Age Constant Dollar Normal Cost with Life Annuity Change Percent of Salary No Assumption Change Life Annuity Change Age 32

41 The normal costs under the entry age constant dollar method under the new assumptions differ from the costs under the original assumptions most greatly during early years. The new assumptions result in approximately a one percentage point greater normal cost in the first years Entry Age Constant Percent Method The normal costs as a percent of salary under the new assumptions are still constant but have the value of 3.9 as opposed to 3.45 under the original assumptions. This is the smallest change in normal costs due to the differing assumptions under any of the methods we have explored. A graph illustrating the difference is shown here Entry Age Constant Percent Normal Cost with Life Annuity Change Percent of Salary No Assumption Change Life Annuity Change Age 33

42 6.2 Inflation Rate Change Next we consider the changes in normal costs that result from changing the inflation rate to 3% but keeping the interest rate at the original 8%. This will isolate the changes that occur when the difference between the salary-based increase and the interest-based increase is altered. Changing the inflation rate changes our salary scale as well as the benefit table. However, because the interest rate remains the same, the discount factor values remain the same as those under our original assumptions. For a clearer comparison, we will also be assuming that a life annuity with 8% interest is purchased at the time of retirement to cover benefits. The differing salary values and benefit values are given below. Attained Age Salaries as a Percent of Entry Age-30 Salary After a Change in Inflation Age Salary Age Salary Age Salary Age Salary

43 Salary with Change in Inflation Multiple of Entry Age Salary Age No Assumption Change Inflation Change 35

44 Unit Credit Benefit Accrual and Accrued Benefit Functions As a Percent of Entry Age-30 Salary After Inflation Change Age Benefit Accrual Accrued Benefit Unit Credit Method The normal costs for the unit credit method under the 3% inflation rate are given below with a corresponding graph. 36

45 Unit Credit Normal Costs for Entry Age 30 With a Change in Inflation Assumption Age Cost Age Cost Age Cost Age Cost Unit Credit Normal Costs with a Change in the Inflation Assumption Percent of Salary Age No Assumption Change Change in Inflation Again the change in normal costs is most drastic in later years, this time with the cost under the new inflation assumption being about 2.5 percentage points lower than the cost under the original assumptions. 37