Commutation Functions. = v x l x. + D x+1. = D x. +, N x. M x+n. ω x. = M x M x+n + D x+n. (this annuity increases to n, then pays n for life),

Transcription

1 Commutation Functions C = v +1 d = v l M = C + C +1 + C +2 + = A = M 1 A :n = M M +n A 1 :n = +n R = M + M +1 + M +2 + S = (this S notation is not salary-related) 1 C = v +t l +t µ ( t)dt 0 ω M = C + C +1 + C +2 + = v +t l +t µ ( t)dt R = M + M +1 + M +2 + A = M A :n = M M +n + +n a = a :n = +n n a = +n+1 0 ( I a ) = S ( I n a ) = S S +n ( I a ) :n = S S n +n +n (this annuity increases to n then pays n for life) ( m ) = 1 m D + 1 m D m D m m ( m a ) = ( m) = α D UDD ( m) β m ( m) a ( m) = UDD ( ) α m β m = α m UDD ( ) ( m = ) ( m) +1 = α m UDD m ( ) = α ( m) ( ) = α ( m) β m β m ( ) ( ) ( ) β ( m) ( ( ) β ( m) ) + β ( m) +1 ( m ) = 1 m D m D m D + ( ~ may have other meanings) + m m m ( m a ) = ( m) = α D UDD ( m)a + γ ( m) ( ~ may have other meanings) ( m) m = UDD ( ) +1 + γ ( m) ( ~ may have other meanings) α m ( ) = ( m ) ( m) +1 = UDD ( ) + α ( m) γ ( m) γ m ( ) +1 ( ~ and γ may have other meanings)

2 Salary scale function We use a function S to describe salary growth in calculations of values of pension benefits and pension contributions when those are related to a worker's salary That function is also called a salary scale and its role is to project a salary from now into the future especially near the time of retirement because the final salary or average of salaries several final years of work are commonly used for calculation of pension benefits But the salary scale can also be used to figure out past salary based on current salary The salary scale function works this way S Salary received at age ie between eact age and eact age +1 = S y Salary received at age y ie between eact age y and eact age y +1 We also consider the derivative of the salary scale function and call it salary rate However if the salary scale function uses only integers ie salary does not change between two consecutive eact ages then we just consider a difference quotient for an increment of 1 ie Salary earned in age interval tt + h ( ) h for h = 1 In order to use the salary scale function to project salaries we simply take Salary received at age ie between eact age and eact age +1= = S S y Salary received at age y ie between eact age and eact age y +1 Pension plans A defined benefit pension plan is a promise by the plan sponsor (typically an employer) to plan participants of some pension benefit (typically based on past salaries years of service and possibly age) A defined contribution pension plan is an arrangement whereby plan sponsor (also typically an employer) sets up an investment account for each plan participant where either or both of these two parties can contribute funds to be invested and later on used to pay for the cost of retirement of the plan participant ote that a defined benefit plan promises a certain benefit so it defines a benefit for a plan participant and the plan participant often does not know how much money the plan sponsor actually contributes to fulfill that promise A defined contribution pension plan on the other hand states (defines) a contribution to be made by plan sponsor (and possibly also a plan participant) but does not make any promise of any specific retirement benefit If a plan participant is required to make contributions to a pension plan the plan is called contributory In the United States a qualified (ie accepted by the regulators and the ta agency as legitimate and approved for ta eemption) defined benefit plan must be non-contributory But in Canada there are defined benefit pension plans approved by regulators and ta authorities that are contributory On the other hand most defined contribution pension plans in the United States are contributory The percentage of pay paid into a defined contribution plan by the employer is called the contribution rate The replacement ratio of a pension plan is defined as the ratio of the pension income in the year after retirement to the final year (last year of work) salary In a defined benefit pension plan there is always a formuls for the amount of benefit If it is only based on the final salary the plan is called a final salary plan If the formula is

3 based on the salary throughout the years of employment the plan is called a career average salary plan Because defined contribution pension plans do not establish the amount of benefit it becomes important to calculate the amount of contribution required to achieve a desired replacement ratio Study ote MLC-09-1 Problem o 295 An employee aged eactly 62 on January has an annual salary rate of 000 on that date Salaries are revised annually on December 1 each year Future salaries are estimated using the salary scale given in the table below where S y y > denotes the S ratio of salary earned in the year of age from y to y + 1 to the salary earned in the year of age to + 1 for a life in employment over the entire period ( y + 1) S The multiple decrement table below models eits from employment: ( 1) (i) d denotes retirements ( 2) (ii) d denotes deaths in employment (iii) There are no other modes of eit l ( 1) d The employee has insurance that pays a death benefit equals to 4 times his salary at death if death occurs while employed and prior to age 65; and pays 0 otherwise The death benefit is payable at moment of death Assume deaths occur at mid-year The annual effective rate of interest is 005 Calculate the actuarial present value of the death benefit A 489 B 4414 C 4472 D 4518 E 4585 Solution ote that the applicable probability of death is ( 2) ( 2) d q 62 = 62+k = d ( 2) 62+k l 62+k 1 k l 62 l 62+k 1 l 62 τ for k = And strictly speaking the cohort size should have been written as l instead of l The actuarial present value of the death benefit is ( 2) d ( )

4 Answer E ( 2) S 62 d S d ( 2) S d ( 2) = S 62 l 62 = May 2012 Course MLC Eamination Problem o 20 Jenny joins XYZ Corporation today as an actuary at age 60 Her starting annual salary is and will increase by 4% each year on her birthday Assume that retirement takes place on a birthday immediately following the salary increase XYZ offers a plan to its employees with the following benefits: A single sum retirement benefit equal to 20% of the final salary at time of retirement for each year of service Retirement is compulsory at age 65; however early retirement is permitted at ages 6 and 64 but with the retirement benefit reduced by 40% and 20% respectively The retirement benefit is paid on the date of retirement A death benefit payable at the end of the year of death equal to a single sum of % of the annual salary rate at the time of death provided death occurs while the employee is still employed You are given that δ = 5% and the following multiple decrement table (w =withdrawal; r = retirement; and d = death): Age ( τ ) l Calculate the epected present value of Jenny s total benefits ( w) d ( r) d A B C D E Solution In order to receive a benefit Jenny can retire at ages 6 64 or 65 or die at ages (without having retired previously) or 64 (without having retired previously) The following table shows the calculation of the actuarial present value of retirement benefits (note that the actuarial present value is calculated as salary times discount factor times probability times (number of years of service times 20%) corrected for reduction and note that the benefit is paid at the beginning of the year of retirement (as opposed to the death benefit which is paid at the end of the year of death): Age at retirement Salary Discount for present value calculation Probability umber of years of service times 20% Reduction ( d) d Actuarial present value

5 e 005 ( r q ) 60 = 6 60% 40% e ( r 4 q ) 60 = 5 80% 20% e ( r 5 q ) 60 = 4 % 0% The sum of these actuarial present values is = For the death benefits the actuarial present values are calculated as follows (death benefit which is salary at death times the discount factor times probability) Age at death Salary Discount for present value calculation Probability Actuarial present value e 005 q ( d ) 0 60 = e q ( d ) 1 60 = e 005 q ( r ) 2 60 = e q ( r ) 60 = e q ( r ) 4 60 = The total actuarial present value of death benefits is = The total actuarial present value of all benefits is = Answer E Valuation and Funding of a Defined Benefit Pension Plans Valuation is the process of establishing the value of accrued liabilities (AL) of a plan and the concept of AL is an analogue of a life insurance reserve as well as the ormal Cost (C) which is an analogue of the premium in life insurance Additionally AL is compared to fund value F and Unfunded Actuarial Liability (UAL) = AL F is calculated Methods considered by us are: - Individual methods in which total AL is equal to the sum of individual participants AL - Aggregate methods Individual Methods Traditional Unit Credit: AL is the value at the valuation date of the pension benefit accrued from the date of entry into the plan to the date of valuation Salary scale

6 functions is not used for evaluation of future wages The normal cost at the beginning of each year is the cost of the pension benefit that is earned in that year Eercise A pension plan uses the Traditional Unit Credit funding method The valuation interest rate is 6% There are no pre-retirement decrements other than death The normal retirement age is 65 The benefit is 20 per month for each year of service On January there are 10 active participants all age 50 The total normal cost for 200 as of January is We also know that p 50 = 09 Calculate total normal cost for 2004 as of January Assume that the assumed mortality is eactly the mortality eperience in 2004 Solution The normal cost per participant at age 50 is: C 50 = ν p 50 a (12) 65 At age 51 the normal cost per participant is: C 51 = v p 51 We see therefore that C 50 = v p C p = vp so that C 51 = (1+ i)c 50 p 50 Because there are 10 participants on January normal cost per participant at that time is C 50 = 2000 Therefore the normal cost on January per participant is C 51 = There are 9 participants surviving as of January thus the total normal cost is: = Projected Unit Credit This method adds the use of salary scale to the Traditional Unit Credit method Eercise: Sample 1999 SOA Course 5 Eamination Problem o 17 You are the actuary for a non-contributory pension plan that provides a pension equal to 15% of final -year average salary for each year of service up to a maimum of 5 years You are given the following The actuarial cost method used is Projected Unit Credit with the valuation interest rate of 7% annual compensation increases of 5% and no decrements before retirement The retirement age is 65 Life annuity factor for age 65 (payable monthly) is 100 There are two participants as of January : Adam and Eve Adam s date of birth is January his date of hire is January and

7 current annual salary is $40000 Eve s date of birth is January her date of hire January and her current annual salary is $50000 You are asked to consider two different approaches for the funding of the plan Approach 1 has benefits prorated on accrued service Approach 2 has benefits prorated on projected total service until retirement Calculate the ormal Cost and the Actuarial Liability for both participants as of January under Approach 1 and Approach 2 Describe the similarities and the differences between these two approaches Solution The two employees are Adam = 60 e = 25 S = Eve = 40 e = 5 S = Approach 1 Projected annual benefit of Adam (note that he already has 5 years of service as he entered at age 25 and is now 60) B Adam = Based on this the Accrued Liability is AL Adam B Adam and the ormal Cost is based on benefit earned in current year C Adam b Adam Projected annual benefit of Eve (she has now 5 years of service and has 25 years till retirement) B Eve = Based on this the Accrued Liability is AL Eve = B Eve and the ormal Cost is C Eve b Eve The total accrued liability is AL = AL Adam + AL Eve and the total ormal Cost is C = C Adam + C Eve = = Approach 2 Benefit of Adam B Adam =

8 Benefit of Eve B Eve = The ormal Cost calculation is the same as in Approach 1 C Adam b Adam C Eve b Eve C = C Adam + C Eve = = Accrued liabilities need to be allocated to the years of service up to date as a fraction of the entire service and for Adam this is full allocation as he has served 5 years so his Accrued Liability is AL Adam 5 5 B Adam but for Eve Accrued Liability is allocated to the five years served as a fraction of total service of 0 years AL Eve = 5 0 B Eve For Adam the two approaches are the same because he has already worked at this company for 5 years which is the maimum amount of acknowledged years of service The main difference is for Eve who has only worked there for 5 years so she has not accrued as many years of service and there is a difference in her projected benefit Entry Age ormal In this method normal cost is assumed to be paid from a participant s entry date to retirement date If the plan is established after the entry date supplemental liability is created for the past service and it is paid off separately amortized over a longer period of time The normal cost can be calculated as a level dollar amount or a level percentage of salary Eercise You are given the following information about a pension plan The plan effective date is January ormal retirement benefit is $200 per month Actuarial cost method is entry age normal Assumed interest rate is 6% There are no preretirement terminations and nobody dies before retirement Retirement age is 65 As of January the plan has one participant aged 45 who was hired January You are also given that ) a 65 = 10 s29 = s 9 = What is the normal cost and actuarial liability as of January ?

9 Solution ) Benefit value at age 65 equals $200 times 12 months times a 65 = 10 so it is $24000 Based on this normal cost equals (note that there are no preretirement terminations and nobody dies before retirement) $ = $0746 s 29 Therefore actuarial liability (retrospectively) equals $0746 s 9 $74511 Individual Level Premium This method funds each participant s projected benefit with level normal cost It is very similar to Entry Age ormal but it starts funding from date of plan inception Eercise You are valuing a pension plan effective January ormal retirement benefit is $600 per month The valuation method used is Individual Level Premium The valuation date is December The valuation rate is 6% There are no preretirement deaths or withdrawals All retirements occur at age 65 You are given that a ) 65 = 25 There is only one participant who was born January What is the normal cost for 2004 as on December ? Solution We have and therefore ( ) C s 1 = a ( ) C = a s 1 Aggregate Pension Funding Methods The pension funding methods we have studied up to this point all started by calculating the normal cost and the actuarial liability for each plan participant and then assessing those quantities for the entire plan by adding individual participants quantities There is an alternative to this approach although less frequently used Instead of assigning liabilities to individual participants one can assign assets of the plan to such individuals and then assess the funding by the need to pay the liabilities not covered by the assets in an amortization process over an appropriate time interval All methods using this approach are termed aggregate methods There is however a variety of aggregate approaches

10 The simplest is termed individual aggregate In this method one starts by assigning a share of the entire fund of assets held by the plan to each individual participant There are various approaches used in such assignment Assets can be assigned on the pro-rata basis in proportion to the present value of accrued benefits or by the actuarial liabilities as established by another funding method (eg entry age normal or individual level premium) or by some other reasonable approach Once the assets are assigned to each individual that individual s normal cost is established as: where C j C j = APV B j ( ) F j a n is this j-th participant s normal cost APV B j ( ) is that participants actuarial present value of accrued benefits F j is that participant s fund share and a n is an appropriate annuity (assumed here payable at the end of each year for a period of amortization n typically until this individual s normal retirement age) Such individual normal costs can then be summed up for the entire plan s normal cost One very important property of the individual aggregate method is that it never produces any unfunded actuarial liability All funding needs are amortized into the normal cost Thus the normal cost is likely to be larger than the one produced by methods developing unfunded liabilities Some employers may desire lower normal cost if funding of the normal cost is required statutorily while supplementary cost may be delayed On the other hand the aggregate methodology produces realistic results The second aggregate method is usually termed just aggregate but sometimes in contrast with the individual aggregate it is also called aggregate aggregate Its approach is very similar to the individual aggregate method with one key difference: that the normal cost pays the difference between total assets and total plan liabilities in a form of a so-called average annuity Let be the number of plan participants and F be the total assets of the plan Thus the normal cost total for the entire plan under the aggregate method is given by the formula: ( ) APV B j F j =1 C = a where a is the average annuity a concept specific to aggregate methods In case of funding cost defined as a dollar amount (as opposed to percentage of salaries) if j is the j-th participant s age and r j is the j-th participant s retirement age such average annuity is defined as: a j :r j j j=1 a = This concept is also used for plans that define normal cost as a percentage of payroll (ie combined wages of all employees) In that case the average annuity is:

11 a = j=1 ( ) APV All Future Salaries j j=1 APV( Current Salary j ) In practice it is quite common to epress the salary-based calculations with the salary based commutation functions A salary function S is a function giving the salary at age or an appropriate multiple or a fraction of salary Most commonly it is assumed that the salary increases by a constant percentage every year so that we multiply current year s salary by a constant factor 1 + s Another approach is to specify the salary function with the use of another function s such that S s = S y s y for any two ages y While the aggregate method has a desirable property of producing no unfunded actuarial liability it produces a relatively high normal cost Because of this alternative forms of aggregate approaches have been developed which produce a separate initial unfunded liability which is amortized and paid separately with the payment termed supplemental cost The first such method is called Frozen Initial Liability (Entry Age ormal) also abbreviated as FIL (EA) In this method the initial actuarial liability at the pension plan inception is calculated for each plan participant using the entry-age normal method and then summed over all plan participants This sum is treated as a separate Frozen Initial Liability denoted by FIL Then the normal cost is calculated as: C = j =1 ( ) APV B j a F FIL Another alternative is to calculate the Frozen Initial Liability using the unit credit method and the resulting aggregate method is then termed Frozen Initial Liability (Attained Age ormal) or FIL (AA) The normal cost under FIL (AA) is higher than under FIL (EA) because the initial unfunded actuarial liability is smaller while the supplementary cost is smaller than under FIL (EA)