Pension Mathematics. Lecture: Weeks Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 1 / 28

Transcription

1 Pension Mathematics Lecture: Weeks Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 1 / 28

2 Chapter summary Chapter summary What are pension plans? Defined benefit vs defined contribution plans Replacement ratios Valuation of benefits actuarial present value calculations using service tables Methods for funding the benefits Materials on: Chapter 10 (Dickson, et al.) Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 2 / 28

3 Introduction What are pension plans? These are usually employer-sponsored plans designed to provide employees with retirement income. The cost of providing the benefits are either fully paid by the employer, or shared (as often the case) between the employer and employee. The employee is called the plan participant. Why do employers provide retirement plans? to provide incentives for new employees or for eisting employees to stay to facilitate turnover of older employees sometimes, to ease the pressure from employee unions a way to remunerate employees with ta advantages a way to remunerate employees who contributes to the success of the company Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 3 / 28

4 Introduction Defined contribution plans In a defined contribution (DC) plan: employer pays a periodic (e.g every paycheck) contribution for each employee, usually a function of salary employee may also contribute, and employer s contribute may match with possible limits the contributions get invested and employee receives whatever the accumulation amount will be at retirement (or even before retirement, subject to some conditions e.g. taability) employee bears the investment risk Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 4 / 28

5 Introduction Defined benefit plans In a defined benefit (DB) plan: plan specifies the amount of benefit to be paid at retirement benefit is typically a function of salary throughout career (career average plans) or near retirement (final salary plans) and the number of years of service to the employer eample of benefit is: 2% of average of final 5 years of salary, for each year of service cost is either paid by employer only, or shared between employee and employer employer bears the investment risk Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 5 / 28

6 The salary scale function The replacement ratio The benefits in a DB plan, or the target for a DC plan, are usually established in consideration with the replacement ratio. It refers to the percentage of the employee s pre-retirement income that is paid out by a pension plan. It is defined to be: R = pension income in the year after retirement salary in the year before retirement Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 6 / 28

7 The salary scale function The salary scale Because most benefits are epressed in terms of salary at or near retirement, we need to project and estimate future salaries. Denote by s y the salary scale function so that if (CS) is the current annual salary for a participant aged, then his/her salary t years later is estimated (or predicted) to be (CS) +t = (CS) s +t s. Salary projections are made to reflect typical increases to salary due to merit, seniority, increases in productivity, and possibly inflation. Salary scale may be in a table format such as that in Table 10.1 on page 338, or epressed as a function such as s y = 1.03 y. Such a salary scale function implies a rate of increase of 3% each year. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 7 / 28

8 The salary scale function Projecting salaries If epressed in terms of the salary scale function s y = 1.04 y, this would be equal to 85, s 62 + s 63 + s 64 s 40 Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 8 / 28 Projecting salaries A plan provides pension benefit based on final average salary which is defined to be the average salary in the three years before retirement. Assume salary increases at the rate of 4% each year occurring at the beginning of each valuation period. A plan participant aged eactly 40 at the valuation date has a current salary of $ 85,000. Calculate his predicted final average salary assuming retirement is at age 65. Solution: His predicted final average salary is the average of his salaries when he reaches 62, 63 and 64. This is equal to 85, ( ) = 209,

9 The salary scale function Projecting benefit payments Projecting benefit payments Assume that salary increases occur on participant s birthdays. Denote by B(, y, h) the projected annual pension benefit rate to commence at age y for a participant currently age and has accumulated h years of service. 1 Fied amount of k per year of service B(, y, h) = k (h + y ) 2 Fied percentage p of final year salary B(, y, h) = p (CS) s y 1 s 3 Fied percentage p of final year salary for each year of service B(, y, h) = p (h + y ) (CS) s y 1 s. 4 Fied percentage p of final average salary (defined to be the average of final m years immediately preceding retirement) for each year of service. 1 s y m + + s y 1 B(, y, h) = p (h + y ) (CS). m s Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 9 / 28

10 The salary scale function SOA question SOA question #19, Fall 2015 A defined benefit pension plan provides its members, upon retirement at age 65, retirement benefits from one of two options: An annual pension equal to 2% of career average salary for each year of service. An annual pension equal to R% of final average salary for each year of service. Final average salary is defined to be the average salary earned in the 5 years immediately preceding retirement. Colton joins the pension plan at age 35 and his salary increases by 2.5% each year on his birthday. Under either option, Colton would receive the same annual pension benefit. Calculate R%. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 10 / 28

11 The salary scale function SOA question SOA question #19, Spring 2015 You are given the following information for John, eact age 30, who just joined a defined benefit pension plan: The plan provides a retirement pension of 1.6% of final average salary for each year of service. The final average salary is defined as the average salary in the three years before retirement. John s salary is currently 40,000 and is epected to increase 3.5% annually on his birthdays. John will retire at eact age 65. Calculate the replacement ratio provided by his pension. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 11 / 28

12 The service table The service table A multiple decrement table describing the survivorship pattern of the group of pension plan participants. We shall denote these decrements by d, w, i, and r, respectively to describe death, withdrawal, ill-health and age retirement. We use the following symbols: = l = number of active members surviving in service at eact age ; d (d) = d = number of deaths in service in the year of age to + 1; l (τ) = w = number of withdrawals from service in the year of age to + 1; d (w) = i = number of ill-health retirements in the year of age to + 1; d (i) d (r) = r = number of retirements due to attainment of retirement age. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 12 / 28

13 The service table - continued The service table - continued The decrement probabilities may be denoted by q (d) = d(d) l (τ), q (w) = d(w) l (τ), q (i) = d(i) The probability of survivorship is therefore l (τ), and q (r) = d(r) l (τ) p (τ) = 1 q (τ) = 1 (d(d) + d (w) + d (i) + d (r) ) l (τ) = l(τ) (d (d) + d (w) l + d (i) + d (r) ) = l(τ) +1 l (τ) For purposes of illustration, we may use the Illustrative Service Table used for Eam LTAM... Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 13 / 28

14 Actuarial present value calculations Retirement or pension benefits Actuarial present value of pension benefits For illustrative purpose, consider a participant now aged with h years of accumulated service and pension benefit payment of B(, y, h). Suppose normal retirement age is 65 but early retirement at ages 62, 63 and 64 is allowed but pension benefit is reduced by 5% for each year prior to age 65. Also, assume pension benefit is paid as a monthly life annuity. The APV at age of the retirement benefit can be epressed as APV = 65 k=62 v k B(, + k, h)[ (65 k)] q(r) +k l (τ) ä (12) +k. Similar calculations can be made for death and withdrawal benefits. Some details will be eplained in class. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 14 / 28

15 Actuarial present value calculations Contributions towards pensions Actuarial present value of contributions Suppose contribution each year is a fied amount C = C er + C ee consisting of employer contributions (C er ) and employee contributions (C ee ). The APV of contributions may be epressed as 64 APV(contributions) = C v k l(τ) +k l (τ) k=0 64 = C k=0 v k p (τ) k If contribution rate is a fied percentage of salary each year, the APV of contributions will be 64 C % (CS) k=0 s +k s v k l (τ) +k l (τ) 64 = C % (CS) k=0 s +k v k p (τ) s k where C % = C % er + C % ee, the overall contribution rate. Note that sometimes the contributions (or benefits) are made mid-year, then adjust the probabilities and discounting accordingly. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 15 / 28

16 Actuarial present value calculations SOA question SOA question #18, Fall 2015 For a defined benefit pension plan, you are given: The retirement benefit is 25 per month for each year of service. The normal retirement age is 65. Early retirement is allowed begining at age 63 with the retirement benefit reduced by 7.2% per year prior to age 65. Bob is an employee age 50 who was hired at age 30. i = 0.07 Benefits are valued assuming that retirement (r) is the only decrement from employment. Employees retire only on their birthdays and Age q (r) ä (12) Calculate the actuarial present value of Bob s retirement benefit. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 16 / 28

17 Valuation of benefits Accrued benefits Accrued benefits Consider a defined benefit plan which provides for a pension benefit equal to 2% of final average salary for each year of service. Final average salary is defined to be average of earnings in the three years just prior to retirement. Normal retirement age is 65. The percentage 2% is called the accrual rate. For a person now age 50 who joined the plan 20 years ago with a current salary of 100,000, his projected final average salary can be written as s 62 + s 63 + s 64 3 s 50 where z 65 = (s 62 + s 63 + s 64 )/3. = z65 s 50, Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 17 / 28

18 Valuation of benefits Accrued benefits - continued His accrued benefit refers to the portion of the benefit that is attributable to his past service. This would be equal to z65 s 50 and the actuarial present value of these accrued benefits, assuming monthly benefit for life, will be z65 s 50 ä (12) 65 v 15 p This assumes no early retirement. For an eample of calculating APV of accrued benefits that allow for early retirement, please see Eample 10.7 on page 352. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 18 / 28

19 Valuation of benefits Funding methods Approaches to funding accrued benefits Because salary increases are never guaranteed, there is a case for omitting them when calculating accrued benefits. When determining the accrued benefits under the traditional unit credit approach (also referred to as the current unit credit approach), assume salaries are frozen from the valuation date until the retirement date unless otherwise indicated. The approach that uses projected salaries to retirement is called the projected unit method. Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 19 / 28

20 Valuation of benefits Funding methods Funding the benefits Similar to life insurance products, we also need to establish reserves for pension obligations. It is called actuarial liability and this liability or reserve at time t is denoted by t V. The normal contribution, denoted by C t, paid at the start of the year is the amount sufficient to pay any mid-year benefits and the end of year actuarial liability. In effect, we have tv + C t = APV(mid-year benefits) + v p 00 1 t+1v so that C t = APV(mid-year benefits) + v p 00 1 t+1v t V Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 20 / 28

21 Numerical illustration Numerical illustration 1 Numerical illustration 1 - similar to Eample A member eact age 45 has 17 years of past service with a current salary of 100,000 at the valuation date. You are given: Accrual rate: 1.4% Final salary plan with pension benefit based on salary in the year before retirement. Normal retirement age: 65 The pension benefit is to be paid as a life annuity payable annually in advance. There is no benefit due on death while in service. Some valuation assumptions needed: No eits other than by death before normal retirement age. Interest rate: 6% per year effective For projected unit credit method, use an annual salary rate increase of 3% per year. Mortality before and after retirement follows the Illustrative Life Table Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 21 / 28

22 Numerical illustration Numerical illustration 1 - continued Calculate the member s accrued pension benefit and the normal contribution due at the start of the year of the valuation date using: 1 the Traditional Unit Credit (TUC) method 2 the Projected Unit Credit (PUC) method Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 22 / 28

23 Numerical illustration Numerical illustration 1 Solution 1 - TUC method Under this funding method, salaries are frozen at time of valuation. Hence, the member s accrued benefit is given by AB 45 = (CS) 45 = = The actuarial liability at the valuation date (t = 0) is the APV of these accrued benefits 0V = AB 45 v 20 20p45 ä 65 = where 20p45 = l 65/l 45 = / and ä 65 = Similary, it can be shown that 1V = (CS) v 19 so that the normal contribution at valuation date is C 0 = vp 45 1 V 0 V = (1.03) 0V 0 V = 0 V p ä 65 [ (1.03) 1 ] = Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 23 / 28

24 Numerical illustration Numerical illustration 1 Solution 1 - PUC method Under this funding method, salaries are allowed to be projected to retirement so that the member s accrued benefit in this case is AB 45 = (CS) 45 (1.03) 19 = = The actuarial liability at the valuation date (t = 0) is the APV of these accrued benefits 0V = AB 45 v 20 Similary, it can be shown that 20p45 ä 65 = V = (CS) 45 (1.03) 19 v 19 p ä 65 so that the normal contribution at valuation date is C 0 = vp 45 1 V 0 V = 18 ( ) V 0 V = 0 V 17 1 = 0 V 1 17 = Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 24 / 28

25 Numerical illustration Numerical illustration 2 Numerical illustration 2 - TUC method Alice, age 40 on December 31, 2015, is a participant of a defined benefit pension plan. You are given: She was hired on January 1, 2001 with a salary of 125,000. She has received a salary increase of 3% each January 1; the employer epects to continue those increases until her retirement. The retirement benefit, payable annually at the start of the year, is 2% of final salary times the number of years of service. Retirement occurs only at age 65. Other decrements, which are assumed to occur in the middle of the year, are 5% per year. No benefits are paid to those who leave before retirement. i = 0.06 ä 65 = 11.2 Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 25 / 28

26 Numerical illustration Numerical illustration 2 - continued Using the traditional unit credit method: 1 Calculate the actuarial present value of Alice s accrued benefit on December 31, Calculate the normal contribution for Alice for the year beginning January 1, Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 26 / 28

27 Numerical illustration Numerical illustration 3 Numerical illustration 3 - PUC method Kevin, age 35 on December 31, 2015, is a participant of a defined benefit pension plan. You are given: He was hired on January 1, 2011 with a salary of 35,000. He has received a salary increase of 3% each January 1; the employer epects to continue those increases until her retirement. The retirement benefit, payable annually at the start of the year, is 2% of 3-year final average salary times the number of years of service. Retirement occurs only at age 65. Other decrements, which are assumed to occur in the middle of the year, are 5% per year. No benefits are paid to those who leave before retirement. i = 0.04 ä 65 = 11 Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 27 / 28

28 Numerical illustration Numerical illustration 3 Numerical illustration 3 - PUC method Using the projected unit credit method: 1 Calculate the actuarial present value of Kevin s accrued benefit on December 31, Calculate the normal contribution for Kevin for the year beginning January 1, Lecture: Weeks (Math 3631) Pension Mathematics Spring Valdez 28 / 28