Illinois State University, Mathematics 480, Spring 2014 Test No. 2, Thursday, April 17, 2014 SOLUTIONS

Transcription

1 Illinois State University Mathematics 480 Spring 2014 Test No 2 Thursday April SOLUTIONS 1 Mr Rowan Bean starts working at Hard Knocks Life Insurance company at age 35 His starting salary is euros a year Salary increases 25% a year every year until age 60 when the growth rate decreases to 15% per year His annual pension benefit equals 2% of last salary times the number of years of service Normal retirement age is 67 Calculate the annual amount of benefit of Mr Bean when his retirement starts The last age at which Mr Bean will be working will be 66 (as he retires when he reaches age 67) At age 64 his salary will be Since he will have worked for 32 years his benefit will be 32 times 2% of that last salary of euros and that amount equals approximately euros 2 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 2013 as of January is We also know that p 50 = 09 You are given that the assumed mortality is exactly the mortality experience in 2014 Which of the following is the closest to the total normal cost for 2014 as of January The normal cost per participant at age 50 is: NC 50 = ν P 50 a (12) 65 At age 51 the normal cost per participant is: ( ) NC 51 = v p 51 a 65 We see therefore that NC 50 p = v NC p = vp so that NC 51 = (1+ i)nc 50 p 50 Because there are 10 participants on January normal cost per participant at that time is NC 50 = 2000 Therefore the normal cost on January per participant is NC 51 =

2 There are 9 participants surviving as of January thus the total normal cost is: = In a pension plan normal retirement benefit is 50% of final three-year average salary The valuation method used is Entry Age Normal (percentage of salary) Valuation rate is 6% Salary increases are assumed at 5% There are no preretirement deaths or terminations All retirements occur at age 65 You are also given that a ) 65 = 9 There is only one participant born on January The date of hire for her was January Her compensation for 2014 is $50000 What is in dollars the normal cost for 2014 as of January ? As of January the participant is 50 years old and has 15 years of service The retirement benefit is: B 65 = = Since there are no deaths prior to retirement we must have: s NC 35 a 35:65 35 = a ) 65 Define j = Then NC 35 = = 2818 a 30 j This constitutes the following % of salary at age 35: = 1172% At age % of the salary is $ For a defined benefit pension plan plan effective date is January Normal retirement benefit is $10 per month per year of service Attained age normal actuarial cost method is used There are no preretirement terminations other than by death The retirement age is 65 As of January there are 70 active employee participants all aged 45 with twelve years of service There are no retired or terminated vested participants You are also given that a 45 = 15 a 45:20 = 10 N 65 = N ) 65 = What is the normal cost for 2014 as of January ? The future service for each employee is 20 years The actuarial present value of benefits earned after plan effective date is

3 We have and a ) 65 D 65 a 45:20 = N 45 N 65 a 45 = N 45 N 65 = a 45 a :20 = 5 45 = N 65 = Therefore ( ) N ) 70 = N 65 = 475 And the Normal Cost required is = $ A pension plan becomes effective January The valuation method used is Individual Level Premium Normal retirement benefit is 800 per month The valuation date is December The valuation interest rate is 8% There are no pre-retirement deaths or any other decrements All retirements occur at the normal retirement age of 65 You are given that a ) 65 = 8 There are two participants both born January What is the normal cost for 2004 as on December ? The normal cost is calculated as the level annuity payment from the plan inception date to retirement sufficient to fund the projected benefit at retirement In this case as the two participants are both 30 and have 35 years till retirement: NC s 35 8% = a ) 65 and since a ) 65 = 8 we get NC = s 35 8% But this is the cost as of January and its value at the end of the year is s 35 8%

4 6 You are the valuation actuary for the Gallipoli Life Insurance Company headquartered in Sydney Australia Your company issues life insurance coverage to the members of the military which is a fully discrete four-year term insurance You have determined that the force of mortality appropriate for this policy is µ ( x + t) = 001t for 0 t 4 where x is the age of the insured at the time the policy is issued and t is the policy duration Because all expenses for the policy are paid separately by the Ministry of Defence individual policyholders (ie soldiers) only pay level benefit premium Your company still wants to be able to pay a higher amount for expenses in the first policy year and it uses Full Preliminary Term method for reserving Policies are only issued in the amount of $ The valuation interest rate is 0% Calculate the modified (FPT) benefit reserve at policy duration 2 We have t t p = e µ ( x+ s)ds 001s ds 0 0 t x = e = e 0005t 2 Therefore 1 = e = e = e = e 008 Under FPT the level renewal valuation premium is P x+1:3 = A 1 x+1:3 = a x+1:3 v + l x+3 v 2 + l x+3 l x+3 l x+4 l x+3 v 3 1+ v + l x+3 v 2 = + l x+3 + l l x+3 x l = x+3 l x l x = p 4 x p x p 1+ 2 x p + 3 x Therefore the modified (FPT) reserve at policy duration 2 is 1 e 008 e e 002 e e e

5 ( A x+2:2 P 1 x+1:3 a x+2:2 ) l x+3 v + l x+3 l x+3 l x+4 v 2 =1 l x+3 l x+3 + l x+3 l x+4 1 l x l x+3 = = l x+3 v = l x+3 = 1 p 4 x p 3 x e 008 e 0045 e e =1 7 You are the valuation actuary for the Hard Knocks Life Insurance Company offering life annuities and life insurance to guinea pigs The valuation interest rate is 1% and mortality of guinea pigs follows De Moivre s Law with l x = 4 x 0 x 4 You can assume that times after age 4 are ignored Your company issues fixed deferred annuities to guinea pigs The interest rate is guaranteed at 5% for the first two years and 0% thereafter The surrender charge is 10% the first year 5% the second year and 0% thereafter Calculate the reserve for a $1000 single premium deferred annuity at issue sold to a newborn There are no front-end loads Assume that you use Commissioners Annuity Reserve Valuation Method and you are doing this in the state of New York Regulations require that surplus must be 5% of assets at the policy issue date Find the additional surplus investment requirement for this policy Because you are doing business in the state of New York you only need to do the calculations at the beginning of each policy year Time Fund Value Cash Surrender Value PV of Cash Value The reserve is given by the present value of the guaranteed cash value at the beginning of the third policy year and it equals Without a surplus contribution the company only has 1000 worth of assets and would not be allowed to issue the policy If the minimum surplus investment is x then we must have

6 and x = x x = You are the valuation actuary for Glorious Life Insurance Company You are implementing a new principle-based reserving methodology for your company You have created a set of 10 economic scenarios that represent a representative range of possible outcomes for your company s future so that you can treat them as a complete probability space with equal probability assigned to each scenario Your company has just received a $10000 single premium for a deferred annuity policy Your analysis has shown that the present value at time 0 of the accumulated deficiency (ie the excess of future interest credited to customer s account over the amount earned on investments after all expenses) at the end of each year is as given in the table below The regulation requires the reserve to be calculated as the current account balance plus the 70-th percentile Conditional Tail Expectation (CTE) of greatest present value of accumulated deficiency Calculate that reserve Scenario Present value of accumulated deficiency at time t = The greatest present values (GPV) of accumulated deficiencies for scenarios 1 through 10 are: CTE70 calculation requires us to average the largest 30 percent of Scenario GPVs of initial account balance plus GPV of accumulated deficiency That means we need to add the starting asset amount to each GPV for 3 scenarios with greatest GPV and calculate their average and those 3 GPV are and 120 so that:

7 9 In the pension plan under consideration the Entry Age Normal (percentage of salary) method is used Normal retirement benefit is 80% of final two-year average salary The valuation interest rate is 6% Salary increases are assumed to be 3% annually There are no pre-retirement deaths or other decrements All retirements are assumed to occur at the normal retirement age of 65 We are also given that a ) 65 = 10 There is only one participant born on January The date of hire for this participant was January The 2004 salary of the participant is What is the normal cost for 2004 as of January ? As of January the participant is 50 years old and has 15 years of service The projected retirement benefit is: B 65 = Since there are no deaths prior to retirement the normal cost is paid as a deterministic salary-based annuity starting at entry age of 35 and ending at retirement age of 65 so that: NC 35 S a = B a ) 65 If we define then j = S a = = a 30 j Therefore NC 35 = a 30 j This constitutes the following percentage of salary at age 35: % At age % of the salary is A pension plan is being valued on January The actuarial funding method used is the Aggregate Method The valuation interest rate is 6% There are no preretirement decrements Salaries are assumed to increase 1% per year It is known that a ) 65 = 10 The normal cost is 10% of salary There is only one participant age 55 Current salary of the participant is per year The assets of the plan are valued at on January Find the expected monthly benefit for this participant at normal retirement age of 65

8 Under the Aggregate Method in the case of this plan the nominal amount of the normal cost in 2004 is: APV Future Benefits NC = ( ) Assets S a 10 In this case we have: S a 10 = and with j = % we have: S a 10 = a 10 j Therefore APV Future Benefits ( ) = Assets + NC S a Let us write x for the monthly retirement benefit Then: APV Future Benefits ( ) = Therefore: = x a ) x x A pension plan provides an annual benefit equal to 2% of final year compensation for each year of service at normal retirement age of 65 A participant has accumulated 25 years of service at normal retirement age and has the final salary of What is the single benefit premium for the pension annuity if D 65 = 210 and N 65 = 2201? The annual benefit earned by this participant is: = The single benefit premium for the pension annuity is: a 65 = N 65 = D = At the beginning of the year 2003 accrued liabilities of a pension plan for the employees of The Honorable Life Insurance Company were The assets of the plan at the same time were On January a contribution of is made to the plan At the same time is used for purchases of life annuities fully discharging the liabilities for retiring participants although based on the actuarial assumptions the cost of those annuities should have been During the year 2003

9 assets of the plan earn 45% while the actuarial valuation rate is 8% The normal cost of the plan for 2003 is established by the plan actuary as (as of January ) There are no deaths or terminations other than retirement in 2003 Find the actuarial gain for this plan for the calendar year 2003 Initial unfunded actuarial liability is: UAL 2003 = AL 2003 F 2003 = = Assets at the end of the year are: F 2004 = F C P + I = ( ) 045 = = Accrued liabilities during 2003 increase with the valuation rate but are decreasing by the discharge of liabilities of and increased by the normal cost of Therefore the accrued liabilities as of January are: ( ) 108 = The unfunded actuarial liabilities on January are therefore: = The expected unfunded actuarial liability on January on the other hand is: ( UAL NC 2004 )( 1+ i) ( C + I C ) = ( ) = = = The actuarial gain is: Ga = ( 43200) ( 94700) = You can also approach this problem by calculating the actuarial gain on assets and actuarial gain on liabilities separately: Actuarial gain on assets ( ) ( ) = $40700 Actuarial gain on liabilities ( ) 108 = Total actuarial gain = 51500