Actuarial Cost Method Flashcards
How many salary increases between 65-n and 65 if we use current year salary? Prior year Salary?
Current Year Salary is n-1
Prior Year Salary is n
How does ax relate to ax+1 ?
Does the relationship change for a temporary annuity?
Either:
- ax = 1 + vpxax+1
- ax+1 = (ax - 1) * (1+i)/px
Doesn’t change for a temporary annuity
ePVFSt+1
(PVFS - salary) x (1 + i) * (1 + s’)/(1 + s)
What would the PVFS adjustment factor be if there is no experience Gain/Loss and salary scale is s’ instead of s?
(PVFS - salary) x (1 + i) * (1 + s’)/(1 + s)
i.e. adjustment factor of (1 + s’)/(1 + s)
eSalaryt+1
px(T)*(1+s)*(Salaryt)
UAL
AAL - AVA
Actuarial Accrued Liability - Actuarial Value of Assets
ePVFBt+1
PVFB * (1 + i) - (Actual BPs + int)
What is the adjustment factor for the PVFB if there was no experience G/L, but
- salary increased by s’, not s
- Benefit rate was $20 per month instead of $18 per month
(1 + s’)/(1 + s)
$20/$18
When projecting PVFB, do you project service for determining elegibility? For calculating the benefit amount?
Project Service for both
Expected UAL at t+1
eUALt+1 = (UALt + NC)*(1 + i) - C - I
C - Contribution Made
I - Interest made on the Contribution
eAALt+1
Note: 3 items - (NC, IC, BP)
eAALt+1 = (1+i)(NCt+AALt) - (actual BP + int)
For the AAL, do you project service to calculate elegibility?
Do you project service to calculate the benefit amount?
Project Service for Eligibility
Don’t project service to calculate benefit.
For investment gain/loss, do you use the MVA or AVA?
AVA
eAssetst+1
eAssetst+1 = (1 + i) * Assetst + (Contributions + int) - (BP + int)
Note use the actual BPs because actual assets use actual BPs and we simply want the difference in investment G/L
Actuarial Gain/Loss
eUALt+1 - UALt+1
AAL - Retrospective Method
AAL = the sum of all past Normal Costs accumulated with Interest/Mortality
AAL - Prospective Method
PVFB - PVFNC Present Value of Future Benefits - Present Value of Future Normal Costs
What is the difference between Projected Unit Credit and Traditional Unit Credit
PUC projects forward the Salary
Unit Credit AAL
- Calculate the accrued benefit at NRA
- Multiply by annuity factor to get PV at NRA
- Using interest/mortality discount back to valuation date
In formula form, this means:
(ABx+1 - ABx)*ara*(Dra/Dx) or (ABx+1 - ABx)*ara*(vra-x ra-xpx)
Unit Credit Normal Cost
- Calculate the change in the accrued benefit at NRA
- Multiply by annuity factor to get PV at NRA
- Using interest/mortality discount back to valuation date
In formula form, this means:
ABx*ara*(Dra/Dx) or ABx*ara*(vra-x ra-xpx)
T/F If a participant is 100% vested, the withdrawal assumption does not affect the AAL under the Unit Credit Method.
True
T/F if there are no G/L and we use Level $ funding where applicable, the normal cost does not change under all funding methods.
False - Under Unit Credit it would
PTP is hired in 2003 and the plan starts in 2006, when is the entry age for the EAN Actuarial Cost Method?
2003
Entry Age is determined without regard to whether the plan was in existence.
Entry Age Normal NC (benefit not based on pay)
- At Entry Age
- At Current Age
- Calculate PVFBEA * (1 / aEA:(RA-EA))
- NC doesn’t change - NCCA = NCEA
Entry Age Normal NC (benefit based on pay)
- At Entry Age
- At Current Age
- Calculate PVFBEA divided by a temp annuity from EA for RA-EA years i.e. PVFBEA * (1 / aEA:(RA-EA)) where the interest is (1+i) / (1+s) - 1
- NCCA = NCEA * (1 + s)CA - EA
EAN AAL
- if benefit is pay based
- if benefit is not pay based
- PVFBCA * (aEA:(CA-EA))/(aEA:(RA-EA))
- PVFBCA * (aEA:(CA-EA))/(aEA:(RA-EA)) where the interest in the annuity is actually (1 + i)/(1 + s) - 1
Under the Individual Level Premium Actuarial Method, a plan change increasing benefits would not increase the AAL ever.
TRUE - Change would prospectively increase normal cost
T/F Both the ILP and EAN amortization period for the PVFB ignore the plan’s inception date.
False only the EAN does.
ILP NC (no pay)
Initial NC = PVFBIA / aia:(ra - ia)
Δ NC = ΔPVFBCA / aca:(ra - ca)
NCCA = Initial NC + Σ Δ NC
ILP AAL
No shortcut - must use the Prospective or Retrospective method to calculate
Under aggregate cost methods, what factor is multiplied by the PVNC to determine the normal cost if benefit is not based on pay?
Σ Lives / Σ PV of Expected Lives to NRA (sum over each ptp)
Under aggregate cost methods, what factor is multiplied by the PVNC to determine the normal cost if benefit is based on pay?
Σ Salary / Σ PVFS (sum over each ptp)
Under aggregate cost methods, which participants are included in the factor multiplied by the PVNC to determine the normal cost?
Employees with non-zero expected future working lifetime
Aggregate Cost Methods Normal Cost
Assume temporary annuity is the variable “z”
(PVFB - UAL - AAL) * z
Aggregate Cost AAL for the 3 Aggregate Cost Methods
Aggregate Cost Method - Not Defined
Frozen Initial Liability - Entry Age Normal
Attained Age Normal - Unit Credit
Under the aggregate cost methods how is the UAL intitially defined?
AAL is calculated by the respective individual cost method and the UAL is (AAL - AVA) at time 0 for the initial
Under the aggregate cost methods how is the UAL rolled forward year to year?
Each year the UAL = expected UAL
Under the aggregate cost methods how is the UAL revised under a plan change?
UAL revised when there is a plan change by
UALOLD + (UALNew - UALOLD)
Under the individual Aggregate Actuarial Cost Method, how are assets allocated to retirees, term vesteds, and actives?
- Assets first go to Retirees & VTs
- Then allocated proportionally to each active participant based upon the prior year normal cost and asset allocation
Individual Aggregate NC
(PVFB - Allocated Assets) / aCA:(NR - CA)
*Note - Problem will tell you how to allocate assets
How are bases/credit balances handled when determining the normal cost under individual aggregate method?
Before allocating the assets to participants, reduce assets by credit balance and/or bases.
If a participant has an decrement for death and termination, but doesn’t die or terminate, what is the mortality and termination G/L?
Mortality Loss: AAL * qx(d)
Termination Loss: AAL * qx(w)
If a participant has an decrement for death and termination, and terminates, what is the mortality and termination G/L?
Mortality Loss: AAL * qx(d)
Termination Gain: Actaul - Expected - Mortality Loss
Under the Unit Credit method, what is the G/L for Actuarially Equivalent Early Retirement? Why?
$0. The benefit as an active is the PV of an annuity at 65 and the early retirement benefit is Actuarily reduced benefit (i.e. PV of an annuity at age 65) so there is no change to AAL.
What is the expected AAL for a retiree with a monthly annuity that dies?
(1 + i)ax(12) - (actual BP + i)
Note: BP depends on actual date of death
What is the expected AAL for a retiree with a monthly annuity that lives full year?
- pxax+1(12) - (11/24) * qx
- (1 + i)ax(12) - (actual BP + i)
What is the expected AAL for a retiree with a annual annuity that lives full year?
- pxax+1
- (1+i) * (ax - 1)
pxy
- Definition
- Rewrite in terms of px and py
- Both the spouse and participant are alive
- px * py
pxy
- Definition
- rewrite in terms of px py and pxy
- Probability at least one of the spouse and participant are alive
- px + py - pxy
axy
- Definition
Annuity that pays while both participant and spouse is alive
axy
- Definition
- rewrite in terms of ax <span>a</span>y and axy
- Annuity that pays while at least one of the spouse and participant are alive
- ax + ay - axy
aY|X
- Definition
- Rewrite in terms of ax, ay and axy
- Pays $1 to life X after the death of Y
- aY|X = ax - axy
Rewrite a K% JnS (that reduces only on employees death) with reversionary annuities.
ax + K(ay - axy)
Rewrite a K% JnS (that reduces on either death) with reversionary annuities.
axy + K*(ax - axy) + K*(ay - axy)
K% JnS with Pop-Up Formula
Assume that if SLA is $1, the K% JnS with Pop-Up is $B.
B*ax + K(ay - axy) + (1-B)*(ax - axy)
Certain and Life of n years
- Formula (Yes it’s easy)
Social Security Level Income Benefit
- Formula
A ptp’s birthday is 12/31/1900. How many Normal Costs are in the future for the 1940 val year if we use an EOY valuation, so 12/31/1940?
25 years of normal cost left