1. Calculate the accrued benefit at NRA 2. Multiply by annuity factor to get PV at NRA 3. 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)

Entry Age Normal NC (benefit not based on pay) 1. At Entry Age 2. At Current Age

1. Calculate PVFBEA \ (1 / aEA:(RA-EA)) 2. NC doesn't change - NCCA = NCEA

Actuarial Cost Method Flashcards by Jeffrey Trytten

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

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How does a_x relate to a_x+1?

Does the relationship change for a temporary annuity?

Either:

a_x= 1 + vp_xa_x+1
a_x+1 = (a_x- 1) * (1+i)/p_x

Doesn’t change for a temporary annuity

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ePVFS_t+1

(PVFS - salary) x (1 + i) * (1 + s’)/(1 + s)

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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)

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eSalary_t+1

p_x^(T)*(1+s)*(Salaryt)

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UAL

AAL - AVA

Actuarial Accrued Liability - Actuarial Value of Assets

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ePVFB_t+1

PVFB * (1 + i) - (Actual BPs + int)

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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

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When projecting PVFB, do you project service for determining elegibility? For calculating the benefit amount?

Project Service for both

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Expected UAL at t+1

eUAL_t+1 = (UAL_t + NC)*(1 + i) - C - I

C - Contribution Made

I - Interest made on the Contribution

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_eAAL_t+1

Note: 3 items - (NC, IC, BP)

_eAAL_t+1 = (1+i)(NC_t+AAL_t) - (actual BP + int)

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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.

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For investment gain/loss, do you use the MVA or AVA?

AVA

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_eAssets_t+1

_eAssets_t+1= (1 + i) * Assets_t+ (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

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Actuarial Gain/Loss

eUAL_t+1- UAL_t+1

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AAL - Retrospective Method

AAL = the sum of all past Normal Costs accumulated with Interest/Mortality

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AAL - Prospective Method

PVFB - PVFNC Present Value of Future Benefits - Present Value of Future Normal Costs

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What is the difference between Projected Unit Credit and Traditional Unit Credit

PUC projects forward the Salary

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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:

(AB_x+1 - AB_x)*a_ra*(D_ra/D_x) or (AB_x+1 - AB_x)*a_ra*(v^ra-x _ra-xp_x)

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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:

AB_x*ara*(Dra/Dx) or AB_x*ara*(v^ra-x_ra-xp_x)

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T/F If a participant is 100% vested, the withdrawal assumption does not affect the AAL under the Unit Credit Method.

True

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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

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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.

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Entry Age Normal NC (benefit not based on pay)

At Entry Age
At Current Age

Calculate PVFB_EA * (1 / a_EA:(RA-EA))
NC doesn’t change - NC_CA = NC_EA

Entry Age Normal NC (benefit based on pay) 1. At Entry Age 2. At Current Age

1. Calculate PVFB_EA divided by a temp annuity from EA for RA-EA years i.e. PVFB_EA \* (1 / a_EA:(RA-EA)) where the interest is (1+i) / (1+s) - 1 2. NC_CA= NC_EA \* (1 + s)^{CA - EA}

EAN AAL 1. if benefit is pay based 2. if benefit is not pay based

1. PVFB_CA \* (a_EA:(CA-EA))/(a_EA:(RA-EA)) 2. PVFBCA \* (a_EA:(CA-EA))/(a_EA:(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 = PVFB_IA/ a_{ia:(ra - ia)} Δ NC = ΔPVFB_CA / a_{ca:(ra - ca)} NC_CA = 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 UAL_OLD+ (UAL_New - UAL_OLD)

Under the individual Aggregate Actuarial Cost Method, how are assets allocated to retirees, term vesteds, and actives?

1. Assets first go to Retirees & VTs 2. Then allocated proportionally to each active participant based upon the prior year normal cost and asset allocation

Individual Aggregate NC

(PVFB - Allocated Assets) / a_{CA:(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 \* q_x^(d) Termination Loss: AAL \* q_x^(w)

If a participant has an decrement for death and termination, and terminates, what is the mortality and termination G/L?

Mortality Loss: AAL \* q_x^(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)a_x⁽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?

* p_xa_x+1⁽¹²⁾ - (11/24) \* q_x * (1 + i)a_x⁽¹²⁾ - (actual BP + i)

What is the expected AAL for a retiree with a annual annuity that lives full year?

* p_xa_x+1 * (1+i) \* (a_x- 1)

p_xy 1. Definition 2. Rewrite in terms of p_x and p_y

1. Both the spouse and participant are alive 2. p_x \* p_y

p_xy 1. Definition 2. rewrite in terms of p_xp_yand p_xy

1. Probability at least one of the spouse and participant are alive 2. p_x + p_y - p_xy

a_xy 1. Definition

Annuity that pays while both participant and spouse is alive

_a_xy 1. Definition 2. rewrite in terms of a_{x a}_yand a_xy

1. Annuity that pays while at least one of the spouse and participant are alive 2. a_x + a_y - a_xy

a_Y|X 1. Definition 2. Rewrite in terms of a_x, a_y and a_xy

1. Pays $1 to life X after the death of Y 2. a_Y|X= a_x- a_xy

Rewrite a K% JnS (that reduces only on employees death) with reversionary annuities.

a_x + K(a_y - a_xy)

Rewrite a K% JnS (that reduces on either death) with reversionary annuities.

a_xy+ K\*(a_x - a_xy) + K\*(a_y - a_xy)

K% JnS with Pop-Up Formula Assume that if SLA is $1, the K% JnS with Pop-Up is $B.

B\*a_x + K(a_y - a_xy) + (1-B)\*(a_x - a_xy)

Certain and Life of n years 1. Formula (Yes it's easy)

Social Security Level Income Benefit 1. 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