Unfunded Actuarial Liabilities
• For a Defined Benefit Plan:
UALt = ALt – Ft ”Funding Ratio” = Ft / ALt
• We can calculate this Unfunded Actuarial Liability at any point in
time
• It is often useful to project what we think the UAL will be in a year
if all assumptions play out as expected
UALt+1 = ALt+1 – Ft+1
• Next year’s Asset Fund, Ft+1, can be thought of as the following
pieces:
• Ft+1 = Ft + I + C – P, where
I = Interest earnings on the fund during the year
C = Dollars contributed to the fund during the year
P = Purchases of annuities for people leaving the plan during the
year
• ALt+1 = ALt plus the pieces that bring it forward one year (NC,
interest, decrement, etc)
Unfunded Actuarial Liabilities
• We can use the expectations about next year’s UAL to
determine what gains or losses we might have on the
defined benefit plan
• In general, UALt+1 = UALt (1 + i)
- (Difference in interest actually earned LESS expected
interest earned on Fund)
- (Contributions actually made LESS Normal Costs
expected in Actuarial Liability calculations)
- (Reductions in liabilities due to actual terminations due to
death or service LESS
expected reductions for death or service)
- (Reductions in liabilities due to actual retirements LESS
expected retirements)
• So…. If all actuarial assumptions go as planned (interest,
terminations, etc) then UALt+1 = UALt (1 + i)
Unfunded Actuarial Liabilities
•
Typically, one or more of the assumptions will not play out as
expected (or “can’t” due to fractional deaths, etc)
•
So the plan will have an Actuarial Gain or Actuarial Loss. The
Gain or Loss will comprised of the interest and liability
components in the definition of UALt+1
•
Actuarial Gain =
(Difference in interest actually earned LESS
expected interest
earned on Fund)
+ (Reductions in liabilities due to actual terminations due to death or
service LESS expected reductions for death or service)
+ (Reductions in liabilities due to actual retirements LESS expected
retirements)
•
The contribution piece is not part of this since it is primarily
controlled by the plan sponsor.
Unfunded Actuarial Liabilities
• Alternatively, in terms of Unfunded Actuarial
Liabilities,
• UALt+1 = UALt (1 + i)
- (Contributions actually made LESS Normal
Costs expected in Actuarial Liability calculations)
- Actuarial Gain
Actuarial Gain =
UALt (1 + i) - UALt+1 - (C – NCt (1 + i) + IC)
Actuarial Gain = (UALt + NCt) (1 + i) – C – IC – UALt+1
• where IC is any additional interest generated from
contributions made to the fund during the year
Unfunded Actuarial Liabilities
• Example – SOA Course 5 Examination, Problem No. 6
• Benefit =
+
1.5% • FS • YOS for 0 < YOS < 10
2.0% • FS • (YOS – 10) for 10 < YOS
• i = 6%; s = 4%; No pre-retirement decrements; r = 65;
Retirement Annuity Factor = 12
•
•
•
•
•
Assets on 1/1/2011 = $300,000
Assets on 1/1/2012 = $320,000
Contributions on 12/31/2011 = $5,000
PUC Funding Method; Two Employees
Employee A: entered plan on 1/1/2001 with e = 30; x = 40;
S40 = $30,000
• Employee B: entered plan on 1/1/1981 with e = 30; x = 60;
S60 = $50,000
• Actuarial Liability as of 1/1/12 = $350,000
• Determine UAL as of 1/1/2011 & Actuarial Gain during 2011
Unfunded Actuarial Liabilities
• UAL as of 1/1/2011:
Actuarial Liability for Employee A on 1/1/2011
=
B40 • Retirement Annuity • Discount from retirement
back to age 40
=
[1.5% • (30,000)(1.04)24 • 10] • 12 • (1.06) -25
=
$32,351.30
Actuarial Liability for Employee B on 1/1/2011
=
B60 • Retirement Annuity • Discount from retirement
back to age 60
=
[[1.5% • (50,000)(1.04)4 • 10] + [2.0% • (50,000)(1.04)4 •
20]] • 12 • (1.06) -5
=
$288,481.51
Total Plan Liability = $32,351.30 + $288,481.51 = $320,732.81
UAL as of 1/1/2011 = $320,732.81 - $300,000 = $20,732.81
Unfunded Actuarial Liabilities
Actuarial Gain during 2011:
Actuarial Gain = (UALt + NCt) (1 + i) – C – IC – UALt+1
Normal Cost for Employee A during 2011
= 2.0% • (30,000)(1.04)24 • 1 • 12 • (1.06) -25
= $4,300.17
Normal Cost for Employee B during 2011
= 2.0% • (50,000)(1.04)4 • 1 • 12 • (1.06) -5
= $10,490.24
Total Plan Normal Cost = $4,300.17 + $10,490.24 =
$14,790.41
Unfunded Actuarial Liabilities
Actuarial Gain during 2011:
Actuarial Gain = (UALt + NCt) (1 + i) – C – IC –
UALt+1
Actuarial Gain = ($20,732.81 +
$14,790.41)(1.06) – 5000 - 0
- ($350,000 - $320,000)a
Actuarial Gain = $2,654.61
Actuarial Gains and Losses
For a Defined Benefit Plan:
Total Experience Gain = Investment Gain
+ Liability Gain
= Earning more than assumed/expected
on asset fund
+ Having better than expected experience
on mortality, salary, withdrawal
assumptions, etc
Actuarial Gains and Losses
Investment Gain =
Actual Asset Fund at End of Year
– Expected Asset Fund at End of Year
= Act F – Exp F
Liability Gain = Expected Actuarial Liability at End of
Year – Actual Actuarial Liability at End of Year
= Exp AL – Act AL
Total Experience Gain
= (Act F – Exp F) + (Exp AL – Act AL)
Actuarial Gains and Losses
•
•
•
•
Example:
Annual Retirement Benefit = $600 per Year of Service
Funding Method = TUC
Actual fund performance = 10% versus 5% expected in
Commutation Function Sheet
• 2 employees with e = 25 and x = 60; r = 65
• 1 dies during the year
• Asset Fund at BOY = $80,000; Normal Cost paid at BOY =
$6,000
• Calculate Total Experience Gain
•
= (Act F – Exp F) + (Exp AL – Act AL)
Actuarial Gains and Losses
• Total Act AL under TUC at x = 60 is
2*(600)(35)(7.94)(.5393) = $179,845.76
• Expected AL under TUC at x = 61 is
$179,845.76 increased for interest, survival and next
payment of Normal Cost
=
$208,863.33
Actual AL under TUC at x = 61 is
1*(600)(36)(7.94)(.6088) = $104,411.63
Liability Gain = $208,863.33 - $104,411.63 = $104,451.70
Actuarial Gains and Losses
• Act F is (80,000 + 6,000) * 1.10 = $94,600
• Exp F is (80,000 + 6,000) * 1.05 = $90,300
• Investment Gain = $94,600 - $90,300 = $4,300
• Total Experience Gain
•
= $4,300 + $104,451.70
•
= $108,751.70
Individual Level Premium
• We have been talking over the last classes about
people entering a defined benefit pension plan that
already exists
• However, sometimes employees are already in
service when a plan is introduced
• Two ways to handle this:
– Create a initial Supplemental Liability and calculate normal costs from the
entry date (like under EANLD)
– Start with no initial liability and fund at a more rapid pace from the
inception date forward
Individual Level Premium
• The ILP cost method is commonly used for the
second of the two methods
• Normal costs are derived from plan inception
through retirement, and account for already accrued
service
• PVFNC = PVFB
• NC (ä(T)a:r-a|) = Br • (D(T)r / D(T)a) • är(12)
• Rearrange and solve for NC
• NC [(N(T)a - N(T)r) / D(T)a] = Br • (D(T)r / D(T)a) • är(12)
Individual Level Premium
• The ILP cost method is commonly used for the
second of the two methods
• Normal costs are derived from plan inception
through retirement, and account for already accrued
service
• ALx = PVFBx - PVFNCx
• ALx = [Br • (D(T)r / D(T)x) • är(12)] - NC (ä(T)x:r-x|)
• ALx = [Br • (D(T)r / D(T)x) • är(12)] - NC • (N(T)x - N(T)r) / D(T)x
Aggregate Cost Methods
• As we discussed early in this section of the class,
funding can be done on an individual basis or on an
aggregate basis
• All that we’ve discussed so far has been individual
methods:
•
TUC
•
PUC
•
EANLD
•
EANLP
•
ILP
• We’ll take a quick look at a couple aggregate
methods
Individual Aggregate
• IA method (an oxymoron?) has components
that are both individual and aggregate in
nature
• Individual: Normal Cost for entire plan is still
the sum of normal costs of each individual
• Aggregate: Normal Cost for each individual
is affected by the amount of excess funding
that exists in the plan as a whole
Individual Aggregate
• Back to our “Entrepreneurial Company”
exercise:
• We all begin as new employees 10 years ago,
now we are established enough to create a
defined benefit plan
• New DB plan recognizes past service credits
• What are different ways to fund the plan?
Individual Aggregate
• Use a purely Individual Method, calculate current
Actuarial Liability, and gather assets of that amount
– Good funding, but expensive at first
• Use the Individual Level Premium method, and cram
the funding in the shorter time to retirement
– Normal costs can be very high
• Determine how much you can afford to post as plan
assets now (call it “F”), and fund the difference
going forward
– Potentially a balance of current and future funding
Individual Aggregate
• Basics of IA method:
• F = amount of excess funding that exists in
the plan
• FP = amount of the excess funding that is
allocated to participant P
• This amount of excess funding helps to
reduce the amount of liability that must be
funded, so normal costs are reduced
• For each individual:
•
PVFNC = PVFB - FP
Individual Aggregate
• Key question: How do you take F and divide it up in
to pieces in order to figure out FP for each
individual??
• Common ways to do this:
•
Divide the fund equally among all participants:
FP = F / n
•
Prorate F by the present value of the projected
retirement benefit
•
Prorate F by the present value of the accrued
benefit
•
Prorate F by the current AL for each participant
•
Make up your own creative way….
Individual Aggregate
• Let’s work through an example with 2
participants
• F = $1,000
• Retirement benefit = 1.65% of final salary
times Years of Service
• r = 65, s = .03
• Participant 1: e = 25, x = 35, Sx = $80,000
• Participant 2: e = 35, x = 45, Sx = $120,000
• How do we cost the plan?
Individual Aggregate
• Let’s work through an example with 2
participants
• Determing FP by using Present Value of
Projected Retirement Benefit
• F = $1,000
• Retirement benefit = 1.65% of final salary times
Years of Service
• r = 65, s = .03
• Participant 1: e = 25, x = 35, Sx = $80,000
• Participant 2: e = 35, x = 45, Sx = $120,000
• How do we cost the plan?
Individual Aggregate
• Let’s work through an example with 2
participants
• Determing FP by dividing evenly between
participants
• F = $1,000
• Retirement benefit = 1.65% of final salary times
Years of Service
• r = 65, s = .03
• Participant 1: e = 25, x = 35, Sx = $80,000
• Participant 2: e = 35, x = 45, Sx = $120,000
• How do we cost the plan?
Aggregate Cost Method
• Now let’s look at a cost method that truly is
totally aggregate in nature
• The exercise, while incorporating individual
information, is really geared towards figuring
out the Normal Cost for the entire plan,
rather than each person individually
• It incorporates the concept of the “average
working-life annuity”
Aggregate Cost Method
•
•
•
•
•
Average working-life annuity
= ä(T)
= Σ [(sN(T)x - sN(T)r) / sD(T)x] • Sx
Σ Sx
= Present Value of all Future
Salaries Paid
•
Total Current Salaries
Aggregate Cost Method
• Using the average working-life annuity
notation, we then have:
• TNC • ä(T) = Σ PVFB – F
• Read this as “The total normal cost for the
plan paid under the assumption of the
average working-life annuity, equals the total
present value of future benefits for the plan
less any excess funding”
Aggregate Cost Method
• Of course, sometimes we want TNC to be a
level percent of total salary so we use TNC =
U • Σ Sx
• Special note: We can express level dollar as
a subset of level percent of salary…
• What adjustments need to be made?
• What happens if we set s = 0, and Sx = 1?
Other Funding Methods
• For Defined Benefit Plans implemented after
employees already have accrued service:
• Individual Level Premium showed a way to
do fast funding with no initial cash outlay
• Could also immediately fund an amount to
pick up EANLD at the point in scale
• But other “in-between” alternatives exist
Frozen Initial Liability
(EAN)
• FIL (EAN) method calculates for each individual what
the current Actuarial Liability would be under EANLD
• But instead of picking up point in scale, we choose a
time period over which to amortize this liability –
called the “Frozen Initial Liability” (FIL)
• This amortization cost of FIL is called the
“Supplement Cost” (SC)
• Meanwhile, we are using the EANLD Normal Cost to
do the remaining funding
• Total Cost in a year = NC + SC
Frozen Initial Liability using
TUC
• Method calculates for each individual what the
current Actuarial Liability would be under TUC
• Again, we choose a time period over which to
amortize this liability – called the “Frozen Initial
Liability” (FIL)
• This amortization cost of FIL is called the
“Supplement Cost” (SC)
• Meanwhile, we are using the TUC method to do the
remaining funding
• Total Cost in a year = NC + SC