General Principles of
Pension Valuation
• Basis for valuation of pension liabilities
has similar actuarial mathematics as
developed for life insurance
• Both create a liability
– The defined benefit pension plan is clearly a liability of the
plan sponsor – they have made the promise to provide
benefits – need to support with appropriate assets
– Life insurance benefits are clearly a liability of the insurance
company – policy reserves are the measure of that liability on
the insurance company’s balance sheet - they have made the
promise to provide benefits – need to support with
appropriate assets
General Principles of
Pension Valuation
• Mathematics often treated
separately
– Pension laws have grown to develop specific
education standards for actuaries who sign off on
pension valuation
– United States Pension Law requires and Enrolled
Actuary (EA designation) see
http://www.beanactuary.org/about/enrolled.cfm
– Licensed by a Joint Board of the Department of
Treasury & Department of Labor
General Principles of
Pension Valuation
• Other reasons why the mathematics is
often treated separately
– Number of participants in a pension plan typically less than
similar insureds in a life insurance portfolio
• As we saw… Can lead to different variances of benefit
estimates
– Discount rates used for pension valuation have more
discretion
• Life Insurance valuation interest rate set by the Standard
Valuation Law (“SVL”) adopted by the NAIC
General Principles of
Pension Valuation
• Other reasons why the mathematics is
often treated separately
– Benefit design in pension plans much broader and wider
than benefit designs in life insurance plans
• Benefits a function of many variables (service, salary,
factors, etc) while life insurance benefit is often fixed
– “Gains and/or losses” due to experience being “better or
worse” than what the actuarial valuation assumptions can
be a big part of a sponsor’s income statement
• Life insurance statutory valuation done typically on a
very conservative basis so that most “bad scenarios”
already planned for
General Principles of
Pension Valuation
• Main concepts of pension
valuation
– Accrued Liability (or Actuarial Liability) – how
much being set aside for future benefits – Balance
Sheet
– Normal Cost – how much to pay in each year to
fund benefits – Income Statement
– Unfunded Liability – a comparison of the assets
and liabilities – Balance Sheet
– Actuarial Gain / Loss - how much was made or
lost due to actual activity being different than
expected – Income Statement
General Principles of
Pension Valuation
• Actuarial Liability calculations will vary
depending mainly on…
– whether the liability only looks at past service – often called
the traditional methods or accrued method – or whether the
liability looks at potential future service – often called a
projected method
– whether the liability is calculated separately for each
participant with specific data – often called the “individual
method” – or whether blanket assumptions are made –
often call the “aggregate method”
Actuarial Liability
• General description of an
“individual accrued” liability:
– In words….
• Accrued Benefit that will be paid annually
from retirement age forward, times
• Factor that pays the Accrued Benefit every
year in a monthly annuity, times
• Actuarial discount (which takes into
account interest, mortality, and completing
service) from the retirement age back to the
current attained age
Actuarial Liability
• General description of an “individual
accrued” liability:
– In mathematical symbols…
• ALx = Bx • är(12) • (D(T)r / D(T)x)
–where Bx = annual pension benefit
that has accrued to age x
–(D(T)r / D(T)x) represents the
discounting process from the
retirement age back to the current
age, involving interest and all
decrements
Actuarial Liability
•
Concrete Example – COUNTRY Financial DB Plan
•
Normal Retirement Age = 65
•
•
Annual Retirement Benefit
1. Calculate A = Average Annual Earnings during five highest
consecutive years of income out the ten years immediately
preceding retirement
2. Calculate B = Min (Years of Service, 32)
3. Annual Retirement Benefit = 1.65% x A x B
•
•
•
What is the Actuarial Liability under an Individual Accrued Method
(“Traditional Unit Credit”) for an employee age 40 with 15 years of
service where earnings have been constant at $50,000 the last 15
years?
• ALx = Bx • är(12) • (D(T)r / D(T)x)
• ALx = [1.65% (50000) min(15,32)] • ä65(12) • (D(T)65 / D(T)40)
• ALx = [12,375] • ä65(12) • (D(T)65 / D(T)40)
Commutation Functions
• Commutation Functions
– A long time ago, in an actuary’s office, far, far
away…. An actuary got tired of writing out all the
different summations to indicate insurance and
annuity benefits
– So…commutation functions were invented
– Basic idea: take the most common terms in
insurance and annuities and create shorthand for
them
Commutation Functions
• Commutation Functions
– Today’s dilemma: The latest version of the Actuarial
Mathematics textbook used to teach actuarial
principles does not have any reference to
commutation functions
– But…you can’t avoid them in other texts or more
importantly, insurance laws and regulations
– And…Defined Benefit Pension Funding formulas
make use of them
– So…we’ll review briefly
Commutation Functions
• Commutation Function Initial
Definitions
– qx = probability of dying during a year
• Mortality tables all are written in terms on qx’s
– px = probability of surviving during a year
– px = 1 – qx
– lx = number of people out of original group
surviving to age x = p0 * p1 * p2 * …. * px-1
– vx = 1 / (1 + i)x = interest discount for x years
Commutation Functions
• Natural extensions of these
definitions
– ly / lx = probability of surviving from age x to
age y = (y-x)px
– Often, y is described in relative terms to x, such
as
y = x +n, and therefore the notation becomes
lx+n / lx = npx
Commutation Functions
• The next steps in definitions
– Actuaries realized that a lot of what they needed
to do involved discounting future payments for
mortality and interest
– “How much do I need to invest today at a given
interest rate and assuming a given mortality to
provide a benefit down the road?”
– Thus… the birth of the Dx function
– Dx = vx lx
Commutation Functions
• The next steps in definitions
– More appropriately, the discounting needed to be done from
a future attained age to a known attained age today
– “How much do I need to invest today at a given interest rate
and assuming a given mortality to provide a benefit at
known point in time down the road given that the
beneficiary has already survived to a known age?”
– Answer? Simple. Divide one Dx function into another
– Dz / Dx = vz lz / vx lx = vz-x * lz / lx = vz-x *(z-x)px
– Read this as “discounted with interest for (z-x) years, times
probability of surviving from x to z”
– Example….
Commutation Functions
• The next steps in definitions
– More appropriately, the discounting needed to be done from
a future attained age to a known attained age today
– “How much do I need to invest today at a given interest rate
and assuming a given mortality to provide a benefit at
known point in time down the road given that the
beneficiary has already survived to a known age?”
– Answer? Simple. Divide one Dx function into another
– Dz / Dx = vz lz / vx lx = vz-x * lz / lx = vz-x *(z-x)px
– Read this as “discounted with interest for (z-x) years, times
probability of surviving from x to z”
– Example….
Commutation Functions
Interest Rate
Age
60
61
62
63
64
65
5.00%
l
x
1,000,000
990,000
970,200
941,094
903,450
858,278
Fund per life BOY Fund with Interest
67.25
71.32
76.42
82.72
90.48
70.61
74.89
80.24
86.86
95.00
qx
px
Fund per life EOY
0.0100
0.0200
0.0300
0.0400
0.0500
0.9900
0.9800
0.9700
0.9600
0.9500
71.32
76.42
82.72
90.48
100.00
Commutation Functions
• Breaking down the lz / lx part so it
makes logical sense….
– Recall that lx = number of people out of original
group surviving to age x = p0 * p1 * p2 * …. * px-1
– lz / lx = (p0 * p1 * p2 * …. * px-1 * px * px+1 * …. * pz-1) / (p0 * p1 *
p2 * …. * px-1)
– What happens? All the terms from 0 to x-1 fall out
– So… lz / lx = px * px+1 * …. * pz-1 = (z-x)px
Commutation Functions
• Same thing but with hard and fast
numbers!!!
– Let z = 65, x = 35
– For example, discounting a retirement benefit
from age 65 for someone currently age 35
– D65 / D35 = v65 l65 / v35 l35 = v65-35 * l65 / l35 = v30 *
30p35
– Read this as “discounted with interest for 30
years, times probability of surviving from 35 to
65”
Commutation Functions
• Take a look at a general D65 / Dx
– What do you see happening at very young
employee ages, say age 25 – 35?
– Even as x nears 65, does D65 / Dx get close to 1?
– And this would just consider mortality – not
including employees leaving the company
Commutation Functions
• General notation for multiple
decrements
– In pension valuation you have several decrements
to the employee population
• Employees die
• Employees leave the company
– So it is common to distinguish the “mortality
only” Dx from the “total decrement” D(T)x
– (T) denotes that all forces of decrement are being
considered
Commutation Functions
• In pension valuation you have several
decrements to the employee population
• Employees die
• Employees leave the company
• An example with multiple decrements…
• Assume 1000 people in a cohort,
mortality rate of 10% throughout each
year, and 5% of the surviving members
leave the cohort at the end of the year
Commutation Functions
• Expansion on Dx
– After inventing Dx, actuaries realized that often
times they needed to discount a whole series of
future payments
– So Nx came next
– Nx = Dx + Dx+1 + Dx+2 + … (goes on until the end of
the mortality table)
– We can prove that an annuity due (where
payments begin immediately) = Nx / Dx
Commutation Functions
• For defined benefit pension funding, Dx
and Nx are used quite frequently
• You are constantly talking about
discounting future benefits to the
current point in time
• Most pension funding notation, and
especially that which is used on the
current SOA exams, are written in this
format
Commutation Functions
• Monthly versus Annual
– Often times, pension benefits are talked about in
terms of monthly payments
– So annuities due are often the monthly versions
of the more common annual notation
– Example:
• äx is used to describe an annuity of 1 beginning
at age x and paid annually
• äx(12) is used to describe an annuity of 1/12
beginning at age x and paid monthly
Commutation Functions
• Combining äx and Dx
– Remember, we proved that äx = Nx / Dx
– And often times, we are talking about discounting
an annuity stream from retirement age, r, to
current age, x
– So the factor…. Dr / Dx times är … often enters
into the discussion
– Well… (Dr * är) / Dx = Nr / Dx … so that is a common
way to write it
– Or perhaps most frequently … N(12)r / D(T)x
Increasing Cost
Individual Cost Methods
• Individual cost methods calculate the cost that
a company should pay to fund a pension
benefit by looking at each participant one at
a time
• Increasing cost means that costs typically
increase as participants age and are not
levelized across the earnings period
• Each year, the “slice” of the benefit that is
earned is funded in that year
Traditional Unit Credit
(TUC) Method
• Liability under TUC is the value of the
accrued pension benefit as of the valuation
date
• Also called Unit Credit Cost Method (by
ERISA) and Accrued Benefit Cost Method
• No consideration is given to what salary
increases or years of service might occur in
the future – strictly look at what has
happened before the valuation date
Traditional Unit Credit
(TUC) Method
• Actuarial Liability for each
participant:
– ALx = Bx • (D(T)r / D(T)x) • är(12)
– where Bx = annual pension benefit that has
accrued to age x
– Perhaps we can read this as “Accrued Benefit,
times discount from r back to x, times annuity
forward from r”
Traditional Unit Credit
(TUC) Method
• Normal Cost for each participant:
– NCx = bx • (D(T)r / D(T)x) • är(12)
– where bx = annual pension benefit earned during
employment at age x
– Perhaps we can read this as “earned benefit this
year, times discount from r back to x, times
annuity forward from r”
Traditional Unit Credit
(TUC) Method
• Need to do some examples….
• Pull out your handy Commutation
Function sheet
Traditional Unit Credit
(TUC) Method
• There’s a clear relationship between Bx
and the bx’s
– bx’s are the slices of the accrued total benefit, Bx
– Many times bx is constant across all years
– Recall in some plans, benefits are a fixed amount per year of
service
– So Bx changes every year, but by the same amount all the
time
– NCx = bx • (D(T)r / D(T)x) • är(12)… and next year…
– NCx+1 = bx+1 • (D(T)r / D(T)x+1) • är(12)
– But if bx = bx+1, then the only thing different is the discount
factor
Traditional Unit Credit
(TUC) Method
• An interesting twist on this…
– If bx = bx+1, then take a look at NCx+1 / NCx
– NCx+1 / NCx = [bx+1 • (D(T)r / D(T)x+1) • är(12)] / [bx • (D(T)r
/ D(T)x) • är(12)]
– NCx+1 / NCx = D(T)x / D(T)x+1 = vx lx / vx+1 lx+1
– NCx+1 / NCx = 1 / (v • px)
– That boiled down to something pretty
simple…which means that exam constructors can
make a problem look really difficult and have it be
solved fairly quickly!