PENSION MATHEMATICS

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PENSION MATHEMATICS ACT 470H1
Week 2
Lorne F. Cohen January 15, 2007
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2.2 - UNIT CREDIT (U.C.)
Assumptions and Notation w age at
hire B(x) accrued benefit at age x, w ? x ?
y y retirement age want to accumulate
at retirement assume no new entrants (for
now) T Employees who terminate between t and
t 1 R Employees who retire between t and t
1 Active Employees at time t thus,
1 - T - R
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• Accrued Liability
• distinguishing characteristics of UC method
  ideal fund assets should equal
• the D's are calculated using probabilities of
  terminations from all sources NOT just mortality
  - called a service table
• the ideal fund balance is called the accrued
  liability (AL)
• Present Value of accrued benefits (Unit
  Credit Method)
• this definition distinguishes it from other cost
  methods
• describes completely the annual pension costs
• again, assume no new entrants (for now)

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• Normal Cost (NC)
• represents the PV of increase in accrued benefits
  between t and t1
• NC does not usually equal the full cost of the
  plan unless
• the fund balance equals the accrued liability,
  and
• assumptions are realized exactly

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Fund Balance (Ft)

• Let It and Ip be interest on contributions and
  pension purchases at the assumed interest rate i

Investment Gain
Termination Gain
Retirement Gain
the boxed terms above give the gain/loss (G/L)
due to experience different from assumptions
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Summary Pension Cost NC amortization of
UAL - amortization of gain (except first year)
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2.3 - ENTRY AGE NORMAL (EAN)

• under UC method, normal cost tends to rise faster
  than payroll
• in EAN, we define NC directly and let AL be the
  corollary item
• if benefit is unrelated to salary, then NC

the level annual contribution such that PV
(future normal costs at w) PV (future benefits
at w)

• note that NC changes proportional to since
  other terms are constant throughout an employees
  career

PV (prior NC) PV (future benefits) PV
(future NC) PVFB - PVFNC
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Recursive Formulas
the underlined terms represent the components of
the gain / loss
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Summary (EAN not based on salary) Pension Cost
NC (level annual contribution from entry age to
retirement age) amortization of UAL -
amortization of actuarial gain
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2.4 - EXTENSION OF EAN (with salary scale - SS)
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Normal Cost
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Accrued Liability
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• 2.5 - INDIVIDUAL LEVEL PREMIUM (ILP)
• when plan uses a lump sum at retirement to buy
  pensions, then there is potentially a short-run
  solvency problem
• need a cost method which guarantees solvency at
  all times
• especially if lump sums are required at
  retirement, due to liquidity strains
• conceptually, build fund quickly, then let
  surplus decrease
• one method is to use EAN, but with EA max (hire
  age, plan inception age)
• no initial unfunded
• problem if substantial portion of AL is in one
  person and salary losses occur ? will have
  "snowball" effect
• ILP addresses both problems
• short-run liquidity
• losses due to poor experience if AL weighted in a
  few members
• starts with a zero UAL
• ILP funds each person's benefit with "level
  premiums" over years of actual participation in
  the plan

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• Normal Cost
• the first year NC is same as EAN method with
  entry age x (attained age on effective date of
  plan)
• in second year

• increase in projected benefit funded by add'n
  "level premiums"
• Method defines NC and AL then falls out

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Actuarial Liability
In
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Unfunded Actuarial Liability

• underlined items form the actuarial gain
• under ILP, there is no term for loss due to
  salary and/or benefit increases greater than
  expected
• all such losses have been pushed into the NC
• again, Gain

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ILP with Salary Increases NC
change is only for part that is above normal
salary increases
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AL
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• Summary (ILP)
• Pension cost NC minus amortization of previous
  year's gains