1. Pension System Analysis:
Basic Concepts and Identities
Tatyana Bogomolova
World Bank, HDNSP

2. 2
Outline
 General framework for quantitative analysis of
pension systems
 Key factors: demographic, economic, pension
system design
 Simple economics of PAYG DB schemes
 Simple economics of Funded DC schemes

3. 3
Demography
Contributors Beneficiaries
PS revenues PS expenditures
Economy
PS balance
Pension system: main flows
PS design
Accumulated assets/debt

4. 4
Main groups of factors
 Demographic environment
 Economic environment
 Pension system design

5. 5
Demographic factors
 Population, age/gender composition  working
age and old age population , old age dependency
ratio
 Fertility (total fertility rate, replacement level)
 Mortality rates  life expectancy, life expectancy
at retirement
 Disability prevalence rates
 Migration flows, age and gender composition
 World-wide trend – population aging 
decreasing fertility, increasing life expectancy

6. 6
Economic factors
 Macroeconomic indicators
GDP
Inflation
Interest rates
 Labor market indicators
Labor force participation rates
Unemployment rates
Informal sector
Wages, earning profile, income
distribution

7. 7
Pension system design
(system revenues)
 Contributor coverage
 Exemptions
 Contribution rate
 Covered wage (ceilings/floors, basic vs total
compensation)

8. 8
Pension system design
(system expenditures)
 Beneficiary coverage
 Eligibility criteria:
 retirement age, early retirement
 vesting period
 qualifying conditions for disability and
survivorship benefits
 Rules for benefit calculations
 Indexation of post-retirement benefits

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PAYG Defined Benefit systems
 Financing: workers contributions today are used to
pay pensioners today; in return, workers get a
promise that they will receive a pension tomorrow
paid for by workers tomorrow
 Benefits: calculated based on a prescribed
(defined benefit) formula; normally linked to
individual’s wages, years of contributions, accrual
rate

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PAYG DB finances
 Total expenditures: EXP = B*P
 Total revenues: REV = C*E
 Books are balanced when B*P = C*E
where:
C = average contribution
B = average benefit (pension)
P = number of pensioners
E = number of contributors

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PAYG DB finances (cont.)
 Given that:
B = RR*W; C = W*CR; and DR = P/E
 The pension fund balance equation can be presented
as: CR=RR*DR
where:
CR = contribution rate
RR = average replacement rate (relative pension level)
W = average wage
DR = system dependency rate (the inverse – support ratio)

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How to keep the system in balance?
 Adjust contribution rate (CR)
 Adjust average replacement rate (RR)
 Adjust parameters/policy variables affecting the
dependency rate (DR)
 Combination of the above
 More direct control of CR and RR; less control
over DR

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Equilibrium contribution rate
 If average replacement rate is fixed (target RR)
 Contribution rate required to finance a given
average replacement rate is:
CR = RR*DR
 So if the dependency rate grows the contribution
rate has to be increased in order to bring the
pension fund into balance

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Equilibrium replacement rate
 If contribution rate is fixed
 Another way to balance the system is through the
average replacement rate  affordable
replacement rate is:
RR = CR/DR
 So, if the dependency rate grows and the
contribution rate remains unchanged the average
replacement rate has to be reduced in order to
keep the system in balance

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Key determinants of average
replacement rate
 Policy choices about target individual replacement
rate (individual pension/individual wage)
 Benefit formula (entry pensions)
 Policy choices about pension indexation method
(post-retirement pensions)
 Economic factors: wages, wage growth rate
 Behavior: contribution density  years of
contributions at retirement

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Benefit formula: typical structure
 Accrual rate per year of service
 Min/max replacement rates, min/max pensions
 Measure of income (reference wage, pensionable
earning measure)
- ceiling on pensionable wages
- averaging period
- valorization rules
 Penalties for early retirement, increments for late
retirement

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Post-retirement pensions:
indexation methods
 Price indexation: pensions move with the price
level; their real value remains unchanged
 Wage indexation: pensions move with wages;
their relative value remains unchanged
 Combination of price and wage indexation (e.g.
Swiss formula)
 Other indexation rules (ad hoc, discretional, fixed
%, progressive indexation, etc.)

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How to affect finances through
system dependency rate?
 If contribution rate and replacement rate are fixed:
DR = CR/RR
 Dependency rate is not a policy variable, but some
policy choices can change it

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Key determinants of system
dependency rate: numerator
 Number of pensioners (P) 
 Demographic factors (old age population,
mortality rates after retirement  life
expectancy at retirement)
 Policy choices in pension system (retirement
age, rules for early retirement, beneficiary
coverage rate, vesting period, eligibility criteria
for receiving disability pensions, survivors
benefits)

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Key determinants of system
dependency rate: denominator
 Number of contributors (E) 
 Demographic factors (working age population,
 fertility in the past, mortality, migration)
 Economic factors (school-leaving age, labor
force participation, unemployment, size of the
informal sector)
 Policy choices (contributor coverage,
retirement age, rules for early retirement, built-
in incentives (e.g. contribution rate), other)

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Retirement age
 Average retirement age  Normal retirement age
and early retirement arrangements
 Quantitative analysis of various pension systems:
retirement age is the most effective policy variable
to adjust long run dependency rate
 Changes in retirement age affect both the
numerator and denominator in DR=P/E
 If life expectancy increases, retirement age has to
be adjusted to keep the system in balance in the
long run

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Policy choices: how much freedom?
 Basic relationship (CR=RR*DR)  To make a
PAYG DB financially sustainable, policy makers
can change only two of the three key parameters:
- contribution rate
- average replacement rate
- retirement age
 Once two parameters are set, the third is
determined endogenously
 Limits for setting exogenous parameters (e.g.
replacement rate – social and political,
contribution rate – economic, retirement age –
physical, social and political)

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Funded defined contribution systems
 Financing: Contributions are put into individual’s
account  Assets are accumulated and earn
interest  Accumulated capital used to pay for
pensions
 Benefits: Calculated based on accumulated capital

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Capital accumulated by the
year of retirement
AC = C1*(1+r)N + C2*(1+r)N-1 +…+ CN*(1+r)
where
AC = accumulated capital
Ct = CRt * Wt
N = number of working years
r = rate of return (here assumed to be constant)
Ct = contribution in year t, for t = 1, 2, …, N
CRt = contribution rate in year t, for t = 1, 2, …, N
Wt = worker’s wage in year t, for t = 1, 2, …, N

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Benefit payout: annuity
 When worker retires, accumulated capital (AC) is
turned into pension which is set so that:
B0+ B1/(1+d) +… + BM/(1+d)M = AC
 Initial benefit calculation: B0 =AC/AF
where
Bt = Bt -1 * indexation coefficient, t>0
M = number of retirement years
d = discount rate
AF = annuity factor
 No bequest to survivors, longevity risk borne by
annuity provider
 Variety of annuity products

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Annuity factor:
If a person of certain age and gender is promised a
benefit=$1, with specified indexation rules, how
much is such a promise worth in today’s dollars?
1+ ind1* surv1/(1+d)+(ind1* ind2 )* (surv1* surv2)/(1+d)2+…
where
indt = indexation coefficient in year t of retirement
survt = probability of surviving from year t-1 to t
d = discount rate

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Benefit payout:
programmed withdrawals
 The account continues to earn interest while
pensioner withdraws funds
 Benefit is recalculated each year:
Bt = RCt / LEt,a
where
RCt = remaining capital in year t
LEt,a = life expectancy at age a in year t
 If dies early, the remaining balance is turned over
to survivors; if lives long, Bt may become very
low; longevity risk borne by individuals
 Other payout forms (lump sums, required
minimum annuity, etc.)

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Main determinants of benefit levels
 Contribution rate
 Individual’s wages
 Rate of return, rate of return-wage growth
gap
 Passivity ratio (retirement years/working
years  years of service, retirement age, life
expectancy)
 Administrative costs
 Annuity factors (life expectancy, indexation,
single vs joint, discount rate)

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