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

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

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

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

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

9
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

10
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

11
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)

12
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

13
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

14
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

15
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

16
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

17
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.)

18
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

19
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)

20
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)

21
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

22
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)

23
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

24
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

25
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

26
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

27
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.)

28
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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