Notional Defined Contribution
Pension Systems in a
Stochastic Context: Design and
Stability
Alan Auerbach
Ronald Lee
NBER Center for Retirement Research
October 19-22, 2006
The Woodstock Inn
Erin Metcalf and Anne Moore provided excellent research
assistance.
Problems with PAYGO public
pensions
• In many countries, the ratio of elderly to workers will
double or more by 2050.
• Consequently, unfunded (PAYGO) pension plans are not
sustainable without major increases in taxes and/or
reductions in benefits
• Other problems
–
–
–
–
–
Below market implicit rate of return
May reduce saving rates and aggregate capital formation
Distort labor supply incentives, e.g cause early retirement
Inevitable uncertainty about rates of return, as in any system
Political risk; no individual control
Privatization?
• Some propose privatizing, replacing
PAYGO with Defined Contribution
accounts.
• These might solve those problems, but…
• Difficulties of transition to funded system
– Huge implicit debts must be repaid
– Can be 1, 2, 3 or 4 times GDP
– Transitional generations suffer heavy burden
Another idea – unfunded individual
accounts
• Notional Defined Contribution plans, or
Non-Financial Defined Contribution (NDC)
• Mimic regular Defined Contribution plans,
but only minimal assets.
• A different flavor of PAYGO.
NDC might solve some problems
• Might be fiscally stable, depending on details.
• Individual accounts (but not bequeathable)
• Actuarially fair at the NDC rate of return, so less distortion
of labor incentives (contributions not viewed as taxes?)
– Deals fairly with tradeoffs between benefit levels and age at
retirement
• Solvency in the face of longevity shifts; automatically
indexed to life expectancy through annuity.
• No transition cost, because implicit debt is rolled over
• Transparency through explicit rules vs political risk
NDC doesn’t solve other problems
• Pays below market rate of return
• For a generation, IRR should equal growth
rate of wage level (g) plus growth rate of
labor force (n), that is growth rate of
covered wage bill (real) (n+g)
• Still probably displaces savings and capital
Sweden instituted their NDC
system in mid 1990s
• Italy and Latvia have also adopted NDC.
• French and German systems have
elements of NDC
Strategy of this study
• Construct stochastic economic and
demographic environment by modifying
existing stochastic forecasting model for
US Social Security (Lee-Tuljapurkar)
• Study the performance of a pension
system in this stochastic environment.
• Relatively new approach, although see
Juha Alho (2006).
Questions addressed
• Can NDC deliver a reasonably consistent
and equitable implicit rate of return (IRR)
across generations?
• Can NDC achieve fiscal stability through
appropriate choice of IRR and annuitized
benefits sensitive to life expectancy?
• How should system be structured to
perform well on these measures?
Preview of results
• Swedish style system does not
automatically stay on the tracks fiscally;
about 30% of the time it collapses.
• We suggest some modifications
• The implicit rates of return are quite
variable from generation to generation.
Plan of rest of talk
• More detail on the Swedish NDC system
• Background on constructing stochastic
simulations
• Results
• Conclusions
Closer look at Swedish NDC
system, tested in this study
• Two phases: pre-retirement and retirement
• Pre-retirement:
– each year’s payroll taxes are added to stock of
“notional pension wealth” (NPW);
– NPW is compounded annually using growth rate of
average wage, g (they do not use n+g)
– Rate of return earned by surviving individual is higher
than cohort rate of return, because survivors inherit
account of those in cohort who die.
– ri is individual rate of return; r is cohort rate of return
NPWt 1  NPWt (1  rt )  Tt
i
Retirement and beyond
• Worker decides when to retire, above minimum.
• Receives a level real annuity based on trend
wage growth rate, g = .016
• Subsequently the rate of return is adjusted up or
down if actual growth rate of wage is faster or
slower then .016.
• Annuity is based on level of mortality at time the
generation reaches a specified age, such as 65.
• This achieves automatic indexing of benefit levels
to life expectancy – if annuity benefit is adjusted
for post retirement changes in mortality, too.
Initial Comments
• Might wish to use growth rate of wage bill,
rather than wage rate, in computing rate of
return on NPW and for annuity (n+g vs. g)
– n+g is the steady state implicit rate of return,
not g
– Even if average growth rate of labor force is
zero, there are fluctuations
• Most demog variation comes from fertility,
not mortality, and using g ignores this.
Fiscal stability?
• No guarantee that NDC plan as used in
Sweden will be stable, in terms of
evolution of debt-payroll ratio.
• If fertility goes below replacement level
and pop growth becomes negative, then
rate of return g will not be sustainable.
• This is recognized in Sweden, so an
additional “brake” mechanism is included
• Brake will control for demography through
the back door by reducing rate of return.
Fiscal status is assessed without
using projections
• Rules are specified in terms of observable
quantities, so no projections involved.
• This further insulates the system from
political pressures.
• Steady state approximations replace
projections for present value of future tax
receipts, C, in the balance equation.
• Downside: calculations of fiscal health
may be less accurate.
How the Brake Works
• Start with the balance ratio:
F C
b
NPW  P
All can be estimated
from base period data,
no projection.
where: F = financial assets
C = a “contribution” asset
P = an approximation of pension
commitments to current retirees
NPW = notional pension wealth
How the Swedish Brake Works
• If bt < 1, then multiply the
rate of return the basic
formula calls for by bt.
• If bt+1 is still <1.0 then
• So when b<1, the rate of
return is adjusted only
when b is falling or
rising.
• If b gets close to zero,
then the ratio can go wild.
1  rt  1  gt  bt
a
1  rt a1  1  gt 1  bt 1 bt
The Brake is asymmetric
• When the ratio of assets to obligations
is falling, the rate of return is reduced.
• Applies when b < 1, but not when b > 1
• Helps avoid deficits, but allows
surpluses to accumulate without limit
Potential Problems with the Brake
• Is the brake strong enough to head off
fiscal disaster?
• Asymmetry means potential for unneeded
asset accumulation, depressing rate of
return.
We design our own brake
1  rt  (1  gt )[1  A(bt  1)]
a
where A is a scaling factor, which provides
another degree of freedom
a
• A=1 gives 1  rt  (1  gt )bt
• The brake can be applied symmetrically
(that is, also for b > 1)
Now briefly consider the
stochastic simulation model
• Starting point is stochastic forecasting
model for Social Security finances,
developed by Lee and Tuljapurkar.
• This model is rooted in historical context
– Baby boom, baby bust
• We construct a model purged of historical
context and of trends:
– quasi-stationary
– Rationale: aim for more general results
The Basic Model
• Stochastic population projections (Lee and
Tuljapurkar) based on mortality and fertility
models of Lee and co-authors; immigration held
constant
– Eliminate drift term in mortality process to generate
quasi-stationary equilibrium
• Below replacement fertility plus constant
immigration inflow implies stationary equilibrium.
• Stochastic interest rates and covered wage
growth rates as well, modeled as stationary
stochastic processes using VAR
• No economic feedbacks; based on stochastic
trend extrapolations.
Stochastic simulations
• Set initial population age distribution based on
expected values of fertility, mortality,
immigration.
• Generate stochastic sample paths for fertility,
mortality (derive population age distribution),
productivity growth, and interest rates.
• Use these to generate stochastic outcomes of
interest.
• Each sample path is 600 years long.
• Throw out first 100 years to permit convergence
to stochastic steady state.
Figure 1. Ratio of Retirees to Workers, 15 Sample Paths
1.2
Ratio of old (>66) to young (21-66)
1
Spain 2050,
UN Low Proj
0.8
0.6
0.4
0.2
Current US
0
0
50
100
150
200
250
Year
300
350
400
450
500
Incorporating an NDC System centered
on US Soc Sec parameters
• As under Soc Sec (OASI), assume 10.6 percent
payroll tax rate, applied to fraction of total wages
below payroll tax earnings cap.
• Long-run covered wage growth of 1.1 percent;
base system rate of return on realized wage
growth (g) or wage bill growth (n+g)
• Accumulate NPW until age 67; then annuitize; in
simulations shown, update annuities after 67 to
reflect changes in rate of return and mortality
(not a big deal)
Simulation Results
• We consider versions of the NDC system that
vary by
– the rate of return used (g vs. n+g)
– the type of brake (none/asymmetric/symmetric;
Swedish or revised)
• To evaluate stability, look at distribution of ratio
of financial assets to payroll
• Look at summary measures of distributions of
implicit rates of return across paths and cohorts
Table 1. Average Internal Rates of Return
Simulation
Mean IRR
NDC (g) No Brake
Asymm brake
reduces rate of
return
Median IRR
.0107
.0107
NDC (g) Asymmetric Brake (Swedish)
.0098
.0103
NDC (g) Asymmetric Brake (modified)
.0093
.0105
NDC (g) Symmetric Brake (modified)
.0106
.0130
NDC (n+g) No Brake
.0110
.0113
NDC (n+g) Asymmetric Brake (modified)
.0109
.0112
NDC (n+g) Symmetric Brake (modified)
.0133
.0134
Source: Calculated from stochastic simulations described in text.
Table 1. Average Internal Rates of Return
Simulation
Mean IRR
Making brake
symmetric
raises the rate
NDC (g) Asymmetric Brake (standard)
of return
NDC (g) No Brake
Median IRR
.0107
.0107
.0098
.0103
NDC (g) Asymmetric Brake (modified)
.0093
.0105
NDC (g) Symmetric Brake (modified)
.0106
.0130
NDC (n+g) No Brake
.0110
.0113
NDC (n+g) Asymmetric Brake (modified)
.0109
.0112
NDC (n+g) Symmetric Brake (modified)
.0133
.0134
Source: Calculated from stochastic simulations described in text.
Now consider fiscal stability
• Following slides look at ratio of financial
asset, F, to payroll.
• For less stable systems, we look at first
100 years only, since the probability
distributions explode.
Figure 2. Financial
Assets/system
Payroll with no
Financial Assets/Payroll
in an NDC(g)
(r=g , purposes
no brake)
brake – solely for reference
30
0.025
0.167
0.5
0.833
0.975
mean
20
Financial Assets/Payroll
10
0
-10
-20
Range at 100 years = 56
-30
-40
0
10
20
30
40
50
Year
60
70
80
90
100
Figure 3. Financial
Financial Assets/Payroll
in theAssets/Payroll
Swedish System:
(r=g , asymmetric
brake, standard)
NDC(g) with asymmetric
brake
25
0.025
0.167
0.5
0.833
0.975
mean
20
Financial Assets/Payroll
15
10
5
0
-5
Range = 28
-10
-15
0
10
20
30
40
50
Year
60
70
80
90
100
Figure 4. Financial Assets/Payroll
(r=g , asymmetric brake, modified: A= .5)
25
0.025
0.167
0.5
0.833
0.975
mean
Financial Assets/Payroll
20
15
Range = 18; negative
ratios are eliminated.
10
5
0
-5
0
10
20
30
40
50
Year
60
70
80
90
100
Figure 6. Financial Assets/Payroll
(r=g , symmetric brake, modified: A= .5)
2.5
0.025
0.167
0.5
0.833
0.975
mean
2
Financial Assets/Payroll
1.5
1
0.5
0
-0.5
-1
Range = 2.4 after 500 years;
highly stable
-1.5
0
50
100
150
200
250
Year
300
350
400
450
500
Now look at systems with rate of
return = n + g
• These should be more stable, because the
rate of return they pay reflects
demography as well as wage growth.
Figure 7. Financial Assets/Payroll
(r=n+g , no brake)
10
0.025
0.167
0.5
0.833
0.975
mean
Financial Assets/Payroll
8
6
Range = 9.3 after 100 years,
does well even without brake.
4
2
0
-2
0
10
20
30
40
50
Year
60
70
80
90
100
Figure 8. Financial Assets/Payroll
(r=n+g , asymmetric brake, modified: A= .5)
10
0.025
0.167
0.5
0.833
0.975
mean
Financial Assets/Payroll
8
6
Range = 8.8 after 100 years;
asymmetric brake helps slightly.
4
2
0
-2
0
10
20
30
40
50
Year
60
70
80
90
100
Figure 9. Financial Assets/Payroll
(r=n+g , symmetric brake, modified: A= .5)
3
0.025
0.167
0.5
0.833
0.975
mean
2.5
Financial Assets/Payroll
2
1.5
Range = 2.0 after 500 years;
symmetric brake makes highly stable.
1
0.5
0
-0.5
0
50
100
150
200
250
Year
300
350
400
450
500
Conclusions
• Swedish-style NDC system not stable, even with
brake (30% failure over 500 yrs)
• System can be made stable, using brake that is
stronger and symmetric.
• Using growth rate of wage bill (n+g) rather than
of wage rate (g) for IRR is inherently more stable
• A considerable share of instability is attributable
to economic, as opposed to demographic,
fluctuations.
Conclusions
• Next step is to evaluate a stable version of
an NDC plan against a stable version of
traditional social security (with benefit or
tax adjustments providing stability) in
terms of intergenerational risk sharing
Outcome measures for generations
• Measures
– IRR (variance)
– NPV (variance)
– Expected Utility (assuming only income for
workers is net wages, and for retirees is
pension benefits).
• Evaluates entire distribution of outcomes
Figure 5. Internal Rates of Return
(r=g , asymmetric brake, modified: A= .5)
0.025
0.025
0.167
0.5
0.833
0.975
mean
0.02
Internal Rate of Return
0.015
0.01
0.005
0
-0.005
-0.01
0
50
100
150
200
Year
250
300
350
400
Compare NDC outcomes to
balanced budget Soc Sec
• Tax-adjust system achieves continuous
balance by adjusting taxes to benefit
costs.
• Benefit-adjust system adjusts benefits to
tax revenues.
• 50-50 adjust system combines these.
Variance of IRR
• The Swedish NDC(g) system: lowest IRR
variance of any plan—but not fiscally
stable.
• The US 50-50 adjust has the lowest IRR
variance of any stable plan.
• Others have similarly high variance.
Expected Utility with risk aversion
• NDC(n+g) with symmetric brake has
highest EU.
• Swedish system next best, but not fiscally
stable.
• US Benefit adjust is the second best
fiscally stable program.
Expected utility without risk
aversion
• NDC(g) and NDC(n+g) with symmetric
brakes are the best.
Conclusion on variability of
outcomes
• Rank ordering depends on summary
measure used; none dominates on all
measures.
• The NDC(n+g) with symmetric brake looks
best on Expected Utility measures.
END
US Current
Law
NPV
Mean value of NPV
US Models
US - Tax
US Adjust
Benefits
(Scaled)
Adjust
US - 50-50
Adjust
(Scaled)
NDC(g) Models
NDC(n+g)
NDC
NDC(g)
NDC(n+g)
Sweden Symmetric Symmetric
Model
Brake
Brake
0.037
-0.050
-0.042
-0.048
-0.047
-0.042
-0.041
Mean var of NPV (across gens)
0.0077
0.0029
0.0025
0.0026
0.0027
0.0026
0.0027
IRR
Mean value of IRR
0.0328
0.0108
0.0112
0.0109
0.0098
0.0106
0.0133
0.0000099
0.0000515
0.0000543
0.0000433
0.0000325
0.0005911
0.0000530
EU without risk aversion
(across gens)
0.0562
-0.0316
-0.0316
-0.0316
-0.0383
-0.0254
-0.0265
EU with Risk Aversion
(across gens)
0.1114
-0.0281
-0.0164
-0.0241
-0.0134
-0.0151
-0.0087
1st-order AC Coefficient
NPV
0.99858
0.99927
0.99901
0.99920
0.99892
0.99886
0.99880
IRR
0.99999
0.99904
0.99921
0.99912
0.99823
0.99888
0.99895
Mean var of IRR (across gens)