Changing Progressivity as a
Means of Risk Protection in
Investment-Based Social Security
Andrew Samwick
Dartmouth College and NBER
October 21, 2006
That’s Quite a Mouthful
What Does It Mean?
Some proposals to restore solvency
combine a scaled-back traditional benefit
with a personal retirement account (PRA)
invested in financial assets.
 Financial assets, particularly equities,
introduce financial risk.
 To make reform more feasible, PRAs can
be designed to minimize financial risk or
mitigate its consequences.

2
Mechanisms To Minimize Financial Risk

Don’t Invest in Equities


Follow Life Cycle Investment Strategies


Reduce exposure to equity risk as retirement
approaches, by shifting steadily into bonds.
Offer a Third-Party Guarantee


At the cost of lower expected returns and higher
required PRA contributions.
Specify a minimum rate of return (hard) or a minimum
benefit level (easier) that will be achieved by the PRA
portfolio.
Use Options To Protect Against Low Outcomes

Either a put or a put plus a written call
3
Really, All of These Are Just a Version
of “Don’t Invest in Equities.”

Life Cycle funds shift to bonds with age.


With Guarantees:


The guarantor funds the guaranteed benefits with
bonds, then lets the investor have the maximum of the
bonds or the portfolio.
With No-Loss Strategies:


The more interesting question is whether the timing
adds value, conditional on the average allocation to
equities.
The investor earmarks a portion of the contributions for
bonds to return the nominal (or real) principal.
With Pension Collars:

The portfolio is (dynamically) equivalent to specified
fractional ownership levels in the stock, a bond at the
lower limit, and a bond at the higher limit.
4
What If Portfolio Restrictions
Are Not Feasible or Desirable?




Social Security already provides a benefit floor,
and it would continue to do so to some degree in
(almost) any reformed system.
If Social Security were made more progressive,
that benefit floor would increase (in relative
terms).
This, in turn, would allow us to be less concerned
about exposure to equity in the PRAs and to allow
them to be less tightly regulated.
The paper quantifies how much equity risk we
can shed based on how progressive we make the
scaled-back traditional benefit.
5
Varying the Progressivity in the
Scaled-Back Traditional Benefit

Proportional:


Floors at the 10th or 25th percentile:



First move all benefits below the specified percentile up
to that percentile’s benefit.
Then scale all benefits down by whatever amount is
needed to achieve a 40% aggregate reduction.
Progressive:



Reduce all benefits by 40% across the board.
First reduce the AIME-to-PIA replacement rates down
from {90, 32, 15} to {67.5, 16, 8}
Then scale all benefits to achieve a 40% aggregate
reduction.
Uniform at Mean:

Set all benefits equal to 40% of the original mean
benefit.
6
Preview of Key Results

Greater progressivity can substitute for higher
equity allocations. Compared to a proportional
cut in the traditional benefits:




A commonly proposed progressive cut to the traditional
benefit allows the worker to shed half the equity risk.
A maximally progressive cut to a uniform benefit allows
the worker to shed two thirds of the equity risk.
Progressivity is more important when the investor is risk
averse or the equity premium is lower.
But progressivity does not change the desire to
invest in equities much.

We would observe similar amounts of financial risk in
PRA portfolios regardless of how traditional benefits
were cut.
7
Details of the Simulation Model
One Cohort of Workers



Start with the age-specific mean and quartiles of
covered earnings in Kunkel (1996) for the years
from 1980 – 1993.
Scale them up to 2003 levels by the growth in
the national average wage relative to the base
year.
Impute a lognormal cross-sectional distribution:



Median = exp(m)
Mean = exp(m + s2/2)
Draw a 10,000 observation sample of wages for
30-year olds based on that distribution.
8
Details of the Simulation Method
Time-Series Earnings Process
A deterministic component that mimics
the low-education income profile from HSZ
(1995).
 An AR(1) stochastic component to log
earnings with r = 0.95 and s drawn from
a uniform distribution on [0.05, 0.20]
 Backcast to 21 and forecast to 67.
 Even for a single cohort, this is a very
stylized model.

9
Details of the Simulation Method
PRA Investment Returns


For each observation, at each age, assign a
randomly drawn “year” from the Ibbotson (2006)
data of asset returns from 1926 – 2005.
Make a few additional assumptions:




Equity is 75-25 large vs small stocks
Govt bonds are equally long-, medium-, and short-term
Parameter that varies is share in bonds (assumed 50-50
corporate-government) relative to equity.
Keep the variation, but reset the means:


Follow SSA’s assumptions when it scores plans: 6.2%
equity, 3.2% corp bonds, 2.7% govt bonds (net of 30
basis points in administrative costs)
Consider alternative equity means of 5.2% and 4.2%
10
Details of the Simulation Model
Benefit Calculations

Traditional Benefit





PRA Benefit



Project the national average wage based on the average
wage growth for this cohort over its working career.
Use this series and the highest 35 years of earnings to
compute the AIME.
Use this series to update the bendpoints in the PIA-toAIME formula and compute benefits.
Modify as appropriate to increase progressivity.
Accumulate a 2- or 3-percent contribution on each year
of covered earnings.
Convert accumulations to a real annuity benefit based
on the period life table for 2002.
Combine 40% of the first with all of the second.
11
Figure 1: Changing Progressivity
Impact of PAYGO Reductions on Total Income
80
100
50% Bond Portfolios
20
40
60
These differences are
relatively unimportant.
0
These differences are
very important.
10000
20000
30000
40000
Annual Benefits
Proportional
Floor at 25th Percentile
Progressive
Uniform at Mean
12
Figure 2: Shifting from Bonds to Equity
Impact of Portfolio Choice on Total Income
40
60
80
100
Proportional Benefit Reductions
0
20
With SSA’s equity premium,
high equity allocations
aren’t the problem.
10000
20000
30000
40000
Annual Benefits
100% Bonds
50% Bonds
Life Cycle
0% Bonds
13
And with 2% off the equity premium
Impact of Portfolio Choice on Total Income
40
60
80
100
Proportional Benefit Reductions
0
20
Even these differences
are not particularly large.
10000
20000
30000
40000
Annual Benefits
100% Bonds
50% Bonds
Life Cycle
0% Bonds
14
Bond Share
in Portfolio
Table 1
Benefit Distributions under SSA Equity Return Assumptions
Method of Reducing Traditional Benefits to Restore Solvency
Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
Scheduled PIA
Reduced PIA
21785
13071
100
90
80
70
60
50
40
30
20
10
0
17914
18367
18634
19435
20064
20766
21552
22432
23417
24519
25752
Standard Deviations
3289
3150
2797
1644
Cut
by
40%
1973
1890
1678
987
0
0
4980
4910
4722
Increase
Progressivity,
Decreasing4102
Variation 3286
5278
5500
6308
7160
8349
9986
12209
15184
19101
24183
5208
5022
4407
5432
5248
4643
6244
6070
5501
Decrease Bond Share
7100
6939
6411
Increasing
Equity Share
8294
8147
7669
9938
9807
9382
Raising
Expected
Benefits
12167
12053
11682
Raising
Variation
15148
15049
14728
19071
18986
18711
24157
24085
23850
3596
3848
4763
5733
7061
8847
11216
14327
18366
23553
15
Calibrating Risk Aversion
How Much Would You Pay to Avoid a
50-50 Chance of Gaining or Losing 25%?
Coefficient of
Expected
Certainty
Relative Risk Aversion
Utility
Equivalent
1
-0.0323
0.9682
2
-1.0667
0.9375
3
-0.6044
0.9095
4
-0.4804
0.8853
5
-0.4463
0.8651
Risk
Premium
3.18%
6.25%
9.05%
11.47%
13.49%
16
Bond Share
in Portfolio
Table 1
Benefit Distributions under SSA Equity Return Assumptions
Method of Reducing Traditional Benefits to Restore Solvency
Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
100
17914
16202
90
18367
16513
80
18634
16675
70
19435
17130
60
20064
17412
50
20766
17661
40
21552
17867
30
22432
18023
20
23417
18124
10
24519
18170
0
25752
18161
Bond Share Equivalent to Proportional:
Certainty Equivalents with CRRA = 3
16302
16491
16804
16617
16814
17132
16781
16981
17302
17240
17449
17777
17525
17738
18070
17776
17993
18327
17983
18202
18538
18140
18360
18696
18240
18461
18796
18285
18504
18838
18275
18491
18823
27
42
56
How Much Equity Risk Can Be Avoided?
(56 – 10)/90 = 51%
17230
17577
17756
18252
18555
18818
19031
19188
19284
19318
19293
72
(72 – 10)/90 = 69%
17
Table 2
Benefit Distributions under SSA Equity Return Assumptions and Lower Risk Aversion
Method of Reducing Traditional Benefits to Restore Solvency
Bond Share Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
in Portfolio Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
100
17914
17289
90
18367
17687
80
18634
17911
70
19435
18563
60
20064
19031
50
20766
19512
40
21552
19995
30
22432
20470
20
23417
20925
10
24519
21348
0
25752
21728
Bond Share Equivalent to Proportional:
Certainty Equivalents with CRRA = 1
17315
17373
17500
17714
17774
17905
17938
18000
18132
18591
18655
18793
19061
19126
19267
19542
19609
19752
20025
20094
20239
20501
20571
20717
20956
21027
21174
21380
21451
21599
21760
21832
21980
1
3
7
17655
18066
18297
18967
19447
19937
20427
20908
21367
21793
22174
12
With less risk aversion, the all-equity portfolio dominates, and greater progressivity
would not enable investors to shed much equity risk.
18
Table 3
Benefit Distributions under SSA Equity Return Assumptions and Higher Risk Aversion
Method of Reducing Traditional Benefits to Restore Solvency
Bond Share Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
in Portfolio Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
100
17914
15318
90
18367
15569
80
18634
15691
70
19435
16029
60
20064
16215
50
20766
16359
40
21552
16457
30
22432
16506
20
23417
16508
10
24519
16468
0
25752
16391
Bond Share Equivalent to Proportional:
Certainty Equivalents with CRRA = 5
15513
15833
16247
15772
16103
16521
15897
16233
16654
16243
16592
17018
16433
16787
17215
16579
16937
17365
16677
17035
17464
16725
17081
17510
16725
17078
17505
16681
17029
17455
16600
16941
17364
55
72
90
16898
17199
17343
17733
17939
18091
18186
18223
18204
18136
18025
100+
With more risk aversion, the optimal equity portfolio shares fall from 90% to 80% or
70% as progressivity increases, and greater progressivity could enable investors to 19
shed all or almost all equity risk.
Table 4
Benefit Distributions under Lower Equity Returns and Risk Aversion = 3
Method of Reducing Traditional Benefits to Restore Solvency
Bond Share Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
in Portfolio Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
Certainty Equivalents with Equity Return = 5.2%
100
17914
16202
16302
16491
16804
90
18228
16418
16521
16715
17032
80
18478
16570
16675
16873
17192
70
18933
16806
16913
17115
17438
60
19329
16964
17071
17276
17600
50
19758
17087
17195
17401
17726
40
20223
17173
17281
17487
17811
30
20728
17218
17326
17531
17854
20
21274
17222
17329
17533
17853
10
21865
17187
17292
17494
17811
0
22504
17116
17219
17418
17732
Bond Share Equivalent to Proportional:
47
63
79
Knocking 100 basis points off the equity premium has analogous effects as
increasing risk aversion: slightly lower equity allocations are optimal, and all or
almost all equity risk could be shed with higher progressivity.
17230
17471
17640
17895
18061
18187
18270
18307
18298
18246
18154
100+
20
Table 5
Benefit Distributions under SSA Equity Return Assumptions and 3 Percent PRAs
Method of Reducing Traditional Benefits to Restore Solvency
Bond Share
Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
in Portfolio
Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
100
20336
17834
90
21016
18281
80
21416
18506
70
22618
19131
60
23560
19496
50
24614
19800
40
25793
20029
30
27112
20178
20
28589
20245
10
30243
20231
0
32092
20144
Bond Share Equivalent to Proportional:
Certainty Equivalents with CRRA = 3
17955
18179
18525
18407
18641
18994
18634
18872
19228
19265
19514
19877
19633
19888
20254
19939
20197
20566
20170
20430
20800
20319
20579
20950
20384
20643
21014
20369
20625
20994
20279
20532
20898
35
48
60
19049
19540
19784
20457
20845
21162
21397
21543
21600
21569
21459
73
With larger PRAs, optimal equity allocations fall from 90% to 80%, and greater
progressivity facilitates shedding about the same proportions of equity risk (half and
21
two thirds).
Life Cycle Strategies

Start with low bond allocations at young ages,
shifting over time to high bond allocations.



Compared to a uniform 50% allocation in bonds,
these strategies have lower return and lower risk.


5 to 95 in increments of 2% per year
27.5 to 72.5 increments of 1% per year.
The average PRA balance grows with age, so equity
allocations above 50 multiply smaller balances.
In general, these strategies don’t outperform
uniform portfolio allocations.

With 200 basis points off the equity premium, the
second strategy can (mildly) dominate the uniform 5050 portfolio, which was previously optimal.
22
Table 6
Benefit Distributions under Life Cycle Portfolio Allocations, Equity Returns = 4.2%
Method of Reducing Traditional Benefits to Restore Solvency
Bond Share Expected Proportional Floor at 10th Floor at 25th Progressive Uniform at
in Portfolio Benefits
Reductions
Percentile
Percentile
Reductions Mean Benefit
Standard Deviations
5 to 95
18651
6148
6087
5923
5388
4713
27.5 to 72.5
18772
6271
6211
6048
5519
4850
50
18898
6591
6532
6376
5869
5233
Certainty Equivalents with CRRA = 3
5 to 95
18651
16550
16653
16848
17165
17605
27.5 to 72.5
18772
16592
16695
16892
17209
17649
55
54
53
52
51
50
49
48
47
46
45
18789
18810
18832
18854
18876
18898
18921
18943
18966
18988
19011
16577
16579
16581
16582
16584
16585
16585
16586
16586
16585
16585
16680
16682
16683
16685
16686
16687
16688
16688
16688
16688
16687
16876
16878
16880
16881
16882
16883
16883
16884
16883
16883
16882
17191
17193
17195
17196
17197
17198
17198
17198
17198
17197
17196
17628
17630
17631
17632
17632
17632
17632
17631
17630
17629
17627
23
Conclusions


Higher progressivity, in this framework, makes
workers better off.
It also allows them to maintain expected utility
with lower exposure to equity risk.




In the baseline, half to two-thirds of this risk can be
eliminated.
More at higher risk aversion or lower equity premiums,
despite lower optimal equity allocations.
However, greater progressivity does not reduce
their desire to invest in equities much.
Life Cycle strategies are of some, but limited, use
in improving welfare.
24
Possibilities for Further Research

Simulating the portfolio returns


Should I be assigning everyone the same
sequence of “years” and bootstrapping the
results?
More sensitivity tests




More variety in (deterministic) wage profiles
Couples versus single households
Multiple cohorts
Actual versus hypothetical workers
25