Changing Progressivity as a Means of Risk Protection in Investment-Based Social Security Andrew Samwick Dartmouth College and NBER October 21, 2006

Changing Progressivity as a

Means of Risk Protection in

Investment-Based Social Security

Andrew SamwickDartmouth College and NBER

October 21, 2006

2

That’s Quite a MouthfulWhat 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.

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Mechanisms To Minimize Financial Risk Don’t Invest in Equities

At the cost of lower expected returns and higher required PRA contributions.

Follow Life Cycle Investment Strategies Reduce exposure to equity risk as retirement

approaches, by shifting steadily into bonds. Offer a Third-Party Guarantee

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

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Really, All of These Are Just a Version of “Don’t Invest in Equities.” Life Cycle funds shift to bonds with age.

The more interesting question is whether the timing adds value, conditional on the average allocation to equities.

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 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.

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What If Portfolio RestrictionsAre 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.

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Varying the Progressivity in theScaled-Back Traditional Benefit Proportional:

Reduce all benefits by 40% across the board. 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: 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.

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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.

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Details of the Simulation ModelOne 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() Mean = exp( + 2/2)

Draw a 10,000 observation sample of wages for 30-year olds based on that distribution.

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Details of the Simulation MethodTime-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 = 0.95 and 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.

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Details of the Simulation MethodPRA 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%

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Details of the Simulation ModelBenefit Calculations Traditional 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-to-AIME formula and compute benefits.

Modify as appropriate to increase progressivity. PRA Benefit

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.

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020

4060

8010

0P

erce

ntile

of

Dis

trib

utio

n

10000 20000 30000 40000Annual Benefits

Proportional ProgressiveFloor at 25th Percentile Uniform at Mean

50% Bond PortfoliosImpact of PAYGO Reductions on Total Income

Figure 1: Changing Progressivity

These differences are very important.

These differences are relatively unimportant.

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Figure 2: Shifting from Bonds to Equity

020

4060

8010

0P

erce

ntile

of

Dis

trib

utio

n

10000 20000 30000 40000Annual Benefits

100% Bonds Life Cycle50% Bonds 0% Bonds

Proportional Benefit ReductionsImpact of Portfolio Choice on Total Income

With SSA’s equity premium, high equity allocations

aren’t the problem.

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And with 2% off the equity premium

020

4060

8010

0P

erce

ntile

of

Dis

trib

utio

n

10000 20000 30000 40000Annual Benefits

100% Bonds Life Cycle50% Bonds 0% Bonds

Proportional Benefit ReductionsImpact of Portfolio Choice on Total Income

Even these differencesare not particularly large.

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Bond Share Expected Proportional Floor at 10th Floor at 25th Progressive Uniform atin Portfolio Benefits Reductions Percentile Percentile Reductions Mean Benefit

Scheduled PIA 21785 3289 3150 2797 1644 0Reduced PIA 13071 1973 1890 1678 987 0

100 17914 4980 4910 4722 4102 328690 18367 5278 5208 5022 4407 359680 18634 5500 5432 5248 4643 384870 19435 6308 6244 6070 5501 476360 20064 7160 7100 6939 6411 573350 20766 8349 8294 8147 7669 706140 21552 9986 9938 9807 9382 884730 22432 12209 12167 12053 11682 1121620 23417 15184 15148 15049 14728 1432710 24519 19101 19071 18986 18711 183660 25752 24183 24157 24085 23850 23553

Table 1

Standard Deviations

Method of Reducing Traditional Benefits to Restore Solvency

Benefit Distributions under SSA Equity Return Assumptions

Cut by 40%

Increase Progressivity, Decreasing Variation

Decrease Bond ShareIncreasing Equity Share

Raising Expected BenefitsRaising Variation

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Coefficient of Expected Certainty RiskRelative Risk Aversion Utility Equivalent Premium

1 -0.0323 0.9682 3.18%2 -1.0667 0.9375 6.25%3 -0.6044 0.9095 9.05%4 -0.4804 0.8853 11.47%5 -0.4463 0.8651 13.49%

Calibrating Risk AversionHow Much Would You Pay to Avoid a

50-50 Chance of Gaining or Losing 25%?

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100 17914 16202 16302 16491 16804 1723090 18367 16513 16617 16814 17132 1757780 18634 16675 16781 16981 17302 1775670 19435 17130 17240 17449 17777 1825260 20064 17412 17525 17738 18070 1855550 20766 17661 17776 17993 18327 1881840 21552 17867 17983 18202 18538 1903130 22432 18023 18140 18360 18696 1918820 23417 18124 18240 18461 18796 1928410 24519 18170 18285 18504 18838 193180 25752 18161 18275 18491 18823 19293

Bond Share Equivalent to Proportional: 27 42 56 72

Certainty Equivalents with CRRA = 3

Table 1Benefit Distributions under SSA Equity Return Assumptions


How Much Equity Risk Can Be Avoided? (56 – 10)/90 = 51% (72 – 10)/90 = 69%

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100 17914 17289 17315 17373 17500 1765590 18367 17687 17714 17774 17905 1806680 18634 17911 17938 18000 18132 1829770 19435 18563 18591 18655 18793 1896760 20064 19031 19061 19126 19267 1944750 20766 19512 19542 19609 19752 1993740 21552 19995 20025 20094 20239 2042730 22432 20470 20501 20571 20717 2090820 23417 20925 20956 21027 21174 2136710 24519 21348 21380 21451 21599 217930 25752 21728 21760 21832 21980 22174


Table 2



Benefit Distributions under SSA Equity Return Assumptions and Lower Risk Aversion

With less risk aversion, the all-equity portfolio dominates, and greater progressivity would not enable investors to shed much equity risk.

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100 17914 15318 15513 15833 16247 1689890 18367 15569 15772 16103 16521 1719980 18634 15691 15897 16233 16654 1734370 19435 16029 16243 16592 17018 1773360 20064 16215 16433 16787 17215 1793950 20766 16359 16579 16937 17365 1809140 21552 16457 16677 17035 17464 1818630 22432 16506 16725 17081 17510 1822320 23417 16508 16725 17078 17505 1820410 24519 16468 16681 17029 17455 181360 25752 16391 16600 16941 17364 18025

Bond Share Equivalent to Proportional: 55 72 90 100+


Table 3Benefit Distributions under SSA Equity Return Assumptions and Higher Risk Aversion


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 shed all or almost all equity risk.

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100 17914 16202 16302 16491 16804 1723090 18228 16418 16521 16715 17032 1747180 18478 16570 16675 16873 17192 1764070 18933 16806 16913 17115 17438 1789560 19329 16964 17071 17276 17600 1806150 19758 17087 17195 17401 17726 1818740 20223 17173 17281 17487 17811 1827030 20728 17218 17326 17531 17854 1830720 21274 17222 17329 17533 17853 1829810 21865 17187 17292 17494 17811 182460 22504 17116 17219 17418 17732 18154

Bond Share Equivalent to Proportional: 47 63 79 100+

Table 4

Certainty Equivalents with Equity Return = 5.2%


Benefit Distributions under Lower Equity Returns and Risk Aversion = 3

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.

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100 20336 17834 17955 18179 18525 1904990 21016 18281 18407 18641 18994 1954080 21416 18506 18634 18872 19228 1978470 22618 19131 19265 19514 19877 2045760 23560 19496 19633 19888 20254 2084550 24614 19800 19939 20197 20566 2116240 25793 20029 20170 20430 20800 2139730 27112 20178 20319 20579 20950 2154320 28589 20245 20384 20643 21014 2160010 30243 20231 20369 20625 20994 215690 32092 20144 20279 20532 20898 21459




Table 5Benefit Distributions under SSA Equity Return Assumptions and 3 Percent PRAs

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 two thirds).

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Life Cycle Strategies Start with low bond allocations at young ages,

shifting over time to high bond allocations. 5 to 95 in increments of 2% per year 27.5 to 72.5 increments of 1% per year.

Compared to a uniform 50% allocation in bonds, these strategies have lower return and lower risk. 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 50-50 portfolio, which was previously optimal.

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5 to 95 18651 6148 6087 5923 5388 471327.5 to 72.5 18772 6271 6211 6048 5519 4850

50 18898 6591 6532 6376 5869 5233

5 to 95 18651 16550 16653 16848 17165 1760527.5 to 72.5 18772 16592 16695 16892 17209 17649

55 18789 16577 16680 16876 17191 1762854 18810 16579 16682 16878 17193 1763053 18832 16581 16683 16880 17195 1763152 18854 16582 16685 16881 17196 1763251 18876 16584 16686 16882 17197 1763250 18898 16585 16687 16883 17198 1763249 18921 16585 16688 16883 17198 1763248 18943 16586 16688 16884 17198 1763147 18966 16586 16688 16883 17198 1763046 18988 16585 16688 16883 17197 1762945 19011 16585 16687 16882 17196 17627

Benefit Distributions under Life Cycle Portfolio Allocations, Equity Returns = 4.2%



Standard Deviations

Table 6

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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.

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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

Documents

Changing Progressivity as a Means of Risk Protection in Investment-Based Social Security Andrew Samwick Dartmouth College and NBER October 21, 2006