Online Appendix for On the Asset Allocation of a Default Pension Fund

Transcription

1 Online Appendix for On the Asset Allocation of a Default Pension Fund Magnus Dahlquist Ofer Setty Roine Vestman January 6, 26 Dahlquist: Stockholm School of Economics and CEPR; magnus.dahlquist@hhs.se. Setty: Tel Aviv University; ofer.setty@gmail.com. Vestman: Stockholm University; roine.vestman@ne.su.se.

2 Contents Details on LINDA 2 2 Details on the notional account 3 3 Calibration of the equity share to a mix of cohorts 4 4 Life-cycle paths for low- and high-cost types 5 5 A 2% equity premium 5 List of Figures Different cohorts for the calibration Different cost types Aggregate equity risk and inequality when µ = 2% Pension inequality when when µ = 2%

3 Details on LINDA LINDA (Longitudinal Individual Data) is a widely used dataset in economic research. It is a joint endeavor between the Department of Economics at Uppsala University, The National Social Insurance Board (RFV), Statistics Sweden, and the Ministries of Finance and Labor. Edin and Fredriksson (2) provide a detailed account of the data collection process for LINDA. More information on LINDA is also available from the web sites of the Department of Economics, Uppsala University ( and Statistics Sweden ( LINDA is a panel dataset that covers slightly more than three percent of the Swedish population annually. There are approximately 3, individuals in the dataset. The starting point for LINDA is a representative, random sample of the Swedish population in 994 which has been tracked back to 968 and forward to 27. New individuals are added to the database each year to ensure that LINDA remains representative of the crosssection of Swedish individuals. In addition, the dataset contains information on all family members of the sampled individual. Thus, LINDA covers all members of approximately 3, households in each year. The core of LINDA are the income registers (Inkomst- och Förmögenhetsstatistiken) and population census data (Folk- och Bostadsräkningen). Each wave of LINDA contains information on taxable income and social transfers (e.g., unemployment benefits) from the Income Registers in a given year. In addition, LINDA contains information on occupation, wages, and educational attainment from separate registers held at Statistics Sweden. We also use the wealth supplement of LINDA, which is available between 999 and 27. The wealth supplement contains information on the market value of houses, owned apartments (co-ops), cabins, plots of land, and other forms of real estate. It also reports the value of total debt and the value of student loans. When Statistics Sweden compiles LINDA, it lacks the information to assign two people 2

4 that belong to the same household but that are unmarried and without children. Such individuals are treated as two separate households. This leads to under-sampling of this particular kind of household. 2 Details on the notional account As mentioned in the paper, carrying the balance of the notional account as a part of the individual s state is computationally demanding. To avoid this, we proceed as follows. First, we simulate many life-cycle income paths based on our calibration of the labor income process. For each path we calculate the true balance on the notional account according to the exact contribution rate of 6%, subject to the ceiling. Then we regress the account balance on a polynomial of the persistent component of labor income at the time of retirement, i.e. z 65 : n A N i65 = β + β j z j i65 + ε i65 j= We choose n = 5 as higher-order polynomials better capture the ceiling of contributions at SEK 5,8 (i.e., 6% of SEK 344,25). The R-squared in this regression is 8%. We use the estimated coefficients from this regression to approximate the account balance. This approximation of A N i65 is applied unless the approximation falls below an implied floor. Since there is a floor of SEK 85,829 in the annuity payment out of the notional account, and since the actuarially fair annuity factor is 5.6%, the implied floor on A N i65 equals SEK,532,66 (SEK 85,829 divided by.56). For those individuals for which the approximation falls below this floor we add a one-time transfer (from the government) to the individuals notional account. Hence, a jump in the average notional account balance is observed at retirement (see Figure 3 in the paper). 3

5 3 Calibration of the equity share to a mix of cohorts To be able to compute the DC equity share profile of the calibration we need to make a few assumptions on how all birth cohorts in the work force in 27 (the year of our financial wealth data) were affected by the reform of the pension system. Recall that the new pension system was implemented in 2 but contributions had accrued since 997 and where then allocated in 2. Year 997, rather than 2, is therefore a key year in this particular context. Individuals born in 973 is a knife-edge cohort because they were 24 years old in 997, i.e. not in the labor force, when contributions in the premium pension system started to accumulate. All cohorts born after 973 have the same equity share while individuals born before 972 has foregone some contributions into equity. For simplicity, we make the following assumptions about these cohorts. First, we assume that prior to the reform, the same contribution rate as in the premium pension system (2.5%) were invested in a risk-free bond. Second, we assume that funds invested prior to the reform remain invested in the risk-free bond also after the reform. Third, we assume that the occupational pension plans are the same for all cohorts with an allocation into equity equal to 6% and 4% into risk-free bonds. Fourth, we assume that realized equity returns have been equal to 4%. Based on these assumptions, we can compute cohort-specific life-cycle paths for the DC equity share. We then compute averages of these cohorts profiles to obtain a single profile for our calibration. At every given age, we include the piece of the cohort profile that corresponds to year 27 and later. To be concrete, the profile of birth cohort 973 is included for age 34, since this cohort was 34 years in 27, up to age 65. In contrast, birth cohort 942 is only included at age 65 since this cohort was 65 years old in 27. At the other extreme, birth cohort 982 is included in its entirety from age 25 to 65 since this cohort was 25 years in 27. It so happens that cohort 982 has the same profile as cohort 973 but to match the 4

6 weights in our data set it needs to given weight in the calculation. Consequently, the equity share of young cohorts are given a greater weight at early stages of the life-cycle while older cohorts only are given some weight at later stages of the life-cycle. Appendix Figure reproduces this the calibrated path for the DC equity share (see also Figure in the paper) and, for illustrative purposes, the path for some specific birth cohorts. 4 Life-cycle paths for low- and high-cost types We allow for cross-sectional variation in the cost, see equation (7), to obtain a better fit of stock market participation and its profile during the working phase. Appendix Figure 2 reproduces Figure of the paper for three different cost types: SEK 5, SEK,, and 25,. On average, low-cost investors will enter early in life whereas high-cost investors will enter later. With a sufficiently low value of κ some low-cost investors will enter immediately. At the end of life, more high-cost than low-cost investors will remain non-participants. 5 A 2% equity premium To examine the robustness of our model findings we re-calibrate it to an equity premium equal to 2% (µ =.2). In the new calibration, the subjective discount factor is lower (β =.947), the risk aversion is lower (γ = 7), and the participation cost is lower on average with a support from to SEK 75. The fit to the data is just as good as when we set µ =.4. Appendix Figure 3 reproduces Figure 4 of the paper when µ =.2. The panels to the left display how outcomes differ for different equity return realizations. As when µ =.4 the main source of variation is the DC account (the second panel to the left). With ten percent probability the peak of the account balance exceeds SEK 9, and with ten 5

7 percent probability its peak is less than SEK 45,. This difference in turn transcends into differences in the DC equity share which at age 64 is less than 2% with ten percent probability and greater than 5% with ten percent probability. The panels the right show that there is also a great deal of inequality in stock market participation and financial wealth if µ =.2. Hence, there is a great deal of cross-sectional variation in optimal DC equity shares. Appendix Figure 4 reproduces Figure 5 of the paper. It shows that in this setting, too, the positive changes in pension are concentrated among those with low pensions. Specifically, 57% of the individuals receive a higher pension under the optimal asset allocation than under -minus-age rule. For µ =.4 this statistic was 75%. Further, both positive and negative changes are smaller. The average increase is just.3% compared to 3.% when µ =.4. Finally, there is a reduction in inequality for µ =.2 as well. Under -minus-age rule the log of the standard deviation of pensions is.75. It is reduced to.59 under the optimal asset allocation. This is a reduction by 9.%. For µ =.4 the reduction is 2%. To sum up, our results remain the same in a qualitative sense even if the magnitudes are smaller. The corresponding statistics in the paper are SEK,2, and SEK 6,. 6

8 References Edin, Per-Anders, and Peter Fredriksson, 2, LINDA Longitudinal INdividual DAta for Sweden Sweden, Working Paper, Uppsala University. 7

9 Appendix Figure : Different cohorts for the calibration Calibration The figure shows the calibrated path for the DC equity share together with paths for some specific cohorts. 8

10 Appendix Figure 2: Different cost types 5 Financial wealth Participation Equity share (conditional) DC equity share Cost = SEK 5 Cost = SEK 5, Cost = SEK 25, The figure shows the financial wealth, participation, equity share conditional on participation, and the DC equity share for three different cost types. The model simulation is based on 5 economies and each cost type has 5 individuals. Financial wealth is expressed in SEK s. 9

11 Appendix Figure 3: Aggregate equity risk and inequality when µ = 2% DC equity share (equity risk) DC equity share (inequality) Mean.2 9th decile 2nd decile DC account (equity risk) Mean.2 9th decile 2nd decile DC account (inequality) Labor income and pension (equity risk) 5 Labor income and pension (inequality) Financial wealth (equity risk) Financial wealth (inequality) Participation (equity risk) Participation (inequality) The figure shows the aggregate equity risk and inequality for the optimal design. The simulation is based on 5 economies and 35 individuals. The left panels show how the averages vary over 5 economies. The second decile refers to the average of economies 6 (sorted). The ninth decile refers to the average of economies 4 45 (sorted). The right panels show how the averages vary over 35 individuals. The second decile is based on the average of individuals The ninth decile is based on the average of individuals Note that the same economies and individuals are not tracked over time, i.e., the sorting at one age is independent of the sorting at another age. The DC account, labor and pension income, and financial wealth are expressed in SEK s.

12 Appendix Figure 4: Pension inequality when when µ = 2%.25 Pension distribution Pension change The figure on the left shows the distribution of pension income (in SEK s) at age 65 in cross-section under the -minus-age rule. For each bar in the distribution, the figure to the right shows the pension changes (in %) associated with a shift from this asset allocation rule to the optimal allocation.