Social Security Reform: Responding to the Critics

Studies | Social Security

No. 281
Wednesday, November 16, 2005
by Andrew J. Rettenmaier and Zijun Wang


Appendix: Basic Assumptions and Additional Findings

Table A-I - Descriptive Statistics of Returns on U.S. Securities

This paper uses three types of security returns to construct returns on lifetime portfolios: equity returns, six-month money market rates and long term government bond returns (all in real terms deflated by consumer price index. Table A-I provides some descriptive statistics. For the 134-year period, the geometric average return of stocks was 6.77 percent, higher than those of short term money market (2.67 percent) and long term bonds (2.58 percent). Not surprisingly, the volatility in the stock market is higher than that of the two lower return securities. Note that even the six-month money market rate is not risk-free. It has a standard deviation of 6.68 percent, which is a measure of variation around the average return. The long-term government bond’s real return is more volatile. The limitation of the empirical return data is obvious: although it spans a period of 134 years, it contains only three independent 44-year periods, the number of years the hypothetical worker is in the work force. It is a small sample from a statistical viewpoint. Therefore, subsequent to the estimates based on the historical and adjusted historical data, the five portfolios are also evaluated using simulated data.

Results Based on Bootstrap Method. In this section we simulate the stock and bond returns using estimates based on the historical data. We simulate 1,000 series of 44 years worth of bond and stock returns following the using a method known as the bootstrap method. The “bootstrap” method relies on the realized historical distributions of stock and bond returns to generate the simulation data. This method can be thought of as sampling from the sets of historical bond and stock returns. It is as if each of the 134 historical sets of bond, stock, and money market returns are represented on separate slips of paper in an urn. Slips of paper are drawn from the urn (and replaced) 1,000 times to produce the results presented in Table A-II and in Figure A-I.

Figure A-I - Simulation Results Using the Bootstrap Method

The results summarized in Table A-II indicate that the simulated data produce higher failure rates than did the actual historical data summarized in Table A-I. Note that for the four portfolios which include some stocks, the probability of a losing simulation ranges from a low of 6.7 percent for the more aggressive lifecycle portfolio to a high of just over 10 percent for the all stock portfolio. Figure A-I presents an alternative way of summarizing the simulation results for all five portfolios. For each of the 1,000 simulations the benefit offset internal rate of return, the return associated with a 100 percent government bond portfolio, is subtracted from each of the five the portfolio returns. Positive values represent PRA accumulations in excess of the benefit offset account accumulation (winning simulations) while negative values represent PRA accumulations less that the benefit offset rate (losing simulations). As depicted, and as was presented in Table A-II, less than 10 percent of the independent, non-overlapping simulations based on portfolios that include equities result in losing outcomes.These results should be interpreted slightly differently than the results presented in Table II. Recall that the results in Table II represent 91 overlapping 44 year periods, while these results represent 1,000 simulated independent periods. Also, by drawing single years or sets of stock, bond, and money market returns and arranging them randomly may not represent the true data generating process. However, it serves as a benchmark way of simulating the data by assuming that the ordering of returns is random.

Results Based on Bootstrap Method with Five-Year Block. If returns on financial instruments are mean-reverting; that is, if returns have a natural tendency to move towards the mean over extended periods, then the assumption of random ordering may result in a higher failure rate. Table A-III and Figure A-II summarize a simulation in which the order is maintained for five-year blocks of data. Five-year blocks are chosen given that the average business cycle spans approximately five years. In this simulation, beginning years of the five-year block are sampled with replacement from the first 130 years of the historical period. In this way a 44 year period is constructed after making 9 draws. As the summary statistics indicate, this procedure produces similar distributional properties for the rates of return, but the failure rates decline for all of the portfolios that include equities. The percentages of losing simulations range from 5.7 to 8.1 percent for an average reduction in failure rates of 1.8 percent­age points relative to results obtained with the random ordering assumption. Again the more aggressive lifecycle portfolio had the fewest simulations where returns fell below the benefit offset rate.

Table A-II - Simulation Results Based on Bootstrapped Data with One-Year Block

Tables A-IV and A-V summarize simulation results for which the average stock return has been reduced to 4.8 percent and the sampling block is 1 and 5 years, respectively. As with the results based on the adjusted historical data summarized in Table III, no attempt to reduce the variance in the stock returns has been made. This assumption produces an upper-bound failure rate given that the variance in equity returns would be expected to fall with a compression in the equity premium. As Table A-IV indicates, if the ordering of returns is random in the simulation, the failure rates range from 21.2 to 24.4 percent for the portfolios that include stock. The corresponding failure rates range from 17.9 to 22.1 percent when 5-year blocks of returns are sampled. 19

Figure A-II - Simulation Results using the Bootstrap Method

Table A-III - Simulation Results Based on Bootstrapped Data with Five-year Block

Table A-IV - Simulation Results Based on Bootstrapped Data with One-Year Block

Table A-V - Simulation Results Based on Bootstrapped Data with Five-Year Block


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