PII: 1057-0810(92)90014-4 FINANCIAL SERVICES REVIEW, 2(l): 41-49 Copyri&t 0 1993 by JAI Press Inc. ISSN: 1057-0810 All rights of reproduction in any form reserved. Long-Run Returns on Stock and Bond Portfolios: Implications for Retirement Planning Kirt C. Butler Dale L. Domian This paper presents asset returns over long aping periods in a form useful for retirement planning. Time diversi~cat~n, heretofore analyzed for lump-sum investments, still serves to reduce the risk of stock investments whenfunds are accumulatedmonth by month, We consider investments in five stock and bond asset classes as well as various asset allocation strategies. Probability distributions are computed for retirement wealth over a range of investment horizons. 1. I~RORUC~ON Time diversification plays a key role in retirement planning decisions. Because of time diversification, the risk of meeting a particular wealth objective is reduced when risky assets with high expected returns are held over long periods of time. Individuals face investment accumulation periods of up to forty years early in their careers, so information about the long-run returns and risks of various asset classes is essential. Existing research on time diversification must be modified for retirement planning. Levy (1978), Reichenstein (1986), and Butler and Domian (1991) assume “lump-sum” investments at the beginning of the holding period, with no other funds added except for reinvested portfolio earnings. However, retirement investments typically have monthly or yearly contributions which increase portfolio size t~oughout the holding period. Thus, returns in early years are not as important to retirement wealth as returns in later years. Kirt C. Butler l Associate Professor of Finance, Department of Finance and Insurance, The Eli Broad Graduate School of Management, Michigan State University, East Lansing, MI 48824. Dale L. Domian l Assistant Professor of Finance, Department of Finance and Insurance, ‘The Eli Broad Graduate School of Management, Michigan State Univet&y, East Lansing, MI 48824. 42 FINANCIAL SERVICES REVIEW, 2(l) 1wm993 Most defined contribution pension plans give the asset allocation choice to the individual participant. For example, many college and university employees allocate retirement contributions between the CREF common stock fund and TIAA which has assets including bonds and mortgages. Additional choices are available through IRAs, 401(k) plans, and other tax-sheltered arrangements. Investors making peri- odic contributions to their retirement portfolios need to know the impact of their asset choice on ending wealth. We use observed capital market history and an empirical resampling procedure to estimate retirement wealth distributions over various investment horizons. These distributions are inflation-adjusted so that retirement income can be stated in terms of today’s purchasing power. The wealth distributions also reveal the likelihood of one asset class outperforming another over various holding periods. Along with stating total wealth at the beginning of retirement, we also calculate monthly payments from lifetime annuities so that investors can plan for their retirement incomes and lifestyles. The next section reviews the existing time diversification literature. Section III describes the resampling procedure for estimating wealth distributions. Section IV presents and interprets the results from the estimations. Concluding remarks are made in Section V. II. PREVIOUS RESEARCH Levy (1978) demonstrates effects of time diversification by tabulating histori- cal returns over various holding periods. He finds that common stocks outperformed Treasury bills in every 25year holding period over 1926-1977. By using one-year overlaps, Levy constructs twenty-eight 25year periods from the 52 years. However, as Levy observes, overlapping holding periods are not independent.’ If common stocks had risen by loo0 percent in January 1952, this amazing return would be included in all but three of Levy’s 25-year periods. Distributions based on overlap- ping periods thus misrepresent the true 25-year return distributions. A way out of this problem is to use the observed history of stock and bond returns in a mote creative way. One approach is to assume stock and bond returns are normally distributed and then derive the time diversification impact of longer holding periods. Reichenstein (1986) uses this method to show the manner in which portfolio risk depends on the holding period. A simulation model can also prove helpful in time diversification research. This is an especially attractive alternative for researchers convinced of fundamental differences between past and future real return distributions. For example, Leibowitz and Langetieg (1989) construct models assuming a 4 percent stock risk premium over bonds rather than the 7 percent common stock/Treasury bond premium ob- served since 1926. A lower stock/bond risk premium decreases the relative advan- tage of stocks over bonds for both short and long holding periods. The disadvantage Long-Run Returns on Stock and Bond Portfdios 43 of this technique is that simulation models are often ad hoc and reflect the creator’s preconceptions regarding the relevant forces driving asset returns. We follow the empirical resampling procedure of Butler and Domian (199 1). In contrast to the simulation model of Leibowitz and Langetieg, this approach uses only the observed history of stock and bond returns. It requires two conditions for the empirical distribution: 1) random draws must he independent over time, and 2) the moments of the distribution must not change over tin~.~*~ A full description of this procedure is presented in the next section. III.THERESAMPLINGPROCEDURE The standard data source in the time diversification literature is the Ibbotson monthly return indices which date back to 1926 (see Ibbotson 1991). We use Ibbotson’s real (inflation- adjusted) returns for five indices: common stocks, small stocks, corporate bonds, Treasury bonds, and Treasury bills. Our study covers 65 years from 1926 through 1990, a time span of 780 months. Table 1 presents summary statistics of the time series. An empirical resampling procedure is used to construct the distribution of retirement plan returns. We first explain the technique with an illustrative example of a 30-year accumulation period, and then go on to show how this can be modified for other investment horizons. Suppose an individual intends to make monthly retirement contributions for 30 years. At the end of this accumulation period, the total accumulated savings amount is used to purchase a single-premium lifetime annuity with monthly payments. For each dollar of monthly investment during the accumulation period, we want to identify the accumulated wealth and the size of the monthly annuity payment. Based on observed capital market history, probability distributions are deter- mined as follows: TABLE1 The Historkal Return Series common Small Corpomte Treasury Treasury statistic StoCkS Stocks BOlldS Bonds Bills MtXll .0072 .0105 .0019 DO14 .0004 St. Dev. .0589 xl902 .0211 .023 1 .0057 Autocorrelation .lOll .1557 .I992 .0891 .5368 Cross Correlations Small Stocks x494 Carp Bonds .2383 .1895 T-Bonds .1932 .1143 .8481 T-Bills .0804 .O262 .3289 .3323 Note: Real monthly time series for 19264990 from Ibbotson Associates. 44 FINANCLQL SERVICES REVIEW, 2(l) 1992/1993 1. Randomly select one of the 780 months. Record the observed real return for this month for each asset class. 2. Do this random selection 360 times with replacement to construct 30 years of monthly returns. 3. Compound and accumulate the returns assuming additional real monthly investments of the same dollar amount. 4. Do this procedure 10,000 times to generate wealth distributions from the observed history of real monthly returns. 5. Divide the ending totals by an annuity factor to get the monthly payments th~ughout retirement. This procedure can easily be modified. First, the accumulation period in step 2 can be shortened or lengthened. The next section shows results for 10,20,30, and 40-year accumulation periods. Second, various asset allocation strategies can be considered, using different mixes of the asset classes. Examples are given in the next section. Third, step 3 can be modified to allow changes in the monthly ~ves~ent amount, perhaps as real earnings increase in later years of life. Finally, the annuity factor in step 5 can be changed to reflect different rates of return. IV. RESULTS Distributions of ending wealth (i.e., at the end of the accumulation period) are shown in Table 2. Selected percentiles are given for 10, 20, 30, and 40-year accumulation periods. For each of five asset classes, the values are ending wealth per dollar of real monthly contribution. Consider the illustrative 30-year accumulation period discussed in the previous section. At the median (fiftieth percentile), a one-dollar monthly investment in common stocks totals $1,194 after 30 years. Small stocks produce more, $1,680, while corporate bonds, Treasury bonds, and Treasury bills provide substantially less. Because these are medians, there are equal chances of getting either more or less than these amounts, based on observed history. The distributions for common stocks and small stocks are skewed to the right so that their means are substantially higher than their medians.4 This probabilistic interpretation can be carried a step further to make relative comparisons between stocks and bonds. For the 30-year accumulation period, the common stock distribution crosses the corporate bond distribution between the 3rd and 4th percentiles. This implies a less than four percent chance of ending up with lower wealth from diversified common stock investments than from corporate bonds over a 30-year accumulation period. The chance is even smaller for common stocks versus Treasury bonds, since these distributions cross at the second percentile. With the 20-year accumulation period, there is still just a small risk that common stocks will underperform bonds. It is only when the accumulation period is reduced to 10 years that the risk is more substantial. 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Period Qile Bonds: - 25% 50% 75% IO@% 10 Yrs. 20 Yrs. 30 Yrs. 40 Yrs. 5 92 loo 108 113 116 25 133 134 133 130 120 50 173 165 155 144 123 75 228 204 180 161 126 95 345 277 227 189 130 5 202 25 345 50 515 75 782 95 1,458 5 363 25 715 50 1,194 75 2,055 95 4,697 5 582 25 1,346 50 2512 75 4,810 95 13,253 227 242 344 327 462 405 637 511 1,042 729 243 299 z 523 232 244 253 262 274 3% 413 397 670 610 526 989 808 638 1,491 1,080 785 2,820 1,688 1,061 350 373 E 432 665 671 615 470 1230 1,053 845 508 1,980 1,488 1,058 533 3,237 2,094 1,350 560 6,901 3.588 1,923 602 Nore: Ending wealth per dollar of real monthly contribution over n-year (n = IO, 20, 30, 40) accumulation periods. Contributions are allocated between common stock and corporate bonds. Asset allocations between other security classes have less interesting results. Common stocks and small stocks have a .85 correlation, so there is little risk reduction benefit through diversification. Because of this relatively high correlation, wealth at each percentile of the logon stock/small stock dist~bution is approxi- mately equal to the weighted average of wealth at the same percentiles of the single-asset distributions. A similar result is obtained for mixtures of stock and Treasury bills, despite their near-zero correlation. Real monthly T-bill returns have stayed near zero for most of the 1926-1990 period. The small standard deviation in the Treasury bill time series results in a near-linear increase in both expected return and standard deviation as greater proportions of wealth are placed in risky stock. Many brokerage services and financial planners advocate investment in com- mon stock early in one’s career with a gradual shift to bonds as retirement ap- proaches, The ~tirement-we~th effects of several versions of this strategy are illustrated in Table 4 for a 30-year accumulation period. Long-Run Returns on Stock and Rod Porifdbs 47 TABLE 4 Chmging the Stock-Bond Mix Asset Years of contributions to each asset Stock: l-10 I-15 I-20 I-25 I-30 %ile Bo&: 11-30 16-30 21-30 24-30 - 5 393 391 383 373 363 25 564 60s 647 678 715 50 765 867 983 1.087 1,194 75 1,083 1,304 1345 1,787 2,055 95 1,970 2$36 3,185 3,912 4,697 Notes: Ending wealth per dollar of real monthly contribution over 30-year accumulation periods. The asset allocation is initially 100% common stock. In later years, new contributions are invested in corporate bonds and stock accumulations are gradually shifted to bonds. The portion transferred at the end of year n is I/(30 -n + 1). Consider the first column of the wealth dist~b~tions in Table 4. One-dollar real monthly con~butions are invested in common stock during the first 10 years. Beginning in year 1 I, new contributions are put into corporate bonds, and at the end of year 11,1/2Oth of the stock total is moved to bonds. The transfer proportion rises to l/l9 in year 12, l/18 in year 13, and so on until year 30 when all remaining stock holdings are redeemed. As before, figures in the table show percentiles of ending wealth. Other columns in Table 4 extend the stock contribution period to 15,20,25, and 30 years, with the 30-year column taken from Table 2. The wealth distributions cross around the eighth percentile. This means that there is only an eight percent chance of ending up with less wealth in an all-stock strategy than in a strategy which shifts to bonds earlier in the accumulation period. Shifting to bonds early in the accumulation period also gives up much of the upside potential of the all-stock strategy. Table 5 illustrates the effect of changing the retirement age when investment contributions begin at age 30. All contributions are invested in common stock. The figures show monthly annuity payments from a lifetime annuity following retire- ment at age 60,65, or 70. Calculations are based on mo~lity data for men7 in the 1990 Life I~~r~~ Fact Book and a monthly discount rate of .0019, the average real monthly corporate bond return over 1926-1990.8 Monthly annuity payments increase dramatically as the retirement age is increased. This is due to the combined influence of two factors: the higher ending wealth from a longer accumulation period, and a shorter remaining lifetime over which to receive annuity payments. The figures in Tables 2 through 5 assume the monthly contribution amount remains constant throughout the accumulation periods. Adding a growth rate to the con~ibutions increases ending wealth and annuity amounts. For example, consider FINANCIAL SERVICES REVIEW, 2(l) 19!w1993 TABLE 5 Lifetime Annuity Payments for Different Retirement Ages Retirement Age Percentile 60 65 70 5 1.76 2.71 3.97 25 3.47 5.58 9.17 so 5.80 9.92 17.12 15 9.99 18.07 32.78 9s 22.82 44.28 90.33 Nofex Annual payments from a lifetime annuity per dollar of real monthly contribution. Contributions begin at age 30 and continue until retirement at ages 60.65. or 70. Investments are in common stock throughout the accumulation period. the 1,194 median ending wealth for common stock contributions over 30 years (Table 2). The median increases to 1,419 with two percent annual growth in real con~ibutions, and climbs to 2,031 with 5 percent growth. The co~sponding corporate bond totals increase from 496 to 65 1 and then to 1,024. However, growth has little effect on crossing points of the distributions. For example, 30-year common stock and corporate bond distributions still cross around the 4th percentile. The relative shapes and orderings of the distributions presented in Table 2 are retained. V. CONCLUSION The return distributions and the resampling procedure of this paper can be used by both individuals and financial planners. Probability distributions help determine the size of monthly retirement contributions which are needed to meet a long-term goal. The resampling procedure can adapt results to a particular investor’s invest- ment needs. Since all results are in real (inffation-adjusted) terms, retirement incomes are conveniently stated in terms of today’s purchasing power. The most striking result is the large advantage held by common stock invest- ments over bonds for longer accumulation periods. As the accumulation period lengthens, stock investments become increasingly less “risky” than bonds. A given monthly investment in stocks can offer a reasonable chance of meeting a long-term investment goal, while the same amount invested in bonds generally has a smaller chance of meeting that goal. Based on capital market history, we find that over a 30-year accumulation period there is less than a four percent chance that colon stocks will underperform corporate bonds. Despite the short-term risks, the stock market is the best bet for most long-term retirement investments. Acknowledgments: We thank two anonymous referees and participants at the 1990 Academy of Financial Services annual meetings for helpful comments. Lang-Run Returns on Stock and Bond Potifolios 49 5. 6. 7. 8. NOTES This point is emphasized by McEnally (1985). The true moments of the distribution are unknown. The sample mean and variance over 1926-l 990 am only estimates of these parameters. Butler and Domian (1991) show that small positive autocormlations in the historical stock and bond series have little impact on return distributions produced by the resampling methodology. The means of the common stock end-of-period wealth distributions are 190 (10 years), 636 (20 years), 1,705 (30 years), and 4,205 (40 years). Small stock distributions have means of 239 (10 years), 1,086 (20 years), 4,020 (30 years), and 14,082 (40 years). The cross-correlation between common stocks and Treasury bonds is 0.19. resulting in distri- butions which are very similar to those in Table 3. Samuelson (1990) considers a lump-sum portfolio without additional monthly contributions. The amounts held in each asset class must be rebalanced to maintain the SO-50 mix. Lower mortality for women produces lower monthly annuity amounts. Corporate bonds comprise over 40 percent of the assets of U. S. life insurance companies (see the 1990 Life Insurance Fact Book). REFERENCES Butler, Kirt C. and Dale L. Domian. 1991. “Risk, Diversification, and the Investment Horizon,” Tk Journal of Portfolio Management, 17(3): 41-117. Ibbotson, Roger G. 1991. Stocks, Bonds, Bills and Injlation: 1991 Yearbook. R. G. Ibbotson Associates. Leibowitz, Martin L. and Terence C. Langetieg. 1989. “Shortfall Risk and the Asset Allocation Decision: A Simulation Analysis of Stock and Bond Risk Profiles,” The Journal of Porrfolio Management, 16(l): 61-68. Levy, Robert A. 1978. “Stocks, Bonds, Bills, and Inflation over 52 Years,” The Journal of Portfolio Management, 4(4): 18-19. McEnally, Richard W. 1985. “Time Diversification: Surest Route to Lower Risk?,” The Journal of Portfolio Management, 1 l(4): 24-26. 19W Life Insurance Fact Book. Washington, D.C.: American Council of Life Insurance. Reichenstein, William. 1986. “When Stock is Less Risky than Treasury Bills,” Financial Analysts Journal, 42(6): 71-75. Samuelson, Paul A. 1990. “Asset Allocation Could Be Dangerous to Your Health,” The Journal of Portfolio Management, 16(3): 5-8.