bonds and $1.50 for Treasury bills. (Calculated by authors based on data from Ibbotson Associates, 1996). In order to obtain higher rates of return, an investor ...
Optimal Retirement Portfolios for Investors Under 35 Peng Chen, Ph.D. student Sherman Hanna1, Professor Consumer Sciences Department The Ohio State University In saving for retirement, the allocation of asset categories in the portfolio is one of the most crucial decisions. It is likely that many investors focus too much on short term volatility, especially if retirement is many years away. This paper uses 70 years of real rates of return for six types of financial assets to find optimal portfolios for saving for retirement for young households. For periodic contributions of 20 years or more, an all-stock portfolio is superior to all other portfolios composed of the six Ibbotson categories. However, to take into account the possibility of early withdrawal, a five-year time frame (Hanna & Chen, 1995), with an expected utility approach was also used. Based on the distribution of financial assets and wealth, most households under the age of 35 should have 100% of their retirement portfolios in small stocks.
I. Overview: In saving for retirement, the allocation of asset categories in the portfolio is one of the most crucial decisions. Most people are not willing to take above average risks to obtain above average returns on their investments (Avery & E lliehausen, 1986). It is likely that many investors focus too much on short term volatility, especially if retiremen t is many ye ars away. For investors under age 35, retirement is typically at least 20 years away. They should consider their portfolio in the long run, not in the short run. This paper uses 70 years of real rates of return for six types of financial assets (see Appendix for definitions of the categories) to find optimal portfolios for saving for retirement for young American households. Risk Versus Return Over the long run, stocks provide a much higher return than corporate bonds and government bonds in both real and nominal amounts. Between the beginning of 1926 and the end of 1995, a dollar invested in sm all stocks wo uld have grown to $3822, compared to $1114 for large stocks (S&P 500), $34 for government long bonds, $36 for government intermediate bonds, $48 for corporate bonds and $13 for Tre asury bills (Ibbotson Associates, 1996). In real term s, a dollar inve sted in sm all stocks wo uld have grow n to $445, compared to $130 for large stocks (S&P 500), $4.00 for governm ent long bonds, $4.20 for
1
government intermediate bonds, $5.60 for corporate bonds and $1.50 for Treasury bills. (Calculated by authors based on data from Ibbotson Associates, 1996). In order to obtain higher rates of return, an inve stor must accept greater risk, or at least greater volatility. The short term risk or volatility does not necessarily reflect the risk for a long term holding period. Small stocks performed best of six investment categories in 48 out of 51 possible 20 year periods between 1926 and 1995, and large stocks performed best in the other three 20 year periods (Ibbotson Associates, 1996). If all future 20 year periods resem ble these 51 tim e periods, sm all stocks present the least risk to the investor. However, the standard deviations of one year returns of the Ibbotson investment categories range from 34.4% for small stocks to 3.3% for Treasury bills (Ibbotson Associates, 1996, p. 33). How should an investor balance the mean return and the volatility as represented by the standard deviations? Samuelson (1969) listed several reasons why a young businessman can take more risk in the financial market than an old widow: 1) expect higher earnings in the future; 2) can “recoup” any current losses in the future. Samuelson’s statement can be viewed as an application of the life cycle mo del offered by Ando and Modigliani (1963). Malkiel (1990) a lso suggested that investment strategy should be keyed to a life cycle. He suggested
Contact author: Sherman Hanna, Consumer and Textile Sciences Department, 1787 Neil Ave., Columbus, OH 43210-1290. 4584. FAX: 614-292-7536. Internet:hanna.1@osu.edu
614-292-
1
that since young investors can use w ages to cover losses from increased risk, they should be more aggressive than old investors. He further suggested that the portion of the portfolio for stocks should decrea se as a person ages. He suggested stock share goes from 70% for a 25-yearold to 30% for a 70-year-old. Delaney and Reic henste in (1996) discussed a similar idea as the life cycle approach offered by Malkiel. They stated that human wealth should play an im portant role on portfolio allocations and investors will typically have more human wealth at early adulthood, it is reasonable for investors to take more risk in early life cycle stages. Lee and Hanna (1995) found that few U.S. households have financial assets that represen t a large proportion of total wealth. Lee (1995) reported that for 9 0% of American households with household head aged less than 35, financial assets represent less than 3.2% of the household total wealth (including human wealth), which means a 20% loss in the value of the finan cial assets would represent less than 0.7% of the total wealth for most young A merican households. Furthermore, only a portion of financial assets a re held for goals ap propriate for stocks, as some portion of financial assets are for ordinary monthly transactions, em ergencies a nd for shortterm goals. If investment assets are defined as the amount of financial assets in excess of three months income, investment assets represent a tiny portion of wealth for most U.S. households. Because of the high level of hum an w ealth holdings of young investors, they should invest more aggressively for long term goals such as retirement. How m uch risk should young investors take? This study uses 70 years of real rates of return for six types of financial assets to find optimal portfolios for retirement saving for young American households. II. Methods Two approaches w ere used for analysis. First, based on the assumption that no money in a retirement fund would be needed until retirem ent, efficient portfolios were calculated based on sim ulations of all possible combinations of the six Ibbotson categories, for period ic contributions fixed in real terms. Second, ta king into account the possibility that money might need to be withdrawn or borrowed from a retirement fund for an emergency or for an expected investment goal before retirem ent, an expected utility/simulation approach was used, taking into account typical patterns of financial assets in relation to total wealth for U.S. households under the age of 35. Efficient Portfolios. The most common investment plan for saving for
retirement consists of periodic investm ents over a relative ly long time period for young investors. In this study, it is assumed that the household save the same real amount of money for retirement, i.e., the nominal amount the investor saves each year would keep the same increase rate as inflation. Periods of 20 to 40 years were simulated, using all overlapping, consecutive time periods in the Ibbotson database from 1926 to 1995. The efficient portfolio was defined as the portfolio with the highest worst case end value for any level of mean return. There were 51 overlapping 20 year periods in the Ibbotson Associates (1996) da ta. Portfolios w ith all possible combinations (in increments of 1%) of each of the six types of investments were evaluated. For each 20 year period, the real accumulation (end value) resulting from investing one dollar per year was calculated. For each portfolio, the mean accumulation and the minimum accumulated for all 20 year periods were calculated. The same process was used for all 21 to 40 year time period s. Expected Utility/Simulation Approach In terms of investigating optimal portfolio, mean-variance analysis and expected utility maximization are the most widely used approaches. The mean-variance model developed by Markowitz remains one of the cornerstone of the optimal portfolio model. It was used in most studies on optimal portfolio formation (Frankfurter and Larmoureux, 1987; Alexander,1978; Alexander and Resnick, 1985; Burgess and Bey, 1988 ; M arsha ll, 1974). How ever, the mean-variance model requires strict assumptions concerning the distribution of stock returns and the investor’s utility function. Levy and Markowitz (1979) and Kroll, Levy and Markowitz (1984) showed both theoretically and empirically that the mean-variance method and the expected utility approach gave almost the same optimal portfolio composition. The expected utility simulation method used in K roll, Levy and M arkowitz (1984) is very similar with the one used in this paper, i.e., the optimal portfolio is defined as the one that maximizes the expected utility among all feasible portfolios. The expected utility approach w as also com monly used to analyze alternative portfolio strategies (e.g., Knight & M andell, 1995). Utility functions can be characterized in terms of relative risk aversion, w hich is a measure of the concavity of the utility function or the disutility of consumption fluctuations (Grossman & Shiller, 1981). The higher the relative risk ave rsion, the more ra pidly m arginal utility decreases as con sum ption or wealth increases. One type of utility function used for analysis of investment decisions is the constant relative risk aversion utility
2
function (e.g., Samuelson, 1990), which can be specified as shown in Equations 1 and 2. U(W) = W (1-x)/(1-x) for x
1 Equation 1 U(W) = ln(W) for x=1 Equation 2 where x = relative risk aversion level W = total wealth For constant relative risk aversion utility functions, the key parameter is relative risk aversion, which may plausibly 1 have a value under 10 (Fan, Chang, & Hanna, 1993). In considering investm ent portfolios, it is reasonab le to take into account the proportion the investment portfolio is of the total household wealth, including human wealth (Hanna & Chen, 1995; Delaney & Reichenstein, 1996). Therefore, wealth should include not only net w orth but also a measure of human wealth, which is defined as the present value of non-investment income. Wealth The measure of wealth should include human wealth and real estate as well as financial assets. Ideally, an analysis should be conducted of the optim al portfolio inclu ding all com ponents of w ealth. H owever, reliable estimates of real estate and human wealth rates of return are not available. For simplicity, it is assu med that all nonfinancial assets maintain their real value during the investment horizon. This assumption is a limitation of the analysis. H owever, it would be very difficult to create reliable long term estimates of real returns of other major a sset categories. Real Rates of Return The real rate of return is the appropriate basis for evaluating investments. Tax considerations may make the nominal rate of return relevant. In this analysis, the purpose is to study the optimal retirement portfolios for investors under 35. Tax considerations are ignored, as most investors under 35 could have their retirement portfolio tax-sheltered. The results of this ana lysis may not be valid for those with portfolios too large to shelter completely. The nominal rates of return and th e inflation rates were drawn from the Ibbotson A ssociates Stock, Bonds, Bills and Inflation Yearbook, 1996. The real rate of return was calculated as shown in Equation 3.
Large stocks were the best investment in 19 out of the 66 five year time periods, small stocks were best in 36 periods, corporate bonds were best in 7 periods, government long bo nds w ere best in one period, government intermed iate bonds w ere best in 3 periods, and T-bills were never best. Table 1 shows the arithm etic mean, standard deviations, minimum and maximum real annualized rates of return, as well as the number of periods with a positive return for the 66 overlapping five year periods. The only negative correlations among categories that were significant at the 0.01 level were between small stocks and government interm ediate bonds and between sm all stocks and treasury bills. Table 1. Selected Statistics for Annualized Real Rates of Return for Six Investment Categories for 66 Five Year Periods, 1926-1995. standard mean dev.
# of periods with positive min. max. real
Category ret. Large Stocks (S&P 500) 6.8% 7.7% -9.3% 23.5% 52 Small Stocks 9.8 13.5 -23.4 47.1 54 Corporate Bonds 2.3 5.7 -10.4 18.6 38 Government Long Bonds 1.7 5.4 -10.1 17.7 35 Government Intermediate Bonds 2.0 4.3 -5.1 13.2 45 Treasury Bills 0.5 3.2 -6.1 8.4 34 Calculated by authors based on data from Ibbotson Associates (1996). Means are arithmetic means of annualized real rates of return for all possible five year periods.
Calculating Expected Utility of Portfolios What combinations of investm ents in six major financial asset categories provide the highest expected utility for each level of risk tolerance? The approach used is the same as reported by Hanna and Chen (1995), with a five year time fram e, assu ming that all assets other than financial assets remain unchanged during the five years. It was assumed that the asset return patterns of each of the 66 overlapping five year periods betw een 1926 and 1995 would be equally likely to occur in the future (Ibbotson Associates, 1996, p. 25). The probability of each set of rates of return is 1/66 (0.0152.) Leibowitz and Krasker (1988) argued that stocks have risk in the long run, but their analysis was based on an assumption of a particular p robability distribution. T his method requires that there is no cross-sectional correlation between asset classes a nd no serial correla tion w ithin an asset class, how ever, em pirical studies have sh own that Equation cross sectional correlations and3serial correlation do exist among the returns of investment instruments, such as stocks, bonds, and bills. In con trast, all that is assumed 3
in this paper is that all future periods resemble periods from the beginning of 1926 to the end of 1995. A simulation program was developed base d on this assumption, with relative risk aversion level ranging from 1 to 20, and fina ncial assets as a proportion of total wealth set to range from 1% to 10% . The expected utility of all possible portfolios ( in increm ents of 1% for each asset category) was calculated, and the portfolio with the highest expected utility for a particular combination of risk tolerance level and the proportion of financial assets as percentage of total wealth level was recorded as the optimal portfolio. III. Results Efficient Portfolios For periodic investing a constant real amount for periods of 20 to 40 years, a portfolio composed 100% of sm all stocks dominated all other possible portfolios, in that the worst end value for a 100% small stock portfolio was always better than any other portfolio for the time period between 20 and 40 yea rs. Expected Utility/S imulation A nalysis The result of expected utility/simulation analysis for five year time frames supports that a 100% small stocks portfolio for a very w ide range of risk tolerance levels (Table 2). The optimal portfolio for investors with high risk tolerance levels (risk aversion less than 6), if the financial asset is less than 10% of total wealth, is 100% sma ll stocks. Large stocks and corporate bonds should be included in the op timal portfolio for inv estors w ith extremely low risk tolerance and a relative high financial asset level -- more tha n 5% of the total wealth (compared to the majority of young American households). However, stocks should represent at least 73%, and sm all stocks should represent at least 42%, of the total portfolio even for young investors w ith extremely low risk tolerance and 10% of total wealth in financial assets. IV. Conclusion For an investment horizon of 20 years or more, households making constan t real dollar contributions to retirement funds should choose 100% small stocks for their retirement portfolios, as that portfolio composition dominates all other possible portfolios using the 19261995 Ibbotson investment categories. However, to allow for the possibility of withdrawal before the planned retirement age, an expected utility/simulation analysis was conducted. Lee (1995) reported that for 90% of American households with household head aged less than 35, financial assets represent less than 3.2% of the house hold total wealth (including human wealth)
Therefore, based on the analsyes reported in this paper, most household s under the age of 35 should have 100% of their retireme nt portfolios in small stocks. Young investors with extre mely low risk tolerance levels might include a small portion of large stocks and corporate bonds in their optimal portfolios. If an investor under the age of 35 does not plan to meet short term goals such as building an emergency fund and saving for a down payment for a home through tax sheltered retirement plans, a retire ment savings portfolio should be 100% in sma ll stocks (c.f., Table 2). The advice that the percent in fixed income assets should equal one's age does not seem to be valid for the retirement portfolio of investors under the age of 35. Table 2. Optimal portfolio for re lative risk aversion level ranging from 6 to 20, and financial assets as a proportion of total wealth set to range from 3% to 10%. Financial Relative assets as risk % of total aversion wealth
Large stocks
Small stocks
Corporate All stocks bonds
10%
6
0%
100%
0%
100%
9%
7
0%
100%
0%
100%
10%
7
2%
98%
0%
100%
8%
8
0%
100%
0%
100%
9%
8
5%
95%
0%
100%
10%
8
13%
87%
0%
100%
7%
9
0%
100%
0%
100%
8%
9
6%
94%
0%
100%
9%
9
15%
85%
0%
100%
10%
9
22%
78%
0%
100%
7%
10
4%
96%
0%
100%
8%
10
15%
85%
0%
100%
9%
10
23%
77%
0%
100%
10%
10
29%
71%
0%
100%
6%
11
0%
100%
0%
100%
7%
11
12%
88%
0%
100%
8%
11
22%
78%
0%
100%
9%
11
29%
71%
0%
100%
10%
11
35%
65%
0%
100%
6%
12
7%
93%
0%
100%
7%
12
19%
81%
0%
100%
8%
12
27%
73%
0%
100%
9%
12
34%
66%
0%
100%
10%
12
40%
60%
0%
100%
5%
13
0%
100%
0%
100%
6%
13
14%
86%
0%
100%
7%
13
24%
76%
0%
100%
8%
13
32%
68%
0%
100%
9%
13
39%
61%
0%
100%
4
10%
13
40%
56%
4%
96%
5%
20
31%
69%
0%
100%
5%
14
6%
94%
0%
100%
6%
20
41%
59%
0%
100%
6%
14
19%
81%
0%
100%
7%
20
38%
53%
9%
91%
7%
14
29%
71%
0%
100%
8%
20
36%
48%
16%
84%
8%
14
37%
63%
0%
100%
9%
20
33%
45%
22%
78%
9%
14
41%
57%
2%
98%
10%
20
31%
42%
27%
73%
10%
14
39%
53%
8%
92%
5%
15
11%
89%
0%
100%
6%
15
24%
76%
0%
100%
Financial assets as % of total wealth
Relative risk aversion
Large stocks
Small Corporate All stocks stocks bonds
Note: portions of results deleted represent cases where it is obvious that the optimal portfolio is 100% small stocks. Example: Relative risk aversion less than 11 AND financial assets represent less than 7% of total wealth.
Endnotes
7%
15
33%
67%
0%
100%
8%
15
41%
59%
0%
100%
9%
15
39%
55%
6%
94%
10%
15
36%
51%
13%
87%
3%
16
0%
100%
0%
100%
4%
16
0%
100%
0%
100%
5%
16
16%
84%
0%
100%
6%
16
28%
72%
0%
100%
7%
16
37%
63%
0%
100%
8%
16
41%
56%
3%
97%
9%
16
38%
52%
10%
90%
10%
16
35%
49%
16%
84%
3%
17
0%
100%
0%
100%
4%
17
4%
96%
0%
100%
5%
17
20%
80%
0%
100%
6%
17
32%
68%
0%
100%
7%
17
40%
60%
0%
100%
8%
17
39%
54%
7%
93%
9%
17
36%
50%
14%
86%
10%
17
34%
47%
19%
81%
3%
18
0%
100%
0%
100%
4%
18
9%
91%
0%
100%
5%
18
24%
76%
0%
100%
6%
18
35%
65%
0%
100%
7%
18
41%
57%
2%
98%
8%
18
38%
52%
10%
90%
9%
18
35%
48%
17%
83%
10%
18
33%
45%
22%
78%
3%
19
0%
100%
0%
100%
4%
19
13%
87%
0%
100%
5%
19
28%
72%
0%
100%
6%
19
38%
62%
0%
100%
7%
19
39%
55%
6%
94%
8%
19
36%
50%
14%
86%
9%
19
34%
46%
20%
80%
10%
19
31%
44%
25%
75%
3%
20
0%
100%
0%
100%
4%
20
17%
83%
0%
100%
1.
A modified version of K imball's (1988) ex ample developed by H anna (1988 ), could explain the concept of relative risk aversion in the context used. Assume that you have one year to live, and may choose an investment to provide you with your consumption for the next year. Once you choose, it will be im possible for you to ob tain income from any other source. You have no assets of any kind. You may choose one of two plans: A or B. Plan A pays you a tax free real income of $50,000 p er year, w hile plan B involves a gamble. If you choose plan B, the government in effect flips a coin, and there is a fifty percent chance of having a real income of $100,000 tax-free, and a fifty percent chance of some lower income I. A t what level of I would you be indifferent betw een P lan B and Plan A . The table below shows how your answer corresponds to your level of relative risk aversion (RRAv). RRAv I(thousands) 1 25.00 2 33.33 3 37.80 4 40.55 5 42.38 6 43.66 7 44.60 8 45.31 9 45.86 10 46.30 11 46.65
R RA v 12 13 14 15 16 17 18 19 20
I(thousands) 46.95 47.19 47.40 47.58 47.74 47.88 48.00 48.11 48.21
For instance, if you had a relative risk aversion level of 18, you would be indifferent between Plan A, the certainty of having $50,000, and Plan B, which would be a 50% chance of $100,000 and a 50% chance of $48,000. The utility loss of the $2,000 reduction in consumption from $50,000 to $48,000 would be valued the same as the $50,000 increase in consum ption from $50,0 00 to $100,000, representing a 25 to 1 ratio of value of utility 5
change to consumption change between a loss and a gain. For a relative risk aversion level of 10, the ratio is 13.5. Someone who w as that risk averse (10) would not be willing to give up $3,700 for a 50% chance of a gain of $50,000. At the other extreme, a relative risk aversion level of 1 means that one is willing to take a 50% chance of having consumption cut in half ($50,000 to $25,000) in exchange for a 50% chance of doubling consumption. That low a level of risk aversion probably exists, but even entrepreneurs do not typically expose themselves to that high a risk of cutting consumption in ha lf, only wealth. In this scenario, there is no second chance and no social safety net, so it is harsher than the real world of the United States. Appendix The Six asset categories provided by Ibbotson Associates are defined as, Large Company Stocks: S&P 500 composite with dividends reinvested. (S&P 500, 1957 - present; S&P 90, 1926-1957) Small Company Stocks: Fifth capitalization quintile of stocks on the NYSE for 1926-1981. Performance of the Dimensional Fund Advisors (DFA) Small Company fund 1982-Present. Corporate Bonds: Salomon Brothers Long-term High grade Corporate Bond Index. Long-Term Government Bonds: 20 year U.S. Bonds. Intermediate-Term Government Bonds: Government Bonds with 5 year maturities. Treasury Bills: 30 day T-bills. Inflation: Consumer Price Index.
Yearbook. Chicago: Ibbotson Associates. Knight, J. R. And Mandell, L. (1995). Optimal holding period for assets that must be liquidated: A certainty wealth equivalent approach. Financial Services Review, 4(2), 97-108. Kroll, Y., Levy, H. & Markowitz, H. M. (1984) Mean-Variance versus direct utility maximization. The Journal of Finance, 39, 47-61. Lee, H.-K. (1995). Direct and Indirect Determinants of Stock Ownership. Ph.D. dissertation, The Ohio State University. Lee, H.-K. & Hanna, S. (1995). Investment portfolios and human wealth, Financial Counseling and Planning, 5, 147-152. Leibowitz, M. L. & Krasker, W. S. (1988: Nov.-Dec.) The persistence of risk: Stocks versus Bonds over the long term. Financial Analysts Journal. 40-47. Levy, H. & Markowitz, H. M. (1979) Approximating expected utility by a function of mean and variance. The American Economic Review, 69, 308-317. Malkiel, B. G. (1990). A Random Walk Down Wall Street. New York: W.W. Norton & Co. Marshall, S. (1974). Capital market imperfections and the composition of optimal portfolios. Journal of Finance, 29, 1241-1253. Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics, 51, 239-246 Samuelson, P. A. (1990). Asset allocation could be dangerous to your health: Pitfalls in across-time diversification, Journal of Portfolio Management, (Spring) 5-8.
Presented at the Academy of Financial Services meeting, New Orleans, October, 1996.
REFERENCES Alexander, G. J. (1978). Optimal mixed security portfolios with restricted borrowing. Journal of Economics and Business. 31, 2228 Alexander, G. J. & Resnick, B. G. (1985). More on estimation risk and simple rules for optimal portfolio selection. Journal of finance, 40, 125-133. Ando, A. & Modigliani, F. (1963). The "life cycle" hypothesis of saving: Aggregate implications and tests. The American Economic Review, 53, 55-74. Avery, R. B., & Elliehausen, G. E. (1986, March). Financial characteristics of high-income families. Federal Reserve Bulletin, 163-177. Burgess, R. C. & Bey, R. P. (1988). Optimal portfolios: Markowitz full covariance versus simple selection rule. Journal of Financial Research. 11, 153-164. Delaney, C. & Reichenstein, W. (1996). An expanded portfolio view includes real estate and human capital, American Association of Individual Investors Journal, (July, 1996), 7-11. Fan, X. J., Chang, Y. R. & Hanna, S. (1993). Real income growth and optimal credit, Financial Services Review, 3(1), 45-58. Frankfurter G. M. & Lamoureux, C. G. (1987). The relevance of the distributional form of common stock return to the construction of optimal portfolios. Journal of Financial and Quantitative Analysis. 22, 505-511. Grossman, S. J., and Shiller, R. J. (1981). The determinants of the variability of stock market prices, American Economic Review: Papers and Proceedings,71 (2), May, 222-227. Hanna, S., & Chen, P. (1995). Optimal portfolios: An expected utility/simulation approach. Presentation at the Academy and Financial Services meeting. Ibbotson Associates (1996). Stocks, Bonds, Bill, and Inflation 1995 6
