Social Policy Brief

Portfolio Theory, Life-Cycle Investing, No. 2007-02 and Retirement Income October 2007

Introduction ment securities and could incorporate There has been much life-cycle funds. Life-cycle funds are a relatively new discussion recently about approach to retirement investing and This brief explores some of the life-cycle funds and their have gained popularity in recent years. theoretical and empirical foundations role in providing a secure Although the characteristics of these for life-cycle funds by reviewing the retirement income for funds vary, the general life-cycle fi nance literature on optimal portfolio older Americans. These proposition calls for investment port- theory. It also discusses actual life- funds, which gradually folios that hold a decreasing propor- cycle plans and the advantages and shift account assets from tion of assets in equities (associated disadvantages of these types of funds. broad-based funds with higher risk) and a greater pro- to funds as a par- portion in fi xed-return investments Portfolio Theory ticipant ages, are becom- (associated with lower risk) as an ing an important vehicle originated individual ages. Those types of plans for retirement savings. with the work of Markowitz (1952), seek to limit potential losses from This policy brief explores who recognized that by combining market fl uctuations as an individual the economic rationale assets that are not perfectly correlated approaches retirement. The building behind the life-cycle (for example, assets whose returns blocks of life-cycle funds are typi- approach and the advan- do not move in complete unison with cally broad-based index funds, such tages and limitations of each other) an could reduce as a stock fund tied to the S&P 500 or life-cycle funds. his or her investment risk without to a corporate that tracks reducing expected returns. It is theo- the Lehman Brothers Corporate Bond retically possible to derive a portfolio Index. Broad-based index funds lower of risky assets that returns the smallest risk to retirement income that would amount of risk for a given return. arise from undiversifi ed investments Repeating this procedure many in individual companies. times for different levels of expected Life-cycle funds are an increasingly return produces the Mean-variance or important topic in discussions about Markowitz effi cient frontier (Chart 1). retirement income. Vanguard (2004) Intended solely for illustrative pur- reports rapid growth in the number poses, the chart shows individual of private-sector retirement plans that with their respective ex-ante offer life-cycle funds. In addition, the expected mean returns and standard defi ned contribution plan offered to deviations. The standard deviation, federal employees (the Thrift Savings which measures how much a stock’s Plan) now includes life-cycle funds. annual return deviates from its - Some Social Security reform pro- term historical average, is used as posals call for the creation of personal an estimate of risk or variability of retirement accounts. Such accounts investment outcomes (the standard would allow individuals to invest in deviation is the square root of the equities, corporate bonds, and govern- variance). The curved line denotes the

www.socialsecurity.gov/policy Life-Cycle Funds Chart 1. Mean-variance The life-cycle funds described here create portfolios that are heavily concentrated in stocks at the beginning Mean return of the work life and gradually shift holdings to bonds as retirement nears. While this brief primarily focuses ¨ on the decision of how to invest accumulated savings, r* ¨ ¨ it is important to note that , in reality, face ¨¨ a set of complex and interrelated decisions over the life-cycle. The discussion here abstracts from many ¨ of those decisions (such as how much schooling to acquire, when to begin working, how much to save each period). ı* Several demographic and economic factors provide Standard deviation some rationale for life-cycle funds. The fi rst deals with

SOURCE: Author's derivation. how the value of human capital varies over time as a fraction of total wealth. Human capital is composed of elements that are fi xed (innate ability), that are largely effi cient frontier, which is derived by combining the fi xed after a certain point (formal schooling), and that individual stocks and taking into account the degree vary with time (experience). A good proxy for measur- of covariation between them. In reality, thousands of ing the value of human capital is the present value of triangles and a very large number of computations wages over an individual’s remaining working life. would be needed to determine the effi cient frontier. This is generally considered much less variable or The implications of the effi cient frontier are quite “stochastic” than equity returns because its determi- signifi cant. All portfolios falling on this frontier will nants are, to some extent, fi xed. Therefore, to maintain provide the highest return for a given level σ* of risk a constant level of variability (risk exposure) over the (vertical line) and, conversely, the lowest risk for a life-cycle, relatively more of one’s total fi nancial assets given level r* of return (horizontal line). The effi - should be held in stocks when young and less as one cient frontier does not eliminate risk but reduces it to gets older (Bodie, Merton, and Samuelson 1992). the lowest level possible for a given expected rate of To illustrate the constant risk of exposure approach, return. Investors wanting a portfolio of stocks with consider someone who wishes to hold 60 percent of low risk could choose one from the bottom left portion total wealth at any given age in riskless assets (for of the effi cient frontier; those wishing to seek higher example, infl ation-protected bonds) and 40 percent in returns by taking on more risk could choose a portfolio risky assets (for example, stocks). Suppose, further, from the upper right portion of the frontier. All points the person’s human capital at a young age is equivalent below the effi cient frontier are suboptimal in the sense to a riskless asset valued at $300,000. If the person that investors could increase their expected rate of has $200,000 in fi nancial assets, they should all be return without assuming any additional risk. held in risky assets (such as stocks) so that the 60/40 Although the discussion up to this point has focused balance is achieved. As the person ages, the value of on stock portfolios, the concept of an effi cient frontier human capital falls (because only a few working years can be generalized to incorporate other asset types, remain) but the fi nancial assets grow. To maintain the including bonds. Bonds typically have both lower 60/40 balance, fi nancial assets must increasingly be returns and lower standard deviations than stocks. shifted out of risky assets. Thus, in Chart 1, a bond-heavy portfolio would be Although human capital is a determinant of the located on the lower left portion and a stock-heavy present value of lifetime earnings, it is not the only portfolio on the upper right portion of the frontier. factor. Individuals have discretion over whether to Life-cycle funds can be thought of as moving along work and how much to work. This issue—labor sup- the frontier (from the upper right to the lower left) as ply fl exibility—is also important in discussions of one ages. life-cycle funds. Intuitively, because younger workers

2 ♦ Policy Brief No. 2007-02 have greater labor supply fl exibility and have just distribution of ending wealth becomes more skewed, begun their working lives, their age can act as a buffer with the mean ending wealth being signifi cantly against market downturns since additional work can greater than the median. Thus, while a younger person make up for lost wealth. Jagannathan and Kocherla- may need to hold only 50 percent in stocks to achieve kota (1996) argue that labor supply fl exibility makes an “expected” ending wealth (the mean distribution) life-cycle funds desirable, but only if equity returns that matches the replacement rate target, such a portfo- are relatively uncorrelated with labor income. Avenues lio would not have a high probability of generating the for future research (Poterba and others 2006) focus on appropriate ending wealth. In general, Booth argues creating life-cycle portfolios that differ for singles as that if individuals desire a high probability of achiev- opposed to married couples—an approach that takes ing the target replacement rate, they may need to hold into account the added labor supply fl exibility mar- a greater share of their portfolio in stocks when they ried couples have because of the potential of having are younger. two earners. Hence, married couples might be more inclined to invest a relatively larger fraction of their Specifi c Approaches wealth in stocks later in life. This section discusses specifi c approaches to life-cycle The work of Jagannathan and Kocherlakota (1996) investing as opposed to the general concept of holding and Poterba and others (2006) examines labor supply a smaller percentage of assets in equities as one ages. fl exibility in the specifi c context of life-cycle funds, Four examples of life-cycle investment approaches are while other research on the same general topic offers considered: insight into the foundations of life-cycle funds. Bodie • a popular rule of thumb known as the “100-minus (2001) examines labor supply fl exibility and portfolio age” rule; choice in terms of retirement age. He assumes a fi xed saving rate and predictable earnings, from which he • the Malkiel approach (1990); determines a baseline retirement income assuming • the Shiller plan (2005); and retirement at age 65 and investments in riskless Trea- • the new “L Fund” plan offered to federal employ- sury securities. He then considers whether the baseline ees through the Thrift Savings Plan, a defi ned con- retirement income could be achieved at an earlier tribution retirement plan similar to a 401(k) plan. retirement age with alternative portfolios: one invested Clearly this is not an exhaustive list of possible life- 50 percent in stocks and 50 percent in riskless Trea- cycle investment plans or approaches, but specifi c sury securities and the other invested completely in examples will help sort out ideas and illustrate impor- stocks. If the future risk premium (the expected return tant concepts. on stocks minus the return on riskless Treasury bonds) is assumed to be 4 percent and the standard deviation Table 1 presents the percentage held in stocks at of stock returns is 20 percent, then the portfolio with specifi c ages under the four approaches. The simplest half of its assets invested in stocks and the other half of these approaches is the “100-minus age” rule, which invested in Treasury bonds has an expected retirement dictates that the percentage invested in the stock mar- age of 61, but this comes with a small probability of ket equal 100 minus one’s age. For example, at age 55, having to postpone retirement to age 67 to achieve the desired level of retirement income. The all-stock portfolio has an expected retirement age of 57, but Table 1. again carries a small probability of having to postpone Illustrations of life-cycle fund allocations retirement until age 68. Although Bodie’s focus is on in equities, by age (in percent) retirement age, his general point is that labor supply 100- fl exibility can offset market losses. With life-cycle minus Malkiel Shiller L fund funds, market losses will tend to be concentrated at Age age approach plan (TSP) relatively younger ages—ages at which health and 25 75 70 85 85 labor market opportunities may be more favorable. 35 65 60 71 75 Booth (2004) fi nds support for life-cycle investing 55 45 50 26 50 using a model that examines replacement rates. He argues that as the investment horizon increases, the SOURCE: Author's calculation.

www.socialsecurity.gov/policy ♦ 3 45 percent of investments should be in a broad-based stock and 55 percent in bonds. A somewhat Table 2. Retirement , by age group more involved approach was suggested in Malkiel (in percent) (1990). Malkiel’s approach suggests stock alloca- tions that are roughly similar to those derived from Age group Bonds Stocks the “100-minus age” rule. The Shiller plan (2005) has 25–44 73460 a baseline life-cycle portfolio that is somewhat more 45–54 63657 aggressive at young ages and less so at later ages than is the “100-minus age” rule or Malkiel’s plan. 55–64 74350 Recently, the federal Thrift Savings Plan began to 65 or older 95537 offer life-cycle products. These products, which are SOURCE: Bodie and Crane (1997). referred to as L funds, are based on planned retire- NOTE: Totals may not sum to 100 percent because of rounding. ment age rather than current age. They are composed of combinations from fi ve underlying funds: the S fund (Wilshire 4500 index), the C Fund (S&P 500 whether they refl ect inherent preferences of individu- index), the I Fund (EAFE international stock index), als or whether they are in response to advice given by the G Fund (federal fund), and the fi nancial planners. F Fund (Lehman Brothers U.S. bond market index). Bodie and Crane (1997) found similar results for To illustrate, consider an individual born in 1980 retirement asset categories. Using data from TIAA- who expects to retire after 2035 (such a person would CREF, the proportion of retirement assets held in use the L 2040 Fund). At age 25, the worker’s port- stocks declined with age (Table 2). folio would be composed of 85 percent stocks and The authors found a negative relationship between 15 percent bonds. By age 55, about 50 percent of the age and stock market investing roughly consistent with portfolio would be held in stocks. The L Fund plan the “100-minus age” rule. To be exact, for each addi- is comparable with the Shiller plan at the beginning, tional year one ages, the fraction invested in stocks but the Shiller plan drops the equity component much declines by 0.6 percent. One caveat to interpreting more quickly at later ages. these results as “age” effects is that investing prefer- Although the specifi c allocations of the four life- ences may change across cohorts or generations (for cycle approaches vary, they are roughly comparable. example, today’s younger workers may be willing to They specify holding a majority of assets in stocks in accept a greater level of fi nancial risk than previous the early years and then shifting to bonds as retirement generations). nears (Table 1). Life-cycle products appear to be increasing in popu- larity. Vanguard (2004) noted that between 2000 and Preferences 2003 the number of plans offering specifi c life-cycle There is evidence that life-cycle investment strate- funds has more than doubled. gies refl ect people’s general preferences. Several researchers have found investment behavior to be Limitations broadly consistent with the life-cycle advice. Schooley Some researchers, however, have questioned whether and Worden (1999), using data from the Survey of life-cycle approaches are appropriate for retirement Consumer Finances, found that investors with long saving. Several research studies show that these funds investment horizons lean more heavily toward stocks. still expose investors to signifi cant risk while eliminat- Specifi cally, those with planning horizons of 5 or more ing most upside potential. Shiller (2005) simulated years had roughly 50 percent of their assets invested ending wealth balances of hypothetical life-cycle in equities compared with less than 12 percent for accounts using historical data for the S&P 500 and those with horizons of 1 year or less. More generally, bond market returns and found that the life-cycle one-half of those with planning horizons of less than fund failed to outperform a 3 percent real return in a year were unwilling to take any fi nancial risk with 32 percent of trials, but a 100 percent investment their assets, in contrast to only 25 percent of those in the S&P 500 would have beaten such a return in with horizons of over 10 years being unwilling to take 98 percent of trials. any risk. One issue these results cannot address is

4 ♦ Policy Brief No. 2007-02 Hickman and others (2001) simulated outcomes and riskier assets (such as stocks) at later ages. Viceira under Malkiel’s approach, the “100-minus age” rule, (2001) and Lynch and Tan (2004) also consider the and 100 percent investment in the S&P 500 index role of labor income in optimal portfolio holdings. fund. Using a 30-year holding period, the two life- cycle approaches (Malkiel and “100-minus age” rule) Conclusion yielded very similar outcomes and produced median This policy brief provides policymakers with an over- wealth at retirement that was approximately one-half view of portfolio theory and the advantages and disad- that associated with the index fund. However, in about vantages of a life-cycle investing approach. Portfolio 15 percent of the simulations the life-cycle approaches theory suggests life-cycle investing can be optimal did outperform the S&P 500, which suggests that occa- in some circumstances. Empirical work has found sionally the shift to bonds at later ages will be a cor- confl icting results. Life-cycle investing is consistent rect strategy. However, the authors question whether in many cases with actual portfolio choices individu- protection against this relatively rare outcome warrants als make, but some researchers question whether the the large reduction in expected ending wealth. protection that such funds provide against market In a similar fashion, Butler and Domian (1993) downturns is worth the lower average returns produced simulated ending balances for stocks, bonds, and life- by shifting assets from stocks to bonds. cycle accounts and derived probability distributions for ending wealth. Their conclusion was consistent References with Hickman and others in that common stocks are Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. the best vehicle for long-term retirement savings (life- Goldstein. 2005. Portfolio choice over the life-cycle in cycle accounts outperformed a portfolio of all stocks the presence of “trickle down” labor income. NBER in only about 8 percent of their simulations). Ho, Working Paper No. 11247, National Bureau of Economic Milevsky, and Robinson (1994) also emphasize the Research, Cambridge, MA. importance of stocks, arguing that higher risk/return Bodie, Zvi. 1995. On the risk of stocks in the long run. investments may be necessary to minimize the chances Financial Analysts Journal 51(3): 18–22. of outliving one’s assets in old age. ———. 2001. Retirement investing: A new approach. It is important to emphasize, however, that the Working Paper No. 2001-03, Boston University School of literature on the limitations of life-cycle funds is based Management. on historical returns and often uses data from periods Bodie, Zvi, and Dwight B. Crane. 1997. Personal investing: when stock returns were strong. While riskless bonds Advice, theory and evidence. Financial Analysts Journal may generally underperform stocks, they also protect 53(6): 13–23. against extremely negative outcomes (Bodie 1995). Bodie, Zvi, Robert C. Merton, and William F. Samuelson. It is questionable whether the historical period for 1992. Labor supply fl exibility and portfolio choice in a which data are available is long enough to refl ect these life-cycle model. Journal of Economic Dynamics and extreme outcomes. Thus, one rationale for life-cycle Control 16: 427–449. funds is that they offer protection against extreme Booth, Laurence. 2004. Formulating retirement targets and outcomes near retirement that occur with a very low the impact of time horizon on asset allocation. Financial probability. Services Review 13(1): 1–17. Finally, some recent work questions the theoreti- Butler, Kurt C., and Dale L. Domian. 1993. Long-run returns on stock and bond portfolios: Implications for cal and intuitive underpinnings of life-cycle funds. retirement planning. Financial Services Review 2(1): Benzoni, Collin-Dufresne, and Goldstein (2005) argue 41–49. that changes in labor income tend to be more heavily Hickman, Kent, Hugh Hunter, John Byrd, John Beck, correlated with stock returns over long horizons. In and Will Terpening. 2001. Life-cycle investing, hold- other words, rather than being “bondlike” in its vari- ing periods, and risk. Journal of Portfolio Management ability, labor income is “stocklike” over long horizons. 27(2): 101–111. In this framework, individuals should diversify by Ho, Kwok, Moshe Arye Milevsky, and Chris Robinson. holding less risky assets (such as bonds) when young 1994. Asset allocation, life expectancy and shortfall. Financial Services Review 3(2): 109–126.

www.socialsecurity.gov/policy ♦ 5 Jagannathan, Ravi, and Narayan R. Kocherlakota. 1996. Vanguard Center for Retirement Research. 2004. How Why should older people invest less in stocks than America saves: A report on Vanguard defi ned contribu- younger people? Federal Reserve Bank of Minnesota tion plans. The Vanguard Group. Quarterly Review 20(3): 11–23. Viceira, Luis. 2001. Optimal portfolio choice for long- Lynch, Anthony, and Sinan Tan. 2004. Labor income horizon returns with non-tradable labor income. Journal dynamics at business-cycle frequencies: Implications of Finance 56: 433–470. for portfolio choice. Working Paper, Stern School of Business, New York University. Malkiel, Burton. 1990. A random walk down Wall Street. New York: W.W. Norton & Co. Markowitz, H. 1952. Portfolio selection. Journal of Finance 7(1): 77–91. Poterba, James, Joshua Rauh, Steven Venti, and David Wise. This brief was prepared by Dale Kintzel of the Social 2006. Life-cycle asset allocation strategies and the distri- Security Administration. bution of 401(k) retirement wealth. NBER Working Paper No. 11974, National Bureau of Economic Research, Questions about the analysis should be directed to Cambridge, MA. the author at 202-358-6218. For additional copies of this brief, e-mail [email protected] or call Schooley, Diane K., and Debra D. Worden. 1999. Investors’ 202-358-6274. asset allocations versus life-cycle funds. Financial Analysts Journal (55)5: 37–43. Social Security Administration Shiller, Robert J. 2005. The life-cycle personal accounts Offi ce of Policy proposal for Social Security: An evaluation. NBER 500 E Street, SW, 8th Floor Working Paper No. 11300, National Bureau of Economic Washington, DC 20254 Research, Cambridge, MA. SSA Publication No. 13-11702

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