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Unsafe at Any Speed? The Designed-In Risks of Target-Date Glide Paths

by Zvi Bodie, Ph.D.; Richard K. Fullmer, CFA; and Jonathan Treussard, Ph.D.


Executive Summary

  • The U.S. Department of Labor currently lists target-date funds (TDFs) as a qualified default investment alternative (QDIA) for defined contribution retirement plans, but this is under review due to recent poor performance.
  • TDFs are designed with some level of investment risk that declines over time as the target retirement date approaches. This is usually accomplished by altering the asset allocation according to a predetermined schedule over time. The "glide path" of the investment risk level varies considerably among TDF providers.
  • It is possible to design a safe TDF in terms of market risk—although in terms of meeting a target level of wealth at retirement (or income in retirement), it would only be safe if matched with a large enough contribution (employee savings) rate.
  • There is currently no generally accepted method of measuring TDF risk. This presents a challenge to policymakers, plan sponsors, plan participants, and financial advisers.
  • The TDF industry today is analogous in many ways to the automobile industry of the 1960s, and we note a number of issues that this analogy brings to light with regard to QDIA policy matters. The public policy discussion that occurred in the automobile industry then provides a useful framework for interested parties to discuss QDIA policy matters today.
  • Risk measurement is not a simple matter. To be useful (and not harmful), the risk measure must be scientifically sound. We discuss the desired attributes of such a risk measure and propose that a standardized method of disclosure and risk measurement would be beneficial to consumers.

Zvi Bodie, Ph.D., is a finance professor at Boston University. He holds a Ph.D. from the Massachusetts Institute of Technology and is coauthor (with Kane and Marcus) of the widely used textbook Investments.

Richard K. Fullmer, CFA, is a senior portfolio strategist at Russell Investments and chairman of the methodologies committee for the Retirement Income Industry Association.

Jonathan Treussard, Ph.D., is an economist whose work has focused on issues in lifecycle finance, derivatives pricing, financial engineering, and risk management. He holds a Ph.D. in economics from Boston University and lives in New York City.


Given the poor investment performance that so many defined contribution plan participants experienced by investing in target-date funds (TDFs) in 2008 and early 2009, it is not surprising that the U.S. Department of Labor is reviewing their suitability as a "safe harbor" qualified default investment alternative (QDIA). These funds are marketed as simple solutions for plan participants who find it too difficult, unpleasant, or time-consuming to choose among investment alternatives. Their objective is to provide individual investors who plan to retire at a specific date with a prudent strategy for managing their retirement assets. This does not mean that they are without risk. Many carry a substantial amount of risk, as evidenced by their recent performance. Table 1 shows 2008 performance for a sample of 2010 target-date fund providers. The risk level in the 2010 funds is especially relevant because their investors were only two years from their targeted retirement date.


 
Lawmakers, regulators, and plan sponsors have expressed concern with TDF safety. The level of investment risk in these funds is primarily a function of their asset allocation glide paths. The term "glide path" refers to the (usually predetermined) schedule for changing the proportion of assets invested in stocks, bonds, or cash over time. It can also refer to the changing duration of the bonds. Thus, the shape of the glide path has a significant effect on the risk and return characteristics of a fund. Glide paths vary substantially among fund manufacturers. Figure 1 shows the range of this variation. This figure reflects the universe of glide paths as of the end of 2008. It includes 2005 funds because most TDFs continue to be managed after the target date has passed.


 
A perfectly natural question to ask is, "If TDFs have similar objectives, why do they have such different glide paths?" Be careful before answering, because this is a trick question. It falls victim to the fallacy of false premise. Contrary to the similarity implied by their names, TDFs can have quite different objectives. One way they can differ lies in the amount of post-retirement consumption that is sought after the target date, or stated a different way, in desired funding level at the target date.1 Some funds seek to fund higher levels of consumption than do others. Another way they can differ lies in the time horizon … even though their target dates may be the same. Some glide paths seek only to manage to the target date, while others go further—to manage through the target date, presumably for the rest of the investor's life. A better-formed question to ask, then, is, "What differences exist in the objectives of TDFs that cause their glide paths to differ?"
 
Whatever their differences, the objectives of every TDF that enjoys QDIA status should align closely with that of the defined contribution system itself. Defined contribution (DC) plans are meant to provide a supplemental source of funding for the post-retirement consumption needs of people in the workforce. The idea is that the stream of cash from the DC plan plus Social Security (and possibly other defined benefit plans that may be available) should replace a specified portion of participants' labor income, enabling them to maintain the same standard of living. That funding needs to be in place when a participant retires—at the target date. What happens after the target date has no bearing on a participant's balance at the target date. Either the necessary amount of post-retirement funding will be there or it will not.
 
The ultimate objective of a defined contribution plan is essentially the same as a defined benefit plan. The theoretical construct and functional equivalent is that of a deferred real annuity contract that begins making payments immediately after the target date and continues doing so for the rest of the investor's life. As with any life annuity, its cost depends on prevailing interest rates—the real yield curve—and projected mortality rates. This objective can be addressed without market risk in the theoretical construct by investing the contributions in a portfolio of duration-matched inflation-protected government bonds—specifically, Treasury Inflation-Protected Securities (TIPS).2 We can think of this portfolio of bonds as a "safe portfolio" and its glide path as a "safe glide path." In terms of its asset allocation mix, the safe glide path is flat over time because it contains only bonds. In terms of the duration of these bonds, however, the safe glide path is not flat because its duration decreases over time.3
 
To obtain any particular funding level at the target date, one can use the real yield curve to compute the necessary amount of contributions to make into the safe portfolio along the way. We can think of this as the "safe contribution rate." We stated earlier that some TDFs seek to fund a higher level of post-retirement consumption than do others. Naturally, the higher the desired funding level, the higher the necessary contribution rate. For example, a fund that seeks to provide $50,000 per year in post-retirement consumption requires twice the contribution rate of a fund that seeks to provide $25,000 per year at the same risk level.4 Regardless of the desired funding level, there exists a safe contribution rate associated with it, assuming it is invested in the safe portfolio. At all contribution rates below the safe contribution rate, the desired funding level objective cannot be achieved without taking risk, which means the glide path must necessarily include risky assets.
 
Because a safe glide path can always be constructed and matched with a safe contribution rate, it is evident that the fact that TDFs have different objectives is not a reason per se for their glide paths to differ. It is only by introducing risk that a fund's objectives will influence its glide path. We could therefore say that risk is "designed into" these funds.
 
Risk can be introduced by one or more of the following methods:

  1. Assuming a contribution rate lower than the safe contribution rate (requiring participants to take risk to have any reasonable chance of meeting the objective)
  2. Setting a target rate of return higher than the TIPS rate (a form of the previous item, in which TIPS will appear risky because the chance of reaching the target wealth funding level by using them will be virtually nil)
  3. Assuming the participant wants to take risk (assigning a non-zero risk tolerance parameter in the objective function of a glide path optimization model)

Our earlier statement that the level of risk in target-date funds is primarily a function of their glide paths is true. Of course, the reverse is also true: Their glide paths are primarily a function of the level of risk that was designed into them in the first place.
 
As the burden of funding retirement shifts further away from institutions (both social and corporate) and onto individuals, it is increasingly important for policymakers (both governmental and corporate) to consider the QDIA suitability issue very carefully. They must ask themselves several questions. Should QDIAs be safe or should they be risky? If risky, then how risky? What is a prudent way to measure risk? To this end, we offer our perspectives in three areas. First, we consider an outline for consumer advocacy borrowed from the automobile industry but analogous in many respects to the defined contribution plan industry. Next, we provide warnings on the traps that often arise from fallacious reasoning and improper risk measurement. Finally, we provide some insights into what can be done to help policymakers, plan sponsors, and participants make better decisions on the subject of retirement savings.

On Safety, Suitability, and Consumer Advocacy

Empirical evidence suggests that TDFs should be an improvement over the investment choices often made by uninformed investors.5 Still, recent history reveals that TDF products are not immune to the question of suitability. How are policymakers and others to approach such a complex question? For perspective, we are reminded of a pioneering work of consumer advocacy published four decades ago.
 
In Unsafe at Any Speed, Ralph Nader's discourse on automotive safety, the author describes numerous examples of "the designed-in dangers of the American automobile." Safety is a multifaceted issue and the author organizes his arguments accordingly. For example:

  • Preference for style over safety. Automotive product design was predominately concerned with styling rather than safety.
  • Lack of disclosure. Known safety issues were not clearly disclosed to automobile sellers, buyers, and owners. Manufacturers and others involved in their distribution networks had little economic incentive to highlight product safety issues to prospective buyers.
  • Added cost and reduced profitability. The additional cost to provide safety improvements made manufacturers unwilling to incorporate them into product designs.
  • Deflecting the responsibility/blame. The industry often deflected the responsibility and blame for safety issues away from its cars and onto the drivers of those cars and those who build and maintain roads.
  • Inattention to the scientific body of knowledge. A large body of "crash science" research was available and known to product design engineers, but largely ignored in their final products.

Regardless of whether one agrees with Nader's position on these issues, he does provide a thought-provoking outline for policymakers and others pondering the question of product safety and suitability.

The Analogy with Target-Date Glide Paths

A number of parallels seem to exist between the automobile industry of yesteryear and the target-date fund industry of today.
 
Preference for style over safety. Many TDF providers appear more concerned with style than safety. Here, "style" refers to marketing efforts to tell the story about how their glide paths work and why their methodology is superior to that of competitors.
 
To develop a glide path, financial engineers must make two types of decisions. One type involves making assumptions about the world, such as the behavior of the capital markets, participant salaries, the retirement age, mortality rates, etc. Another type involves establishing the post-retirement consumption goal in some quantitative way, such as a final-year income replacement ratio (for example, to provide a lifetime income stream equivalent to 40 percent of one's final pre-retirement salary). After making these decisions, the engineers use sophisticated methods for constructing the glide path. These methods may involve capital market forecasting, human capital valuation, surplus optimization, etc.
 
Such glide path styling is fine and indeed good. But recall that the objective of defined contribution plans is to provide a supplemental source of funding for the post-retirement consumption needs of the participants. The glide path is merely a means to an end. The end is the objective to fund some desired level of post-retirement consumption.
 
Providers may even avoid the concept of safety entirely. They may emphasize the "need" to take risk in order to achieve the higher investment returns "necessary" to achieve one's retirement goals. Rarely do they acknowledge that these goals can be funded without taking any investment risk at all by using risk-free assets (TIPS). Yes, a safe glide path may require a higher contribution rate to achieve the same ending value as expected of a risky glide path, but that is simply the trade-off for safety. The point is that plan participants do not "need" higher, and riskier, returns any more than they "need" their car to have long tail fins, a chrome dashboard, and a super-high-powered engine. Many will want this, and this is fine if they are willing to accept the risk, but what they actually need is a safe mode of financial transportation to their retirement date.
 
Lack of disclosure. Like the automobile industry of the 1960s, there seems to be little economic incentive for TDF manufacturers and others involved in their distribution networks to highlight product safety issues to prospective buyers.
 
Even though the methodology behind a manufacturer's glide path styling may be advertised, the amount of post-retirement consumption that it seeks to deliver may not be so readily available. Is the fund's goal to provide post-retirement consumption equal to 40 percent of final year salary? Sixty percent? What contribution rate is required in order to meet the goal? Does the glide path even have a goal?
 
Even if the objectives are fully disclosed, the degree of risk inherent in achieving them may not be. Such disclosures are clearly in the participant's interest.6
 
Added cost and reduced profitability. Guarantees are not without cost, and manufacturers may be reluctant to add them if they think the added cost may reduce sales revenue or cut into profit margins.
 
Consider also how the industry typically charges fees. Fund manufacturers typically earn fees based on a percentage of assets under management. The rate of return earned on these assets directly affects their value. Thus, the rate of return directly affects the manufacturer's profits. Manufacturers therefore have an economic incentive to incorporate risky assets into their glide paths. Due to the risk premium, they may expect higher cumulative returns to the fund over time, leading to higher revenues for themselves.
 
Ordinarily, the economic incentive of the manufacturer to favor risky assets would be balanced by the fact that informed investors would only invest in such a fund if it coincided with their tolerance for risk taking. Recall, however, that (1) many plan participants are not well-informed investors and (2) TDFs are marketed to those who find the task of portfolio selection too difficult, unpleasant, or time-consuming to do themselves. As a result, defined contribution plan assets are considered relatively sticky. This is particularly true of the assets invested in a plan's default investment option. Knowing that these assets are sticky, a manufacturer might be tempted to accept the risk of substantially poor performance during bad markets for the expectation of higher asset valuations (and therefore higher revenues) in the infinitely long run. Because the assets are sticky, the risk of poor performance to the manufacturer in terms of the potential for lost assets and reduced revenue is limited. However, the risk of poor performance to the participant in terms of a reduced post-retirement standard of living is always borne in full.
 
This is not to say that fund manufacturers act contrary to the interest of their investors. It is only to say that disincentives may exist to offering safe glide paths as an alternative.
 
Deflecting the responsibility/blame. Manufacturers and distributors of TDFs have been called to task on the bear market performance of their products. It is not always easy to differentiate between those responses that truly address the issue and those that deflect it. For example, a question seeking justification for the allocation to risky assets may be deflected by pointing out how unusually the market behaved in 2008. While this may be true, it does not answer the question.
 
Inattention to the scientific body of knowledge. There is reason to worry that some practitioners may have either misunderstood or otherwise ignored the scientific body of knowledge in the area of lifecycle finance. One area of particular concern has to do with risk measurement. Consider the following quote from the Investment Company Institute (the italics are ours):

Some providers place higher priority on producing immediate income and preserving assets at retirement age, while others emphasize the need to earn higher returns at and after retirement age to increase assets and generate income later in retirement to address longevity risk.7

The italicized portion of this statement is startling. Yet, its sentiment is reflected in numerous marketing and educational materials claiming that risky portfolios are necessary in the post-retirement period because people often live for another 30 to 40 or more years and therefore "need" to earn higher returns as a means to "address" (reduce, manage) longevity risk. Such statements are frequently supported by the results of Monte Carlo simulation showing that for higher post-retirement withdrawal rates, increasing the equity allocation can increase the probability that the withdrawals can be sustained over long horizons. The trouble is that probability-based risk measures such as this have been shown to be deficient and misleading. They are dangerous because they do not fully measure risk in financial terms. Reliance on such measures can lead to unexpectedly bad outcomes for investors. We examine this issue further in the next section.

Traps, Fallacies, and Worst-Practices

It is incumbent on TDF manufacturers and the glide path engineers that they employ to understand the nature of risk in the capital markets. The academic literature on this topic is rich. A few of the traps that lie in wait for unsuspecting engineers include the false notion that stocks are an effective hedge against inflation, the fallacy of time diversification of risk, and reliance on probability statistics as a measure of risk.
 
We are also concerned that investors today must make decisions without a standardized means of evaluating the risk/ reward trade-off of the many choices available to them.
 
Stocks as a hedge against inflation. While many have argued that investors with long time horizons should own stocks as a means of hedging against inflation, there is no evidence that stocks offer an effective hedge, even in the long run. In fact, empirical studies show that stock returns are largely uncorrelated with inflation.8 Not only that, but stocks have often performed very poorly during periods of high inflation, such as experienced in the 1970s. The idea that stocks should be included in a glide path as an effective hedge against inflation is not justified by the facts.
 
The fallacy of time diversification. The idea that the risk of holding risky assets somehow decreases with the length of the holding period has perhaps been around as long as investing itself. That this is a fallacy is well documented.9 A simple way to understand this is to consider the riskiness of an asset, or portfolio of assets, in terms of the cost to insure that it will earn at least the risk-free rate of return over time. Bodie (1995) shows that the cost of this insurance increases with the time horizon, and the empirical evidence supports this conclusion. Such insurance can be replicated by purchasing a put option, and the actual prices of put options traded in the capital markets do in fact increase with the length of their horizons.
 
Reliance on probability statistics as a measure of risk. Probability theory has strongly influenced modern economics, including the area of lifecycle finance. In fact, we can learn much about its application—and limitations—from its 17th century founders, Blaise Pascal and Pierre de Fermat. It was Pascal who so famously reasoned that knowing the probability of an event was not enough. The consequences of the event matter, too. Thus, risk has two dimensions. One involves the probabilities of certain events. The other involves the consequences of those events. In terms of the defined contribution plan objective, we can think of this decomposition of risk in terms of (1) the probability that a given funding level objective will be met and, if not, then (2) the magnitude by which it could fall short of its objective. Any risk measure that does not address both dimensions is flawed.10 Risk measurement is not the same thing as probability measurement.
 
Consider the argument that increasing the level of equities in the glide path can increase the probability of meeting its objective, or as the ICI statement suggests, can "address longevity risk." This is an argument that is frequently given in support of higher equity allocations at the target date and thereafter in the post-retirement period. Note that whether or not this probability-based argument is true depends on the assumed rate of post-retirement consumption to be funded by (withdrawn from) the portfolio. Studies by Kaplan (2006) and Fullmer (2009) reveal that increasing the level of equity allocations can increase the probability of sustaining an inflation-adjusted spending plan, but only at withdrawal rates so high as to be relatively unsafe no matter what asset allocation is used! The opposite is true at lower (that is, safer) withdrawal rates, in which case, conservative portfolios provide a higher probability of success than do riskier portfolios. What is going on here is simple. At low withdrawal rates, conservative portfolios can provide the necessary cash flows with little risk to achieving them, while riskier portfolios only add to the chance that a severe downturn in the markets could result in financial ruin at some point in the future. At high withdrawal rates, however, conservative portfolios can scarcely hope to provide the additional returns necessary to sustain the higher cash flows. Instead, the investor must "roll the dice" with a risky portfolio in the hope that future equity markets will be favorable.11
 
The argument that equities can address longevity risk already appears weak—and so far, we have only considered the probability dimension of risk. It looks even weaker when we also consider the magnitude dimension. For any given withdrawal rate, if an investor has the good fortune to retire into a long bull market, her portfolio might never be entirely consumed during her lifetime. If, instead, she has the misfortune to retire into a bear market, her portfolio might run dry. The higher the equity allocation, the sooner this financial ruin could occur. This is true regardless of the selected withdrawal rate. Obviously, running out of money sooner is worse than running out of money later. The magnitude matters. Even in terms of their ability to sustain a lifetime income stream, risky portfolios are still risky.
 
Lack of standards. Nader highlighted a concern with the lack of standardization in the shifting patterns of cars with automatic transmissions. Some used a pattern of "P R N D L" that put Reverse between Park and Neutral, with Low at the bottom. Others used a pattern of "P N D L R" that put Reverse at the bottom, just after Low. A driver of the second type of transmission who was used to the first type of transmission and who wanted to put the car into its lowest gear could accidentally shift into Reverse, sending the car backward unexpectedly. The same thing could happen to a driver who was used to manual transmissions, which typically also placed the lowest gear at the bottom.
 
Because of a lack of standardization and disclosure, a similar type of misunderstanding could easily happen to plan participants who move from one TDF to another. These moves are likely to be common, for example, when a participant changes jobs and is presented with a new DC plan that offers a different set of TDFs. The investor could easily move from a relatively safe TDF to a relatively risky TDF and not even realize it.

What Can Be Done?

These issues make the evaluation of TDF suitability difficult for policymakers and financial advisers. We offer the following perspectives on the subject.
 
One set of issues has to do with fund objectives. For one thing, they should be fully disclosed to investors, and these disclosures should use a common base for measurement. The logical base measure is that of a deferred real life annuity contract starting at the target date. This way, plan sponsors, participants, and their advisers could readily compare the objective of the various TDFs available to them to assess their suitability. In addition, the important assumptions used in creating the glide path should also be disclosed, such as the contribution rate expected of the participant.
 
Another set of issues has to do with risk measurement. Currently, there is no generally accepted standard for measuring target-date fund risk. This has led to much uncertainty over just how risky they are. A standard is needed, one that meets the criteria discussed earlier. Treussard (2007) provides the mathematical foundation for the attributes required of this risk measure. It must be risk-neutral, addressing both the probability and magnitude dimensions of risk. The method for constructing this risk measure is to evaluate the cost of an insurance policy that guarantees a fund will meet its investment objective at the target date.12 Such a risk-neutral measure provides a complete and unbiased representation of the riskiness of any TDF. It allows for a direct comparison of the relative riskiness of one TDF to another. Additionally, it allows for a direct comparison of the riskiness of a particular TDF relative to a safe glide path.
 
Establishing fundamentally sound standards for (1) measuring the investment objective and (2) measuring its risk provides plan sponsors, participants, and their advisers with the means to evaluate their options and make informed decisions. Depending on their risk aversion, some will prefer safe TDFs while others will prefer riskier TDFs.
 
In the same way that standards exist for food safety and labeling, so too standards can be developed for QDIA safety and labeling. The former standards aid society by promoting better dietary decision-making, leading to better physical health. The latter standards aid society by promoting better investment decision-making, leading to better financial health.

Endnotes

  1. The funding level is defined as the amount of wealth necessary at the target date to provide the amount of post-retirement consumption that is sought.
  2. We include the qualifier "in the theoretical construct" because these TIPS may not be available in the necessary durations. This does not limit the usefulness of such a portfolio as a baseline for risk measurement, as we propose later. Additionally, this statement assumes that mortality projections, which typically anticipate a level of mortality improvement, are stable over the investment lifecycle. Longevity swaps are designed to hedge such risk.
  3. The safe portfolio is discussed further in Bodie and Treussard (2007). Recall that our objective is stated as a real annuity commencing at the target date. The price of this annuity depends on real interest rates. Because interest rates are stochastic, it is not possible to prepackage a perfectly duration-matched safe glide path. Duration matching can be managed dynamically, however.
  4. This consumption occurs with after-tax dollars and therefore depends on the system of taxation in place once the investor retires. In the typical case of a progressive income tax, if doubling the contribution rate puts the investor into a higher tax bracket, then the effect on the investor's post-retirement consumption level will be less than double.
  5. This is because there is a tendency for uninformed investors to overinvest in company stock (exposing them to a diversifiable risk) or in money market funds (exposing them to reinvestment risk).
  6. Funds and fund disclosures are regulated by governmental and self-regulatory bodies such as the Financial Industry Regulatory Authority (FINRA). Any new types of disclosures would have to be approved by these bodies.
  7. From the Investment Company Institute's "Frequently Asked Questions About Target Date or Lifecycle Funds" (June 2009): http://www.ici.org/faqs.
  8. See, for example, Bodie (1976).
  9. This fallacy is explained in Samuelson (1963, 1971, 1989, 1994), Merton and Samuelson (1974), and Bodie (1995).
  10. This point is explained further in Harlow (1991), Treussard (2007), Fullmer (2007), and Fullmer (2009).
  11. The idea that a QDIA might be designed under an assumption that the investor should engage in such a gamble with their future financial security is unsettling. TDFs that qualify as QDIAs should be constructed using reasonably safe withdrawal rate or annuitization assumptions.
  12. Merton and Perold (1993) and Bodie (1995) show that this insurance is replicated by a put option on the portfolio. The risk may therefore be measured using option pricing models.

References

Bodie, Zvi. 1976. "Common Stocks as a Hedge Against Inflation." Journal of Finance 31, 2 (May): 459–470.

Bodie, Zvi. 1995. "On the Risk of Stocks in the Long Run." Financial Analysts Journal 51, 3 (May–June): 18–22.

Bodie, Zvi, and Jonathan Treussard. 2007. "Making Investment Choices as Simple as Possible, but Not Simpler." Financial Analysts Journal 63, 3 (May–June): 42–47.

Fullmer, Richard K. 2007. "Modern Portfolio Decumulation: A New Strategy for Managing Retirement Income." Journal of Financial Planning 20, 8 (August): 40–51.

Fullmer, Richard K. 2009. "Mismeasurement of Risk in Financial Planning." Russell Research Report (October).

Harlow, W. V. 1991. "Asset Allocation in a Downside Risk Framework." Financial Analysts Journal 47, 5 (September–October): 28–40.

Kaplan, Paul. 2006. "Asset Allocation with Annuities for Retirement Income Management." The Journal of Wealth Management 8, 4 (Spring): 27–40.

Merton, Robert C., and André F. Perold. 1993. "Theory of Risk Capital in Financial Firms." Journal of Applied Corporate Finance 6, 3 (Fall): 16–32.

Merton, Robert C., and Paul A. Samuelson. 1974. "Fallacy of the Log-Normal Approximation to Optimal Portfolio Decision-Making over Many Periods." Journal of Financial Economics 1, 1 (May): 67–94.

Nader, Ralph. 1965. Unsafe at Any Speed: The Designed-In Dangers of the American Automobile. New York: Grossman Publishers.

Samuelson, Paul A. 1963. "Risk and Uncertainty: A Fallacy of Large Numbers." Scientia (April–May): 1–6.

Samuelson, Paul A. 1971. "The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling." Proceedings of the National Academy of Science 68, 10 (October): 2493–2496.

Samuelson, Paul A. 1989. "The Judgment of Economic Science on Rational Portfolio Management: Timing and Long-Horizon Effects." Journal of Portfolio Management 16, 1 (Fall): 4–12.

Samuelson, Paul A. 1994. "The Long-Term Case of Equities." Journal of Portfolio Management 21, 1 (Fall): 15–24.

Treussard, Jonathan. 2007. "The Non-monotonicity of Value-at-Risk and the Validity of Risk Measures over Different Horizons." The ICFAI Journal of Financial Risk Management 4, 1 (March): 7–18.