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Strategic Asset AllocationPortfolio Choice for Long-Term Investors$

John Y. Campbell and Luis M. Viceira

Print publication date: 2002

Print ISBN-13: 9780198296942

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198296940.001.0001

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Introduction

Introduction

Chapter:
(p.1) 1 Introduction
Source:
Strategic Asset Allocation
Author(s):

John Y. Campbell (Contributor Webpage)

Luis M. Viceira (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/0198296940.003.0001

Abstract and Keywords

The mean‐variance paradigm has the strong implication that all investors should hold risky assets in the same proportion. Financial planners typically advise conservative investors to tilt their risky portfolios towards bonds and away from stocks; this has been called the “asset allocation puzzle” since it contradicts standard mean‐variance analysis. Financial planners also argue that long‐term investors can afford greater exposure to stock market risk. This book will show how financial planners’ advice can be justified by an inter‐temporal model of a rational investor. The model ignores some important real‐world issues, including diversification of individual stocks, transactions costs, taxation, and the biases identified by research in behavioural finance.

Keywords:   behavioural finance, diversification, equity premium, financial planning, inter‐temporal model, mean‐variance analysis, risk aversion, taxes, transaction costs

One of the most important decisions many people face is the choice of a portfolio of assets for retirement savings. These assets may be held as a supplement to defined‐benefit public or private pension plans; or they may be accumulated in a defined‐contribution pension plan, as the major source of retirement income. In either case, a dizzying array of assets is available.

Consider for example the increasing set of choices offered by TIAA–CREF, the principal pension organization for university employees in the United States. Until 1988, the two available choices were TIAA, a traditional nominal annuity, and CREF, an actively managed equity fund. Funds could readily be moved from CREF to TIAA, but the reverse transfer was difficult and could be accomplished only gradually. In 1988 it became possible to move funds between two CREF accounts, a money market fund and an equity fund. Since then other choices have been added: a bond fund and a socially responsible stock fund in 1990, a global equity fund in 1992, equity index and growth funds in 1994, a real estate fund in 1995, and an inflation‐indexed bond fund in 1997. Retirement savings can easily be moved among these funds, each of which represents a broad class of assets with a different profile of returns.

Institutional investors also face complex decisions. Some institutions invest on behalf of their clients, but others, such as foundations and university endowments, are similar to individuals in that they seek to finance a long‐term stream of discretionary spending. The investment options for these institutions have also expanded enormously since the days when a portfolio of government bonds was the norm.

Mean–Variance Analysis

What does financial economics have to say about these investment decisions? Modern finance theory is often thought to have started with the mean–variance analysis of Markowitz (1952); this (p.2)

                   Introduction

Figure 1.1. Mean–standard deviation diagram.

makes portfolio choice theory the original subject of modern finance. Markowitz showed how investors should pick assets if they care only about the mean and variance—or equivalently the mean and standard deviation—of portfolio returns over a single period.

The results of his analysis are shown in the classic mean–standard deviation diagram, Figure 1.1. (A much more careful mathematical explanation can be found in the next chapter.) For simplicity the figure considers three assets: stocks, bonds, and cash (not literally currency, but a short‐term money market fund). The vertical axis shows expected return, and the horizontal axis shows risk as measured by standard deviation. Stocks are shown as offering a high mean return and a high standard deviation, bonds a lower mean and lower standard deviation. Cash has a lower mean return again, but is riskless over one period, so it is plotted on the vertical zero‐risk axis. (In the presence of inflation risk, nominal money market investments are not literally riskless in real terms, but this short‐term inflation risk is small enough that it is conventional to ignore it. We follow this convention here and return to the issue in the next chapter.)

The curved line in Figure 1.1 shows the set of means and standard deviations that can be achieved by combining stocks (p.3) and bonds in a risky portfolio. When cash is added to a portfolio of risky assets, the set of means and standard deviations that can be achieved is a straight line on the diagram connecting cash to the risky portfolio. An investor who cares only about the mean and standard deviation of his portfolio will choose a point on the straight line illustrated in the figure, that is tangent to the curved line. This straight line, the mean–variance efficient frontier, offers the highest mean return for any given standard deviation. The point where the straight line touches the curved line is a ‘tangency portfolio’ of risky assets, marked in the figure as ‘Best mix of stocks and bonds’.

The striking conclusion of this analysis is that all investors who care only about mean and standard deviation will hold the same portfolio of risky assets, the unique best mix of stocks and bonds. Conservative investors will combine this portfolio with cash to achieve a point on the mean–variance efficient frontier that is low down and to the left; moderate investors will reduce their cash holdings, moving up and to the right; aggressive investors may even borrow to leverage their holdings of the tangency portfolio, reaching a point on the straight line that is even riskier than the tangency portfolio. But none of these investors should alter the relative proportions of risky assets in the tangency portfolio. This result is the mutual fund theorem of Tobin (1958).

Financial Planning Advice

Financial planners have traditionally resisted the simple investment advice embodied in Figure 1.1. This resistance may to some extent be self‐serving; as Peter Bernstein points out in his 1992 book Capital Ideas, many financial planners and advisors justify their fees by emphasizing the need for each investor to build a portfolio reflecting his or her unique personal situation. Bernstein calls this the ‘interior decorator fallacy’, the view that portfolios should reflect personal characteristics in the same way that interior decor reflects personal taste.

There are however many legitimate reasons why different portfolios of risky assets might be appropriate for different investors. The complexity of the tax code creates many differences across investors, not only in their tax brackets but also in their opportunities for sheltering income from taxation. Beyond this, investors differ in their investment horizons: some may have (p.4) relatively short‐term objectives, others may be saving to make college tuition payments in the medium term, yet others may be saving for retirement or to ensure the well‐being of their heirs. Investors also differ in the characteristics of their labor income: young investors may expect many years of income, which may be relatively safe for some and risky for others, while older investors may need to finance the bulk of their consumption from accumulated financial wealth. Investors often have illiquid assets such as family businesses, restricted stock options granted by their employer, or real estate.1 We shall argue in this book that the traditional academic analysis of portfolio choice needs to be modified to handle long investment horizons, labor income, and illiquid assets. The modified theory explains several of the patterns that we see in conventional financial planning advice.

One strong pattern is that financial planners typically encourage young investors, with a long investment horizon, to take more risk than older investors. The single‐period mean–variance analysis illustrated in Figure 1.1 assumes a short investment horizon. In this book we shall explore the conditions under which a long investment horizon does indeed justify greater risk‐taking.

A second pattern in financial planning advice is that conservative investors are typically encouraged to hold more bonds, relative to stocks, than aggressive investors, contrary to the constant bond–stock ratio illustrated in Figure 1.1. Canner, Mankiw, and Weil (1997) call this the asset allocation puzzle. Table 1.1, which reproduces Table 1 from Canner et al.'s article, illustrates the puzzle. The table summarizes model portfolios recommended by four different investment advisors in the early 1990s: Fidelity, Merrill Lynch, the financial journalist Jane Bryant Quinn, and the New York Times. While the portfolios differ in their details, in every case the recommended ratio of bonds to stocks is (p.5)

Table 1.1. The Asset Allocation Puzzle of Canner et al. (1997)

Advisor and investor type

% of portfolio

Ratio of bonds to stocks

Cash

Bonds

Stocks

(A) Fidelity

Conservative

50

30

20

1.50

Moderate

20

40

40

1.00

Aggressive

5

30

65

0.46

(B) Merrill Lynch

Conservative

20

35

45

0.78

Moderate

5

40

55

0.73

Aggressive

5

20

75

0.27

(C) Jane Bryant Quinn

Conservative

50

30

20

1.50

Moderate

10

40

50

0.80

Aggressive

0

0

100

0.00

(D) New York Times

Conservative

20

40

40

1.00

Moderate

10

30

60

0.50

Aggressive

0

20

80

0.25

higher for moderate investors than for aggressive investors, and higher again for conservative investors.

One possible explanation for this pattern of advice is that aggressive investors are unable to borrow at the riskless interest rate, and thus cannot reach the upper right portion of the straight line in Figure 1.1. In this situation, aggressive investors should move along the curved line, increasing their allocation to stocks and reducing their allocation to bonds. The difficulty with this explanation is that it only applies after the constraint on borrowing has started to bind on investors, that is, once cash holdings have been reduced to zero; but the bond–stock ratio in Canner et al.'s Table 1 varies even when cash holdings are positive.2

(p.6) Long‐Term Portfolios

This book argues that it is possible to make sense of both the asset allocation puzzle and the tendency of financial planners to recommend riskier portfolios to young investors. The key is to recognize that optimal portfolios for long‐term investors need not be the same as for short‐term investors. Long‐term investors, who value wealth not for its own sake but for the standard of living that it can support, may judge risks very differently from short‐term investors. Cash, for example, is risky in the long term even though it is safe in the short term, because cash holdings must be reinvested in the future at unknown real interest rates. Inflation‐indexed bonds, on the other hand, provide a known stream of long‐term real payments even though their capital value is uncertain in the short term. There is considerable evidence that stocks, too, can support a stable standard of living more successfully than their short‐term price variability would indicate. For these reasons, a long‐term investor may be willing to hold more stocks and bonds, and less cash, than a short‐term investor would do; and a conservative long‐term investor may hold a portfolio that is dominated by bonds rather than cash.

Labor income is also important for long‐term investors. One can think of working investors as implicitly holding an asset—human wealth—whose dividends equal labor income. This asset is non‐tradable, so investors cannot sell it; but they can adjust their financial asset holdings to take account of their implicit holdings of human wealth. For most investors, human wealth is sufficiently stable in value to tilt financial portfolios towards greater holdings of risky assets.

At a theoretical level, these points have been understood for many years. Samuelson (1963, 1969), Mossin (1968), Merton (1969), and Fama (1970) first described the restrictive conditions under which long‐horizon investors should make the same decisions as short‐horizon investors. Modigliani and Sutch (1966) asserted that long‐term bonds are safe assets for long‐term investors, and Stiglitz (1970) and Rubinstein (1976a,b) built (p.7) rigorous theoretical models that validate and illustrate this point. Mayers (1972) and Fama and Schwert (1977b) asked how human wealth influences portfolio choice. The seminal work of Merton (1971, 1973) provided a general framework for understanding the portfolio demands of long‐term investors when investment opportunities vary over time. Rubinstein (1976b) and Breeden (1979) showed how Merton's results could be interpreted in terms of consumption risk, an idea that has had major influence in macroeconomics through the work of Lucas (1978), Grossman and Shiller (1981), Shiller (1982), Hansen and Singleton (1983), Mehra and Prescott (1985), and others.

Until recently, however, empirical work on long‐term portfolio choice has lagged far behind the theoretical literature. Perhaps for this reason, there has been very slow diffusion of understanding from the academic literature to institutional investors, asset managers, financial planners, and households. Most MBA courses, for example, still teach mean–variance analysis as if it were a universally accepted framework for portfolio choice.

One reason for the slow development of the field has been the difficulty of solving Merton's intertemporal model. Closed‐form solutions to the model have been available only in a few special cases, in which it turns out that long‐term portfolios should be the same as short‐term portfolios. Recently this situation has begun to change as a result of several related developments. First, computing power and numerical methods have advanced to the point at which realistic multi‐period portfolio choice problems can be solved numerically using discrete‐state approximations.3 Second, financial theorists have discovered some new closed‐form solutions to the Merton model.4 Third, approximate (p.8) analytical solutions have been developed (Campbell and Viceira 1999, 2001a). These solutions are based on perturbations of known exact solutions, so they are accurate in a neighborhood of those solutions. They offer analytical insights into portfolio choice in models that fall outside the still limited class that can be solved exactly. This book offers examples of all three approaches to long‐term portfolio choice, but the major emphasis is on approximate analytical solution methods and on the development of empirical models that can be used by practitioners as well as academics.

Utility Theory and Behavioral Finance

Throughout the book, we derive portfolio decisions using standard finance models of preferences. We assume that investors derive utility from consumption, which in turn is supported by financial wealth (and possibly labor income). The utility function takes a standard form in which relative risk aversion is constant.5 Our contributions are in modelling the financial environment, and developing the link between that environment and portfolio choice, rather than in modelling investor preferences.

Some investment practitioners are uncomfortable with the academic use of utility functions. However, investor preferences—in particular impatience and aversion to risk—do influence optimal portfolios. Students of portfolio choice have no alternative but to model preferences, no matter how hard that task may be. We regard utility functions as a convenient way to capture intuitive notions such as the distinction between conservative and aggressive investors. While it may be hard to interpret numbers such as the coefficient of relative risk aversion in the abstract, our experience is that these numbers are easy to use and to understand in the context of specific empirical models.

(p.9) Recently there has been great interest in the field of behavioral finance, which postulates that some investors have non‐standard preferences. Often these preferences are modelled using experimental evidence from psychology. The prospect theory of Kahneman and Tversky (1979), for example, models utility as depending on wealth relative to a reference point; the utility function has a kink at the reference point, and different patterns of curvature above and below this point. The theory of mental accounting (Thaler 1985) goes further by suggesting that investors care directly about the values of individual stocks in their portfolios rather than their total wealth. These and many other behavioral models are surveyed by Shiller (1999).

We regard behavioral finance as a promising research area with the potential to explain some types of investor behavior and possibly some patterns in asset pricing. However, we do not believe that it provides a sound basis for a normative theory of asset allocation. First, the experimental evidence that motivates many behavioral models is based on individuals' reactions to risks that are necessarily small. It is impossible to design an experiment that subjects individuals to the large risks they face when saving over a lifetime. Standard theory probably applies better to large risks, with potentially serious consequences for lifetime well‐being, than to small risks in an experimental setting.6 Second, even if behavioral finance describes how investors actually do behave, it may not describe how they should behave. That is, investors may abandon their behavioral biases once they have the benefit of financial education and financial planning advice.7

More specifically, we believe that any normative model should judge a portfolio by its total value, rather than by the values of (p.10) the individual assets it contains, and should ultimately be based on the standard of living that the portfolio supports. That is, both wealth and expected asset returns are relevant for an investor because these variables determine the consumption that the investor can afford. This normative use of standard finance theory does not contradict behavioral finance as a positive description of investor behavior; in fact, the motivation for a normative analysis is much stronger if investors are subject to behavioral biases than if they are already successfully making optimal portfolio decisions.

The Equity Premium Puzzle

Our thesis in this book is that long‐term portfolio choice theory can be implemented empirically to deliver specific quantitative advice about asset allocation. To illustrate the potential of the theory, we solve for the portfolios that are implied by particular preference parameters, together with particular beliefs about the stochastic processes driving asset returns and income. Although our framework can be used by investors with arbitrary beliefs, for illustrative purposes we want to assume beliefs that are reasonable given historical data. A natural approach is to estimate a parametric model for asset returns on historical data, and then to use the estimated parameters to calculate optimal portfolios. In Chapter 3 we use an affine term structure model in this way, in Chapter 4 we use a vector autoregression, and in Chapter 5 we use a stochastic volatility model.

This approach runs into difficulties when we compute the optimal holdings of equities. Historical average returns on US equities have been extraordinarily high relative to returns on bills or bonds, and equities are correspondingly attractive to investors who believe that these returns will continue in the future. Investors with low or moderate risk aversion will hold highly leveraged equity portfolios if they are permitted to do so, and their consumption will be extremely volatile. Only the most conservative investors will shy away from equity risk, given historically based beliefs about stock returns. Many of the considerations we emphasize in this book, such as mean‐reversion in stock returns (Chapter 4) and the need to take account of labor income (Chapters 6 and 7), only increase investors' enthusiasm for stocks. Factors that might dampen this enthusiasm, such as (p.11) volatility risk and uncertainty about average returns (Chapter 5), appear to be somewhat less important empirically.

This raises the question of how such high average stock returns can be consistent with equilibrium in the stock market. Why don't enthusiastic investors bid up the prices of stocks so that their average returns fall to more reasonable levels? In the asset pricing literature, this issue is known as the equity premium puzzle. Mehra and Prescott (1985) first identified high average stock returns as a puzzle, building on the theoretical work of Rubinstein (1976b), Lucas (1978), Breeden (1979), and Grossman and Shiller (1981). Economists have worked intensively on the puzzle for over 15 years but have not agreed on an explanation.8 Certainly the factors discussed in this book tend to exacerbate the puzzle rather than solve it, as pointed out by Campbell (1996).

One reasonable response to the equity premium puzzle is that historical US data are a misleading guide to future stock returns. Perhaps the twentieth century was an unusually lucky period in which favorable shocks produced stock returns that were well above the true long‐run average. Economists such as Blanchard (1993), Campbell and Shiller (2001), Fama and French (2000), and Jagannathan, McGrattan, and Scherbina (2001) have all argued that average stock returns are likely to be lower in the future than they have been in the past.

We do not try to settle the issue in this book; it is something that each investor must decide for himself. Instead, we illustrate alternative responses to the puzzle. In Chapters 3, 4, and 5 we use historically estimated parameters without modification. In Chapters 6 and 7, by contrast, we assume a fixed equity premium of 4%, considerably below the historical average but still large enough to make equities an attractive asset class for long‐term investors.

Limitations of the Book

This book has several limitations that should be acknowledged at the outset. First, we provide an analysis of asset allocation rather than a complete analysis of portfolio choice. That is, we work with broad asset classes such as cash, equities, inflation‐indexed (p.12) bonds, and nominal bonds, and we say nothing about the choice of individual assets within these broad classes. This is not because the choice of individual assets is unimportant—in fact, the benefits of diversification are enormous—but because we have nothing to add to the traditional mean–variance analysis on this subject.

In effect, we assume that investors have already formed diversified portfolios within each asset class and face a final decision about the amounts to invest in each portfolio. Thus, our results can help individual investors allocate their retirement savings among broadly diversified mutual funds, but not investors trying to pick individual stocks. Similarly, our results are relevant for pension funds and endowments determining their strategic policy portfolios, but are not directly relevant for active asset managers who are judged on their short‐term performance relative to a benchmark index within their asset class.

A second limitation is that we do not consider international issues. In our empirical work, we adopt a traditional closed‐economy perspective and look at US data to measure the returns on asset classes. Our models could be applied without difficulty to an integrated global economy, in which each asset class contains asset from different countries. We do not explore models with international state variables such as real exchange rates.

Third, for most of the book we assume that investors know with certainty the parameters of the process generating asset returns, including the means, variances, and covariances of returns and the evolution of these moments over time. As we have already discussed, either we can assume that these parameters equal historically estimated values (as we do in Chapters 3, 4, and 5) or we can assume arbitrary parameters reflecting beliefs that the future will differ from the past (as we do in Chapters 6 and 7). It is more challenging to incorporate parameter uncertainty into the asset allocation decision. If an investor anticipates that he will learn by observing asset market data, changing his beliefs in response to returns, this introduces a new type of intertemporal hedging demand into the portfolio decision. We provide a very brief introduction to the literature on optimal learning in Chapter 5, but can do no more than scratch the surface of this important topic.

(p.13) Fourth, we assume that assets can be traded without cost. Transaction costs are extremely important for investment strategies that require frequent trading, but are less critical for the long‐term asset allocation decisions that are the subject of this book.

Fifth, we ignore taxation. Our results apply directly to tax‐exempt investors such as endowments or foundations. In a setting where each asset class return has a given tax rate, which is not affected by the length of time the asset class is held, our results apply to taxable investors if we model after‐tax returns rather than pre‐tax returns. It is much more difficult to handle taxes like the capital gains tax, whose burden depends on whether an asset is held or sold.9

Sixth, we have nothing to say about housing. For many individuals this is an important asset that plays multiple roles: it is a large component of wealth, a fixed commitment that is costly to reverse, and a source of housing services and tax benefits. Real estate is equally important in the portfolios of many institutional investors.

Finally, we present a series of partial models rather than a single integrated model that captures all the important aspects of strategic asset allocation. In the first part of the book, for example, we consider time‐varying investment opportunities but ignore labor income, while in the last two chapters we assume fixed investment opportunities and concentrate our efforts on realistically modelling labor income. We follow this approach both because it delivers the clearest insights about each aspect of strategic asset allocation, and because the academic literature has not yet developed a reliable, generally accepted model of the complete portfolio problem.

For all these reasons, our book is no substitute for traditional financial planning advice. Nor do we provide cookbook formulas for optimal portfolio weights. Instead, we try to explain carefully what it means to have a long investment horizon. Our objective is to provide a rigorous, empirically relevant framework that (p.14) long‐term investors can use to understand the trade‐off of risk and return.

Organization of the Book

The rest of the book is organized as follows. Chapter 2 first reviews the traditional mean–variance analysis, showing how it can be founded on utility theory. The chapter argues that the benchmark model of utility should assume that relative risk aversion is independent of wealth. With this assumption, there are well‐known conditions under which long‐term investors should invest myopically, choosing the same portfolios as short‐term investors. Chapter 2 explains these conditions, due originally to Merton (1969) and Samuelson (1969). Myopic portfolio choice is optimal if investors have no labor income and investment opportunities are constant over time. If investors have relative risk aversion equal to one, then myopic portfolio choice is optimal even if investment opportunities are time‐varying. Although these conditions are simple, they are widely misunderstood, and the chapter makes an effort to address fallacies that commonly arise in popular discussion.

Legitimate arguments for horizon effects on portfolio choice depend on violations of the Merton–Samuelson conditions. Chapters 3–7 explore such violations. Chapter 3 argues that investment opportunities are not constant because real interest rates move over time. Even if expected excess returns on risky assets over safe assets are constant, time variation in real interest rates is enough to generate large differences between optimal portfolios for long‐term and short‐term investors. The chapter shows that conservative long‐term investors should hold portfolios that consist largely of long‐term bonds. These bonds should be inflation‐indexed if possible; however, nominal bonds may be adequate substitutes for inflation‐indexed bonds if inflation risk is modest, as it has been in the United States since the early 1980s.

The assumption of constant risk premia in Chapter 3 implies that optimal portfolios are constant over time for both short‐term and long‐term investors. Chapter 4 allows for time variation in the expected excess returns on stocks and bonds. This generates time variation in optimal portfolios: Both short‐term and long‐term investors should seek to ‘time the markets’, holding more risky assets at times when the rewards for doing so are high. (p.15) But in addition, long‐term investors with relative risk aversion greater than one should increase their average holdings of risky assets whose returns are negatively correlated with the rewards for risk‐bearing; for example, they should increase their average allocation to stocks because the stock market appears to mean‐revert, doing relatively poorly after price increases and relatively well after price declines. These findings are an empirical development of Merton's (1973) theoretical concept of intertemporal hedging by long‐term investors.

Chapter 5 seeks to relate the results of Chapters 3 and 4 more closely to the extensive theoretical literature set in continuous time. Explicit solutions for optimal portfolios are provided in Chapters 3 and 4 by the use of loglinear approximations to discrete‐time Euler equations and budget constraints. Chapter 5 clarifies the conditions under which such approximate solutions hold exactly, and shows how equivalent approximations can be used in continuous time. This chapter also explores optimal portfolio choice in the presence of time‐varying stock market risk and parameter uncertainty. Chapter 5 is the most technically demanding chapter in the book, and less mathematical readers can skip it without loss of continuity.

Chapters 6 and 7 introduce labor income into the long‐term portfolio choice problem. Chapter 6 discusses the effect of human wealth—a claim to a stream of labor income—on the optimal allocation of financial wealth by investors with constant relative risk aversion. Drawing on an important paper by Bodie, Merton, and Samuelson (1992), this chapter first uses a stylized two‐period model to show that exogenous labor income reduces the aversion to financial risk by reducing the proportional sensitivity of consumption to asset returns. If investors can flexibly adjust their labor supply, varying their labor income endogenously in response to circumstances, their aversion to financial risk is further reduced. This chapter also considers the possibility that investors have subsistence levels—required minimum levels of consumption—and that they derive utility only from consumption that exceeds subsistence needs. The chapter shows that subsistence levels work like negative labor income, subtracting utility‐generating resources from financial wealth in the same way that labor income adds resources. Finally, the chapter extends the analysis to a stylized long‐horizon model in which (p.16) employed investors have an exogenous probability of retirement each period.

Chapter 7 embeds labor income in a life‐cycle model and asks how investors should adjust their portfolios as they age. The chapter shows that typical households should be willing to invest heavily in risky financial assets in early adulthood, but should scale back their financial risk‐taking in late middle age. This conclusion is consistent with the typical advice of financial planners. The chapter also shows that different households can have very different risk properties of their labor income, and that this has large effects on optimal portfolio allocations. Self‐employed, college‐educated households, for example, have labor income that is particularly volatile and highly correlated with stock returns; these households should be cautious about equity investments.

Chapter 7 also reviews the existing empirical evidence on how households actually do invest over the life cycle. The evidence is fragmentary, and it is clear that household investment strategies are extraordinarily diverse; but a few patterns do appear consistent with the normative recommendations of the life‐cycle model. For example, there is some tendency for household risk exposure to decline with age in later years, and wealthy households that own private businesses tend to own fewer publicly traded equities than other wealthy households. The chapter ends by drawing general conclusions that summarize the message of this book.

Notes:

(1) Perhaps some of these differences underlie the PaineWebber advertisement that ran in the New Yorker and other magazines in 1998: ‘If our clients were all the same, their portfolios would be too. They say the research has been sifted. The numbers have been crunched. The analysts have spoken: Behold! The ideal portfolio. We say building a portfolio is not “one size fits all”. It begins with knowing you—how you feel about money, how much risk you can tolerate, your hopes for your family, and for your future. By starting with the human element, our Financial Advisors can do something a black box can't do—take the benefits of what PaineWebber has to offer and create an investment plan unique to you.’

(2) Elton and Gruber (2000) reconcile the asset allocations in Table 1.1 with mean–variance analysis by arguing that short‐term investments are risky in real terms, short sales are restricted, and 5% of the portfolio should be held in cash as a special liquidity reserve. One problem with their analysis is that much of the variability in short‐term real returns is predictable, as we show in Chapter 3; thus, it does not represent risk to a short‐term investor. Note that for advisors other than Merrill Lynch, the stock–bond ratio in Table 1.1 varies at cash holdings that are considerably greater than 5%, so a 5% liquidity reserve by itself does not resolve the asset allocation puzzle.

(3) Balduzzi and Lynch (1999), Barberis (2000), Brennan, Schwartz, and Lagnado (1997, 1999), Cocco, Gomes, and Maenhout (1998), and Lynch (2001) are important examples of this style of work.

(4) In a continuous‐time model with a constant riskless interest rate and a single risky asset whose expected return follows a mean‐reverting (Ornstein–Uhlenbeck) process, for example, the model can be solved if long‐lived investors have power utility defined over terminal wealth, or if investors have power utility defined over consumption and the innovation to the expected asset return is perfectly correlated with the innovation to the unexpected return, making the asset market effectively complete, or if the investor has Epstein–Zin utility with intertemporal elasticity of substitution equal to one. We discuss these results in detail in Ch. 5.

(5) We do allow the elasticity of intertemporal substitution to differ from the coefficient of relative risk aversion, following Epstein and Zin (1989), while assuming that both parameters are constant. Also, in Ch. 6 we briefly discuss models in which utility is derived from the difference between consumption and a subsistence level. Such models imply that relative risk aversion is constant if consumption and the subsistence level both increase in proportion, but is decreasing if consumption increases relative to the subsistence level. Gollier and Zeckhauser (1997) and Ross (1999) explore horizon effects in models where utility is defined over terminal wealth and can take a general form.

(6) Consistent with this view, Bowman, Minehart, and Rabin (1999) embed small‐scale behavioral elements in a more standard large‐scale model of preferences.

(7) There is also some recent work in decision theory that develops axiomatic foundations for models with reference points (Gul 1991). Such models have been applied to equilibrium asset pricing problems by Epstein and Zin (1990) and Bekaert, Hodrick, and Marshall (1997b), and to portfolio choice problems by Ang, Bekaert, and Liu (2000). It is conceivable, but in our view unlikely, that such models will displace the constant relative risk‐averse model used in this book as the standard paradigm of finance theory. We believe that a more promising approach is to incorporate reference points within the constant relative risk‐averse paradigm by modelling subsistence levels in the manner discussed in Ch. 6.

(8) Kocherlakota (1996) and Campbell (1999) survey the enormous literature on the equity premium puzzle.

(9) Constantinides (1983, 1984) discusses optimal investment, strategies for taxable investors in the presence of capital gains taxes. Constantinides derives restrictive conditions under which portfolio choice is unaffected by such taxes. Dammon, Spatt, and Zhang (2001) discuss capital gains taxes in a life‐cycle model.