The conventional capital asset pricing model (CAPM) seeks to describe the cost of capital for firms and asset allocation choices by investors. In its canonical formulation, the CAPM relies principally upon the optimization of mean return relative to the variance of the market-wide portfolio (Fama and French, 2004). Among its many flaws, however, the CAPM fails to reflect human behavior (Shefrin and Statman, 1994).

A higher-moment capital asset pricing model may be derived from the Taylor series expansion of the logarithm of expected returns. Higher-moment CAPM (or CAPM+), once paired leading behavioral accounts of economics, explains seemingly “irrational” phenomena such as skewness preference and the bonds-and-bullets structure of financial decision-making.

A four-moment variant of CAPM+ is expressed in terms of mean, variance, skewness, and kurtosis (Jurczenko and Maillet, 2012). It can be derived from the Taylor series expansion of logarithmic returns from a continuously compounded financial series (Harvey and Siddique, 2000, p. 1269; Jondeau and Rockinger, 2006, p. 33; Harvey et al., 2010, pp. 469–470):

where o[(x−μ)⁵] represents the fifth order and other remaining terms.

The formulation in ¶ 3 exhibits an alternating pattern of positive and negative signs. Modest assumptions such as positive marginal utility and decreasing risk aversion support this summary of CAPM+: Humans prefer high values for odd-numbered moments (mean and skewness), but low values for even-numbered moments (variance and kurtosis) (de Athayde and Flôres, 2004, p. 1336; Estrada, 2004, p. 241; Jondeau and Rockinger, 2006; Brunnermeier et al., 2007; Bali et al., 2011, p. 33). This trait enables higher-moment asset pricing models to provide effective guidance in advanced portfolio design and hedged trading applications (Brooks et al., 2012; Knif et al., 2020).

Exploring moments beyond variance explains many of the descriptive failures of conventional financial theory. The welfare implications of higher-moment asset pricing stem from disparate investor reactions to odd- and even-numbered moments. Behavioral departures from strict rationality begin with skewness, the first odd-numbered moment beyond variance. Kurtosis is properly associated with epistemic failures, with the inability to predict (let alone adapt to) previously unobserved phenomena. Consequently, skewness and kurtosis heavily influence environmental and resource economics.

Skewness preference arises when investors privilege skewness (the third moment) over expected return (the first). This departure from conventional rationality may represent the most obvious application of CAPM+. A wide range of behaviors of interest to various bodies of financial regulation reflects skewness preference: lotteries, prize-linked savings, private equity, crowdfunding, and initial public offerings. A preference for skewed outcomes, especially when the expected return is zero or negative, underlies many economic conditions thought to warrant regulatory intervention.

Especially in the cumulative formulation that acknowledges first- and second-order stochastic dominance, prospect theory gives behavioral meaning to skewness preference and its fourth-moment counterpart, leptokurtic blindness or insensitivity (Kahneman and Tversky, 1984; Tversky and Kahneman, 1992). Although Daniel Kahneman and Amos Tversky relied on a two-piece utility function to define prospective theory's value function, the cumulative distribution function of a right-skewed distribution such as the lognormal or the log-logistic illustrates all of that function's important properties.

Figure 1 displays three important properties of human decision-making under uncertainty. First, humans evaluate all decisions according to a fixed reference point. Second, humans are averse toward losses. All else being equal, losing hurts worse than winning feels good. Third, humans over time become less sensitive to changes in utility, no matter whether such changes are gains or losses. “If prospect theory had a flag,” the banner in Figure 1 would depict those three principles (Kahneman, 2011, p. 282).

Although volatility figures prominently in nearly every model in mathematical finance, even-numbered moments are harder to interpret. Starting with variance, however, finite higher moments cannot be assumed. If, as has been hypothesized for nearly six decades, financial returns follow a stable Paretian distribution (Fama, 1963, 1965; Ortobelli and Rachev, 2001), even variance (and, a fortiori, higher moments) may be infinite. This analytically debilitating mathematical property stems from the definition of a generalized Pareto distribution (Castillo and Hadi, 1997; Gençay and Selçuk, 2004, p. 291–292).

Leptokurtosis may be the most tractable statistical representation of tail risk and epistemic blindness. It provides a statistical basis for the longstanding distinction between probabilistic risk and aleatory uncertainty. Leptokurtosis likewise describes prospect theory's phenomenon of diminishing sensitivity at each extreme. These treatments of the fourth moment provide a mathematical bridge between rational and behavioral accounts of economic decision-making. This unity arises because human perception becomes duller precisely where information, as an empirical matter, becomes less attainable.

Combining these insights with behavioral finance explains the prevalence of “bonds and bullets” wealth allocations in numerous economic circumstances. Bifurcating even-numbered moments reveals the mathematical congruence between two seemingly divergent economic instincts. When forced to confront the loss of basic means of survival, humans do focus on downside risk. But once hope meets fear, even risk averse individuals will entertain upside gambles. Merging these insights expands the mathematical toolkit of finance and environmental economics.

Part 2 suggests how higher-moment asset pricing might affect environmental and resource economics. The Taylor series expansion of logarithmic returns counsels against simplistic reliance on the naked magnitude of expected gain or loss. Higher-moment pricing and valuation models reveal the opposite effects of odd- and even-numbered moments. But another boundary looms between mean and variance, on one hand, and the paucal moments of skewness and kurtosis. The most potentially treacherous decisions under uncertainty respond to internal asymmetry and extremity within the distribution of returns.

Ernst Haeckel is credited with introducing the term ecology from OiKOÇ, the ancient Greek word for house (Gould, 1977, p. 76 n.^*). Economics and economy share the same root. By uniting human economy with natural ecology, environmental economics defines housekeeping in both social and biological terms (Caradonna, 2014, pp. 112–113).

Law and policy give voice to the idea of ecology as housekeeping through sustainability and the precautionary principle (Cameron and Abouchar, 1991; McIntyre and Mosedale, 1997; Sand, 2000). The definition of sustainability, at least, is contestable. The narrowest definition of environmental sustainability stems from strict notions of the human ecological footprint, which in turn dictate a definition of sustainability according to physical flows of energy and matter (Heal, 2012).

By contrast, the Hartwick principle holds that renewable environmental resources, non-renewable resources, and capital investments are subject to exchange (Hartwick, 1977). Because the Hartwick principle directly compares physical energy flows with financial returns, it is the starting point for any application of financial economics to environmental topics (Gowdy and McDaniel, 1999).

Contingent valuation of ecosystem services is central to policy regarding climate change mitigation and related questions of natural resource economics (Carson et al., 2001; Champ and Bishop, 2001; Poe et al., 2002). Skewness preference and pricing premiums associated with uncertainty and kurtosis beset ecological valuation.

Perhaps the most striking application of higher-moment asset pricing to the valuation of ecosystem services involves biodiversity conservation, including the politically salient and controversial subfield of bioprospecting (Chen, 2014). A more outlandish instance of skewness preference in natural resource economics can scarcely be imagined. If the logic of bioprospecting is stretched to its absurd extreme, Costa Rica's biodiversity is worth saving only to the extent that endemic organisms with pharmaceutical potential can be profitably exploited.

Disputes over bioprospecting and its rhetorically rude cousin, “biopiracy” illustrate an extension of the rank effect from behavioral finance to resource economics. According to the “rank effect,” investors are likelier to sell their extreme winning and losing positions, even without considering the economic fundamentals of any firm in the portfolio (Hartzmark, 2015). Focusing exclusively on the best and worst positions, wholly without regard to the actual level of returns effectively ignores the rest of the portfolio (ibid.). Biodiversity, to say the least, vastly exceeds the genomic profitability of species of greatest commercial interest to humans.

Disaster law as a specialized branch of environmental law emphasizes downside risk and uncertainty (Chen, 2011). This emphasis highlights an anomaly in environmental economics. Stated preference studies often strive to quantify either respondents' willingness to pay (WTP) for environmental enhancements or their willingness to accept (WTA) compensation for environmental degradation.

Neoclassical economic theory presumes that WTP and WTA quantities should be equivalent. Experimental outcomes suggest otherwise. There is a considerable premium for willingness-to-accept compensation in cases of degradation, relative to willingness-to- pay bids in cases of enhancement (Hanemann, 1991; Shogren et al., 1994; Sayman and Öncüler, 2005). In addition, both WTA and WTP distributions are quite wide, in the sense their standard distributions are quite often multiples of the mean amount. This appears to be an artifact of numerous 0 responses in WTA surveys (Amigues et al., 2002, p. 25) and high bids in WTP surveys (Sillano and de Dios Ortúzar, 2005, p. 540).

The obvious behavioral explanation for the WTA/WTP premium lies in the endowment effect (Kahneman et al., 1990) and the closely related notion of myopic loss aversion (Benartzi and Thaler, 1995). Because losing hurts worse than winning feels good, willingness to accept on the downside should be expected to exceed willingness to pay on the upside. Behavioral accounts of finance, however, should never be detached from economic fundamentals (Zhang, 2005, p. 69). That admonition suggests that the WTA/WTP premium reveals more nuanced human judgment. That judgment is consistent with higher-moment asset pricing and related ideas of liquidity preference and comovement among asset classes.

Static, unconditional models of finance assume that agents live no longer than a single period (Merton, 1973). As a descriptive matter, this assumption is demonstrably false. Worse, the single-period assumption is normatively deficient, even morally repellent. One need not embrace notion of “deep ecology” to reach this prescriptive conclusion (Naess, 1988). Even Hartwick's rule of weak sustainability demands a commitment to compensate future generations for immediate consumption of exhaustible resources.

The WTA/WTP premium is most pronounced in two settings. First, there is a significant premium for public and non-marketable goods relative to “ordinary” goods readily available in private markets (Horowitz and McConnell, 2002). Second, consumers demand a high premium for goods whose future, contingent value is currently uncertain (Zhao and King, 2004).

In concert, the presence of a large premium in these contexts reveals an awareness (or at least an intuition) that assets have value only relative to the broader state of the economy. These principles suggest that instruments of exchange and storehouses of value within the human economy have worth only relative to the biological and abiotic condition of global ecology.

Under conditions of relative abundance and stability, Hartwick's assumptions regarding exchangeability and frictionless intergenerational bargaining may hold. In a manner of speaking, Merton (1973) meets Coase (1960). But finite carrying capacity and the potential disruption of physical flows within ecosystem services serve stern notice that the ecological basis of human economy cannot be treated as static and permanent. The premium for willingness to accept over willingness to pay thus represents the environmental equivalent of the liquidity and equity risk premiums in behavioral finance.

The balance of this article will address two additional aspects of higher-moment thinking in environmental economics. At this pivotal stage, a brief preview of parts 4 and 5 is warranted.

First, innate reactions to dispersion, ambiguity, and uncertainty are the domain of even-moment effects within CAPM+. In environmental economics, these effects explain the progression from the Arrow-Lind theorem of risk-neutral public investment to Kenneth Arrow's own partial repudiation of his own work in later work with Anthony Fisher. The tension in these treatments of information uncertainty leads naturally to the dismal theorem, which arises from the work of William Nordhaus, Martin Weitzman, and other economists evaluating the costs of anthropogenic climate change.

Second and perhaps even more pressingly, the ultimate question is how humans will handle ecosystem services and the terrestrial life support systems under attack in the Anthropocene. Highly risk-seeking behavior has been observed in settings such as subsistence farming and diamond mining. Wealthy actors are engaging in similar “shots-at-greatness” behavior with respect to fossil fuel and climate change policy. Because these preferences reflect expectations of high levels of kurtosis, higher-moment asset pricing helps explain why “bonds-and-bullets” portfolios have such universal appeal. Less optimistically, CAPM+ suggests that this heuristic approach to managing risk may disserve humanity in a moment of existential exigency.

Purchases and sales within an exchange economy constitute a “central nervous system” (Supreme Court of the United States, 1940, p. 225, n. 59). Finance analyzes the market for capital to support speculative undertakings (Supreme Court of the United States, 1935, p. 689 [Stone, J., dissenting]). Prices as tools for transmitting economic knowledge within a collective “wisdom of prices” (Hayek, 1937, 1945; Grossman and Stiglitz, 1980).

An efficient capital market's very raison d'être is to reward investors who assume the risk of entrepreneurial failure (Ross, 1976). Indeed, the “first law of finance” dictates that excess return over a risk-free asset should correspond to volatility (Anderson et al., 2009, p. 233). Legal authorities recognize that abnormal returns are associated with elevated risk (Supreme Court of the United States, 1909, p. 49).

In environmental settings as elsewhere, the basic problem of finance becomes difficult, perhaps even intractable, when the investment horizon stretches into an indefinite future. Even without regard to temporal scales, risk management becomes virtually impossible where risks are poorly perceived and probabilities cannot be accurately estimated (Farber, 2011, p. 906).

A useful point of departure is “the impact of uncertainty on the behavior of investors and, ultimately, on market prices” (Campbell et al., 1997, p. 3). Knight (1921, pp. 19–20) and Keynes (1937, pp. 213–214) first recognized the theoretical difference between quantifiable, statistical risk and unknowable uncertainty. Situations where statistical probabilities can influence decision-making stand apart from truly aleatory circumstances where information is so vague that it eludes quantification (Epstein and Wang, 1994, p. 283; Runde, 1998, p. 539).

Uncertainty affects all economic activity (Bloom, 2009; Bachman et al., 2013; Baker et al., 2016), from household savings (Giavazzi and McMahon, 2012) and government borrowing (Pástor and Veronesi, 2012, 2013) to investment across the real economy (Born and Pfeifer, 2014; Fernández-Villaverde et al., 2015). When uncertainty clouds the economic outlook, risk averse consumers are the likeliest to realize option value from publicly supplied goods and services (Weisbrod, 1964; Cichetti and Freeman, 1971).

Ambiguity surrounding information affecting firm valuation has a powerful tendency to cast capital markets into uncertainty (Zhang, 2006, p. 105). Uncertainty exacts a far steeper toll on the downside, and not merely because the prospect of loss ground terrifies human decisionmakers. Coercion, after all, arises from “[t]hreat of loss” and not from “hope of gain” (Supreme Court of the United States, 1936, p. 82 [Stone, J., dissenting]). Economic retreat, whether attributable to an economy-wide recession or to bad news affecting an isolated sector or even a single firm, necessarily throttles the flow of information among buyers and sellers (Bloom, 2014, p. 162).

In all settings, economic agents prefer “known rather than unknown or vague probabilities” (Epstein and Wang, 1994, p. 284). Difficulty in judging the quality of information leads agents to “treat signals as ambiguous” (Epstein and Schneider, 2008, p. 197). All risk premiums rise alongside information uncertainty as investors ponder the probability of default, the amount at stake in potential business failures, transaction costs associated with bankruptcy, and even the size of the default premium itself (Christiano et al., 2014).

A specification of information uncertainty proceeds in two steps. First, an observed financial signal, or s, can be defined simply as s = v + e, where v indicates fundamental value implied by future cash flows or dividends, and e represents error or noise (Zhang, 2006, p. 105 n.2).

The second step consists of measuring the variance of the observed signal. Combining the variance of the firm's underlying volatility, or var(v), with var(e), the variance of the error term as an indicator of informational quality, enables information uncertainty to be expressed formally: var(s) = var(v) + var(e) (ibid.). This second formula recognizes the possibility that variance in cash flow or a series of returns may reflect not only fundamental economic variance, but also an additional premium based on information uncertainty.

The human reaction to uncertainty profoundly affects valuation and pricing. When agents face information uncertainty on top of risk, “they demand a higher premium” (Anderson et al., 2009, p. 234). This expression formalizes the relationship between risk and uncertainty (ibid., pp. 234–235):

where E designates the expectation operator, r_e indicates excess return over the risk-free baseline, V indicates market-wide conditional volatility, and M measures uncertainty throughout the economy. The temporal indexing variable t governs all of these variables as well as the expectation operator.

γ and θ, the coefficients in the foregoing formula, indicate aversion, respectively, to risk and uncertainty. Positive values for γ as well as θ imply that a positive premium for both risk and uncertainty (ibid., p. 234). In other words, investors will demand compensation bearing unknowable uncertainty as well as predictable risk. The expression, E_t r_{e, t+1} = γV_t+θM_t, should therefore be understood as a special instance of the more general formula, var(s) = var(v) + var(e).

In all events, it is crucial to distinguish between a fundamental economic signal (s = v + e) and information uncertainty as the sum of variance in informational quality and variance in the signal itself [var(s) = var(v) + var(e)]. The variability in many signals may stem from different sources of information, some less quantifiable than others. Uncertainty along economic, legal, scientific, and technological dimensions raises the cost of investing, by private actors as well as the government, in low- or zero-carbon generation and other responses to climate change.

On the other hand, severe uncertainty can drive variability, either in valuable flows of ecological services or in the quality information regarding those flows, effectively toward infinity. In other words, either var(s) → ∞ or var(e) → ∞. Alternatively, the value of those flows may implode within a foreseeable timeframe, such that v, s → 0. These are apocalyptic circumstances. A comparably cataclysmic approach to economic analysis is warranted.

The enormity of the Anthropocene catastrophe invites even more extreme approaches to uncertainty. When climate change inflicts an infinite amount of expected loss, the dismal theorem disables “standard economic analysis” altogether (Nordhaus, 2011, p. 240). More formally, since no amount of learning can prepare humanity for unlimited exposure to a fat-tailed risk, ordinary actuarial details such as risk assessment, social discounting, and the calibration of premiums to permit the smoothing of consumption all fall by the wayside (Weitzman, 2009, pp. 10–12, 18).

Risks contributing to the fat, leptokurtic tails associated with the dismal theorem bear many names. Whether it is described as variance risk (Carr and Wu, 2009; Bali and Zhou, 2016), tail risk (Bollerslev and Todorov, 2011; Kelly and Jiang, 2014), jump risk (Todorov, 2010; Dreschler and Yaron, 2011), or rare disaster risk (Gabaix, 2012), this risk resides at extremes where human epistemology exceeds its limits and outcomes observe no finite limits.

Translating bonds-and-bullets portfolio construction into the language of higher-moment asset pricing produces a convenient shorthand for this sort of risk-taking: Kurtosis preference. Affluent investors build layered portfolios according to opposite ends of Maslow's pyramid. While the bottom layer preserves capital as a bulwark against penury, the top layer takes “a shot at riches” (Shefrin and Statman, 2000, p. 141). This split portfolio assumes that the tails at either extreme will be fatter than the rest of the distribution of returns (ibid., p. 145). The combination of caution and optimism underlying this approach reflects the rank effect in behavioral finance (Hartzmark, 2015). It overestimates probabilities associated with the worst outcomes—and with the best (Shefrin and Statman, 2000, p. 141).

Kurtosis preference and bonds-and-bullets risk-taking appear to be innate frames for making decisions under conditions of extreme preference. But the innate optimism of the odd-numbered moments, especially skewness, lurks as a treacherous pitfall. Another existential threat to humanity illustrates the problem.

The Covid-19 pandemic, the greatest public health crisis in living memory, has killed millions around the world. Covid arguably poses a more immediate threat than climate change. At the very least, Covid-19 infection happens at the personal level and can reveal itself in hours rather than decades. Yet large swaths of the population perceive neither risk as urgent (Ruiu et al., 2020; Botzen et al., 2021). Indeed, at least in the United States, denying the threat has arguably become a badge of political allegiance. The same logic that urges Covid deniers to await deliverance through hydroxychloroquine, ivermectin, or some other miraculous therapeutic motivates a political preference to defer climate action. Justice delayed, as it were, is simply waiting for the deus ex machina of solar radiation management or geoengineering on the cheap.

In the opening passage to Their Eyes Were Watching God, Hurston (2006) distinguished those fortunate few whose ships “come in with the tide” from perpetual dreamers whose ships of dreams “sail forever on the horizon, never out of sight” (p. 1). Less wistfully, Nick the Greek lauded thrill-seeking gamblers who await the single “streak of luck” that might make up “for all the bad times” (Thackrey, 1968, p. 67).

Once the prospect of infinite loss has entered the casino, though, the dismal theorem counsels complete reconsideration of all approaches to risk management. That same principle also offers no guidance of its own. Humanity is consequently left to rely on its own instincts.

Those instincts may be quite destructive. In the context of Covid and other pandemic diseases, such instincts may defeat cooperative public health measures, as humans defer and avoid perceived risks associated with vaccination in favor of last-second therapeutic measures. To like effect, behaviorally influenced environmental decision-making often disfavors measures for mitigating climate change. The urgency of immediate sacrifices diminishes in the shadow of miraculous deliverance through future responses such as solar radiation management and other grandiose feats of geoengineering.