Skewness-seeking behavior and financial investments

Skewness is a measure of the asymmetry of a distribution, and it is usually captured by the skewness coefficient, which is equal to the third standardized moment: \({\widetilde{\mu }}_{3}=E\left[{\left(\frac{X-\mu }{\sigma }\right)}^{3}\right]\). Preference for skewness is a concept used to refer to the preference for a positively skewed distribution over a negatively skewed one. Arditti (1967) first related skewness to expected utility theory. His approach relied on approximating expected utility with a truncated Taylor series after three terms. Under three reasonable assumptions, he “proved” that utility should be increasing in skewness.¹ However, this approach neglected all the other higher moments of the distribution of outcomes. For this reason, it was criticized by Brockett and Kahane (1992) and Brockett and Garven (1998), who proved that, under EUT, utility is not necessarily increasing in skewness.²

An individual with a positive third-order derivative of the utility function, \({U}^{\mathrm{^{\prime}}\mathrm{^{\prime}}\mathrm{^{\prime}}}\left(w\right)>0\) is said to be prudent, a concept introduced by Kimball (1990). Prudence is prevalent in several experimental works (Deck and Schlesinger 2010; Ebert and Wiesen 2011; Fairley and Sanfey 2020; Heinrich and Shachat 2020; Noussair et al. 2014). Skewness-seeking behavior is defined as the preference for a distribution of outcomes with a higher skewness coefficient over another with the same expected value, variance, and kurtosis (Ebert and Wiesen 2011). From a theoretical perspective, a prudent individual should be a skewness-seeker, but the opposite may not necessarily be true. Ebert and Wiesen (2011) provided experimental evidence in support of this.

One of the first studies of skewness preferences was Mao’s (1970): he asked managers to decide between two binary lotteries with the same mean and variance but differing skewness. Managers were almost equally split between the alternatives if the investment represented a small portion of the company resources. However, when the lotteries represented the whole outcome of the company, they all picked the positively skewed distribution because of its better downside (i.e., a better outcome in the negative state). Brünner et al. (2011) studied skewness-seeking behavior using pairs of binary prospects with the same mean and variance but differing skewness and once again found evidence of skewness-seeking behavior, with about 60% of the subjects choosing the prospect with larger skewness more than half the time. Ebert and Wiesen (2011) used Mao lottery pairs (i.e., sets of two binary lotteries with the same mean, variance, and kurtosis, but different skewness) and found skewness-seeking behavior for about 75% of the subjects. Åstebro et al. (2015) tested skewness preferences with a variation of Holt and Laury’s (2002) multiple price list format and found that subjects made, on average, skewness-seeking choices. Similarly, Ebert (2015) found that 64% of subjects make skewness-seeking choices, and Dertwinkel-Kalt and Köster (2020) found a preference for positive skewness.

Grossman and Eckel (2015) used a variation of Eckel and Grossman lotteries (2002, 2008) and found that more than 80% of the subjects were skewness-seeker. Taylor (2020) used a slight modification of Grossman and Eckel’s (2015) design aimed at reducing the impact that loss aversion may have had on skewness-seeking behavior and indeed found that while this behavior was still prevalent, it was less frequently observed compared to Grossman and Eckel’s study. Unlike the previous studies, both Grossman and Eckel (2015) and Taylor (2020) did not use binary prospects but three outcome prospects. Bougherara et al. (2021) found mostly skewness-avoidance in an experiment where they elicited certain equivalents of three-outcome prospects.³ In a subsequent study with similar experimental settings, Bougherara et al. (2022) found mostly skewness-seeking behavior.

While using binary prospects seems to lead to skewness-seeking behavior, introducing other outcomes makes the situation less clear-cut. The impact of a shift from finite outcome prospects to continuous distributions is even more dramatic. Vrecko et al. (2009) found that the form used to represent an investment affects the decision: subjects were found to be skewness-seeker when a cumulative distribution function was used to represent the alternatives, whereas they were skewness-avoider when a probability density function was utilized instead. Skewness-avoidance in the density treatment was rooted in anchoring (Tversky and Kahneman 1974): the peak of the distribution would serve as an anchor for the estimation of the unknown mean (Summers and Duxbury 2006). As a result, when the expected value of the distribution is unknown, it is overestimated for negatively skewed distributions and underestimated for positively skewed ones (Vrecko et al. 2009). Consider Fig. 1: as the skewness coefficient increases, the distribution’s mode decreases (moving to the left and passing from positive to negative territory), even if the expected return is the same for all four distributions. Thus, using probability density functions may discourage skewness-seeking behavior due to biased risk perception and incorrect estimation of the expected return. Holzmeister et al. (2020) elicited risk perception and investment propensity of continuous distributions represented by histograms of samples of 200 draws from such distributions.⁴ The distributions differed in variance, skewness, and kurtosis, and they found that positively skewed distributions are generally perceived as riskier than negatively skewed ones by both financial professionals and laypeople, with this phenomenon being driven by the higher probability of a loss. Likewise, investment propensity was negatively associated with risk perception, and thus, positively skewed distributions of outcomes were less likely to be chosen.

The experiment aims to study skewness preferences under several perspectives using continuous distributions of outcomes. What we mean by skewness preferences are the choices over the third moment of the distribution of returns, in isolation from mean and variance. When we study the interactions between skewness preferences and risk-taking, we consider the joint choice over the second and third moments of the distribution. In other words, we operationalize the concept of skewness preferences with the choice of the third standardized moment (skewness coefficient), and the concept of risk-taking with the choice of the second central moment (variance). In line with Ebert and Wiesen (2011), we consider an individual to be a skewness-seeker if, among the choices available, she selects a positively skewed distribution instead of a negatively skewed distribution with equal mean, variance, and kurtosis.

Within our experimental setting, we address four main research questions about skewness preferences.

The first research question concerns the relationship between skewness and risk-taking: “How do skewness and risk-taking interact?”. We address this question with the Skew-risk and Skew-risk-reference tasks (see Sect. 4.1).

The second research question concerns skewness preferences: “Do subjects exhibit skewness-seeking behavior when opportunities are shown using continuous distributions of outcomes?”. We address this question with the Binary-base, Multiple-base, Multiple-partial, Multiple-full tasks (see Sects. 4.2 and 4.3).

The third research question regards the trade-off between skewness and expected return: “Do subjects trade off skewness with expected returns, or do they exhibit mean–variance preferences?”. We address this question with the two Binary-adjustment tasks (see Sect. 4.4).

The fourth research question is about the drivers of skewness preferences: “What are the characteristics driving skewness preferences?”. We address this question by combining the choices of the Binary and Multiple tasks with measures collected outside part one of the experiments, which concern speculative behavior, risk, and loss preferences (see Sect. 4.5).

The online experiment was programmed and executed with oTree (Chen et al. 2016). We present here the different parts of the experiment (see “Appendix A” for a detailed description). After an introductory non-incentivized part, participants were given a tutorial on the experimental framework. The tutorial reviewed the concepts of probability density function, variance, and skewness, and it offered the chance to see how changes in these moments would affect the distribution of outcomes. The tutorial was followed by a mandatory comprehension check, and then subjects were asked about their general aspirations about investments. Then, the first part of the experiment began: subjects played eight rounds, making incentivized decisions over distributions differing in skewness. In the second part of the experiment, subjects’ risk preferences were elicited using a scaled version of Holt and Laury (2002) multiple price list, with one of the ten choices randomly selected for payment. Finally, a demographic and a financial-behavior questionnaire were administered.

Like Ebert (2015), we framed our alternative distributions in terms of returns rather than outcomes: the final payment from part one of the experiment was computed, compounding the returns obtained in two of the eight rounds. The distributions were skew-normal with parameters (ξ, ω, α) appropriately chosen so that the expected value would be equal to 6%, the standard deviation would be equal to 20%, and the skewness coefficient equal to some target level in the [1, 1] interval. The skew-normal distribution has been widely used in financial applications involving risk management, capital allocation in financial markets, and insurance (Adcock et al. 2015; Adcock and Azzalini 2020; Bernardi 2013; Bodnar and Gupta 2015; Vernic 2006) because it handles the asymmetry in a simple and flexible way.

We chose a graphical representation of outcomes, with the possibility to initially sample from the displayed distributions,⁶ to make information provision easier to process for participants relative to a static numerical representation (on this see Kaufmann et al. (2013) and Bradbury et al. (2015)). Furthermore, we decided to present the probability density functions instead of histograms (like Holzmeister et al. (2020)) because their smoothness improves the comparability between different distributions.

In the Binary-base and Binary-adjustment tasks, subjects faced a binary decision between two distributions differing in skewness: they visualized both distributions (like those in Fig. 1) and clicked on a button to choose the distribution they preferred. In the Multiple-base, Multiple-partial, and Multiple-full tasks, subjects made a decision among eleven distributions differing in skewness: they could visualize one distribution at a time and change the displayed distribution by horizontally dragging the slider placed below the distribution (see Fig. 1).

In the Skew-risk and Skew-risk-reference tasks, subjects could move two sliders to change the skewness and standard deviation levels. The skewness slider worked like in the previous tasks, while the standard deviation slider was vertical. The interface also provided some information about the probability of experiencing some outcomes (see Fig. 2).

Table 1 briefly summarizes the rounds and the associated tasks. For a full description of the tasks, please refer to “Appendix A”.

We collected 180 valid observations in the month of February 2022 on the platform Prolific.⁷ Due to the complex nature of our tasks, we set some restrictions on the Prolific subject pool. Eligible subjects needed to be at least 21 years old, be fluent in English, have completed high school, and specialized in either a STEM subject or economics/finance. Moreover, the sample was gender-balanced. On average, subjects took 22 min (with a SD of 10 min) to complete the experiment and earned about £4.6. The final payment was composed of a fixed participation fee, a variable fee for part one (mean £1.1 and SD £0.31), and a variable fee for part two (mean £0.78 and SD £0.38).

In the Skew-risk (SR) and Skew-risk-reference (SRR) tasks, subjects could manipulate both skewness and standard deviation. In one round, they were assigned to the positive skewness treatment, meaning they could choose only a positively skewed distribution. In the other round, they were assigned to the negative skewness treatment, meaning they could choose only a negatively skewed distribution. The order was randomized at the subject level. For both treatments, the subjects could choose three levels of standard deviation: 0.16, 0.20, or 0.24, and five levels of skewness,⁸ for a total of fifteen possible distributions. Moreover, in the Skew-risk-reference (SRR) task, subjects were also provided with a reference point: their current return.⁹

We found that—as shown in Fig. 3—in both rounds, risk-taking (i.e., the level of chosen standard deviation) was significantly higher for the subjects assigned to the negative treatment than for those assigned to the positive treatment (p-value of Wilcoxon rank sum test is 0.002 in the SR task, and 0.018 in the SRR task).

Furthermore, we analyzed the level of skewness and standard deviation within treatment. Although the level of these two variables was positively correlated in both tasks (controlling for treatment), Kendall’s Tau correlation coefficient was significantly different from zero only in the two treatments of the Skew-risk-reference task.¹⁰

In the Binary-base task, the choice was between two alternatives. Depending on the randomly assigned treatment, the choice could be between (1) a positively skewed and a symmetric distribution, (2) a negatively skewed and a symmetric distribution, or (3) a positively and a negatively skewed distribution. All investments had the same expected return and volatility but differed in skewness (equal to − 0.75, 0, and 0.75, depending on the distribution). Subjects could generate random samples from the displayed distributions to enhance familiarity with probability density functions.¹¹ In the Multiple-base task, the choice was between eleven alternatives differing in skewness—five positively skewed, one symmetric, and five negatively skewed—and the choice was made dragging a slider.¹²

Like in Brünner et al. (2011), higher skewness implies third-degree stochastic dominance. Thus, in an EUT framework, for any decision maker with utility function \(U\left(w\right)\) such that \({U}{\prime}\left(w\right)>0\), \({U}^{{\prime}{\prime}}\left(w\right)<0\), and \({U}^{{\prime}{\prime}{\prime}}\left(w\right)>0\), the distribution with higher skewness coefficient should be preferred (Levy 1992). Menezes et al. (1980) define downside risk as “a leftward transfer of risk, keeping mean and variance the same”. Given two distributions with densities f(x) and g(x), if f dominates g by third-degree stochastic dominance (TSD), and they have the same mean and variance, then f has less downside risk than g, and it is more right-skewed than g (Menezes et al. 1980). Thus, the downside risk is decreasing in skewness in our framework, and a sufficient condition to prefer the distribution with a higher skewness coefficient is \({U}^{{\prime}{\prime}{\prime}}\left(w\right)>0\), or downside risk aversion (Menezes et al. 1980, theorem 2). Finally, our distributions satisfied the skewness comparability criteria (Chiu 2010). “Appendix B” expands on skewness comparability, third-degree stochastic dominance, and downside risk increase.

In our settings, an individual is considered a skewness-seeker if she selects a positively skewed distribution over a negatively skewed one. Our definition is consistent with Ebert and Wiesen’s concept of skewness-seeking behavior: indeed, in the Multiple tasks and in the positive–negative treatment of the Binary-base task, every time an individual selects a positively skewed distribution, she is revealing she prefers it over another distribution with the same mean, variance, and kurtosis, but opposite skewness coefficient.

In the Binary-base task, the proportion of subjects who did select the distribution with the larger skewness coefficient was not significantly different from 50%, neither at the aggregate level nor dividing by the three treatments. Therefore, we did not find evidence of the prevalence of prudence or skewness-seeking behavior.¹³ Moreover, we cannot reject the hypothesis that choices were made randomly in this task.

In the Multiple-base task, we found evidence of skewness-seeking behavior: the proportion of subjects investing in a positively skewed alternative was significantly different from 50% and equal to about 68%.¹⁴ Moreover, we can reject the hypothesis that choices were made randomly (p-value of Chi-squared test < 0.001).

In the Multiple-base task, subjects could decide among eleven investment opportunities differing in skewness. Still, they had no numerical information about them, except that they all had the same expected return and volatility. All they could do was infer some information from the probability density functions. In the Multiple-partial task, subjects faced the same decision environment but received one piece of information regarding the displayed distributions. Depending on the treatment, they may visualize (1) the probability of a loss, (2) the probability of a loss larger than 20% (“large loss”), or (3) the probability of a gain larger than 40% (“large gain”). As they moved the slider, the distribution displayed changed, as well as the displayed probability and the area associated with the displayed probability, as shown in Fig. 4.

First, we reject the hypothesis that choices were random (Chi-squared test, p-value < 0.001). If subjects could perfectly (and somewhat unrealistically) infer probabilities from the pictures of the distributions, then none of the treatments should impact their decisions, which should be equal to the decision of the Multiple-base. On the contrary, if subjects had no understanding of the distributions, they would rely just on the probabilities shown and pick an edge choice—either the most positively or the most negatively skewed alternative—optimizing for the probabilities displayed. Assuming subjects fall somewhere in between, that is, they have some understanding of the probability density functions, and they incorporate the new piece of information, we should expect the “probability of a loss” treatment to reduce skewness-seeking. In contrast, the other two treatments should increase it. Statistical tests do indeed indicate that treatments were effective in orienting decisions. Since the median skewness level across treatments is statistically different (Kruskal–Wallis test, p-value < 0.001), we perform a pairwise comparison of the three treatments using a Wilcoxon rank sum test and find that the difference in median skewness is not statistically significant only between the “probability of a large gain” and “probability of a large loss” (p-value = 0.23; all other p-values < 0.001). Two-sample Kolmogorov–Smirnov tests on the distribution of skewness choices across the treatments confirm the previous results (p-value = 0.42 for the “probability of a large gain” and “probability of a large loss” comparison; all other p-values < 0.001). Despite the effectiveness of the treatments in orienting decisions, the choices in the Multiple-partial task were still consistent with those made in the Multiple-base task: for all three treatments, subjects previously classified as skewness-seeker were still more skewness-seeking than the others (Wilcoxon rank sum test, p-value < 0.01 for all three treatments).

In the Multiple-full task, subjects were again asked to choose one of the eleven distributions, but they were provided with all three pieces of information of the Multiple-partial task. This task was more complex than the previous because the three probabilities displayed provided contrasting cues: the probability of a large gain and the probability of a loss increased in skewness, while the probability of a large loss decreased in skewness. The increase in the complexity was indeed perceived by the subjects, who moved the slider (i.e., explored the environment) significantly more times. We can reject the hypothesis that decisions were affected by the treatment assigned in the Multiple-partial task (Kruskal–Wallis test, p-value = 0.63) and that the choices were random (Chi-squared test, p-value < 0.014). This suggests that subjects incorporated new information when making their decision. We found that even if the skewness-seeker group was still the largest, the proportion of skewness-seeker reduced significantly from the Multiple-base to the Multiple-full task. The prevalence of skewness-seeking behavior largely disappeared: skewness-seeker were not significantly more than 50% of the sample anymore (Proportion test, p-value = 0.69); they were about 52% versus 48% skewness-avoider.¹⁵ Subjects classified as skewness-seeker and skewness-avoider based on the Multiple-base task were still, on average, skewness-seeker and skewness-avoider, respectively. Still, the median skewness chosen for both groups was closer to zero. This process resulted in a more uniform (but not random) distribution of choices (see the bottom row of Fig. 5).

In the two Binary-adjustment tasks, subjects faced the same decision environment of the Binary-base task. In one round, they could receive a bonus if they decided to invest in the opportunity they had not selected in the Binary-base task (“bonus treatment”). In contrast, in the other, they would pay a penalty if they decided to invest in the opportunity they had selected in the Binary-base task (“penalty treatment”). Both treatments were played during the two consecutive rounds, with the order of treatments being randomly assigned at the subject level. The value of the adjustment (bonus/penalty) ranged between 0 and 1%, extremes included, with 0.10% increments. In the first three columns of Table 2, we report the generalized linear mixed-effect models analyzing the probability of changing distribution with respect to the Binary-base task. The results show that subjects traded off skewness with expected returns: the probability of changing increases in the magnitude of the adjustment. Moreover, subjects were less likely to change when the shift would have been from a positively to a negatively skewed distribution (or vice versa) than when the change involved a symmetric distribution: the further away the new level of skewness from the initial favorite level, the less likely the subjects to change, given the adjustment.

We now combine the decisions taken in several tasks to study the determinants of skewness preferences. Considering the decisions in the three Binary tasks (see Table 2, Models 4 and 5), we find that, in general, the higher the upside score, the more likely a subject to pick the distribution with the higher skewness coefficient in a binary choice. The upside score is an indicator of the willingness to achieve better upside opportunities or speculative gains, which is computed based on the answers given in the final questionnaire.

Furthermore, we pool together the decisions made in the three Multiple tasks (Table 3). The upside score was a driver of skewness preferences also in these tasks. Moreover, risk perception of positive skewness was a driver of decisions: the higher the risk perception (of positively skewed distributions vis-a-vis negatively skewed distributions), the lower the skewness level and the lower the likelihood of choosing a positively skewed distribution. In the Multiple-partial and Multiple-full tasks, subjects were given different pieces of information about the distributions. We conjecture that subjects gave different weights to the three probabilities and, depending on these weightings, made a decision in the Multiple-full task. We use information collected in the aspirations phase to classify subjects into three groups, with each group expected to give more weight to one of the three probabilities. On the aspirations page, we asked subjects whether they would rather combine ownership of a stock with a financial instrument that enhances returns in case the stock performs well (i.e., a call option) or with a financial instrument that reduces losses in case the stock performs badly (i.e., a put option). We assume that those who chose the call option are relatively more focused on the upside, while those who chose the put option are relatively more focused on the downside. Therefore, more upside-focused subjects are expected to give more weight to the probability of a large gain, so they are expected to be relatively more skewness-seeker. As for the downside-focused subject, we distinguish between subjects concerned about avoiding losses and subjects concerned about avoiding large losses. Hence, we consider the maximum self-reported threshold for losses in an investment: those who are willing to bear a loss up to a given low threshold τ are expected to give more weight to the probability of a loss, so they are expected to be relatively less skewness-seeker.

The others, who can tolerate losses larger than τ, are expected to attribute more weight to the probability of a large loss, so they are expected to be relatively more skewness-seeker. A natural threshold could be 10%, that is the mid-point between the two loss levels provided, 0% and 20%. Since the probability of a loss has been found to be salient in risky decisions (Holzmeister et al. 2020; Zeisberger 2022), we also used a 5% threshold, which is closer to the salient value of 0%.

The regressions in Table 3 are consistent with our conjectures. Skewness preferences seem to be ultimately driven by two channels: first, the speculative channel (operationalized through the “Upside score” and the “Call preference” dummy) indicates that speculators tend to choose more positively skewed distributions than non-speculators. The second channel is the risk perception of positively skewed distributions versus negatively skewed distributions, a variable that we label “Risk perception”.

While risk perception plays an important role in skewness preferences, traditional risk preferences do not seem to play a role within our framework. In part two of the experiment, we elicited risk preferences using a modified version of the multiple price list¹⁶ (Holt and Laury 2002), and classified subjects as risk-averter, risk-neutral, and risk-seeker. We found no systematic difference in skewness preferences across the tasks among the three groups. This result is not surprising since skewness concerns downside risk preferences, and both risk lovers and risk averters can be downside risk averse (Menezes et al. 1980). Indeed Haering et al. (2020) did not find differences across these two groups in their preferences for higher-order odd moments, including skewness.

Risk perception plays a relevant role in explaining skewness preferences. Within our framework, the relationship between skewness and risk would depend on the operationalization of the latter. Volatility σ was constant across the distributions, and the behavioral risk measure σ_B² (Davies and De Servigny 2012)¹⁷ was decreasing in skewness, as well as semi-variance. The probability of experiencing a loss—which, according to Holzmeister et al. (2020), is the main channel through which skewness translates into risk perception—was increasing in skewness. Finally, since the skewness comparability criteria (Chiu 2010; Oja 1981; Van Zwet 1964) were satisfied, a distribution with a higher skewness coefficient could be considered a downside risk decrease with respect to any other distribution with a lower skewness coefficient (see “Appendix B” for more details).

In the Multiple tasks, risk perception was correlated with actual choices. In particular, in the Multiple-full, i.e., when information became “fully available”, the median skewness level for subjects who perceived positive skewness as less risky was significantly higher than zero (p < 0.003), while it was significantly lower than zero for subjects who perceived positive skewness riskier (p < 0.032). Only about 35% of our subjects perceived positively skewed distributions as riskier than negatively skewed distributions, while 26% perceived them as safer, and the remaining 39% of the subjects believed they bear about the same risk. This result aligns with the idea that risk can be measured in different ways, but it contrasts with previous literature specifically focused on risk perception and skewness of continuous distributions (Holzmeister et al. 2020). This difference could stem from the elicitation procedure: while the measurement of risk perception was a key element of Holzmeister et al.’s study—elicited for every distribution—we only asked our subjects at the end of the study to indicate their level of agreement with the statement that positively skewed distributions are riskier than negatively skewed ones. Our subjects had thus already experienced the treatments and were aware of the relationship between skewness and the probability of losses, large losses, and large gains. However, at least part of the risk perception was already formed before the visualization of pieces of information about probabilities, as risk perception was significantly correlated with choices made before probabilities were provided (Multiple-base task). Like Holzmeister et al. (2020), we found that investment propensity and risk perception are inversely related.

Most experimental evidence indicates a positive relationship between skewness and risk-taking (Åstebro et al. 2015; Brünner et al. 2011; Dertwinkel-Kalt and Köster 2020; Ebert and Wiesen 2011; Ebert 2015). Grossman and Eckel (2015) also found that risk-taking increases as skewness available increases, but Taylor (2020) while confirming this result, attributed part of this effect to loss aversion.

The justification of the positive relationship between skewness and risk-taking can be found in Amaya et al. (2015): “As positive asymmetry increases, volatility is welfare increasing as it implies a larger probability of an extremely good state of the economy. The opposite is true for the case of negative skewness since higher volatility increases the likelihood of a left tail event”. Behavioral components may enhance this mechanism: according to Åstebro et al. (2015), skewness-seeking behavior is driven by optimism and likelihood insensitivity. Similarly, Dertwinkel-Kalt et al. (2020) relate skewness-seeking behavior to salience theory (Bordalo et al. 2012). On the contrary, Bougherara et al. (2021) found that variance and skewness do not interact in a positively skewed environment, while the difference between the certain equivalent of a highly negatively skewed prospect and a low negatively skewed prospect is higher when variance is higher. Differently, Bougherara et al. (2022) did not find a significant interaction between skewness and risk-taking.

We contribute to this literature with two findings. First, we show that subjects took more risk when forced (by the treatment) to choose negatively skewed distributions. Such distributions have a longer left tail and a shorter right tail, so the probability of obtaining large positive outcomes is relatively lower. A subject interested in improving her upside may be forced to increase risk-taking because changing only skewness may not be satisfactory enough. This is unnecessary in a positively skewed environment, where increasing skewness may be sufficient. Since a satisfactory upside may be achieved either by increasing the skewness coefficient in a positively skewed environment or increasing variance in a negatively skewed environment, we claim that the direction of skewness—positive or negative—and risk-taking can be seen as substitutes.

Secondly, we find that—especially when a reference point is provided—skewness and risk-taking are positively correlated: individuals react to the reference point by selecting a corner distribution more often. We define a corner distribution as a distribution where skewness and standard deviation are either maximized or minimized within the available set of distributions, like those shown in Fig. 2. In this sense, skewness and risk-taking acted synergically as complements to achieve a goal: in the maximization case, subjects maximized the probability of a large gain, whereas in the minimization case, they minimized the probability of a loss.¹⁸ For each heatmap in Fig. 6, the top-right and bottom-left squares represent the corner distributions: for both treatments, they were more frequent in the Skew-risk-reference tasks (heatmaps on the right) than in the Skew-risk tasks (heatmaps on the left).

We defined skewness-seeking behavior as the preference for a positively skewed distribution over a symmetric and a negative one when multiple alternatives with the same mean, variance, and kurtosis were available. Our definition is consistent with Ebert and Wiesen (2011): when a subject picked a positively skewed distribution in the Multiple tasks, she was not indeed picking another distribution with the same mean, variance, and kurtosis, but negative skewness coefficient. In the Binary-base task, subjects faced a binary decision, and skewness-seeking could only be tested for the treatment where a positively and a negatively skewed distribution were available. In the other two treatments, only prudence could be tested. In the Binary-base task, in none of the three treatments, we found evidence of subjects choosing the distribution with the largest skewness coefficient with a probability significantly larger than 50%. Therefore, we rejected the prevalence of both prudence and skewness-seeking behavior. However, a few considerations are worth mentioning. Firstly, our results already contrast Vrecko et al. (2009) and Holzmeister et al. (2020), who found evidence of skewness-avoidance when probability density functions are used. Secondly, the preference for skewness may not arise immediately: in Brünner et al. (2011), the proportion of skewness-seeker in the first two rounds was about 40%, while in the last rounds, it was around 67%, and since the order of the rounds was randomized, they attributed this phenomenon to some form of learning. Finally, since Ebert and Wiesen (2011) suggest that prudent individuals are mostly skewness-seeker, but the opposite is not necessarily true, we could see the proportion of prudent individuals in the Binary-base task as a lower bound for skewness-seeker since the imprudent skewness-seeker should be more than the prudent skewness-avoider. Indeed, when we tested for skewness-seeking behavior in the Multiple-base, 80% of prudent decision-makers (i.e., those who made a prudent choice in the Binary-base task) were skewness-seeker. At the same time, only 55% of imprudents were skewness-seekers.¹⁹ Overall, in the Multiple-base task, about two-thirds of the subjects were skewness-seekers, a proportion in line with the existing experimental literature. Considering the five pairs of distributions with the same mean, variance, and kurtosis, the proportion of skewness-seeker in each pair ranged between 65 and 72%. The prevalence of skewness-seeking behavior can be attributed to at least two elements. First, compared to the Binary-base task, subjects earned some additional experience with distributions differing in skewness, and the round properly allowed the testing of this behavior. Second, in the Multiple-base task, subjects could visualize only one distribution at a time, and by dragging the slider, they could change the displayed distribution: the impact of increasing skewness on the probability density function became extremely evident and salient. Thus, the “dynamic” presentation of the alternatives may have made the comparisons easier compared to the traditional “static” presentation.

Furthermore, we found that skewness is traded off directly with the expected return, suggesting that it may play a major role in financial decisions. Ebert and Karehnke (2021) discuss skewness-seeking behavior in the context of binary lotteries and analyze the order of skewness preferences under different theories. In the expected utility (EUT) framework, skewness preferences are shown to be second or third order, whereas in the cumulative prospect theory (CPT) framework (Tversky and Kahneman 1992), skewness preferences are first order. Therefore, our results do not conform well to EUT, while they could be consistent with CPT. Indeed, thanks to the introduction of probability weighting, CPT allows skewness to play a more prominent role. Moreover, the different parametrizations of the value function as well as the probability weighting function do not define ex-ante a single relationship between skewness and utility, allowing individuals to be both skewness-seeker or skewness-avoider (Barberis 2012). Our findings of significant heterogeneity in skewness preferences are consistent with this view.

Since subjects traded off skewness for returns in the two Binary-adjustment tasks, mean–variance preferences do not fit their preferences well, and skewness must also be considered. However, a linear relationship like the one suggested by Arditti (1967) does not seem consistent with our results, either. Indeed, our results also suggest a convoluted relationship between variance, skewness, and the skewness environment (i.e., the direction of skewness of the available options). Consider the classification of the subjects into the three groups skewness-seeker, skewness-avoider, and skewness-neutral based on their decisions in the Multiple-full task, and analyze their choices in the two subsequent rounds: the Skew-risk and Skew-risk-return tasks. Not surprisingly, for both treatments, the subjects in the skewness-seeker group were significantly more skewness-seeking than those in the skewness-avoider (p-value of Wilxon rank sum test < 0.001), indicating the robustness of skewness preferences in both directions. The few skewness-neutral subjects were somewhere between the two groups for both treatments.²⁰ However, there is a difference between the groups in the level of variance in the two treatments: while they all take about the same risk in the positive treatment (p-value of Kruskal–Wallis test equal to 0.63), subjects in the skewness-seeker group take significantly more risk in the negative treatment than those in the skewness-avoider group (p-value of Kruskal–Wallis test and subsequent pairwise Wilcoxon test < 0.05). Furthermore, both skewness-seeker and skewness-avoider select a distribution with higher variance in the negative treatment (p-values of the paired Wilcoxon tests < 0.001 and < 0.05 for the first and second group, respectively). Interpreting these results under a moment-preferences light suggests that individuals exhibit heterogeneous preferences for the third moment of the distribution. These preferences, combined with the availability of alternatives with skewness in the preferred direction, influence the decisions involving the second moment. First, in general, regardless of skewness preferences, individuals take more risk in a negatively skewed environment than in positively skewed environments. This is because, in this environment, the upside opportunities are more limited. Therefore, increasing variance is a solution to the bounded upside. This is not necessary in a positively skewed environment because the positive skewness alone can allow a more satisfactory upside, and only a limited number of subjects seek to maximize both skewness and variance. Second, in a negatively skewed environment, skewness-seekers are more aggressive than skewness-avoiders in taking more risks by increasing variance. This is probably because one reason to be a skewness-avoider is to reduce the probability of a loss, and increasing risk-taking would harm this objective. The specification of a theoretical model to account for these phenomena is beyond the scope of this paper, and it is left to future research.

We conclude this section by showing that subjects exhibited consistency in their revealed preferences throughout the rounds: we conduct seemingly unrelated regressions (SUR) of the skewness levels of the six tasks in which the distributions had the same expected return (i.e., all the tasks except the two Binary-adjustment tasks), and analyze the correlation matrix of the residuals. The residuals are all positively correlated. This correlation is statistically significant in most cases, indicating that unobservable idiosyncratic traits influenced the decisions over the rounds (see “Appendix C” for more details), confirming once again the fact that preferences over the third moment consistently play a first-order role in financial decisions.