Risk and Uncertainty at the Outbreak of the COVID-19 Pandemic

Abstract

The classic paradigm in finance maintains that asset returns are paid as a compensation for bearing risk. This study extends the literature and explores whether asset prices are also affected by uncertainty. This research invokes the Expected Utility with Uncertainty Probabilities Model and utilizes the natural experiment conditions of the COVID-19 pandemic outbreak, in order to determine whether investors’ behavior during the sharp economic decline was driven by risk, or uncertainty. We limit this research only to the outbreak of the pandemic, since the recovery of the markets suggests investors have adjusted to the unexpected nature of the crisis. Using high-frequency data of the S&P 500 Index, we estimate the investors’ risk and ambiguity aversions, finding that in the pre-pandemic period investors exhibited risk aversion as well as an ambiguity-seeking attitude, while during the pandemic they demonstrated risk- and ambiguity-neutral behavior. The implications of these findings could suggest that in regular times, the financial markets are operated by risk-averse investors with decreasing risk-averse behavior.

1. Introduction

The Expected Utility Model [1] sets the rigorous foundations of economic theory by describing the behavior of economic agents under risk conditions. At the basis of the model lies the investor’s preference, which satisfies Reflexive, Complete, Transitive, and Continuity Axioms over a set of outcomes. One of the challenges facing economic theory is designing experiments that reliably test its models. For example, the Expected Utility Model was examined extensively in laboratory experiments, which led to the development of Prospect Theory [2,3].

However, do economic agents only operate under risk conditions? Alternatively, are they exposed to uncertainty as well? The answer to these two questions lies in the difference between risk and uncertainty. Risk is defined as a state in which the event to be realized is a priori unknown, but all its probabilities are perfectly known. However, ambiguity (i.e., Knightian uncertainty) is a state in which not only the event to be realized is a priori unknown, but its probabilities are also unknown [4].

In order to clarify the difference between risk and ambiguity, consider these two events: (a) the Central Bank’s decision regarding interest rates; and (b) the declaration of the World Health Organization (WHO) regarding a worldwide pandemic. In event (a), the investors, i.e., the economic agents, do not know what the actual interest rate will be, but they estimate the probabilities of each possible interest rate. Thus, in this case, the investors operate only under risk conditions. However, in event (b), prior to the outbreak of the COVID-19 pandemic, investors most likely did not factor the effect of a possible worldwide pandemic on financial assets in their estimations; therefore, in this case, the investors operate under ambiguity conditions. Furthermore, after the outbreak of the COVID-19 pandemic, investors regarded the appearance of pandemic variations as an affine combination of both risk and ambiguity conditions.

The Expected Utility with Uncertainty Probabilities Model [5,6] was developed on the solid foundations of the Expected Utility Model [1], describing the behavior of economic agents under both risk and uncertainty conditions. In other words, this model presents predictions regarding the economic agents’ decision-making, under both risk and uncertainty. But how can this model’s predictions be tested, and what would an unexpected event look like?

This study argues that the outbreak of the COVID-19 pandemic serves as a natural experiment that allows us to examine investors’ behavior, under both risk and uncertainty conditions in the financial markets.

Figure 1 shows the evolution of the S&P 500 Index between 1/2/2018 and 5/29/2020. The steep decline in the S&P 500 Index during the first quarter of 2020 expresses a loss of 31.8% in the index value, which is followed by a recovery of the index.

This study offers a unique combination of three elements: a progressive theoretical methodology—the Expected Utility with Uncertainty Probabilities Model; the opportunity of a natural experiment—the outbreak of the global COVID-19 pandemic; and high-frequency data—five-minute observations of the S&P 500 Index. Thus, it presents a rare glimpse into the investors’ decision-making process under severe risk and uncertainty conditions in the financial market.

We postulated that, under these severe conditions, inventors display both risk and ambiguity aversion. However, our findings suggest that the capital market is normally operated by risk-averse and ambiguity-seeking investors, while during the pandemic outbreak the capital market was governed by risk- and ambiguity-neutral investors.

The contribution of this research holds even more importance with regards to the immediate future (the emergence of possible new variants of the COVID-19 virus, and their impact on the global economy), or even to the more remote and unknown future, by attempting to understand the inherent nature of the investors’ behavior under both risk and uncertainty conditions. These insights may be of high value for investors themselves, as well as for policy makers, when determining intervention policies during unexpected crises.

2. Theoretical Background

The Expected Utility with Uncertainty Probabilities Model was extensively studied by Izhakian [5,6] in order to rigorously incorporate it into economic theory. The significant steppingstones of its evolution include: the Expected Utility Model [1]; the Choquet Expected Utility Model [7]; the Max-Min Expected Utility Model [8]; Cumulative Prospect Theory [9]; the Alpha-Max-Min Expected Utility Model [10]; and the Smooth Model of Ambiguity [11]. These advancements eventually resulted in two outcomes: (a) the ambiguity measurement; and (b) a pricing equation based on investors’ preferences regarding both risk and ambiguity.

In order to further clarify the difference between risk and ambiguity, consider the following example [12], in which a decision maker faces a gamble where they have a 50% chance of gaining 20%, or a 50% chance of losing 10% of their bet money. The expected value of this gamble is gaining 5% of their bet money, the standard deviation (risk) of this gamble is 15%; therefore, since the probabilities of the outcomes (not the outcomes themselves) are perfectly known, the ambiguity of the gamble is equal to zero. Next, consider the same decision maker facing a similar gamble, where the probabilities of each outcome have themselves two probabilities. This means that instead of assuming a 50%:50% chance for each outcome, there is a 60%:40% chance of gaining 20% and a 60%:40% chance of losing 10%. Although the expected value and standard deviation (risk) of this new gamble are identical to those of the old one, the ambiguity of the new gamble is equal to 10% (the standard deviation of the probabilities). While the two gambles are identical in their mean and risk, the new gamble has an unaccounted uncertainty or ambiguity, which must be priced into the bet.

In this subsection, we briefly review the main results of the Expected Utility with Uncertainty Probabilities Model [6]. Let $\Omega$ be an infinite primary space (i.e., the outcomes set) with a sigma-algebra $\Omega \in ℰ$ (i.e., the events set), and let $x\in \mathrm{X} =\left[0,1\right]$ be a convex set (i.e., the consequences set); then, a primary act of mapping $f:\Omega \to \mathrm{X}$ is a function from outcomes to consequences. Let $\mathsf{𝓟}$ be a nonempty secondary space (i.e., the probabilities of the outcomes set), and let $P\in \mathsf{𝓟}$ be a first-order probability measure, which can be thought of as a state in the secondary space; then, a secondary act of mapping $\mathsf{𝒈}:\mathsf{𝓟}\to \mathrm{X}$ is a function from the probabilities to consequences. Then, $P_f\left(x\right)=P\left(\left\{\omega \in \Omega |f\left(\omega \right)\le x\right\}\right)$ and ${\phi }_f\left(x\right)$ is the uncertainty probability density function.

$${\mho }^2\left[f\right]=\underset{\mathrm{X}}{\overset{}{\int }}E\left[{\phi }_f\left(x\right)\right]Var\left[{\phi }_f\left(x\right)\right]dx$$

(1)

Following [6], Equation (1) defines ${\mho }^2$ (mho²) as the degree of ambiguity associated with beliefs in isolation of attitudes toward ambiguity and risk. In other words, this measure is associated with the probabilities of the events set and not the events set itself. Hence, the ambiguity measure could also be thought of as the expected volatility of the probabilities.

The relation between risk and ambiguity in the financial markets has an elegant analogy, according to which if risk is defined as the volatility of the outcomes [13], then the ambiguity is defined as the volatility of the returns’ probabilities.

The Expected Utility with Uncertainty Probabilities Model states that, given a utility function $U:\Omega \to ℝ$ that is a continuous, strictly increasing and concave function, normalized by a riskless reference asset’s return, so that: $U\left(x_F=0\right)=0$; and an ambiguity preference function ${\rm Y}:\mathrm{X}\to ℝ$ that is a continuous, non-constant, bounded function; then there exists a function $V:\Omega \to ℝ$ that describes the investor’s preferences under both risk and ambiguity conditions:

$$\begin{gathered} V\left(f\right)=\underset{z\le 0}{\overset{}{\int }}\left[{{\rm Y}}^{-1}\left(\underset{\mathcal{P}}{\overset{}{\int }}{\rm Y}\left(P\left(\left\{\omega \in \Omega |U\left(f\left(\omega \right)\right)\ge z\right\}\right)\right)d\xi \right)-1\right]dz \\ +\underset{z\ge 0}{\overset{}{\int }}\left[{{\rm Y}}^{-1}\left(\underset{\mathcal{P}}{\overset{}{\int }}{\rm Y}\left(P\left(\left\{\omega \in \Omega |U\left(f\left(\omega \right)\right)\ge z\right\}\right)\right)d\xi \right)\right]dz \end{gathered}$$

(2)

Equation (2) characterizes the investor’s behavior under risk and ambiguity conditions using Choquet integration over the set of outcomes. This equation is an equivalent version of the von Neumann-Morgenstern utility function in the Expected Utility Model, but only for the Expected Utility with Uncertainty Probabilities Model. It follows from Equation (1) that the ambiguity measure satisfies the monotonicity property, which means that an investor prefers a lower ambiguity [6], which leads to Equation (3) that was developed [6]. Equation (3) states the asset pricing model that is based on a Taylor series approximation around the riskless asset’s return. Hence, $h\left(x\right)$ represents the remainder of the high orders of the series, which is proved negligible [4]. The functions $U^{\prime }$ and $U^{″}$ as well as ${{\rm Y}}^{\prime }$ and ${{\rm Y}}^{″}$ (${\rm Y}$ upsilon) represent the utility and the ambiguity preference functions of the investor, respectively.

$$\{\begin{array}{c}E\left[x\right]-x_F=-\frac{1}{2}\frac{U^{″}\left(E\left[x\right]\right)}{U^{\prime }\left(E\left[x\right]\right)}Var\left[x\right]-E\left[\frac{{{\rm Y}}^{″}\left(1-E\left[P\left(x\right)\right]\right)}{{{\rm Y}}^{\prime }\left(1-E\left[P\left(x\right)\right]\right)}\right]E\left[\left|x-E\left[x\right]\right|\right]{\mho }^2\left[x\right]+h\left(x\right)\\ h\left(x\right)=o\left(E\left[{\left|x-E\left[x\right]\right|}^2\right]\right) , \left|x-E\left[x\right]\right|\to 0\end{array}$$

(3)

Equation (3) is similar to the methodology that was developed in the Arrow-Pratt risk aversion coefficient [14], and its asset-pricing model. This equation states that under optimal decision-making conditions, an asset’s premium return is comprised independently of risk and ambiguity premiums. In other words, the risk premium is composed of the Arrow-Pratt risk aversion coefficient (a subjective risk preference) and the returns’ variance (an objective risk measure); and the ambiguity premium is composed of the ambiguity measure, the expected absolute deviation from the mean returns (an objective risk measure), and an ambiguity aversion coefficient (a subjective risk preference). Similarly to the Arrow-Pratt risk aversion coefficient results, which distinguish between risk aversion ($-U^{″}/U^{\prime }>0$), risk neutrality ($-U^{″}/U^{\prime }=0$), and risk seeking ($-U^{″}/U^{\prime }<0$), the ambiguity aversion coefficient distinguishes between ambiguity-averse ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }>0$), ambiguity-neutral ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }=0$), and ambiguity-seeking ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }<0$) attitudes toward uncertainty.

Equation (3) describes the asset-pricing equation of the Expected Utility with Uncertainty Probabilities Model, from which we can quantify the main hypothesis of this study. First, in order to be coherent with economic theory, we test for risk-averse investors; therefore, the Arrow-Pratt risk aversion coefficient should have a positive value ($-U^{″}/U^{\prime }>0$). Second, in addition to their risk-averse behavior, we postulate that in severe uncertainty conditions the investors exhibit an ambiguity-averse ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }>0$) attitude. However, if our conjecture does not hold, then the investors can exhibit ambiguity-neutral ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }=0$) or ambiguity-seeking ($-{{\rm Y}}^{″}/{{\rm Y}}^{\prime }<0$) attitudes, which suggest that they are either not reacting to the severe conditions or operating in the capital market for a higher return, respectively.

Although the theoretical development of the Expected Utility with Uncertainty Probabilities Model is somewhat complicated, its empirical application is considerably simpler, as it is based on an econometric model.

3. Empirical Analysis

The Expected Utility with Uncertainty Probabilities Model develops a theory that describes an economic agent’s behavior, under risk and uncertainty conditions. But, how can this theory be tested? After all, this theory requires the occurrence of an unexpected event.

Whether the sharp decline in the S&P 500 Index was caused due to the investors’ reaction to risk or uncertainty, this drastic drop expresses anything but stability. Thus, we argue that the outbreak of the worldwide COVID-19 pandemic can be considered as an unexpected event. Thus, we limit our study to the outbreak of the pandemic, when investors first had to adjust to new circumstances. The fast recovery of the financial markets indicates that the investors quickly understood the implications of the pandemic and learned to quantify them in their expectations. The timeframe of this study is, therefore, clearly bounded by a natural experiment environment. In other words, the outbreak of the pandemic provides a rare glimpse of the investors’ behavior under extreme conditions. Hence, extending the sample of the outbreak period might mean losing that rare opportunity. Thus, our main hypothesis postulates that the sharp decline in the financial market during the pandemic outbreak derived from both risk and ambiguity aversion.

Our study utilizes high-frequency data obtained from the Pi-Trading database. Our data consists of 47,268 five-minute observations of the S&P 500 Index between 1/2/2018 and 5/29/2020. We considered the full sample, meaning we did not exclude any periods of low trading activity (such as bank holidays or weekends), since the extreme conditions also continued during these days; therefore, the investors had to continually assess the circumstances. This period covers the outbreak of the pandemic and two years before it.

The empirical analysis includes two steps: First, we calculated the daily risk and ambiguity measures as described below. Second, we implemented a one-step regression model, to estimate the risk and the ambiguity based on the results of the first step.

We employed discrete versions of the expected value, risk, and ambiguity measure as well as the asset pricing equation under both risk and uncertainty. The ambiguity measure estimator, as given in [6], is noted with ${\widehat{\mho }}^2\left[x\right]$ which corresponds to Equation (1).

$${\widehat{\mho }}^2\left[x\right]=\frac{1}{\sqrt{w\left(1-w\right)}}\sum _{k=1}^K\widehat{E}\left[P\left(B_k\right)\right]\widehat{Var}\left[P\left(B_k\right)\right]$$

(4)

Equation (4) defines the daily ambiguity measure estimator. At this point, the high-frequency data plays a crucial part as it enables us to regard each daily return as an independent random variable, and thus it has a probability density function. Hence, $\widehat{E}\left[P\left(B_k\right)\right]$ and $\widehat{Var}\left[P\left(B_k\right)\right]$ are the daily mean and the variance of the hourly probabilities, which quantify the expected value of the probability of the density and its volatility. The ambiguity measure is defined as the expected volatility of the probabilities; therefore, in order to employ it in an econometric model, we divided the inter-day data to six hours per day and the returns’ range was divided into $K$ bins (equal intervals, $B_k$ where k =1, …, K). The $P\left(B_k\right)$ was calculated as the number of hourly returns within bin $B_k$ divided by T, the number of returns measured per hour; and $w$ was the increment of the bins’ density. Moreover, in order to eliminate the measure’s dependency on the bins’ intervals, the measure was adjusted using Sheppard’s correction, which is equal to: $1/\sqrt{w\left(1-w\right)}$, where $w\in \left(0,1\right)$.

We present the detailed estimation methodology for daily mean, variance, and the ambiguity measure with an Algorithm 1.

Algorithm 1: Algorithm for estimating the daily mean, variance, and the ambiguity measure (the code is available in the Supplementary Materials of this research):

(1) Read input: x (the entire data set as a time-series). (2) Read input: K (the number of bins). (3) For each day t: (3.1) $\mathrm{Denote} x_{max}$$\mathrm{and} x_{min}$ as the maximum and minimum values in day t and $\mathrm{calculate} \mathrm{the} \mathrm{increment} w=\left(x_{max}-x_{min}\right)/K$. (3.2) $\mathrm{Calculate} \mathrm{Sheppard}’\mathrm{s} \mathrm{correction}, \mathrm{so} \mathrm{that}: 1/\sqrt{w\left(1-w\right)}$. (3.3) $\mathrm{Build} \mathrm{an} \mathrm{array} \mathrm{of} K \mathrm{bins} (B_1,\dots ,B_K$$) \mathrm{to} \mathrm{determine} \mathrm{the} \mathrm{ranges} \mathrm{of} \mathrm{the} \mathrm{returns}, \mathrm{which} \mathrm{are} \mathrm{later} \mathrm{used} \mathrm{in} \mathrm{order} \mathrm{to} \mathrm{calculate} \mathrm{the} \mathrm{probability} \mathrm{for} \mathrm{each} \mathrm{bin}. \mathrm{The} \mathrm{bins} \mathrm{have} K \mathrm{equal} \mathrm{intervals}, \mathrm{which} \mathrm{range} \mathrm{between} x_{min}$$\mathrm{by} \mathrm{increasing} \mathrm{with} \mathrm{the} \mathrm{increment} w$$\mathrm{up} \mathrm{to} x_{max}$. (3.4) Repeat for each hour in the inter-day data return: (3.4.1) $\mathrm{For} \mathrm{each} \mathrm{bin} k (1, \dots , K), \mathrm{count} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{returns} \mathrm{in} \mathrm{the} \mathrm{current} \mathrm{hour}, \mathrm{which} \mathrm{ranged} \mathrm{within} \mathrm{the} \mathrm{current} \mathrm{bin} (B_k$$) \mathrm{and} \mathrm{set} P\left(B_k\right)$$\mathrm{to} \mathrm{be} \mathrm{that} \mathrm{number}, \mathrm{divided} \mathrm{by} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{returns} \mathrm{measured} \mathrm{per} \mathrm{hour} \left(T\right) (\mathrm{note} \mathrm{that} \mathrm{this} \mathrm{step} \mathrm{results} \mathrm{in} K \mathrm{probabilities} P\left(B_k\right)$ for each hour). (3.5) $\mathrm{For} \mathrm{each} \mathrm{bin} k (1, \dots , K), \mathrm{calculate} \mathrm{the} \mathrm{daily} \mathrm{probability} \mathrm{mean}, \widehat{E}\left[P\left(B_k\right)\right]$$, \mathrm{and} \mathrm{variance}, \widehat{Var}\left[P\left(B_k\right)\right]$ (i.e., calculate the mean and variance over all hourly probabilities for each bin k in day t). (3.6) $\mathrm{Calculate} \mathrm{the} \mathrm{daily} \mathrm{ambiguity} \mathrm{measure} {\widehat{\mho }}^2\left[x_t\right]$ according to Equation (4). (3.7) $\mathrm{Calculate} \mathrm{the} \mathrm{daily} \mathrm{mean}, \widehat{E}\left[x_t\right]$$, \mathrm{and} \mathrm{variance}, \widehat{Var}\left[x_t\right]$ for day t. (3.8) $\mathrm{Calculate} \mathrm{the} \mathrm{daily} \mathrm{absolute} \mathrm{mean} \mathrm{deviation}, \mathrm{so} \mathrm{that}: \widehat{E}\left[\left|x_t-\widehat{E}\left[x_t\right]\right|\right]$ for day t. (4) $\mathrm{Present} \mathrm{output}: \mathrm{an} \mathrm{array} \mathrm{of} \widehat{E}\left[x_t\right], \widehat{Var}\left[x_t\right], \widehat{E}\left[\left|x_t-\widehat{E}\left[x_t\right]\right|\right]$$\mathrm{and} {\widehat{\mho }}^2\left[x_t\right]$.

Figure 2 shows the risk (the variance) and the ambiguity measure estimators of the S&P 500 Index between 1/2/2018 and 5/29/2020. There is an overlap between the risk and the ambiguity; i.e., periods with increased risks are followed by periods with increased ambiguities. This relation is quite clear in the first two quarters of 2020, where both risk and ambiguity measures exhibit a reverse correlation with the S&P 500 Index value.

In order to examine our main hypothesis, we employ the one-step prediction econometric model [12] that corresponds with Equation (3), where: $Retur{n}_{t+1}=\widehat{E}\left[x_t\right]-x_{F,t}$ is the return’s premium; $Ris{k}_t=\frac{1}{2}\widehat{Var}\left[x_t\right]$ is the return’s variance (multiplication by half according to the Taylor series approximation); $Ambiguit{y}_t=\widehat{E}\left[\left|x_t-\widehat{E}\left[x_t\right]\right|\right]{\widehat{\mho }}^2\left[x_t\right]$ is the expected absolute deviation of the return from its mean multiplied by the ambiguity measure; and ${ϵ}_t$ is the regression error.

$$Retur{n}_{t+1}=\alpha +\beta \times Ris{k}_t+\gamma \times Ambiguit{y}_t+{ϵ}_t$$

(5)

Equation (5) describes the extended asset pricing equation as a linear regression with the following parameters: $\alpha$ is the intercept and the normalization parameter of the riskless asset; $\beta$ is the Arrow-Pratt risk aversion coefficient (i.e.,: $\beta >0$ represents risk aversion; $\beta =0$ represents risk neutrality, $\beta <0$ represents risk seeking); and $\gamma$ is the ambiguity aversion coefficient (i.e.,: $\gamma >0$ represents ambiguity aversion, $\gamma =0$ represents ambiguity neutrality, $\gamma <0$ represents ambiguity seeking).

We divided the data into three periods: full period (2/1/2018–5/29/2020), which includes the full sample of the research; the pre-pandemic period (2/1/2018–3/2/2020); and the pandemic period (3/3/2020–5/29/2020) act as a “control sample” and a “test sample” in this natural experiment.

Table 1. Summary of the results.

Period Criteria	Full Period (2/1/2018–5/29/2020)	Pre-Pandemic Period (2/1/2018–3/2/2020)	Pandemic Period (3/3/2020–5/29/2020)
Descriptive statistics	Table A1—Panel A	Table A1—Panel B	Table A1—Panel C
One-step prediction model	Table A2—Panel A	Table A2—Panel B	Table A2—Panel C
Robustness checks using different intervals (bins)	Table A3—Panel A	Table A3—Panel B	Table A3—Panel C

reports the summary of the results divided by three criteria according to the periods, such that:

Table A1. Descriptive statistics. Panel (A)—Full period (2/1/2018–5/29/2020). Panel (B)—Pre-pandemic period (2/1/2018–3/2/2020). Panel (C)—The pandemic period (3/3/2020–5/29/2020).

(A)
	Return	Risk	Ambiguity
Mean	0.020	0.002	1.662
Med	0.085	0.0005	0.661
Max	8.968	0.0981	16.387
Min	−12.765	0.0000	0.000
Stdev	1.540	0.0069	2.307
Skew	−0.938	8.773	2.201
Kurt	19.393	96.427	8.688
JB	6874	228170	1306
#Obs	606	606	606
(B)
	Return	Risk	Ambiguity
Mean	0.017	0.0008	1.223
Med	0.083	0.0004	0.517
Max	4.840	0.019	10.856
Min	−4.517	0.000	0.000
Stdev	0.968	0.001	1.699
Skew	−0.815	6.179	2.292
Kurt	6.957	61.266	8.863
JB	414	80266	1253
#Obs	543	543	543
(C)
	Return	Risk	Ambiguity
Mean	0.048	0.0120	5.440
Med	0.392	0.0046	5.201
Max	8.968	0.0981	16.387
Min	−12.765	0.0004	0.623
Stdev	3.868	0.0183	3.239
Skew	−0.490	2.7987	0.755
Kurt	4.598	11.2418	3.735
JB	9.23	260.55	7.40
#Obs	63	63	63

Notes: Each table illustrates the descriptive statistics of the key variables for the full period (Panel A), pre-pandemic period (Panel B), and the pandemic period (Panel C). JB is the Jarque-Berra test statistic.

reports the descriptive statistics according to the three periods;

Table A2. One-step prediction model. Panel (A)—Full period (2/1/2018–5/29/2020). Panel (B)—Pre-pandemic period (2/1/2018–3/2/2020). Panel (C)—The pandemic period (3/3/2020–5/29/2020).

(A)
	Reg. 1	Reg. 2	Reg. 3
Intercept	−0.006 [−0.48]	0.006 [0.36]	0.017 [1.06]
Risk	4.72 ** [2.60]		8.826 ** [3.70]
Ambiguity		−0.002 [−0.29]	−0.019 ** [−2.64]
Adj-R2	0.011	0.0001	0.0224
F-Stat.	6.77 **	0.088	6.91 **
#Obs	605	605	605
(B)
	Reg. 1	Reg. 2	Reg. 3
Intercept	−0.001 [−0.02]	0.017 [1.61]	0.014 [1.32]
Risk	4.98 [0.81]		30.32 ** [3.44]
Ambiguity		−0.011 * [−2.12]	−0.029 ** [−3.97]
Adj-R2	0.00	0.12	0.0295
F-Stat.	0.65	4.50 *	8.21 **
#Obs	543	543	543
(C)
	Reg. 1	Reg. 2	Reg. 3
Intercept	−0.085 [−0.73]	−0.119 [−0.62]	−0.033 [−0.16]
Risk	6.595 [1.84]		8.31 [1.76]
Ambiguity		0.02 [0.69]	−0.013 [−0.30]
Adj-R2	0.025	0.01	0.0264
F-Stat.	1.53	0.48	0.80
#Obs	62	62	62

Notes: “**”and “*” indicate statistical significance at the 1% and 5% levels, respectively. All results utilize the Newey-West estimator. The t-Statistic values appear in brackets.

reports the estimation results of the one-step prediction regression model; and,

Table A3. Robustness checks using different intervals (bins). Panel (A)—Full period (2/1/2018–5/29/2020). Panel (B)—Pre-pandemic period (2/1/2018–3/2/2020). Panel (C)—The pandemic period (3/3/2020–5/29/2020).

(A)
	Bin = 200	Bin = 100	Bin = 50	Bin = 40	Bin = 30	Bin = 20
Intercept	0.015 [0.76]	0.017 [1.06]	0.018 [1.18]	0.017 [1.17]	0.026 [1.74]	0.021 [1.41]
Risk	6.31 ** [2.96]	8.82 ** [3.70]	10.95 ** [4.06]	12.05 ** [4.20]	14.69 ** [5.16]	13.97 ** [4.72]
Ambiguity	−0.009 [−1.42]	−0.019 ** [−2.64]	−0.032 ** [−3.11]	−0.037 ** [−3.27]	−0.059 ** [−4.50]	−0.063 ** [−3.93]
Adj-R2	0.014	0.022	0.026	0.028	0.043	0.036
F-Stat.	4.40 *	6.91 **	8.27 **	8.81 **	13.63 **	11.18 **
#Obs	605	605	605	605	605	605
(B)
	Bin = 200	Bin = 100	Bin = 50	Bin = 40	Bin = 30	Bin = 20
Intercept	0.022 [1.62]	0.014 [1.32]	0.007 [0.65]	0.005 [0.51]	0.006 [0.56]	0.005 [0.5]
Risk	16.31 * [2.13]	30.32 ** [3.44]	23.75 * [2.43]	20.99 * [2.12]	21.52 * [2.22]	20.54 * [2.11]
Ambiguity	−0.014 * [−2.45]	−0.029 ** [−3.97]	−0.028 * [−2.46]	−0.026 * [−2.06]	−0.032 * [−2.20]	−0.037 * [−2.06]
Adj-R2	0.012	0.029	0.012	0.009	0.010	0.010
F-Stat.	3.33 *	8.21 **	3.34 *	2.45	2.75	2.44
#Obs	543	543	543	543	543	543
(C)
	Bin = 200	Bin = 100	Bin = 50	Bin = 40	Bin = 30	Bin = 20
Intercept	−0.273 [−0.85]	−0.033 [−0.16]	0.077 [0.45]	0.053 [0.36]	0.118 [0.82]	0.067 [0.48]
Risk	3.56 [0.49]	8.31 [1.76]	16.26 [1.78]	18.01 [1.90]	23.73 * [2.61]	21.55 * [2.25]
Ambiguity	0.037 [0.63]	−0.013 [−0.30]	−0.07 [−1.30]	−0.075 [−1.45]	−0.129 * [−2.29]	−0.127 [−1.86]
Adj-R2	0.031	0.026	0.052	0.058	0.104	0.079
F−Stat.	0.95	0.80	1.62	1.83	3.44 *	2.53
#Obs	62	62	62	62	62	62

Notes: “**” and “*” indicate statistical significance at the 1% and 5% levels, respectively. All results utilize the Newey-West estimator. The t-Statistic values appear in brackets.

reports the robustness check for the main regression model using different bins.

Table A1—Panels A, B, and C (see Appendix A) report the descriptive statistics for each one of the periods, respectively. The mean returns, risk, and ambiguity are higher during the pandemic (Panel C) compared to pre-pandemic period (Panel B). To wit, the mean returns during the pandemic period are 2.8 times higher compared to the pre-pandemic period (0.048 vs. 0.017); the risk during the pandemic period is 15 times higher than in the pre-pandemic period (0.012 vs. 0.0008); and the ambiguity during the pandemic period is 4.5 times higher than in the pre-pandemic period (5.44 vs. 1.223). These findings paint the picture of extreme conditions, to which the investors had to adjust during the first wave of the pandemic in the US capital market.

Table A2—Panels A, B, and C (see Appendix B) report the estimation results for the one-step prediction model in the form of Equation (5) with 100 bins. In each panel, we estimate the three possible models: only risk, only ambiguity, and both risk and ambiguity. In Panel C the multiple regression for risk and ambiguity is insignificant (F-stat = 0.8, p-value > 0.05). Therefore, the investors exhibit a risk-neutral attitude (estimator = 8.31 [t-statistic = 1.76, p-value > 0.05]) as well as an ambiguity-neutral attitude (estimator = −0.013 [t-statistic = −0.30, p-value > 0.05]) during the pandemic period, as opposed to Panels A and B, which report that the multiple regression for risk and ambiguity is significant (Panel A: F-stat = 6.9, p-value < 0.01; Panel B: F-stat = 8.21, p-value < 0.01). Therefore, Panels A and B report strong risk-averse and ambiguity-seeking attitudes during both the full period (estimator for the coefficient of risk is 8.826 [t-statistic = 3.70, p-value < 0.01] and estimator for the coefficient of ambiguity is −0.019 [t-statistic = −2.64, p-value < 0.01], respectively) and pre-pandemic period (estimator for the coefficient of risk is 30.32 [t-statistic = 3.44, p-value < 0.01] and estimator for the coefficient of ambiguity is −0.029 [t-statistic = −3.97, p-value < 0.01], respectively).

Table A3—Panels A, B and C (see Appendix C) report the robustness checks using different intervals (bins). Although the ambiguity measure estimator is adjusted using Sheppard’s correction, we invoked robustness checks on the number of bins, which affected the density of the probabilities. Technically, we divided the range of returns, from −9% to +6%, into 20, 30, 40, 50, 100, and 200 intervals (bins), calculated the ambiguity measure in Equation (4), and estimated the one-step prediction model according to Equation (5). The overall picture of the robustness checks supports the one-step prediction model. While in Panel A all the robustness checks support the previously mentioned results, in Panel B only the robustness check of 20, 30 and 40 bins turned out insignificant; in Panel C only the robustness check of 30 bins turned out significant. From these findings, we concluded that robustness checks are consistent with the results for 20, 30, 40, 50, and 100 bins. However, the robustness check of over 200 bins in this range renders the ambiguity measure negligible; therefore, the ambiguity aversion coefficient is insignificant.

Moreover, in addition to the robustness checks, our findings highlighted that the model’s intercept is statistically insignificant, which means that the riskless asset’s return is a valid reference point for investors in the capital market.

Our results can be summarized as follows: during regular times and the pre-pandemic period, we found that investors exhibit risk aversion as well as ambiguity seeking. In addition, during the pandemic period we found that investors exhibit risk and ambiguity neutrality. Hence, our findings state that investors are risk-averse with ambiguity-seeking preferences during regular times, but shift to risk and ambiguity neutrality during unexpected events.

These results are consistent with the classical paradigm, which states that assets are priced according to their risks [15], thus investors exhibit risk-averse behavior [16,17]. Furthermore, our findings are also coherent with a recent study [12,18,19] that showed that the investors display an ambiguity-seeking attitude.

Although our main result is supported with other empirical evidence, on the surface it might seem illogical that if assets are priced according to their risks, they should also be priced according to their ambiguities [20,21,22]. Yet, the results of this study suggest that investors are risk-averse, as well as ambiguity-seeking, i.e., investors are willing to take chances for the possibility of increased returns. The theoretical meaning of this finding is that investors are risk-averse with decreasing risk aversion, which was already empirically documented [23]. This empirical result is based on the Lorenz curve ranking method [24], in conjunction with the Schechtman-Shelef-Yitzhaki-Zitikis test [25].

Furthermore, this methodology could be used to examine the relationship between financial markets and energy commodities, in both regular as well as severe times of risk [26] and uncertainty conditions, or in financial efficiency [27,28].

4. Conclusions

We argued that the outbreak of the COVID-19 pandemic can be thought of as a severe risk and uncertainty condition in the financial market, since it was an unexpected event caused by non-economic factors that could not have given any previous indication as to its future effect. We employed the Expected Utility with Uncertainty Probabilities Model in order to examine the model’s predictions of investors’ behavior under severe risk and uncertainty conditions.

Our data utilized high-frequency returns of the S&P 500 Index during the years 2018–2020. We divided the data into three periods: full sample period, pre-pandemic period, and the pandemic period. Then, we calculated the risk and the ambiguity measures for each period. For each of these periods, we estimated the parameters of the risk and ambiguity aversions using a one-step prediction econometric model.

Our findings indicated that during the pandemic period, investors did not exhibit any significant risk and ambiguity aversion in the prevailing conditions in the capital market. However, under the full period and the pre-pandemic period, we found that the investors exhibited significant risk aversion, as well as ambiguity-seeking behavior. Our findings are consistent with similar results documented in the literature.

While evidence of risk-averse behavior had already been published in the literature, ambiguity-seeking behavior means that investors are willing to take chances for higher rewards, which is equivalent to indicating that investors are risk-averse with decreasing risk aversion. This high-degree risk-aversion property has been theoretically predicted in the Expected Utility Model, using the general index of absolute risk attitude, which was also documented empirically.

The outbreak of the COVID-19 pandemic clearly illustrates the difference between risk and uncertainty, but it is not the first time that the financial markets experienced uncertainty events. One might argue, for example, that the September 11 attacks (2001), the global financial crisis (2008), or the Tōhoku earthquake (2011) could also be considered as significant uncertainty events, which can also be used to estimate the risk aversion and ambiguity attitude under severe conditions using the methodology proposed in this study.

Although the outbreak of the COVID-19 pandemic occurred a few years ago, it highlighted the fragile stability of the financial markets during unexpected events. The pandemic might be over soon, but we are almost sure that it is not the last unexpected event that could affect the financial markets (an unexpected event by definition cannot be anticipated). Thus, this research demonstrates that while in regular times investors exhibit risk-averse behavior and an ambiguity-seeking attitude, during times of severe risk and uncertainty their behavior regresses toward risk- and ambiguity-neutral behaviors. These findings have significant implications concerning the resiliency of the financial markets, affected both by the investors themselves, and by regulatory policies.

The observed behavior of investors during regular times and extreme periods is important for both the investors themselves, as well as for regulatory authorities. Investors display a risk-averse preference with ambiguity seeking in regular times, but exhibit risk- and ambiguity-neutral preferences in extreme periods. This means that in extreme periods the market decreases, and investors adjust their portfolios quickly to these extreme risk and uncertainty conditions. Thus, investors recover swiftly. Regarding regulatory authorities, these results carry even more weight and draw attention to the role of central banks and governments. Understanding the nature of uncertainty can help central banks and governments design operational frameworks and strategies to mitigate it. Indeed, authorities across the world have taken an active role in containing the uncertainty during the Great Recession of 2008 using different tools such as inhibiting uncertainty by going down the path of short-term interest rates, quantitative easing, and increasing government expenditure. Hence, we believe that the findings of this research can help policy makers to better react to the market’s fluctuations, following unexpected events.

We expect that further theoretical research should facilitate bridging between ambiguity seeking and the marginal utility of high-degree risk aversion. In addition, more empirical studies are required in order to further establish the investors’ behavior under both severe risk and uncertainty conditions.