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Ambiguity and Abnormal Returns in Virtual Currencies Di Luo, Tapas Mishra, Larisa Yarovaya, and Zhuang Zhang November 2020 Abstract Rationally justifying Bitcoin markets huge price uctuations has remained a persistent challenge for both investors and researchers in this eld. A primary reason is our po- tential weakness towards a robust quanti cation of unquanti able risks or ambiguity in Bitcoin returns. This paper introduces a behavioural channel to regard the degree of ambiguity aversion as a prominent source of abnormal returns from investment in Bitcoin markets. Using daily data over a period of ten years, we show that on average, Bitcoin investors depict an increasing aversion to ambiguity. Further, we nd that Bit- coin investors earn abnormal returns only when ambiguity is low. Robustness exercise reassures validity of our results. Di Luo, Tapas Mishra, Larisa Yarovaya, and Zhuang Zhang are from the University of Southamp- ton. We are grateful to conference participants at Cryptocurrency Research Conference, UK, 2020 for their helpful comments and suggestions. All remaining errors are our own. Emails: d.luosoton.ac.uk; t.k.mishrasoton.ac.uk; l.yarovayasoton.ac.uk; zhuang.zhangsoton.ac.uk Electronic copy available at: 1. Introduction But Bitcoin is an example of ambiguity, and the e cient market theory does not capture what is going on in the market for this cryptocurrency. | Robert Shiller1 Bitcoin valuation is exceptionally ambiguous . | Robert Shiller2 1.1. Context These quotes from Robert Shiller could hardly be more accurate in describing the aim of the present study, in which we attempt to answer the broad question of how ambiguity determines abnormal returns in virtual currencies, such as Bitcoin. Virtual currencies represent both the emergence of a new form of currency and a new payment technology to purchase goods and services. Among virtual currencies, Bitcoin has undoubtedly emerged as the most prominent new form of currency and a new payment technology to purchase goods and services (Dwyer, 2015; White et al., 2020). Due to its importance to the nancial institutions, its frequent sus- ceptibility to large-scale price manipulations, and investors increasing tendency of its choice over other established theory-backed assets (Trimborn and H ardle, 2017), ambiguity appears to be central in quantifying the magnitude of abnormal returns. This paper lls a gap in the 1 2 1 Electronic copy available at: literature by rigorously studying the impact of ambiguity in Bitcoin returns. Our empirical foundation rests on Brenner and Izhakian (2018).3 As the leading cryptocurrency, Bitcoin continues to draw high attention from investors, entrepreneurs, regulators and the general public. Many of the recent public discussions on Bitcoin, triggered by the substantial changes in their prices (Garc a-Monle on, et al., 2021), claim that the market for Bitcoin is a bubble without any fundamental value, and also concerns about evasion of regulatory and legal oversight (Akyildirim, 2020; Alexander and Heck, 2020). A large strand of literature attempts to understand market phenomena through the lens of the traditional neoclassical nance theories (Borri, 2019; Corbet et al, 2020l). Speci cally, Urquhart (2016) shows that Bitcoin returns do not follow the random walk model, based on which he concludes that the Bitcoin market exhibits a signi cant degree of ine ciency, particularly in the early years of its existence. In the time and frequency domains, Corbet et al. (2018) analyze the relationship between the return of three di erent cryptocurrencies and a variety of other nancial assets, showing lack of relationships between crypto- and other assets. Gillaizeau et al. (2019) consider cross-market spillover of volatility in Bitcoin markets and identify net receivers and net givers of shocks. Liu and Tsyvinski (2020) investigate whether cryptocurrency pricing bears any similarity to stocks: however, none of the risk factors explaining movements in stock prices applies to cryptocurrencies in their sample. Moreover, 3Camerer and Weber, 1992 in an early e ort provided evidence, theoretical explanations, and applications of research on ambiguity and subjective expected utility. Recent e orts include a design of a survey module by Cavatorta and Schroeder, 2019 to experimentally validate ambiguity preference that has wider applications for economics and nance. 2 Electronic copy available at: movements in exchange rates, commodity prices, or macroeconomic factors of traditional signi cance for other assets play little to no role for most cryptocurrencies4. All that apart, Bitcoin is an example of uncertainty and ambiguity, and the neoclassical theory fails to explain the behavior in the market for this cryptocurrency.5 There has not been enough daily information coming in to rationally justify Bitcoins huge price uctuation. This type of uncertainty may arise for two reasons: (1) the technology is rather complicated and opaque to unsophisticated traders, and (2) the fundamental value of cryptocurrencies is unclear. As we highlighted above, even if it is strictly positive, it is likely to be derived from intangible factors and, as such, is rather uncertain. Therefore, we wish to extend our understanding of this cryptocurrency market from a behavioral nance perspective. This paper examines the role of the perspective of unquanti able risk, or ambiguity in Bitcoin returns. The notion of uncertainty has been investigated in the literature since the seminal works of Keynes (1921) and Knight (1921) from two perspectives: risk and ambiguity. While risk is a situation in which the beliefs of a decision maker (DM) are captured by a unique probability measure, ambiguity is a situation in which a DMs beliefs are not pinned down by a unique probability measure because of a lack of information(Snow 2010; Cavatorta and Schroder 2018). When investors choose between di erent assets, their knowledge of future returns is 4In a di erent context, Duan et al. 2019 show how macroeconomic variations can account for ambiguity and volatility in real estate market. 5A recurrent issue in nancial theories is to study how agents make decisions on investments under risk. This is di erent from the concept of ambiguity, which is the subject of our study. While risk refers to situations where the perceived likelihoods of events can be represented by a unique probability distribution, ambiguity refers to situations where an agents subjective knowledge about likelihoods of contingent events is consistent with multiple probability distributions. Importantly, the agent does not know what the precise distribution is. 3 Electronic copy available at: critical. When they are fully con dent about the return of the investment, we can consider it a safe asset. If di erent returns are possible, but investors know the distribution over these returns, the asset is risky. When di erent returns are possible, but investors have only incomplete knowledge of probabilities, we would classify the asset as ambiguous. Epstein and Schneider (2008) investigate the e ects of bad news and good news on investors behaviors and show that, under ambiguity, investors overvalue negative information and undervalue positive information (Trautmann et al., 2008). Kelsey et al., (2011) investigate the pro tability of momentum strategies (buying past winners and selling past losers) in stock trading under ambiguity. Using the US stock market and accounting data, Kelsey et al., (2011) identify that negative momentum is greater than positive momentum in terms of magnitude and persistency of portfolio returns, and that such asymmetric patterns depend on ambiguity. In another recent study, Driouchi et al. (2018) investigate the behavior of US index put option holders during the pre-crisis and credit crunch period 20062008. They nd evidence of ambiguity in the US index options market during 20062008 and measure the e ect of ambiguity on realized index volatility that is implied directly from observed option prices. Based on portfolio data from a large nancial institution in France, Bianchi and Tallon (2018) show that ambiguity averse investors are relatively more exposed to the French stock market than to the international stock market. This result implies that ambiguity aversion plays a signi cant role in explaining home bias in equity markets. Most research on ambiguity focuses on traditional nancial assets while a few studies explore the role of ambiguity in the upcoming 4 Electronic copy available at: digital currency such as Bitcoin.6Using an incentivized survey, Anantanasuwoung et al. (2019) investigate ambiguity attitudes toward traditional nancial assets and cryptocurrency. Their nding suggests that individuals perceived ambiguity levels di er depending on the type of asset. In this paper, we refer to ambiguity as uncertainty over the probability of potential fu- ture outcomes, while risk refers to uncertainty over those outcomes following Knight (1921). Speci cally, we estimate ambiguity using ve-minute Bitcoin returns based on the model of Brenner and Izhakian (2018). Our ndings show that ambiguity plays an important role in Bitcoin returns; that is, investors take into account ambiguity when they price ambiguity. Our evidence further implies that investors show an increasing aversion to ambiguity. We conduct a battery of robustness tests to verify our ndings. For example, we use the forward-looking implied volatility from the S Ulrich 2013; Antoniou, Harris, and Zhang 2015; Williams 2015; Brenner and Izhakian 2018). Antoniou, Harris, and Zhang (2015) show that increases in ambiguity will lead to out ows from equity market. Brenner and Izhakian (2011) document the role of ambiguity in the US Equity markets, their nding suggested that ambiguity-averse investors will request a premium for bearing ambiguity. 5 Electronic copy available at: (1965), the FamaFrench (1993) three-factor model (FF3FM), the Carhart (1997) momentum- extended FF3FM, and the FamaFrench (2015) ve-factor model (FF5FM). We rst con rm their ndings and further show that investors earn the abnormal returns of Bitcoins only when ambiguity is low but not when ambiguity is high. 1.2. Contribution We contribute to the literature in several ways. First, we make a behavioural attempt at identifying the potential impact of ambiguity on asset pricing and the risk-return relationship. This is useful, because the use of Bitcoin, in a wider portfolio management strategy, has been shown to provide hedging bene ts (Kajtazi and Moro, 2018; Atsalakis et al. 2019; Ma et al., 2020; Thampanya et al., 2020); yet Bitcoin markets are typically characterized by crashes (Fry and Cheah, 2016), excessive volatility (Katsiampa, 2017), and positive returns when the fundamental value is shown to be zero (Cheah and Fry, 2015). It is well known that traditional asset pricing models have di culties in explaining the Bitcoin returns. Our study extends our understanding of the cryptocurrency market from a behavioral nance perspective, and we nd that ambiguity plays an important role in explaining the abnormal returns of Bitcoin. Second, our study is related to general studies which have focused mainly on the theoretical aspects of attitudes toward (aversion to) ambiguity, rather than on the actual measurement of ambiguity. Only a few studies used market data to measure ambiguity; for example, Ulrich (2013) uses entropy of in ation and Williams (2015) uses the Volatility Index (VIX). Following 6 Electronic copy available at: Brenner and Izhakian (2018), we explore the importance of ambiguity in the cryptocurrency market using Bitcoin data. Our study has important implications for sustainability. By studying the unique ambiguous feature of Bitcoin, we aim to at least partially take into account the dynamics of this highly volatile currency. This way, we aim to empower investors | small or big, to be able to make informed decisions regarding their choices. Moreover, our proposal has practical importance too. Not only individual investors but various funds|such as Crypto Fund AG|have risk exposure to Bitcoin. This paper helps to shed light on their investment decisions on Bitcoin. If investors can indeed earn the risk premium after adjusting for systematic risk, then it is helpful to allocate their wealth to Bitcoin. However, if the risk premium is conditional on ambiguity as shown in our results, caution should be exercised by investors in real-time trading because the risk premium becomes insigni cant during periods of high ambiguity. Our work also has important implications for policy makers. While Bitcoin markets are largely unregulated under current market conditions, policy makers can use our study to guide regulations if they plan to implement these in the future. For example, policy makers can use our method to estimate the ambiguity of Bitcoin which can help to identify potential market bubbles. They can also use the ambiguity of Bitcoin to cool-o trading in the Bitcoin markets. The remainder of the paper proceeds as follows. Section 2 discusses the construction of the ambiguity measure. Section 3 describes the data while section 4 reports the main empirical results and performs various robustness tests. Section 5 concludes the paper. 7 Electronic copy available at: 2. Measurement of Ambiguity As we noted before, ambiguity refers to situations where an agents subjective knowledge about likelihoods of contingent events is consistent with multiple probability distributions, there has been an evolution in the way we measure ambiguity, focusing in particular, on the way we embed information. For the purpose of our paper, we follow Izhakian (2020) and de ne ambiguity as f2r = Z E(r)Var(r)dr; (1) where r is the Bitcoin return, (r) is the marginal probability, E is the expectation, and Var is the variance. While risk can be measured by the volatility of returns,f2r captures the fact that ambiguity can be measured by the volatility of probabilities (Rothschild and Stiglitz, 1970). By construction, f2r is independent of risk, attitudes towards risk and/or attitude towards ambiguity and takes into account the variance of all the moments of the outcome distribution (Brenner and Izhakian, 2018). In line with Andersen et al. (2001), we use ve-minutes intervals price to compute returns to minimize microstructure e ects. For each day we use ve-minute returns to compute the normalized (by the number of intraday observations) daily mean ( ) and variance of the return ( ), respectively. Following Scholes and Williams (1977), we estimate by taking into account the adjustment for nonsynchronous trading. Speci cally, is computed as 8 Electronic copy available at: 2t = NtX i=1 (ri;t Eri;t)2 + NtX i=2 (ri;t Eri;t)(ri;t 1 Eri;t 1); (2) where there are Nt ve-minute returns, ri;t, in day t. Following Br
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