The Cross-Section of Cryptocurrency Returns∗

Nicola Borri Kirill Shakhnov

LUISS Surrey

This version: January 26, 2021

Abstract

At a given point in time, bitcoin prices are diﬀerent on exchanges located in diﬀerent

countries, or against diﬀerent currencies. While existing literature attributes the largest

price diﬀerences to frictions, like market segmentation, trading platforms advertize how

to execute trades based on this information. We provide a novel risk-based explanation

of these price diﬀerences for a sample containing the most reputable exchanges and after

accounting for all transaction costs and limitations to trade. Bitcoin prices for more

“expensive” pairs are riskier because they depreciate more in bad times for cryptocurrency

investors, when aggregate liquidity is lower, investors’ sentiment is lower, and counter-

party risk is higher.

Keywords: cryptocurrencies; common factors; liquidity; momentum; counter-party risk JEL Classiﬁcation: G12, G14, G15, F31

∗Borri: Department of Economics and Finance, LUISS University, Viale Romania 32, 00197 Rome, Italy; [email protected]; https://sites.google.com/site/nicolaborri. Shakhnov: School of Economics, University of Sur- rey, Guildford, Surrey, GU2 7XH, UK; [email protected]; https://sites.google.com/site/kshakhnov/. We thank Bob Chirinko, Sergei Kovbasyuk, Gabriele Zinna, Paolo Santucci de Magistris, Aleh Tsyvin- ski, Yukun Liu and seminar and conference participants at LUISS, EIEF, Crypto Valley Conference on Blockchain Technology, Technische Universit¨atBerlin, Bank of Italy, University of Navarra, ICEF (HSE), CFE-CFStatistics 2018, CFE-CFStatistics 2019. Aleksandr Schneider provided excellent research assistance. Some of the material in this paper previously circulated under the title “Cryptomarket Discounts.” 1 Introduction

Investors buy bitcoins on a multitude of exchanges located in different countries and against different currencies. At one point in time, and in a frictionless world where investors could trade instantaneously across exchanges, the bitcoin price, converted in a common currency, should be the same everywhere and for any currency pair. Instead, there are large differences in bitcoin prices across exchanges located in different countries, or for different currency pairs, and these differences are known to investors, who discuss them extensively on cryptocurrency blogs.

Two common explanations for these large and persistent price differences, or bitcoin discounts, are limits to arbitrage (Makarov and Schoar, 2019), and market inefficiency (Krückeberg and Scholz, 2020). While various frictions and the opacity of many exchanges explain the largest differences in bitcoin prices, we document smaller, but economically significant, differences also in a sample where these frictions are likely to be small, and for the most reputable exchanges. This casts doubts over the possibility that these stories alone could provide a comprehensive explanation. In this paper, after accounting for all the transaction costs, we propose a novel explanation that relates deviations in bitcoin prices to aggregate risk in cryptocurrency markets.

With the rapidly growing importance of bitcoin and blooming of dozens of new cryptocurrencies, it becomes crucial to study the efficiency and structure of cryptocurrency markets. We focus on bitcoin because it was the first cryptocurrency, and it currently accounts for two-thirds of the total market capitalization and one-third of the trading volume. Cryp- tocurrencies are used as means of payment, store of value, to diversify portfolios, and to raise funds in so-called initial coin offers (ICOs). Therefore, the efficient allocation of capital depends on well-functioning markets. Further, the growing interest in digital currencies and concern about inflation, related to the massive increase in the balance sheet of central banks and government debt in advanced economies in response to the Covid-19 pandemic, have contributed to the eight-fold rise in bitcoin price since March 2020. Finally, cryptocurrency

2 markets could be a laboratory to understand markets for diﬀerent assets that share a similar structure. In fact, the diﬀusion of electronic trading platforms has multiplied the venues where investors trade traditional assets, like foreign currencies.

We carefully select the sample of exchange-currency pairs to minimize the impact of various frictions, like market segmentation, data reliability and liquidity; then, we form portfolios to isolate the possible common components, and accommodate the large time-variation in the number of exchanges and currencies pairs. We select exchanges based on several indicators of “quality”, including ratings, web traffic data, and liquidity. For example, the exchanges in our sample are not among those singled out for “fake volume and/or non-economic wash trading” in a widely circulated report by Bitwise, the creator of the first crypto index fund (Bitwise, 2019). Furthermore, we use the capital control index constructed by Fernándezet al.(2016) to select exchange locations and currency pairs available to international investors. While this dramatically reduces the number of exchanges and pairs in our sample, it addresses the issue of data quality and market segmentation.

We take the perspective of investors that observe bitcoin discounts and form portfolios according to two investable strategies. The ﬁrst, which we denote as cross strategy, is based on the persistence of the discounts and is designed to buy bitcoin using a benchmark exchange- currency pair and to sell bitcoin using relatively “expensive” exchange-currency pairs. The second, which we denote as within strategy, is based on the mean-reversion of the discounts and is designed to buy bitcoin using relatively “cheap” exchange-currency pairs and to sell bitcoin at a later date using the same pair.

The first portfolio contains the pairs with the smallest average discounts, while the last portfolio contains the pairs with the largest average discounts. For both strategies, we obtain large, and significant, cross-sections of excess returns before transaction costs. Specifically, for cross portfolios, the average excess return of a long/short strategy that goes long in the last portfolio and short in the first portfolio is equal to 181 basis point per day. Similarly, for within portfolios, the average excess return of the long/short strategy is equal to 127 basis

3 points per day. Although these returns are extremely large, they are mostly absorbed by transaction costs, which significantly reduce average portfolio returns by approximately 50 basis points for each of the cross portfolios and by 90 basis points for each within portfolios. In fact, within portfolio returns are not statistically significant after accounting for all the transaction costs and fees. In contrast, cross portfolio returns, after transaction costs, are also smaller but significantly larger than zero. Since within portfolio returns are negative after transaction costs, we concentrate our attention on net returns for cross portfolios.

To account for transaction costs, we collect data on withdrawal, deposit, margin, and trading fees for all the exchanges in the sample, and on bid/ask spreads for a subset of 33 pairs for which we were able to collect reliable data. We document large heterogeneity in bid/ask spreads across exchanges, currency pairs, and time. We further extend the sample of bid/ask spreads using the estimates from predictive regressions of the bid/ask spreads on intra-day price volatility and trading volume and, for robustness, using the methodologies proposed by Roll(1984) and Abdi and Ranaldo(2017). Further, we drop illiquid pairs.

The average net excess return of a long/short cross strategy that goes long in the last portfolio and short in the first portfolio is equal to 31 basis points per day for a daily (non- annualized) Sharpe ratio of 13%. As a reference, over the same period and frequency, the Sharpe ratio of bitcoin returns is equal to approximately 6%, and the Sharpe ratio of U.S. market excess returns is equal to 4.5%. Bid/ask spreads are responsible for most of this reduction. To get a rough estimate of the contribution of the different transaction costs, note that withdrawal and deposit fees are typically small and lump sum and the average trading fee is 15 basis points per trade. Investors pay these costs twice for the within strategy, but only once for the cross strategy. This is because in the cross strategy, one of the two trades always involves the benchmark pair, the dollar-to-bitcoin pair on the Kraken exchange, for which these costs are negligible, especially for larger investors. Bid/ask spreads bite particularly long/short strategies, as they require four trades to be completed. However, we find that the short leg of the strategy does not contribute much to the overall returns. Then, investors

4 could reduce transaction costs by executing only the long leg.

Both strategies are risky because investors, in order to form portfolios by bitcoin discounts, must transfer balances across exchanges and, then, buy and sell bitcoin at diﬀerent times. The latter assumption captures the uncertain waiting times required by the architecture of both the blockchain (i.e., the time required for the proof-of-work) and the exchanges (i.e., the number of conﬁrmations from the blockchain required to settle transfers of balances), which imply that pure, instantaneous, arbitrage strategies are impossible. Since transaction costs fully absorbs within portfolio returns, they are a measure of the magnitude of frictions impeding arbitrage activity.

A principal component analysis indicates that two common components explain most of the variation in portfolio cross returns. The ﬁrst principal component is a level factor because all portfolios load similarly on this factor. It is essentially the average bitcoin return, i.e., the increase in the bitcoin price, and we call it the bitcoin factor, or Btc. The second principal component is a slope factor whose weights decrease from positive to negative from low to high bitcoin discount portfolios. Hence, average returns on the bitcoin portfolios line up with portfolio loadings on this second component. This slope factor is highly correlated with the return spread between the last and ﬁrst portfolio. We denote this excess return as the bitcoin

carry factor, or CarryBtc. Motivated by the results of the principal component analysis, we estimate a two-factor model to identify the common factors that could explain this cross-

section. We ﬁnd that CarryBtc explains a large fraction of the cross-sectional variation in portfolio returns.

We show that these excess returns compensate investors for taking on more aggregate risk for cryptocurrency markets. Bitcoin prices for more “expensive” pairs depreciate more in bad times for cryptocurrency investors, when aggregate liquidity is lower, investors’ sentiment about bitcoin is lower, and counter-party risk is higher. The basic ﬁnance logic we use for any other asset can be applied to cryptocurrencies. If an asset oﬀers low returns when investors’ marginal utility is high, it is risky, and then investors require a compensation through a

5 positive excess return. We identify bad times for investors by projecting CarryBtc on a

large set of crypto and non-crypto factors. We ﬁnd that CarryBtc is unrelated to non-crypto factors, but that a large fraction of its variation is signiﬁcantly and positively related to bitcoin aggregate liquidity risks, which we proxy with the cross-sectional volatility in bid/ask spreads on all pairs, the cross-sectional sum and volatility of the volume of transactions; to bitcoin sentiment, which we proxy with changes in the Google Trend index for the query “bitcoin”; to bitcoin momentum; and to bitcoin counter-party risk, proxied by the fraction of

temporarily inactive exchanges. Formal tests show that CarryBtc is a relevant factor in the time series and the cross-section of portfolio returns both in sample and out-of-sample. For the latter, we split the bitcoin pairs in our sample into two non-overlapping groups and show

that the CarryBtc risk factor obtained from the ﬁrst group is priced in the cross-section of portfolio returns of the second group.

We evaluate the robustness of our results along several dimensions. First, we consider all transaction costs, like trading and margin fees, and show that the cross-section of portfolio returns remains large and significant, but trade size can substantially reduce portfolio returns. Second, we find that our results hold in a more recent sample, and in sample that only includes crypto-to-crypto pairs, like ethereum-to-bitcoin. Note that, while governments can use domestic legislation to restrict flows of fiat currencies across borders, they have few instruments to restrict flows of cryptocurrencies. Finally, we find a cross-section of cross portfolio returns on a shorter sample that starts in 2018, indicating that our results also hold in the more recent period, after the bitcoin frenzy of 2017.

This paper contributes to the recent and growing literature on empirical asset pricing of cryptocurrencies1. One strand of the literature studies the risk-return characteristics of cryptocurrencies, typically considering returns before transaction costs. Liu and Tsyvinski

1Yermack(2013), Velde et al.(2013), and Dwyer(2015) are excellent primers that describe the functioning of the blockchain and cryptocurrencies. Catalini and Gans(2016), Biais et al.(2019), Cong et al.(2018), and Ma et al.(2018) analyze from the perspective of economic theory how blockchain technology and cryptocurrencies will inﬂuence the rate and direction of innovation and the incentives and equilibria behind the “proof of work” protocols.

6 (2018) ﬁnd that only crypto-speciﬁc risk factors, like investors’ attention and momentum, can explain the time-series risk-return relation for cryptocurrencies. Liu et al.(2019) show that the cross-section of returns for a large set of cryptocurrencies is explained by three crypto-

specific factors, unrelated to traditional risk factors. These factors are unrelated to CarryBtc which, thus, contains additional pricing information. A second strand of the literature studies the efficiency and pricing of cryptocurrencies. Makarov and Schoar(2019) use bitcoin price differences across exchange-currency pairs to study the efficiency of cryptocurrency markets. In a small sample of pairs characterized by small bid/ask spreads, they document large arbitrage opportunities across exchanges in different locations and explain them with capital controls and weak financial institutions. In a companion paper, we expand Makarov and Schoar’s sample to include all the fiat and cryptocurrencies and decompose the variability in bitcoin prices in components associated to, in order of importance, time, location of the exchanges, and currency pair (Borri and Shakhnov, 2018). Krückeberg and Scholz(2020), instead, attribute these price differences to market inefficiencies and untapped arbitrage opportunities. This paper builds a bridge between these two strands of the literature: it accounts for all the transaction costs and limitations to trade, and provides a risk-based explanation of bitcoin price differences across exchanges, currency pairs, and time. This paper is also related to the large finance literature on market efficiency and anomalies as well as limits to arbitrage. Some papers argue that market inefficiencies can explain differences in the prices of homogenous assets, others attribute them to differences in risk. Examples of the former, are Lee et al.(1991); Chen et al.(1993) for closed-end funds; Lamont and Thaler (2003) for tech stock carve-outs; Gagnon and Karolyi(2010) for cross-listed stocks, such as ADRs; Burnside(2011) for the forward premium; Du et al.(2018) for the deviations from the covered interest parity. Examples of the latter are Cochrane(2002) for tech stock carve-outs; Krishnamurthy(2002) for on the run and off the run bonds; Lustig and Verdelhan(2007) for the forward premium. Pontiff(1996) and Tuckman and Vila(1992) argue that idiosyncratic risks limit arbitrage opportunities. This paper is the first to propose a risk-based explanation

7 for bitcoin price diﬀerences across liquid ﬁat-to-bitcoin pairs traded on exchanges located in countries with no capital controls.

The rest of the paper is organized as follows: Section2 begins by describing the data, the method used to build the bitcoin portfolios, and the main characteristics of these portfolios; Section3 shows that one risk factor, related to bitcoin liquidity, momentum, sentiment, and counter-party risks, explains most of the cross-sectional variation in bitcoin excess returns; Section4 considers several extensions and robustness analysis; Section5 presents our conclusions.

2 Bitcoin Portfolios

In order to investigate how far a risk-based explanation can go in explaining bitcoin price differences in different markets and against different currencies, we take the perspective of investors who form portfolios according to two strategies based on the persistence or mean-reversion of bitcoin discounts. By forming portfolios, we accommodate the large time- variation in the number of exchanges and currencies pairs and isolate common factors. For both strategies, we obtain a large and significant cross-section of returns before transaction costs. In this Section, we focus on a sample in which frictions are small, containing the most liquid pairs and exchanges located in countries with limited constraints for foreign investors and satisfying a set of constraints in terms of “quality”. Section4 accounts for execution risks, additional transaction costs, and different samples.

2.1 Building portfolios

Data There are two types of exchanges where investors can trade bitcoin and other cryptocurrencies. The ﬁrst, denoted crypto-to-crypto exchanges, are exchanges on which only cryptocurrency pairs are traded (i.e., bitcoin for ethereum), and where investors can deposit and withdraw only cryptocurrencies; the second, denoted ﬁat-to-crypto exchanges, are instead

8 exchanges where investors can trade fiat currencies for cryptocurrencies (i.e., U.S. dollar for bitcoin), and deposit and withdraw both fiat and cryptocurrencies. While the second type of exchanges is subject to country regulations, and, for example, U.S. based fiat-to-crypto exchanges are registered as a money service business with the Financial Crimes Enforcement Network of the U.S. Department of Treasury, the first type of exchanges is mostly unregulated, and this has recently raised concerns about their reliability (Bitwise, 2019; Gandal et al., 2018) and risk of price manipulation using pump-and-dump schemes Li et al.(2020). 2

The two types of exchanges also differ in the number of currency pairs traded: while investors can trade many, often in the hundreds, crypto-to-crypto currency pairs on crypto- to-crypto exchanges, they can trade only a few, mostly fiat-to-crypto, currency pairs on fiat-to-crypto exchanges. The majority of transactions takes place on exchanges that function like standard equity markets where investors submit buy and sell orders that are cleared by a centralized order book. Instead, other exchanges offer order-matching services and match buyers’ and sellers’ orders when they overlap. Finally, note that all exchanges operate 24/7, including Saturdays, Sundays, and holidays, and use the UNIX time-stamp to track time and ensure the immediate comparability of market prices.

We start by collecting daily bitcoin price and volume data for all exchanges and currency pairs available on cryptocompare.com. In our baseline analysis, we restrict our sample along several dimensions in order to focus on reliable exchanges and currency pairs that are likely to be available to foreign investors.

First, we only consider ﬁat-to-crypto pairs on ﬁat-to-crypto exchanges (for example, the U.S. dollar-to-bitcoin pair on Coinbase, but not the ether-to-bitcoin pair on Coinbase), because of the lower reliability of crypto-to-crypto exchanges. In what follows, we treat each

2According to the report by Bitwise(2019), 95% of the volume in CoinMarketCap is fake and/or non- economic in nature. While this claim has not been independently verified, it is reasonable to assume that exchanges have an incentive to appear at the top of the lists used by media organizations to attract listing fees from ICOs and altcoins. We note that all the exchanges described in the report as fabricating data are crypto-to-crypto exchanges and, thus, not in our sample of bitcoin-to-fiat pairs. Li et al.(2020) argue that pump-and-dump schemes occur mostly on crypto-to-crypto exchanges, last only few minutes, and that fiat-to-crypto exchanges, like Bittrex, banned them. In Section B.I of the Online Appendix we report, for each exchange in the baseline sample, various “quality” indicators (see Table A3).

9 fiat-to-crypto currency pair on a given exchange as a different asset. For example, we treat the dollar-to-bitcoin pair on Coinbase and the dollar-to-bitcoin pair on Kraken differently. Similarly, the dollar-to-bitcoin and euro-to-bitcoin pairs, both on Coinbase, are also different assets.

Second, we consider only fiat-to-crypto exchanges that satisfy several indicators of “quality”, including ratings and web traffic data produced by Nomics and Cryptocompare, two data companies that provide market data to institutional crypto investors and exchanges; the new liquidity metric produced by CoinMarketCap, which is specifically designed to detect fake volume; and the list of exchanges providing bitcoin quotes for Bloomberg.

Third, in order to focus on pairs likely available to foreign investors, we include in the sample only pairs that satisfy both of the two following criteria to minimize the role of restrictions to capital flows. The exchanges must be located in one of the following countries: Australia, Canada, Denmark, the Eurozone, Hong Kong, Israel, Japan, Poland, Switzerland, the U.K., and the U.S. The fiat currencies must be one of the following: Australian dollar, Canadian dollar, Danish krone, euro, Hong Kong dollar, Israeli shekel, Japanese yen, Polish zloty, Swiss franc, British pound, the U.S. dollar. We select this set of countries and currencies based on the tightness in capital controls with respect to the U.S. using the capital control index constructed by Fernándezet al.(2016).

Fourth, we compute U.S. dollar prices for all ﬁat-to-bitcoin pairs using daily spot rates from WM/Reuters and available on Datastream. For this reason, we exclude non-business days because of the unavailability of spot rates. Fifth, to focus on the most liquid pairs, we consider the following additional restrictions. We exclude pairs with less than 30 days of observations; pairs traded on peer-to-peer platforms, which typically have very low volume; pairs with an average daily bid/ask spread larger than 2%; and observations corresponding to the 5-day average of daily volume smaller than 10 bitcoins.

Our baseline sample starts on May 26, 2015, and ﬁnishes on October 31, 2019. The starting date of the sample depends on the availability of a minimum number of pairs to

10 build portfolios and the possibility of shorting the dollar-to-bitcoin pair. Our raw sample contains 258 ﬁat-to-bitcoin pairs, while our baseline sample contains, at most, 47 pairs traded in 20 exchanges. Table1 contains descriptive statistics of the baseline sample. The number of exchanges and currency pairs increases over time because pairs enter and exit the sample as new exchanges open and close, and new pairs are introduced. Speciﬁcally, we start with 18 pairs traded on 11 exchanges in May 2015 and end with 40 pairs traded on 20 exchanges in October 2019. Over the sample, the daily median trading volume increased from 250,000 U.S. dollars to approximately 1.52 million of U.S. dollars. In the same period, the bitcoin price increased from approximately 430 U.S. dollars to more than 14,000 U.S. dollars in 2017 and then declined to about 9,000 U.S. dollars at the end of October 2019.

In the robustness analysis of Section4, we relax the restrictions that define the baseline sample. First, we consider pairs traded against additional fiat currencies and on exchanges located in countries not included in the baseline sample (for example, Korean won-to-bitcoin traded on a Korean exchange). Second, we consider pairs traded against additional cryptocurrencies on both fiat-to-crypto and crypto-to-crypto exchanges (for example, bitcoin-to- ethereum traded on Binance).

Table 1: Baseline Sample

price (Kraken) daily volume (USD millions) daily volume (BTC) baseline all ﬁat-to-crypto year USD per BTC median median p10 p90 # exchanges # pairs # exchanges # pairs 2015 431.31 0.25 580.37 3.24 15340.62 11 18 24 36 2016 960.70 0.55 662.24 5.76 7973.35 11 20 28 45 2017 14402.20 44.49 3085.81 18.02 23992.61 14 27 33 69 2018 3691.90 2.63 696.68 7.52 13429.96 20 42 62 117 2019 9146.70 1.52 182.45 6.90 5673.22 20 40 52 97

Notes: The Table reports descriptive statistics for the baseline sample. For each year, from 2015 to 2019, we report the bitcoin price in U.S. dollars on the Kraken exchange (our benchmark pair); the median daily volume in millions of U.S. dollars, and in units of bitcoin; the number of exchanges and pairs. For the volume expressed in bitcoins units, we also report the 10th and 90th percentile values (respectively, p10 and p90). The last two columns report the total number of exchanges and pairs in the sample containing all the ﬁat-to-crypto pairs after eliminating peer-to-peer and illiquid exchanges (i.e., Panel C of Table 9 is based on this sample). Descriptive statistics are for the last available day of each year. For 2019, they are for October 31, 2019, the last day of our sample. Note that pairs enter and exit the sample, and the total number of available pairs in the baseline sample is 47. Data are from cryptocompare.com and Thomson Reuters.

11 Bitcoin discounts Investors can trade bitcoin in a set of m = 1,...,M markets (i.e.,

? exchanges), using a set of j = 1,...,J currencies. We denote with Pm,j the units of currency j = 1,...,J required to buy one bitcoin in market m. We take the perspective of U.S. dollar- based investors trading bitcoin in these markets and currencies. We denote with Sj the spot exchange rate expressed in units of currency j per U.S. dollar. The U.S. dollar price of one bitcoin in market m and currency j is

P ? P = m,j m,j Sj

If investors could trade instantaneously across exchanges and in the absence of frictions, according to the law of one price they should get the same dollars per bitcoin in each market m and currency j. In fact, there exist large, persistent, and time-varying price diﬀerences across both markets and currencies, and over time (Makarov and Schoar, 2019; Borri and Shakhnov, 2018)3.

We denote the price in market m = 1, the Kraken exchange, and currency j = 1, the U.S. dollar, as the numeraire. This choice is motivated by the fact that Kraken is one of the oldest exchanges, launched in September 2013, with large bitcoin trading volume and low transaction costs. In addition, Kraken is one of the ﬁrst exchanges to introduce margin trading and the possibility of short-selling the dollar-to-bitcoin pair. At a given point in time, we deﬁne the bitcoin discount in market m and currency j as

Pm,j Dm,j = − 1 (1) P1,1

If Dm,j < 0, then investors get a smaller number of dollars per bitcoin in market m and

3Given the current blockchain technology, there is a minimum of 10 minutes wait for confirmation in the bitcoin blockchain which is hardcoded into the program, and exchanges typically require more than one confirmation before deposit cryptocurrencies in investor’s accounts. Therefore, investors cannot trade instantaneously across exchanges, unless they have the possibility to buy and short-sell bitcoin and keep balances in different exchanges. However, short-selling is not possible in all exchanges and maintaining balances imply inventory costs. For these reasons, although higher-frequency (i.e., intra-day) data are available, we do not use them in the construction of returns.

12 currency j than in the reference market. On the contrary, if Dm,j > 0, then investors get a larger number of dollars per bitcoin in market m and currency j than in the reference market.

Finally, when Dm,j = 0, then investors get the same number of bitcoins in all markets and currencies.

Figure 1 summarizes the characteristics of the bitcoin discounts for the pairs in our sample. Specifically, the top panel plots the median (red mark), along with the 25% and 75% percentiles (blue box); the middle panel plots their time series; finally, the bottom panel depicts the first-order autocorrelation coefficients estimated, for each pair, over the longest available sample. Discounts are, on average, different from zero and usually positive and volatile. Specifically, for the average pair, almost 17% of the daily observations have discounts larger than 1% in absolute value. Within the same market, discounts are time-varying and can be positive and negative and can reach absolute values of 30%. Finally, discounts are very persistent and mean-reverting. For most pairs, the autoregressive coefficients are statistically different from zero, and according to standard augmented Dickey-Fuller tests, we reject the null of unit root for all pairs at standard significance levels.

Bitcoin excess returns We assume investors can borrow at the dollar risk-free rate Rf and denote with lower case letters the log of any variable (i.e., x = log(X)). We consider the following two strategies based on observed bitcoin discounts. The ﬁrst is a cross strategy that exploits the persistence of bitcoin discounts. The second is a within strategy, based on the mean reversion of bitcoin discounts.

Figure 2 exemplifies the timing of the cross strategy. At time t, an investor borrows one U.S. dollar at the risk-free rate to buy bitcoin traded against currency j = 1 (i.e., the U.S. dollar) in the reference market m = 1 (i.e., Kraken) using the benchmark bitcoin-to-dollar pair. Note that, in order for the investor to complete the transaction, she first needs to deposit the initial dollar amount on the Kraken exchange (or she must have an existing balance). The investor then transfers her bitcoin balance to a different exchange, for example, Coinbase. In

13 Figure 1: Bitcoin Discounts

% 0

-2

0 %

-20 01/16 01/17 01/18 01/19

0.5

Notes: The top panel of the Figure contains a boxplot for the bitcoin discounts (Dm,j ) for all the pairs in our baseline sample. For each box, the central red mark indicates the median discount; and the bottom and top edge of the box indicate the 25th and 75th percentiles. The middle panel of the Figure plots the time-series of the bitcoin discounts (Dm,j,t) defined according to (1) and multiplied by 100. The bottom panel reports the estimated autoregressive coefficients obtained by fitting a first-order autoregressive model to the bitcoin discounts (ρAR(1)). For the autoregressive coefficients, the black dashed line denotes the cross-sectional average (approximately equal to 0.45). Note that according to standard augmented Dickey-Fuller tests, we reject the null of unit root for all pairs at standard significance levels. Data are daily from cryptocompare.com and Thomson Reuters for the period 5/26/2015–10/31/2019.

the Figure, the transfer, which could take up to 24 hours before conﬁrmation, is denoted by the vertical dashed line. At time t + 1, the investor ﬁrst trades bitcoins for euros on Coinbase and then exchanges euros for U.S. dollars at the spot rate on the foreign exchange (FX) market. Finally, the investor pays back the initial loan plus any accrued interest.

Cross excess returns, expressed in U.S. dollars, are then

cross f rxm,j,t+1 = pm,j,t+1 − p1,1,t − rt

? j ? f = pm,j,t+1 − st+1 − p1,1,t − rt (2)

Note that, for the cross strategy, the transaction at time t always occurs using the benchmark dollar-to-bitcoin pair on Kraken. Despite some similarities with an arbitrage strategy, this is

14 actually a risky trade. In fact, investors are exposed to the risk of price changes during the transfer of bitcoins from Kraken at time t to market m at time t + 1 (in the figure Coinbase). In addition, investors are further exposed to the exchange rate risk at t+1, with the exception of trades against other dollar-to-bitcoin pairs. Because of the uncertainty around the confirmation time, we make the assumption of a period to be one day and we show that results are robust to a slower weekly frequency. The exchange Kraken advertises an average confirmation time of 1 hour. However, according to data we collect on Blockchain, the confirmation time varies over time: the 99 percentile of the average confirmation time for a given day is approximately equal to 24 hours, and the average is equal to one hour.

Figure 2: Cross strategy

Kraken Kraken-Coinbase Coinbase FX market -US$ BTC BTC EUR US$ deposit conﬁrmation time withdrawal

time t time t + 1

Notes: This Figure exemplifies the timing of a cross strategy. At time t, the investor buys bitcoins with U.S. dollars in the reference market, i.e., Kraken, using the benchmark bitcoin-to-dollar pair. In order for the investor to complete the transaction, she first needs to deposit the initial dollar amount on the Kraken exchange (or she must have an existing balance). The investor then transfers her bitcoin balance to a different exchange (Coinbase in the Figure). The transfer could take up to 24 hours before confirmation. At time t + 1, the investor first trades bitcoins for euros on Coinbase, and then exchanges euros for U.S. dollars at the spot rate on the foreign exchange (FX) market. Because of the uncertainty around the confirmation time, we make the conservative assumption of a period to be equal to 1 day.

Figure 3 exemplifies the timing of the within strategy for a dollar-based investor considering the euro-to-bitcoin pair on Coinbase. At time t, the investor borrows dollars and exchanges them for euros on the spot market, and then buys bitcoins for euros on Coinbase. In order for the investor to complete the transaction, she first needs to deposit the initial euro amount on Coinbase (or she must have an existing balance). The investor holds the investment for one period and, in the following period, at time t + 1, she first trades bitcoins for euros, and then exchanges euros back into U.S. dollars at the spot rate and pays back

15 the initial loan. Similarly to the cross strategy, also for the within strategy we take a period to be one day. Diﬀerently with respect to the cross strategy, for the within strategy both the transaction at time t and t + 1 occur on the same exchange.

Figure 3: Within strategy

time t time t + 1

FX Coinbase waiting Coinbase FX -US$ EUR BTC BTC EUR US$ time

Notes: This Figure exempliﬁes the timing of a within strategy for a dollar-based investor considering the euro-to-bitcoin pair on Coinbase. At time t, the investor exchanges dollars for euros on the spot market, and then buys bitcoins for euros on Coinbase. In order for the investor to complete the transaction, she ﬁrst needs to deposit the initial euro amount on Coinbase (or she must have an existing balance). The investor holds the investment for one period and, in the following period, at time t + 1, she trades bitcoins for euros, and then exchanges euros back into U.S. dollars at the spot rate. Both the transaction at time t and t + 1 occur on the same exchange.

Within excess returns, expressed in U.S. dollars, are then

within f rxm,j,t+1 = pm,j,t+1 − pm,j,t − rt

? ? j f = pm,j,t+1 − pm,j,t − ∆st+1 − rt (3)

In the construction of returns for both strategies, we need to account for the possibility of exchange shutdowns: i.e., the fact that investors cannot complete their desired trade at t + 1 because of the non-availability of a given ﬁat-to-bitcoin pair. Temporary shutdowns are relatively frequent, for example because of distributed denial-of-service attacks (DDos) or software malfunction. We make the assumption that investors, in case of a shutdown, can sell their bitcoins at the median price prevailing at t + 1 across all pairs. In Section4 we consider additional execution risks and evaluate the robustness of our assumption.

Bid and ask prices We obtain daily bid-ask spreads data from Bitcoinity for a subset of 33 ﬁat-to-bitcoin pairs in our sample. In addition, we estimate bid and ask prices for all the

16 remaining ﬁat-to-bitcoin pairs using the predicted values from the following panel regression.

BAm,j,t = αj + γt + B(L)vm,j,t + A(L)hlm,j,t + m,j,t, (4)

where B(L) and A(L) are ﬁfth-order lag polynomial, v is the log trading volume in

bitcoins, hl is the lag of the high-low spread, and αj and γt are currency and time ﬁxed eﬀects4. Returns net of bid-ask spreads are

cross,net ask bid rm,j,t+1 ≡ pm,j,t+1 − p1,1,t = pm,j,t+1 − 0.5BAm,j,t+1 − (p1,1,t + 0.5BA1,1,t) (5)

within,net ask bid rm,j,t+1 ≡ pm,j,t+1 − pm,j,t = pm,j,t+1 − 0.5BAm,j,t+1 − (pm,j,t + 0.5BAm,j,t) (6)

ask bid ask where BAm,j = (pm,j − pm,j)/pm,j is the bid-ask spread for market m and currency j, and we assume that the price without transaction costs is halfway between the bid and ask prices.

Portfolios. At the end of each day t, investors form two sets of seven portfolios sorted by

bitcoin discounts Dm,j,t. Portfolios are ranked from the lowest negative to the highest positive bitcoin discount. For the first set of portfolios, investors use the cross strategy. For the second set of portfolios, investors use the within strategy. These strategies are implementable, in the sense that they are based on signals available to all investors. In order to execute a cross strategy investors must transfer crypto and fiat currencies across exchanges. The within strategy, on the other hand, requires transfers of cryptocurrencies only to adjust balances. In fact, we note that the cross strategy is popular among cryptocurrency investors, and websites like bitsgap or tokenspread offer the possibility to identify in real-time the largest discounts

4The (within) R-square of our panel estimation is 20% and most of the coefficients are significant at standard confidence levels. Specifically, higher trading volume and lower high-low spread are associated with smaller bid/ask spreads. Brauneis et al.(2021) argue that low frequency liquidity measures, for example based on high and low prices, are good estimates of actual liquidity measured with high-frequency indicators. In Section B.II of the separate Online Appendix we estimate bid/ask spreads with alternative estimators based on daily high, low, and close prices. Note that while the bid-ask spreads for the most liquid pairs in our sample are small and consistent with other studies (Dyhrberg et al., 2018), those for the less liquid pairs are an order of magnitude larger. Finally, note that some exchanges do not offer a limit order book, but only a matching of buy/sell orders. For these exchanges, then, our bid/ask estimates are a proxy of liquidity, as they do not explicitly post bid/ask prices.

17 and immediately start cross trades.

In the cross strategy, investors always start at time t by buying bitcoin using the benchmark pair. Recall that negative (positive) discounts denote pairs that have a low (high) price relative to the benchmark. Portfolio 1, then, groups the returns from selling, at time t + 1, the pairs with the lowest discounts, while portfolio 7 the pairs with the highest discounts. In the within strategy, investors buy and hold one pair for one period. Portfolio 1, then, groups the returns from investing in the pairs with the lowest discounts, while portfolio 7 in the pairs with the highest discounts.

For both sets of portfolios, we compute the bitcoin excess returns rxk,t+1 for portfolio k by taking the average of the bitcoin excess returns in each portfolio k

1 X rxk,t+1 = rxm,j,t+1 Nk {m,j}∈Pk

The total number of pairs in our portfolios varies over time. We have a total of 10 assets at the beginning of the sample on May 26, 2015, and 34 at the end of October 31, 2019. The maximum number of pairs attained during the sample is 38. Note that we start building portfolios in May 2015 because of the availability of a minimum number of pairs and the possibility of shorting bitcoin on Kraken, our benchmark exchange.

2.2 Returns to bitcoin speculation for U.S. investor

Table 2 oﬀers a quick snapshot of the properties of the two sets of seven portfolios sorted by bitcoin discounts. For each portfolio k, we report the average and standard deviation of discounts (panel A); cross returns (panel B); and within returns (panel C). For returns, we also report the average of long/short returns. For cross returns, the long/short returns correspond to a high-minus-low strategy that goes long in portfolio 7 and short in portfolio 1 (i.e., 7 − 1). For within returns, the long/short returns correspond to a low-minus-high strategy that goes long in portfolio 1 and short in portfolio 7 (1 − 7). For returns, we also

18 report standard errors, computed by bootstrap, and Sharpe ratios computed as the ratio of mean to standard deviation of daily excess returns. All moments are daily, and Sharpe ratios are not annualized.5

Panel A reports the average bitcoin discounts. Portfolio 1 contains pairs with the largest negative discounts and portfolio 7 pairs with the largest positive discounts. Recall that when discounts are negative (positive), investors get a smaller (larger) number of dollars per bitcoin than they would in the case of the benchmark pair on Kraken. Across our seven portfolios, discounts increase monotonically from -108 basis points to 198 basis points per day.

Panel B shows that, by sorting pairs by their discounts, we obtain a monotonic increasing cross-section of gross and net cross returns. Specifically, gross excess returns increase, across the seven portfolios, from 8 basis points to 189 basis points per day. Standard deviations of excess returns are similar across portfolios so that we also obtain a cross-section of Sharpe ratios. Specifically, daily Sharpe ratios increase from 2% to 41%. The standard errors indicate that for all portfolios except the first, average returns are significantly different from zero as they are more than two standard errors from zero6. A zero-cost high-minus-low strategy that goes long in portfolio 7 and short in portfolio 1 produces large and statistically significant excess returns of approximately 181 basis points.

For many pairs, bid/ask spreads are not negligible. In fact, net returns are 50 basis points smaller for the ﬁrst portfolio, and almost 65 basis points smaller for the last portfolio. The average return of the long/short strategy is substantially smaller after accounting for transaction costs and approximately equal to 68 basis points for a Sharpe ratio of 29%. Net long/short returns are not simply equal to the diﬀerence between the corner portfolios’ net returns. As back of the envelope calculation, consider that long/short net returns are instead

5In Section F.V and F.I of the separate Online Appendix we show that the properties of the bitcoin portfolios are robust to smaller and larger numbers of portfolios and provide evidence of no clustering of subsets of the pairs in any portfolio using the k-means clustering test. 6We reject the null of equal returns, at standard significance levels, for the corner portfolios using Newey and West(1986) heteroskedasticity and autocorrelation consistent standard errors. In addition, we also find that the Patton and Timmermann(2010)’s tests for a monotonic relationship ( MR and MRall) reject the null of flat or weakly decreasing pattern against the alternative of strictly increasing pattern.

19 equal to the long/short gross returns net of the difference between the averages of the gross and net returns of the corner portfolios (i.e., 1.81 − (1.89 − 1.26) − (0.08 + 0.42) = 0.68). In order to go short in portfolio 1, investors must be able to short bitcoin on Kraken, our benchmark exchange. Kraken introduced margin trading and the possibility to open short positions starting on 5/25/2015, thus at the beginning of our sample. While, at the time of this writing, several of the exchanges in our sample offer margin trading, in 2015, investors could short bitcoin only on Kraken and Bitfinex7. In Section4, we show that the mean long/short return remains positive and statistically different from zero also after accounting for exchange and margin fees.

Panel C of Table 2 shows that, by sorting pairs by their discounts, we also obtain a monotonic decreasing cross-section of gross and net within returns. Specifically, gross returns decrease, across the seven portfolios, from 120 basis points to -7 basis points per day. However, net returns are negative or not statistically different from zero for all portfolios. In fact, while the gross long/short average return is equal to 127 basis points per day, the corresponding average net return is equal to -83 basis points per day. Therefore, the within strategy is not economically profitable and captures the magnitude of the frictions that impede arbitrage activity. Further, in order to implement the long/short within strategy over time, investors need to be able not only to transfer either bitcoin or fiat currency across exchanges, but they also have to be able to short pairs on all exchanges. The latter is not always possible. Alternatively, for the short position, investors could keep balances of fiat currencies on all exchanges; and, for the long position, investors could keep balances of bitcoin on all exchanges. These balances would grow over time, exposing investors to inventory costs related to changes in bitcoin and fiat currency prices.

For both the cross and within long/short strategies, the long position accounts for most of the return while the contribution of the short position is negligible. This is important

7Alternatively, investors could hold balances on all exchanges. Speciﬁcally, for the short position, investors could keep balances of ﬁat on all exchanges and bitcoin on Kraken. At the end of each period, investors should adjust their balances by transferring bitcoin to Kraken.

20 for two reasons. First, as discussed above, short positions are not available on all exchanges or for the entire sample. Second, the short position accounts for one-half of the transaction costs and, additionally, requires the payment of margin fees (see also Table 8). Therefore, we consider the net returns from the long/short strategies as conservative estimates of the returns available to investors.

Table 2: Bitcoin portfolios: US investor

Portfolio 1 2 3 4 5 6 7 long/short Panel A: discounts Mean -1.08 -0.28 -0.04 0.16 0.41 0.76 1.98 Std 1.28 0.60 0.55 0.59 0.75 1.01 1.89 Panel B: cross returns 7-1 gross returns Mean 0.08 0.41 0.51 0.62 0.79 1.09 1.89 1.81 Std 4.52 4.53 4.57 4.55 4.55 4.61 4.65 2.21 SE 0.13 0.13 0.14 0.14 0.13 0.14 0.14 0.07 SR 0.02 0.09 0.11 0.14 0.17 0.24 0.41 0.82 returns net of bid/ask Mean -0.42 0.13 0.27 0.39 0.52 0.77 1.26 0.68 Std 4.55 4.54 4.57 4.57 4.58 4.63 4.72 2.33 SE 0.14 0.14 0.14 0.14 0.14 0.14 0.13 0.07 SR -0.09 0.03 0.06 0.09 0.11 0.17 0.27 0.29 Panel C: within returns 1-7 gross returns Mean 1.20 0.70 0.55 0.46 0.38 0.33 -0.07 1.27 Std 4.74 4.58 4.57 4.53 4.50 4.52 4.37 2.08 SE 0.14 0.13 0.14 0.13 0.14 0.14 0.13 0.06 SR 0.25 0.15 0.12 0.10 0.08 0.07 -0.02 0.61 returns net of bid/ask Mean 0.28 0.22 0.14 0.08 -0.09 -0.24 -1.24 -0.83 Std 4.73 4.61 4.60 4.56 4.57 4.59 4.77 1.48 SE 0.14 0.14 0.14 0.13 0.14 0.13 0.13 0.06 SR 0.06 0.05 0.03 0.02 -0.02 -0.05 -0.26 -0.79

Notes: This Table reports, for each portfolio k = 1,..., 7, the mean and standard deviation for the average discounts (panel A); the mean (Mean), standard deviation (Std), standard error (SE), and Sharpe ratio (SR) for the cross returns (panel B) and for the within returns (panel C). Standard errors are by bootstrap, Sharpe ratios are computed as ratios of daily means to daily standard deviations and are not annualized. Only for returns, the table reports mean, standard deviation, standard error, and Sharpe ratio for long/short excess returns, deﬁned as the returns of a zero-cost strategy that goes long in portfolio 7 and short in portfolio 1 for cross returns; and long in portfolio 1 and short in portfolio 7 for within returns. For both the cross and within investment strategies, we consider both gross and net returns. Portfolios are constructed by sorting assets into seven groups at time t by their discounts Dm,j,t. The ﬁrst portfolio contains pairs with the lowest negative discounts. The last portfolio contains pairs with the highest positive discounts. Data are from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

While Table 2 documents the unconditional properties of the bitcoin portfolios, Figure 4,

21 for the cross strategy, provides a snapshot of the time-variation in discounts and returns. Specifically, it plots the time-varying discounts for the corner and median portfolios (top panel); the time-varying portfolio’s returns for the corner portfolios, along with the bitcoin return (middle panel); and long/short 7-1 gross and net returns (bottom panel), using a rolling window of 100 days. The Figure shows that the spread in bitcoin discounts for the corner portfolios are correlated with the bitcoin price. For example, they are larger in 2017, when the bitcoin price was at its peak. The Figure also shows that discounts have not significantly declined over time. Furthermore, the Figure shows that portfolio returns are correlated with bitcoin returns, and always larger for portfolio 7 than for portfolio 1; and that the difference between gross and net returns has also not declined over time. This is because new, less liquid, pairs have entered the sample. Therefore, net long/short returns have been mostly nonnegative throughout the entire period. The contribution of transaction costs to net long/short returns is almost constant with the exception of the beginning of the sample. In fact, all the time variation in net long/short returns is due to the time variation in gross returns.

3 Common Factors in Bitcoin Returns

We have uncovered two different cross-sections of bitcoin excess returns related to strategies based on observed bitcoin discounts. After accounting for transaction costs, portfolio within returns are not statistically different from zero. These returns, then, are likely a manifestation of frictions on cryptocurrency markets that prevent investors from absorbing price differences across markets and pairs. In contrast, we have found a significant cross-section of portfolio cross returns after accounting for transaction costs. The latter is a popular strategy among investors and facilitated by trading platforms which advertize bitcoin discounts.

In this Section we show that the large cross-section of cross portfolio returns captures the compensation that investors demand to bear aggregate risk in cryptocurrency markets.

22 Figure 4: Time-varying Returns

discounts 6 mean (cross) 1 4 7 2 % 0 -2

2016 2017 2018 2019 returns 6 1 (net) 1 (gross) 4 7 (net) 7 (gross) btc 2 % 0 -2

2016 2017 2018 2019 long/short: 7-1 6 gross net 4 2 % 0 -2

2016 2017 2018 2019

Notes: This Figure plots the time-varying averages of the bitcoin discounts, portfolio cross returns, and 7-1 long/short returns, computed using a rolling window of 100 days. Speciﬁcally, the top panel plots the average discounts for portfolio 1 (black solid line) and 7 (red dashed line) along the cross-sectional average (blue dotted line). The middle panel plots the average returns for portfolio 1 (black) and 7 (red). We report average returns both gross (dashed line) and net (solid line) of bid/ask spreads. Additionally, we also report the average bitcoin returns (dotted green line). The bottom panel plots the average 7-1 long/short returns, both for gross (dashed line) and net (solid line) of bid/ask spread returns. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

Diﬀerences in portfolio cross returns are matched by covariances with just two risk factors: the bitcoin return (Btc), and the excess return form a long/short zero-cost strategy that goes long in the last portfolio and short in the ﬁrst (7 − 1). We refer to this second factor as

CarryBtc. The ﬁrst factor is a level factor that captures the returns of investing in bitcoin using the benchmark pair, i.e., the dollar-to-bitcoin pair on Kraken. The second is a slope

factor that explains the cross-section of portfolio returns. We show that CarryBtc is related

23 to bitcoin aggregate liquidity, momentum, and counter-party risks, while is not related to traditional risk factors, like the Fama and French(1993)’s factors for equity markets.

In this Section, we always consider portfolio returns net of bid/ask spreads, and not gross returns. In Section4 we additionally account for exchange trading and margin fees.

3.1 Methodology

Linear factor models predict that average returns on a cross-section of assets can be attributed to risk premia associated with their exposure to a small number of risk factors. In the arbitrage pricing theory (APT) of Ross(1976), these factors capture common variation in individual asset returns.

Table 3 illustrates the results of a principal component analysis of the cross portfolios, which reveals that two factors explain approximately 97% of the variation in portfolio returns. Because the loadings on the first principal component are similar across the seven portfolios, we interpret this first component as a level factor. On the contrary, the loadings on the second component decrease across the seven portfolios, from 0.71 on the first portfolio to -0.62 on the last portfolio. We interpret the second component as a slope factor. Since average excess returns increase monotonically across portfolios, the second principal component is a plausible candidate risk factor that might explain the cross-section of portfolio excess returns.

Motivated by the principal component analysis, we construct a two-factor model. The ﬁrst factor, denoted Btc, is simply the return of an investment in the bitcoin-to-dollar pair

on Kraken, our benchmark pair. The second factor, denoted CarryBtc, captures the excess returns of a zero-cost strategy that goes long in portfolio 7 and short in portfolio 1.

Table 4 presents descriptive statistics for the two risk factors, CarryBtc and Btc, as well as a third factor, denoted Cross, which is the average excess returns across all seven cross

portfolios. The average return on CarryBtc is equal to 0.68% per day. CarryBtc returns are also volatile (2.33%), positively skewed (0.50), and with a very large kurtosis (14.51). The

mean return on Btc is approximately one-half the mean CarryBtc, with a larger standard

24 Table 3: Principal Components

Portfolio 1 2 3 4 5 1 0.36 0.71 0.57 0.08 -0.01 2 0.38 0.14 -0.18 -0.16 -0.23 3 0.38 0.06 -0.43 -0.39 -0.47 4 0.38 0.09 -0.27 -0.12 0.35 5 0.38 -0.08 -0.22 0.15 0.70 6 0.38 -0.27 -0.03 0.80 -0.35 7 0.38 -0.62 0.58 -0.37 -0.00 % Var. 95.40 2.02 1.15 0.52 0.36

Notes: This Table reports the principal component coefficients of the cross portfolio net returns presented in Panel B of Table 2. The last row reports (in %) the share of the total variance explained by each of the first five principal components. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

deviation. Therefore, while the Sharpe ratio of CarryBtc is approximately equal to 29%,

the Sharpe ratio of Btc is approximately 6%. CarryBtc is highly correlated with the second principal component (-0.83), while Btc is highly correlated with the ﬁrst principal component (0.99). Note how Cross is de facto the ﬁrst principal component, and therefore is highly correlated with Btc.

Table 4: Descriptive Statistics: Risk Factors

Mean Std Skew Kurt V aR1% Corr(PC1, x) Corr(PC2, x)

CarryBtc 0.68 2.33 0.50 14.51 -5.85 0.07 -0.83

Btc 0.28 4.58 0.34 7.13 -12.93 0.99 0.04

Cross 0.42 4.49 0.40 7.58 -11.82 1.00 0.00

Notes: The Table reports mean (Mean), standard deviation (Std), skewness (Skew), kurtosis (Kurt), value at risk with confidence 1% (V aR1%), and correlation coefficients with respect to the first two principal components for CarryBtc, Btc, and Cross (Corr(PCi, x), with i = 1, 2 and x = Carry, Btc, Cross). CarryBtc denotes the excess returns of a zero-cost cross strategy that goes long in portfolio 7 and short in portfolio 1. Btc denotes the returns of investing in bitcoins using the benchmark dollar-to-bitcoin pair on the Kraken exchange. Cross denotes the returns for an investor that goes long in all the cross portfolios. All returns are net of bid/ask spreads. Data are daily from cryptocompare.com and Thomson Reuters for the period 5/26/2015–10/31/2019.

25 3.2 Results

Table 5 reports the asset pricing results obtained using two procedures applied to the portfolios sorted on bitcoin discounts: a Generalized Method of Moments estimation (GMM) applied to linear factor models, following Hansen(1982), and a two-state OLS estimation following Fama and MacBeth(1973), henceforth FMB (see SectionE in the Online Appendix for details

on the estimation procedures). Note that we always include the CarryBtc portfolio, i.e., our candidate slope factor, in the set of test assets as recommended by Lewellen et al.(2010). As a result, the number of test assets is equal to 8.

Cross-sectional regressions The top panel of the Table reports estimates of the market prices of risk λ and the stochastic discount factor (SDF) loadings b, the adjusted R2, the square-root of the mean-squared error RMSE and the p-value of a χ2 test for the null that all the cross-sectional prices errors are zero (in percentage points). The first risk factor, Btc, has an estimated risk price of 21 basis points, compared with a sample mean of 27 basis points. All portfolios have a beta close to one with respect to this first factor. As a result, the first factor explains none of the cross-sectional variation in portfolio returns. This is why the standard errors on the risk price estimates are large. For example, the GMM standard error is 15 basis points. While the Btc factor does not explain any of the cross-sectional variation in expected returns, it is important for the level of the average returns. The market price

of risk of the second factor, CarryBtc, is equal to 124 basis points per day. This means that

an asset with a beta of one on the CarryBtc, and a beta of zero on the ﬁrst factor, earns a risk premium of 1.24% per day. The GMM standard error of the risk price is 14 basis points. The FMB standard error is only 7 basis points. So, the risk price is more than two standard errors from zero, and highly statistically signiﬁcant.

Since the factors are returns, no-arbitrage implies that the prices of risk for these factors should equal their average excess returns. This condition depends on the fact that the Euler equation applies to the risk factor itself. In our estimation, this no-arbitrage condition is

26 satisﬁed. The average excess return on CarryBtc (last row of the top panel of Table 5) is 68 basis points, with a standard error of 20 basis points. Although the estimated risk price is approximately 50 basis points away from the point estimate implied by linear factor pricing, we cannot reject the null that the estimated risk price is equal to its sample mean.

Although pricing errors are small (the RMSE is just 29 basis points) and the adjusted R2 is approximately 94 percent, the null that all pricing errors are zero is always rejected, as

p−values are lower than 5%. In SectionF of the Online Appendix, we show that CarryBtc is a priced risk factor also in portfolios at the weekly frequency, and in a shorter sample starting in January 2018, after the bitcoin frenzy of 2017. In addition, we show that the

market price of CarryBtc risk is time-varying, and in particular, it is higher in bad times for ﬁnancial markets, when the Vix volatility index is high.

Time Series Regressions The bottom panel of Table 5 reports the intercepts (denoted αj) and the slope coeﬃcients (denoted βj) obtained by running time-series regressions of

k each portfolio’s excess returns Rx on a constant and the Btc and CarryBtc risk factors. The first column reports α’s estimates in percentage points per day. The point estimates are small but statistically different from zero, particularly in the corner portfolios. As a result, the null that the αs are jointly zero is rejected at standard significance levels. This indicates that the true factor structure possibly contains more than two factors. In fact, the principal component analysis of Table 3 indicates that the third component, which captures curvature, has a contribution to the total variance similar to that of the second component. Further, the αs might capture the effect of possible frictions that we were not able to eliminate despite the restrictions on the sample and the use of portfolios. The second and third columns of

the same panel report the estimated βs for the Btc and CarryBtc factors. The βs on the

Btc factor are similar across portfolios and close to one, while the βs on the CarryBtc factor increase from -0.36 for the ﬁrst portfolio to 0.60 for the last portfolio (i.e., portfolio 7). Note

that, since we add the CarryBtc risk factor to the test assets, the corresponding β of portfolio

27 8 is equal to one.

A natural interpretation of our results is that portfolios with higher comovement with

CarryBtc are riskier exactly because, on average, have high returns in good times for cryp-

tocurrency investors, when CarryBtc excess returns are large, and low returns in bad times, when CarryBtc excess returns are small. On the contrary, the Btc factor is a level factor that explains the average excess return but does not price the cross-section of returns. In the

next paragraph, we show that CarryBtc is large when risks associated with bitcoin liquidity, sentiment, momentum, and counter-party risk are higher.

What risks? We investigate whether standard non-crypto risk factors can span the CarryBtc and Btc factors, as well as the Cross risk factor. Speciﬁcally, Table 6 presents the results of linear regressions, at daily frequency, of these three risk factors on a large set of crypto and non-crypto factors.

Panel A groups the crypto factors. We consider the bitcoin (Btc), the CarryBtc, and

the bitcoin momentum (MOMBtc) factors. The latter is the cumulated bitcoin return over

the previous 30 days (MOMBtc). Furthermore, we consider several proxies for liquidity risk and, speciﬁcally, the cross-sectional sum (T V ol) and volatility (σ(V ol)) of trading volume across all pairs, and the cross-sectional volatility in bid/ask spreads on all pairs (σ(BA)). We consider an increase in the Google Trend index for the query “bitcoin” on all geographical areas (∆Goog) as proxy for investors’ bitcoin sentiment, as it is associated with a greater interest for bitcoin. Liu and Tsyvinski(2018) consider this factor a proxy for investors’ attention. Finally, we consider the fraction of inactive exchanges (DDoS) as a proxy for bitcoin counter-party risk.

Panel B groups the non-crypto factors. We consider the standard Fama and French(1993) three factors (MKT,SMB,HML) and the Carhart(1997)’s momentum factor ( MOM). In

addition, we also include a proxy for the currency carry trade (CarryFX ); the log price changes for the Gold Bullion (Gold); the CBOE Vix volatility index (∆V ix). MKT,SMB,HML,

28 Table 5: Asset Pricing: US investor

Panel I: Risk Prices 2 2 λBtc λCarryBtc bBtc bCarryBtc R RMSE χ (%)

GMM1 0.35 1.06 1.34 19.40 63.21 0.26 [ 0.16 ] [ 0.15 ] [ 0.74 ] [ 2.70 ] 0.00

GMM2 0.21 1.24 0.60 22.90 93.34 0.29 [ 0.15 ] [ 0.14 ] [ 0.71 ] [ 2.54 ] 0.00 FMB 0.35 1.06 1.34 19.38 57.34 0.26 [ 0.14 ] [ 0.07 ] [ 0.66 ] [ 1.30 ] 0.00 Mean 0.27 0.68 s.e. [ 0.14 ] [ 0.20 ] Panel II: Factor Betas Portfolio αk βk βk R2 χ2(α) p-value (%) 0 Btc CarryBtc 1 -0.46 0.94 -0.31 90.16 [ 0.07 ] [ 0.01 ] [ 0.04 ] 2 -0.11 0.98 -0.04 96.90 [ 0.03 ] [ 0.01 ] [ 0.03 ] 3 -0.02 0.98 0.03 96.16 [ 0.04 ] [ 0.01 ] [ 0.04 ] 4 0.12 0.98 0.00 96.53 [ 0.03 ] [ 0.01 ] [ 0.02 ] 5 0.19 0.97 0.09 95.46 [ 0.05 ] [ 0.01 ] [ 0.04 ] 6 0.37 0.97 0.19 93.78 [ 0.06 ] [ 0.01 ] [ 0.06 ] 7 0.63 0.95 0.55 94.01 [ 0.07 ] [ 0.01 ] [ 0.06 ] 8 0.00 0.00 1.00 100.00 [ 0.00 ] [ 0.00 ] [ 0.00 ] All 570.82 0.00

Notes: Panel I reports results from GMM and Fama and MacBeth(1973) asset pricing procedures on the seven bitcoin portfolios sorted with respect to bitcoin discounts. Market prices of risk λ, the adjusted R2, the square-root of the mean-squared errors RMSE and the p-values of χ2 tests on pricing errors are reported in percentage points. b denotes the vector of factor loadings. All excess returns are multiplied by 100. Shanken(1992)-corrected standard errors are reported in parentheses. We do not include a constant in the second step of the FMB procedure. Panel II reports OLS estimates of the factor betas. R2s and p- values are reported in percentage points. The standard errors in brackets are Newey and West(1986) standard errors computed 2 0 −1 with the optimal number of lags according to Andrews(1991). The χ test statistic α Vα α tests the null that all intercepts are jointly zero. This statistic is constructed from the Newey and West(1986) variance-covariance matrix (1 lag) for the system of equations (see Cochrane(2009)). Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019 and contains 940 daily observations per test asset.

and MOM are risk factors commonly used to price various types of assets, like equities and bonds, and are from the Kennet French’s website. Gold, like other precious metals, is considered a store of value, and a popular narrative considers cryptocurrencies an alternative to these precious metals. ∆V ix captures movements in aggregate liquidity and investors’ risk

29 aversion. Data on gold prices and the Vix index are from Datastream. We proxy the currency carry trade factor with the returns of the Deutsche Bank G10 currency carry trade ETF, whose prices we obtain from Bloomberg. In brackets, we report Newey and West(1986)’s standard errors.

We summarize our results as follows. First, we ﬁnd that CarryBtc returns, as well as Btc returns, are unrelated to non-crypto factors, which echoes Liu and Tsyvinski(2018)’s results for aggregate bitcoin returns. Second, while it is hard to explain Btc returns, we ﬁnd that a

sizeable fraction of CarryBtc returns can be explained by different crypto factors. Specifically, we find that CarryBtc returns are lower when bitcoin momentum returns are also lower; when bitcoin liquidity risk is higher (i.e., when aggregate trading volume is lower and the dispersion in trading volume and bid/ask spreads across exchanges is higher); when the Google Trend

index is lower; and when the fraction if inactive exchanges is larger. Therefore, CarryBtc returns are risky because they are high (low) in good (bad) times for bitcoin investors. For

CarryBtc, the adjusted R-square is approximately equal to 27.5 percent, in comparison to the adjusted R-square for Btc of -0.15 percent, and the intercept (α) is significantly different from zero at standard confidence levels. Note that these results are robust with respect to a different window in the construction of the bitcoin momentum factor. Finally, Cross is fully explained by the bitcoin return.

Liu et al.(2019) identify three crypto-factors that price the cross-section of cryptocurrencies. The three factors are a cryptocurrency market factor (CMKT), a cryptocurrency size factor (CSMB), and a cryptocurrency momentum factor (CMOM), and are only available at weekly frequency8. They argue that the momentum eﬀect is concentrated among the large coins and is consistent with recent theories of investors’ overreaction; that the size eﬀect is mainly concentrated among the smallest coins and is interpreted as an illiquidity premium; while the market factor is simply the bitcoin return. In Table A9 (SectionF of the Online

8We are grateful to Yukun Liu for sharing the data on the three factors, which are only available at weekly frequency. We refer to Liu et al.(2019) for details on the construction of the three factors and their interpretation.

30 Appendix) we show that CarryBtc is not spanned by either the cryptocurrency size factor

(CSMB) or the cryptocurrency momentum factor (CMOM). Instead, we interpret CarryBtc

as a proxy for good times in crypto currency markets. In particular, we show that CarryBtc return is higher at times of higher liquidity and investors’ sentiment, and at times of lower counter-party risk.

4 Robustness

In this Section, we consider several extensions. First, we report additional characteristics of the seven bitcoin portfolios. Second, we carefully consider all transaction costs, like trading, margin, and exchange fees. Third, we consider alternative samples with more pairs, like additional ﬁat and crypto currencies. Fourth, we consider execution risk, i.e., the risk that a transaction is not executed within the range of recent market prices observed by investors.

The main takeaway of this Section is that CarryBtc returns remain large and signiﬁcant after accounting for all transaction costs and in more recent samples, or samples containing additional ﬁat-to-bitcoin pairs and crypto-to-bitcoin pairs. However, execution risk and trade

size can substantially reduce CarryBtc returns. We further test the robustness of our results by considering an out-of-sample experiment

using two randomly selected subsamples of our data. We use the CarryBtc factor extracted from the ﬁrst random subsample to successfully price the cross-section of portfolio returns obtained using the second random sample.

Additional characteristics We present in Table 7 additional characteristics of the bitcoin portfolios sorted by their discounts using data described in detail in Borri and Shakhnov (2018). We ﬁnd that corner portfolios, i.e., portfolios with the largest positive or negative discounts, are characterized, on average, by lower liquidity and higher counter-party risk.

A lower volume of transactions and higher bid/ask spreads and intra-day volatility indicate lower liquidity; a smaller wallet size, larger fraction of inactive pairs, and a lower rating

31 (measured by the exchange grade points assigned by CoinMarketCap), indicate higher counter- party risk. The wallet size of an exchange captures not only the supply of bitcoins (i.e., the number of bitcoins in circulation on a particular exchange), but also provides a measure of assets for an exchange, similar to deposits for a ﬁnancial institution.

We consider as additional indicators of counter-party risk the mean fraction of inactive pairs and exchange rating for each portfolio. The fraction of inactive pairs is computed as the average number of pairs not available at time t + 1, but in which investors could have invest at time t. As back of the envelope calculation, consider that the average number of pairs per portfolio is equal to 6. Therefore, when the fraction of inactive pairs for a given portfolio is equal to 1%, there is one inactive pair in that portfolio every 16 days (1/(6 × 1%) ≈ 16). Cryptocompare.com assigns a rating to diﬀerent exchanges, summarized by the grade points, where a higher number corresponds to a more reliable exchange.

The high/low spread, deﬁned as the diﬀerence between the high and low daily bitcoin prices as a fraction of their average, also indicates a higher risk for investors because of the higher intra-day volatility. The average degree of capital controls, measured by the overall

restriction index ka from Fernándezet al.(2016), is an indicator of the frictions that might prevent investors to execute their trades. With respect to both of these measures, we do not find significant differences across portfolios9.

Finally, we observe that, by sorting pairs into portfolios by their discounts, we obtain a decreasing cross-section of average exchange rate growth (i.e., the change in the value of fiat currencies with respect to the dollar). Note that the average exchange rate growth is positive and significant for portfolio 1 (i.e., a dollar appreciation), and negative and significant for portfolio 7 (i.e., a dollar depreciation). Although economically small, the effect of the change in the exchange rate reduces the risk of the long/short strategy. For example, an investor long in portfolio 7 can exchange fiat currency for dollars at an increased exchange rate.

9As a reference, ka is equal to 0.13 for the U.S. at the end of 2017, and the average value across the seven portfolios is equal to 0.11 (lower values correspond to looser capital control levels)

32 Transaction Costs We now show how the returns of the two investment strategies based on bitcoin discounts change after accounting for all transaction costs, i.e., bid/ask spreads and trading fees. To get a rough estimate of the impact of transaction costs, note that the mean daily bid/ask spread is 35 basis points (Table 7), and the average trading fee is 15 basis points per trade (Table A3 in Section B.I of the Online Appendix). Investors pay these costs twice for the within strategy, but only once for the cross strategy. This is because the cross strategy always involves one of the two trades on the benchmark pair, the dollar-to-bitcoin pair on the Kraken exchange, for which these costs are negligible.

In what follows, we carefully consider the heterogeneity in bid/ask spreads across portfolios; the market depth; and all trading and margin fees that can potentially aﬀect returns. We focus on both the cross and within strategies, and only on the long/short returns. Recall that, for cross investors, the long/short strategy goes short in the ﬁrst portfolio, with the most negative discount. On the contrary, for within investors, the long/short strategy goes short in the last portfolio, with the most positive discount. Bitcoin investors must additionally pay deposit and withdrawal fees to the exchanges; trading fees; and margin fees when entering short positions.

Table 8 documents the impact of the different transaction costs on the long/short returns. The left side of the table refers to cross returns, while the right side of the table to within returns. Panel A reports gross average portfolios returns, while Panels B to E reports average portfolio returns for different specifications of the transaction costs.

Panel B reports the average portfolio returns net of bid/ask spreads, where the bid/ask spreads are those for the subset of 33 pairs for which we were able to collect data on Bitcoin- ity. As highlighted in the discussion of Table 2, bid/ask spreads have a diﬀerential impact across portfolios. Speciﬁcally, average returns decrease more for corner portfolios. Panel C additionally accounts for trading fees, which uniformly lower returns by 37 basis points for cross portfolios and by 67 basis points for within portfolios. After accounting for bid/ask spreads, net returns of long/short within strategies are negative. For this reason, in what

33 follows we focus on the impact of additional transaction costs and frictions only on cross portfolios.

In order to evaluate the likely impact of trading fees on the cross-section of portfolio returns, we manually collect current trading fees for all the exchanges in our sample. For all exchanges, except Kraken, we fix trading fees to 0.15%, which corresponds to the median taker trading fee. Taker fees are charged to investors who absorb liquidity with market orders, as opposed to maker fees, which are charged to investors who provide liquidity by placing limit orders. Therefore, we implicitly assume investors post their offers which are, then, executed within the horizon of one day. For the exchanges in our sample, the conservative estimates for the median taker and maker feed are, respectively, 0.25% and 0.15%. These values are calculated based on maximum fees for investors, but exchanges typically employ an asymmetric pricing model with fee discounts in order to incentivize higher levels of trading activity. As for Kraken, we fix the trading fee to 0%, which corresponds to the current fee charged to investors with total orders larger than 10 million U.S. dollars in a month. Because, for the cross strategy, investors must every day start on Kraken, we should expect a large number of orders in a given month on this exchange. Table A3 in the Online Appendix presents detailed information on trading fees for all exchanges. Finally, we fix the margin fee required to maintain a short position on Kraken to 0.07%, which corresponds to the current opening fee plus the roll-over positions for 24 hours.

After accounting for both bid/ask spreads and trading fees, long/short returns for portfolio 7 are positive and statistically diﬀerent from zero, and, as for our baseline portfolios, the Patton and Timmermann(2010)’s tests reject the null of ﬂat or weakly decreasing pattern against the alternative of the strictly increasing pattern. Since measures of the historical bid/ask spreads are not available for all our pairs, in Panels B and C, we use gross returns based on all available pairs, while bid-ask spreads are based on the subsample of available pairs. In Panel D we generate synthetic bid/ask spreads for all the pairs estimating the panel equation (4). Also, in this case, the long/short returns for the last portfolio are positive and

34 statistically signiﬁcant.

Finally, Panel E reports average returns net of trading fees and bid/ask spreads at the market depth of 10 BTC (approximately 90,000 U.S. dollars at the end of the sample), to account for sizable trading positions. In this case, returns for all portfolios are large and negative. Therefore, the size of trades matters: once we account for market depth, returns become large and negative at orders larger than 90,000 U.S. dollars, which correspond to approximately 6% of the daily median trading volume and is five times larger than the 10th percentile. Therefore, market depth introduces a constraint on the maximum trade size to obtain non-negative expected returns, which influences the likelihood that large investors would engage in this trade and the opportunity for investors to fully absorb bitcoin price differences. Because it implies that larger orders are likely to have a price impact, poor market depth is also related to execution risk and, specifically, to the high/low spread which we have found to be associated to higher idiosyncratic risk (see section G.II).

Alternative samples In Table 9 we document average portfolio discounts and net cross returns for diﬀerent alternative samples. For returns, we also report the average long/short returns (i.e., 7-1 for cross returns and 1-7 for within returns).

Panel A simply reports our baseline results from Table 2. Panel B considers a shorter sample, starting on January 2, 2018, after the bitcoin frenzy of 2017, for the same pairs of Panel A. Also for this sample, cross returns increase monotonically from the last to the ﬁrst portfolio. However, we observe that returns are now negative for all portfolios, except for the last two portfolios. The average long/short return is equal to 66 basis points per day, and is statistically diﬀerent from zero. As in the baseline sample, within portfolio returns decrease from portfolio 1 to portfolio 7 and are negative for all portfolios.

Panel C considers a larger sample, including all the ﬁat-to-bitcoin pairs available on cryptocompare.com10. In this case, cross portfolio returns are, on average larger and mono-

10The sample includes 183 pairs for 32 different fiat-to-bitcoin currency pairs traded on exchanged located in any country in the world. The additional fiat currencies, with respect to our baseline sample, are Argentinian peso, Brazilian real, Chilean peso, Chinese Yuan, Colombia peso, Czech krona, Indonesian rupiah, Indian

35 tonically increasing from the last to the first portfolio. The average long/short return is equal to 344 basis points per day and statistically different from zero. However, note that this larger sample also contains countries with restrictions on capital flows, limiting investors’ ability to transfer money in, and out of, some of the countries in which the exchanges operate, and less reliable exchanges. Although some of the capital control measures in place are likely to bite more retail than large investors (Baba et al., 2011), they are likely to make the cross strategy difficult to implement. As for the other samples, within returns decline monotonically from the first to the last portfolio, but are negative for all portfolios after accounting for bid/ask spreads.

Finally, Panel D considers a sample containing several crypto-to-bitcoin pairs for the period starting March 31, 2016. Specifically, we include 281 pairs for six of the most-traded cryptocurrencies against bitcoin11. In this case, we consider gross and not net returns, because of the unavailability of the historical bid/ask spreads for most of the pairs. Even though transferring cryptocurrencies, as opposed to fiat currencies, should be less restrictive from the perspective of domestic regulations of international flows, we document a large cross- section of discounts, from -6.51% on the first portfolio to 3.32% on the last portfolio. The cross-section of discounts is matched by a monotonic cross-section of both cross and within returns. For both strategies, we obtain economically large and significant average long/short returns. While for several crypto-to-bitcoin pairs bid/ask spreads (for example, in May 2019 the bid/ask spread on Kraken for the ethereum-to-bitcoin pair was approximately 18 basis points), the magnitude of the long/short returns indicates that, for many other pairs, bid/ask spreads must be much larger, at least for larger orders.

Execution risk Bitcoin investors are exposed to diﬀerent forms of execution risks. While all investors face the risk that they cannot execute a transaction within the range of market rupee, Korean won, Mexican peso, Malaysian ringgit, Nigerian naira, Peruvian sol, Philippine peso, Russian ruble, Singapore dollar, Thai baht, Turkish lira, Ukrainian hryvnia, Venezuelan bolivar, South African rand. 11The six cryptocurrencies are tether (USDT), ripple (XRP), ethereum (ETH), litecoin (LTH), bitcoin cash (BCH), and ethereum cash (ETC).

36 prices observed by investors before sending their orders, cross investors face the additional risks related to the transfer of balances across exchanges.

Table 10 documents the effect on investors of different forms of execution risk. Recall that, in our baseline analysis, when a pair is available at time t but not at t + 1, we assume that investors can close their trade at the median closing price of the day across all pairs. We denote this situation as an “exchange shutdown”. Exchange shutdowns can be temporary, for example, because of a software malfunctioning or denial-of-service attack, or permanent, in case of bankruptcy. While Panel A reports the baseline cross and within portfolio returns, Panel B documents average portfolio returns under the more conservative assumption that all shutdowns are permanent, and that investors must sell their balance at a zero price (i.e., for a -100% return). In this case, average returns are lower. For cross portfolios, the average return on portfolio 7 remains highly significant. In contrast, for within portfolios, the average return on portfolio 1 becomes not statistically different from zero. Table A6 in SectionD of the Online Appendix reports estimates of the returns for investors who had their balances stuck in an exchange because of a long shutdown and show that, on average, they are slightly positive assuming that they can sell their balance at the price when the exchange finally reopens12.

The deﬁnition of excess returns from equation (2) relies on the assumption of near-instant speed of execution at daily closing prices. Completing a trade requires the time to transfer bitcoins across exchanges and to execute a trade within the exchange. It is hard to exactly quantify these amounts of time, as they depend both on the type of investor (i.e., retail or hedge fund) and the state of the network. The time required by the bitcoin blockchain to transfer bitcoins across wallets of exchanges depends on the congestion of the network and

12Moore and Christin(2013) ﬁnd that, by early 2013, 45% of bitcoin exchanges had closed and many of the remaining markets were subject to frequent outages and security breaches, while Vasek and Moore (2015) document several denial-of-service attacks against cryptocurrency exchanges. In our sample, on a given day, approximately 14% of the pairs are not available (see Figure A11). While the latter ﬁgure denotes the unconditional number of inactive exchanges, the numbers reported in Table 7 correspond to pairs that are active at time t but not at t + 1. In addition, Table A5 reports a list of critical events for the largest cryptocurrency exchanges.

37 has recently ranged anywhere between 10 minutes to 24 hours. On the other hand, trade execution within an exchange is much faster, with only 3.95% of trades taking longer than one second to be executed for the largest exchanges (Krückeberg and Scholz, 2020). Panel C and D report returns under different scenarios for the prices at t + 1. Specifically, Panel C assumes that investors always sell at the lowest price of the day for a given pair. In this case, we obtain cross-sections of cross and within portfolio returns, but now the average returns are large and negative for all portfolios, dropping by an average of 300 basis points. Finally, returns reported in Panel D are based on the assumption that investors always sell at the average price between the closing and the lowest price of the day for a given pair. In this case, only the last cross portfolio has a positive and statistically significant average return (equal to 34 basis points), while the average of portfolio returns drops by 150 basis points. To sum up, our results indicate that bitcoin execution risk can significantly lower the returns to bitcoin investors and contribute to explaining bitcoin price differences across exchanges and currency pairs. In SectionG of the Online Appendix we discuss an alternative explanation for the documented cross-section of portfolio returns based on costly arbitrage and idiosyncratic risks.

Out-of-sample We investigate whether our results hold out-of-sample by constructing two non-overlapping random selected groups, of approximately the same size, from the universe of our pairs from the baseline sample. First, we form portfolios based on bitcoin discounts for each of the two groups, as described in Section2. Because of the lower number of pairs per group, we now form only 5 portfolios. We obtain a monotonically increasing cross-section of cross net returns in both samples (see Table 11). Second, we construct the CarryBtc factor using portfolio returns based on the ﬁrst group, and repeat the asset pricing exercise using as test assets the portfolios based on the second group. Note that, as the portfolios are formed

using two non-overlapping sets of pairs, CarryBtc is constructed from pairs diﬀerent with

respect to those contained in the portfolios that we use as test assets. Speciﬁcally, CarryBtc is

38 now the returns from a zero-cost strategy that goes long in portfolio 5 and short in portfolio 1 from the ﬁrst sample.

Table 12 presents our results. The market price of the CarryBtc factor is equal to 90 basis points, statistically different from zero, and very similar to the one we obtain in our baseline asset pricing estimation (i.e., 124 basis points). The no-arbitrage conditions are satisfied, and the market price of the Btc factor is not statistically different from zero. These

results support our conclusion that CarryBtc is a slope factor that prices the cross-section of portfolio returns. In SectionF in the Online Appendix, we show that these out-of-sample results are robust to diﬀerent sample splits using a Montecarlo experiment.

5 Conclusions

Investors buy bitcoins on a multitude of exchanges located in different countries and against different fiat and crypto currencies. At one point in time and in a frictionless world where investors could trade instantaneously across exchanges, the bitcoin price, converted in a common currency, should be the same in all of these exchanges and for any currency pair. Instead, there are sizable differences in bitcoin prices across exchanges and currency pairs.

We document these differences in bitcoin prices also in a sample in which frictions are likely to be small: across reputable exchanges located in countries with limited capital controls and liquid fiat currency pairs. These differences are well known to investors, who discuss extensively how to profit from them on cryptocurrency blogs.

In this paper, we propose a novel risk-based explanation of these price diﬀerences and carefully account for all the transaction costs and limitations to trade. Bitcoin prices for more “expensive” pairs depreciate more in bad times for cryptocurrency investors, when aggregate liquidity is lower and more uncertain, investors’ sentiment about bitcoin is lower, and counter-party risk is higher. Therefore, bitcoin excess returns compensate investors for taking on more aggregate risk.

39 We identify common risk factors in the data by building portfolios sorted on bitcoin past price deviations. By forming portfolios we can focus on the common components in bitcoin returns, accommodate variations in the number of exchanges and currencies pairs, and average out idiosyncratic risks. A two-factor model explains a large fraction of the cross- sectional variation in portfolio returns. Therefore, our results support the conclusion that the documented cross-section of returns represents compensation for risk, and is not just a measure of the inefficiency of cryptocurrency markets. The recent recovery in prices of bitcoin and other cryptocurrencies has revived the interest in these new assets. One view of cryptocurrency markets is that they are just fraud (Roubini, 2018, for example) and extremely inefficient (Makarov and Schoar, 2019, for example). The results in this paper challenge this view and are essential to analyze the efficiency of cryptocurrency markets and, potentially, can help understand markets for different assets that share a similar structure. For the former, this paper is the first to document the common factors in bitcoin investments sorted by their discounts, which is the key ingredient in a risk-based explanation of bitcoin price differences across exchanges and currency pairs. For the latter, the recent growing importance, in terms of trading volume, of cryptocurrency markets where investors trade derivatives, like futures on the bitcoin, makes it even more essential the understanding of bitcoin discounts. In fact, the exchanges where bitcoin derivative contracts are traded, use as underlying assets the value of indices constructed by aggregating bitcoin prices from different spot exchanges. Trading volume and open interest on crypto derivatives exchanges, like BitMex and Deribit, are currently an order of magnitude higher than on spot exchanges. In addition, positions on crypto derivative exchanges are typically highly leveraged, even up to 100 times. Therefore, small price changes in the value of the indices underlying a derivative contract, for example because of temporary deviations in bitcoin prices across exchanges, could cause massive sell-offs. Our results indicate that price differences, of apparently homogenous assets, are the results of differences in the exposure to aggregate risks.

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44 Table 6: The Disconnect between Crypto and Non-Crypto Factors

CarryBtc Btc Cross α 1.663 0.730 0.371 [ 0.267 ] [ 0.494 ] [ 0.102 ] Panel A: crypto factors Btc -0.000 – 0.967 [ 0.019 ] – [ 0.007 ]

CarryBtc – -0.002 0.041 – [ 0.103 ] [ 0.027 ]

MOMBtc 0.016 0.004 0.004 [ 0.003 ] [ 0.006 ] [ 0.001 ] T V ol 0.124 0.023 -0.012 [ 0.029 ] [ 0.043 ] [ 0.007 ] σ(V ol) -1.249 -0.192 0.270 [ 0.443 ] [ 0.493 ] [ 0.106 ] σ(BA) -0.919 -0.129 -0.153 [ 0.153 ] [ 0.245 ] [ 0.061 ] ∆Goog 0.014 0.006 0.002 [ 0.006 ] [ 0.016 ] [ 0.002 ] DDoS -0.058 -0.034 -0.019 [ 0.013 ] [ 0.023 ] [ 0.004 ] Panel B: non-crypto factors MKT -0.185 -0.271 0.034 [ 0.107 ] [ 0.328 ] [ 0.034 ] SMB 0.162 0.275 0.012 [ 0.104 ] [ 0.300 ] [ 0.043 ] HML 0.027 0.050 0.019 [ 0.122 ] [ 0.262 ] [ 0.035 ] MOM 0.023 -0.036 0.010 [ 0.074 ] [ 0.201 ] [ 0.026 ]

CarryFX 0.005 -0.184 0.037 [ 0.125 ] [ 0.294 ] [ 0.039 ] Gold -0.060 0.354 0.016 [ 0.076 ] [ 0.192 ] [ 0.026 ] ∆V ix -0.019 -0.048 0.002 [ 0.012 ] [ 0.050 ] [ 0.004 ] R2 adj. (%) 27.598 -0.149 98.036 F (%) 0.000 75.631 0.000

Notes: This Table presents the results of linear regressions, at daily frequency, of the CarryBtc and Btc factors, as well as the Cross factor, on different non-crypto and crypto factors. CarryBtc is the excess return of a zero-cost cross strategy that goes long in portfolio 7 and short in portfolio 1. Btc is the return of a strategy that buys and sells bitcoins always using the dollar-to- bitcoin pair on Kraken. Cross is the return of a strategy that goes long in all the cross pairs. Panel A groups the crypto factors, while Panel B groups the non-crypto factors. The crypto factors are: the bitcoin (Btc), carry (CarryBtc), and momentum (MOMBtc) factors; the aggregate bitcoin trading volume (T V ol) and the cross-sectional volatility in trading volume across pairs (σ(V ol)); the cross-sectional volatility in bid/ask spreads on all pairs (σ(BA)); the change in the Google Trend index for the query “bitcoin” (∆Goog); the fraction of inactive exchanges (DDoS). Bitcoin momentum is defined as the cumulated bitcoin return over the previous 30 days (up to period t − 1). The non-crypto factors are: the Fama and French(1993) three factors (MKT,SMB,HML); the Carhart(1997)’s momentum factor ( MOM); the currency carry trade (CarryFX ) factor, proxied by the returns of the Deutsche Bank G10 currency carry trade ETF; the log price changes for the Gold Bullion (Gold); and the CBOE Vix volatility index (∆V ix). Standard errors in brackets are Newey and West(1986). The last row reports the p-values (in percentages) of an F-test on all coefficients equal to zero (excluding the constant). Data are daily from cryptocompare.com, Thomson Reuters, Bloomberg, Google, and the Kenneth French data library for the period 5/26/2015–10/31/2019.

45 Table 7: Bitcoin Portfolios: Additional Characteristics

Portfolio 1 2 3 4 5 6 7 volume (btc thousands) Mean 4.17 5.92 6.21 5.74 4.40 3.82 2.76 SE 0.22 0.22 0.26 0.23 0.21 0.18 0.10 bid-ask spread (%) Mean 0.50 0.28 0.25 0.23 0.27 0.32 0.63 SE 0.02 0.01 0.01 0.01 0.01 0.01 0.02 high-low spread (%) Mean 6.00 5.71 5.53 5.59 5.85 5.65 5.91 SE 0.16 0.17 0.15 0.14 0.20 0.16 0.15 wallet (btc thousands) Mean 30.51 76.32 80.66 65.27 42.27 30.43 17.21 SE 2.15 3.25 3.38 3.03 2.79 2.53 1.27 fraction of inactive pairs (%) Mean 0.91 0.13 0.21 0.20 0.32 0.15 0.41 SE 0.17 0.07 0.11 0.08 0.10 0.06 0.08 exchange grade points Mean 47.67 50.56 51.95 51.91 50.74 48.68 45.74 SE 0.20 0.19 0.15 0.16 0.17 0.19 0.14 capital control index Mean 0.11 0.10 0.09 0.09 0.10 0.11 0.13 SE 0.33 0.25 0.19 0.21 0.22 0.27 0.25 exchange rate growth (%) Mean 0.04 0.02 -0.01 0.01 -0.01 -0.03 -0.03 SE 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Notes: This Table reports additional characteristics of the seven portfolios sorted by bitcoin discounts. The additional characteristics are: volume (in thousands of bitcoins); bid/ask spread (in percentage); high/low spread (in percentage); wallet size (in thousands of bitcoins); fraction of inactive pairs (in percentage); exchange grade points; capital control index; and exchange rate growth ((in percentage)). At each point in time t, and for each portfolio k, we compute the equally weighted average for each characteristic. Each panel reports mean, standard deviation, and standard error by bootstrap estimated over the sample 5/26/2015–10/31/2019. The high/low spread is constructed as the difference between the high and low daily bitcoin prices as a fraction of their average. Details on the definition of inactive pairs and wallet size are in section4. Data are daily from cryptocompare.com and Thomson Reuters. The exchange grade points are from cryptocompare.com and are constant for each exchange throughout the sample (a higher number corresponds to a more reliable exchange). The capital control index is the overall restriction index ka from Fernándezet al.(2016)( ka is equal to 0.13 for the U.S. in 2017; a higher number is associated to more restrictions). Note that the ka is available at annual frequency up to 2017. For the sample since 2017 we fix ka to the last available values.

46 Table 8: Bitcoin portfolios: Long/Short Returns

k 1 k 7 cross returns long/short: rxt+1 − rxt+1 within returns long/short: rxt+1 − rxt+1 Portfolio 2-1 3-1 4-1 5-1 6-1 7-1 1-7 2-7 3-7 4-7 5-7 6-7 Panel A: gross returns Mean 0.33 0.43 0.54 0.71 1.01 1.81 1.27 0.77 0.62 0.53 0.45 0.40 SE 0.13 0.14 0.14 0.13 0.14 0.14 0.14 0.13 0.14 0.13 0.14 0.14 Panel B: net of bid-ask Mean -0.29 -0.15 -0.07 0.09 0.34 1.00 -0.20 -0.26 -0.34 -0.48 -0.58 -0.73 SE 0.05 0.05 0.05 0.05 0.06 0.07 0.06 0.04 0.04 0.05 0.04 0.04 Panel C: trading fees & net of bid-ask Mean -0.66 -0.52 -0.44 -0.28 -0.03 0.63 -0.87 -0.93 -1.01 -1.15 -1.25 -1.40 SE 0.04 0.05 0.05 0.05 0.06 0.07 0.06 0.04 0.05 0.05 0.04 0.04 Panel D: net of trading fees & bid-ask (synthetic) Mean -0.82 -0.69 -0.56 -0.44 -0.19 0.31 -1.50 -1.55 -1.64 -1.69 -1.86 -2.01 SE 0.04 0.05 0.05 0.05 0.06 0.07 0.07 0.05 0.06 0.06 0.06 0.06 Panel E: net of trading fees & bid-ask (10 btc depth) Mean -3.24 -2.96 -2.93 -2.85 -2.85 -2.69 -7.06 -5.50 -5.30 -5.56 -5.80 -6.45 SE 0.13 0.12 0.11 0.11 0.12 0.13 0.07 0.05 0.06 0.06 0.06 0.06

Notes: This Table reports long/short portfolio cross and within returns. For cross returns, the long/short strategy goes long in portfolio k = 2,..., 7 and short in the first portfolio. For within returns, the long/short strategy goes long in portfolio k = 1,..., 6 and short in the last portfolio. Panel A reports the gross portfolio returns. Panel B reports returns net of bid/ask spreads for a subsample of the bitcoin-to-fiat currency pairs for which we obtained bid/ask spreads from Bitcoinity. Panel C reports returns that additionally account for trading fees. Panel D reports returns net of synthetic bid/ask spreads and trading fees. The synthetic bid/ask spreads are generated for all the bitcoin-to-fiat currency pairs using the predicted values from panel regressions of the available bid/ask spreads on the historical market data. Panel E reports the return net of trading fees and bid/ask spreads at the market depth of 10 btc. Data are daily from cryptocompare.com, Bitcoinity, and Thomson Reuters for the period 5/26/2015–10/31/2019.

47 Table 9: Bitcoin portfolios: Alternative Samples

Portfolio 1 2 3 4 5 6 7 long/short Panel A: baseline sample discounts Mean -1.08 -0.28 -0.04 0.16 0.41 0.76 1.98 SE 0.04 0.02 0.02 0.02 0.02 0.03 0.06 cross returns Mean -0.42 0.13 0.27 0.39 0.52 0.77 1.26 0.68 SE 0.14 0.13 0.14 0.13 0.14 0.14 0.14 0.07 within returns Mean 0.28 0.22 0.14 0.08 -0.09 -0.24 -1.24 -0.83 SE 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.07 Panel B: baseline sample, since 2018 discounts Mean -0.97 -0.20 -0.04 0.09 0.26 0.53 1.59 SE 0.05 0.01 0.01 0.01 0.02 0.03 0.07 cross returns Mean -1.07 -0.30 -0.20 -0.11 -0.08 0.15 0.77 0.66 SE 0.23 0.23 0.22 0.23 0.23 0.22 0.22 0.10 within returns Mean -0.64 -0.37 -0.38 -0.38 -0.55 -0.60 -1.23 -1.19 SE 0.23 0.22 0.22 0.22 0.24 0.22 0.23 0.12 Panel C: all ﬁat-to-bitcoin pairs discounts Mean -2.09 -0.41 0.01 0.36 0.81 1.60 4.79 SE 0.06 0.02 0.02 0.03 0.03 0.05 0.10 cross returns Mean -1.58 -0.23 0.10 0.26 0.54 1.10 3.51 3.44 SE 0.15 0.13 0.14 0.14 0.14 0.14 0.16 0.13 within returns Mean -0.16 -0.25 -0.30 -0.48 -0.74 -1.11 -2.34 -2.34 SE 0.15 0.14 0.14 0.14 0.14 0.14 0.16 0.13 Panel D: all crypto-to-bitcoin pairs discounts Mean -6.51 -0.71 -0.32 -0.12 0.08 0.50 3.32 SE 0.23 0.04 0.02 0.02 0.02 0.04 0.15 cross gross returns Mean -4.18 0.31 0.36 0.39 0.48 0.68 2.46 6.64 SE 0.29 0.15 0.16 0.16 0.16 0.16 0.22 0.13 within gross returns Mean 2.78 1.07 0.68 0.52 0.40 0.18 -0.55 3.33 SE 0.23 0.16 0.16 0.16 0.17 0.17 0.23 0.13

Notes: This Table reports, for each portfolio j, the means and standard errors by bootstrap for the average discounts, and the average cross and within returns of portfolios sorted by bitcoin discounts and based on diﬀerent samples. Panel A corresponds to the baseline sample for the period 5/26/2015–10/31/2019. Panel B considers a shorter sample, starting on January 2, 2018, for the same set of pairs as in Panel A. Panel C considers a larger sample, including all the ﬁat-to-bitcoin pairs available on cryptocompare.com. Panel D considers a sample containing 281 crypto-to-bitcoin pairs for the period starting March 31, 2016. All returns, with the exception of those of Panel D, are net of bid/ask spreads. The cryptocurrencies traded against bitcoin are tether (USDT), ripple (XRP), ethereum (ETH), litecoin (LTH), bitcoin cash (BCH), and ethereum cash (ETC). Data are daily from cryptocompare.com, Bitcoinity, and Thomson Reuters.

48 Table 10: Execution Risks

cross portfolios within portfolios Portfolio 1234567 1234567 Panel A: baseline gross returns Mean 0.08 0.41 0.51 0.62 0.79 1.09 1.89 1.20 0.70 0.55 0.46 0.38 0.33 -0.07 SE 0.13 0.13 0.14 0.14 0.13 0.14 0.14 0.14 0.13 0.14 0.13 0.14 0.14 0.13 Panel B: sell at zero price when exchange shutdowns Mean -0.84 0.27 0.31 0.42 0.46 0.93 1.48 0.25 0.56 0.34 0.26 0.06 0.17 -0.47 SE 0.21 0.15 0.18 0.16 0.17 0.15 0.16 0.22 0.15 0.17 0.16 0.16 0.15 0.16 Panel C: sell at low price of the day Mean -3.27 -2.75 -2.57 -2.47 -2.36 -2.02 -1.21 -2.21 -2.47 -2.53 -2.63 -2.75 -2.75 -3.10 SE 0.12 0.13 0.13 0.12 0.13 0.13 0.12 0.12 0.13 0.13 0.13 0.13 0.13 0.12 Panel D: sell at mean low/high price of the day Mean -1.60 -1.17 -1.03 -0.93 -0.79 -0.46 0.34 -0.51 -0.89 -0.99 -1.09 -1.19 -1.21 -1.58 SE 0.12 0.12 0.12 0.11 0.12 0.12 0.12 0.12 0.12 0.13 0.12 0.12 0.12 0.12

Notes: This Table reports, for each portfolio j, the mean cross and within returns, and corresponding standard errors by bootstrap, for the seven portfolios sorted by bitcoin discounts. Panel A reports the baseline gross returns. In Panel B, we assume that investors always close their trades at a price of zero if a pair available at time t then drops out of the sample at t + 1. In Panel C, we assume that investors close their trades at t + 1 always at the lowest price of the day for a given exchange-currency pair. In Panel D, we assume that investors close their trades at t + 1 always at the mean between the closing and the lowest price of the day. All returns are before transaction costs. Data are daily from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

49 Table 11: Bitcoin portfolios: two random samples

Portfolio 1 2 3 4 5 Panel A: 1st random sample discounts Mean -1.09 -0.20 0.15 0.53 1.71 Std 1.59 0.69 0.68 1.01 1.96 cross returns (net of bid/ask) Mean -0.48 0.13 0.36 0.67 1.34 Std 4.79 4.74 4.71 4.62 4.80 SE 0.14 0.14 0.14 0.13 0.14 within returns (net of bid/ask) Mean 0.22 0.12 0.04 -0.07 -0.72 Std 4.98 4.92 4.80 4.61 4.52 SE 0.15 0.14 0.14 0.14 0.14 Panel B: 2nd random sample discounts Mean -0.61 -0.10 0.13 0.45 1.54 Std 0.98 0.61 0.62 0.94 1.92 cross returns (net of bid/ask) Mean 0.04 0.28 0.42 0.56 1.21 Std 4.78 4.64 4.69 4.74 4.79 SE 0.14 0.14 0.14 0.14 0.14 within returns (net of bid/ask) Mean 0.43 0.27 0.18 -0.07 -0.70 Std 4.78 4.64 4.69 4.74 4.79 SE 0.15 0.14 0.14 0.15 0.14

Notes: This Table reports, for each portfolio j, the mean and standard deviation for the average discount Dj , the mean and standard deviation for the cross-asset log excess return rxcross,j , the mean and standard deviation for the within assets log excess returns rxwithin,j . All moments, with the exception of those for volume, are reported in percentage points. Portfolios are constructed by sorting assets into five groups at time t based on their discounts Dj defined by equation (1). The first portfolio contains assets with the lowest negative discounts. The last portfolio contains assets with the highest positive discounts. Portfolios are constructed using assets contained in two subsets of the assets considered in section2. Each sample contains 60 assets sampled uniformly with replacement from the full sample containing 126 assets. The fraction of assets common to both subsets is approximately 30%. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

50 Table 12: Asset Pricing: US investor (random sample)

Risk Prices 2 2 λBtc λCarryBtc bBtc bCarryBtc R RMSE χ (%)

GMM1 0.40 0.87 1.25 17.35 17.40 0.40 [ 0.16 ] [ 0.13 ] [ 0.73 ] [ 2.68 ] 0.00

GMM2 0.35 0.90 1.01 17.93 83.29 0.40 [ 0.16 ] [ 0.13 ] [ 0.71 ] [ 2.67 ] 0.00 FMB 0.40 0.87 1.25 17.33 -8.71 0.40 [ 0.15 ] [ 0.07 ] [ 0.69 ] [ 1.46 ] 0.00 Mean 0.30 0.59 s.e. [ 0.15 ] [ 0.22 ]

Notes: Panel I reports results from GMM and Fama and MacBeth(1973) asset pricing procedures on the seven within-asset bitcoin portfolios. Portfolios are constructed using assets contained in two subsets of the assets considered in section2. Each sample contains 40 assets sampled uniformly with replacement from the full sample containing 115 assets. The last portfolio contains assets with the highest positive discounts. The fraction of assets common to both subsets is approximately 27%. Market prices of risk λ, the adjusted R2, the square-root of the mean-squared errors RMSE, and the p-values of χ2 tests on pricing errors are reported in percentage points. b denotes the vector of factor loadings. All excess returns are multiplied by 100. Shanken(1992)-corrected standard errors are reported in parentheses. We do not include a constant in the second step of the FMB procedure. Panel II reports OLS estimates of the factor betas. R2s and p-values are reported in percentage points. The standard errors in brackets are Newey and West(1986) standard errors computed with the optimal number of lags according 2 0 −1 to Andrews(1991). The χ test statistic α Vα α tests the null that all intercepts are jointly zero. This statistic is constructed from the Newey and West(1986) variance-covariance matrix (1 lag) for the system of equations (see Cochrane(2009)). Each sample contains 60 assets sampled uniformly with replacement from the full sample containing 126 assets. The fraction of assets common to both subsets is approximately 30%. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 7/31/2014–10/31/2019.

51 Online Appendix (not for publication)

This is the Online Appendix for Borri, N. and K. Shakhnov. (2021). “The Cross-Section of Cryptocurrency Returns”. The Appendix is organized as follows:

• SectionA presents additional details on the data used in our paper.

• SectionB presents additional information on bitcoin transaction costs for diﬀerent exchanges and currency pairs.

• SectionC discusses measures of exchange “quality”.

• SectionD discusses trade execution risk on cryptocurrency markets.

• SectionE presents the two procedures used to estimate the price of CarryBtc risk. • SectionF presents additional robustness checks.

• SectionG discusses an alternative explanation of bitcoin discounts based on the idea of costly arbitrage.

52 A Data

In this Section, we report additional details on the cryptocurrency data. Specifically, first, we provide details on the major cryptocurrency markets across the globe. Second, we consider bitcoin data on prices and discounts. Third, we provide information on execution times and stylized facts on cryptocurrency cyber-attacks and critical events. Fourth, we provide information on transaction costs for different cryptocurrencies. Finally, we consider exchange wallets and counter-party risk.

A.I Cryptocurrencies and Cryptocurrency Markets Investors trade bitcoin 24/7, every day of the week, including holidays, on several exchanges across the globe. A cryptocurrency exchange can be a market maker that typically takes the bid/ask spreads as compensation for its service, or a matching platform which simply charges fees. Note that the blockchain does not keep track of transactions that occur within an exchange, while it keeps track of deposits to, and withdrawals from, the exchange. This technological feature has important implications. On the one hand, it makes it possible to trade frequently on exchanges without the constraint given by the time required to add a new block to the blockchain. On the other, it exposes investors trading on exchanges to counter-party risks, as their transactions are not veriﬁed and/or tracked by the blockchain. All the exchanges we consider are matching platforms with a limit order book, i.e., a record of unexecuted limit orders maintained by the exchange. Speciﬁcally, we consider two types of exchanges:

1. exchanges on which investors can trade cryptocurrency pairs (i.e., bitcoin for ethereum) and deposit and withdraw only cryptocurrencies;

2. exchanges on which investors can trade ﬁat for cryptocurrencies (i.e., bitcoin for U.S. dollar), and where investors can deposit and withdraw both ﬁat and cryptocurrencies.

Table A1 reports a list of the largest exchanges by trading volume as of January 201813. The third column reports the total number of currency pairs that can be traded, which is as large as 409 for HitBTC, an exchange on which only cryptocurrencies are traded. The exchange with the largest trading volume is Binance. Note that Binance is an exchange on which only cryptocurrency pairs are traded. The second-largest exchange is Bithumb, a Korean exchange, where investors can trade Korean won (KRW) for several cryptocurrencies. The largest U.S. dollar-based exchange is Bitfinex, where investors can use U.S. dollars and euros to buy several cryptocurrencies. Note that our baseline sample does not contain pairs traded on Binance and Bithumb because our sample covers only fiat-to-bitcoin pairs traded on exchanges located in a set of advanced countries with limited or no capital controls and against major fiat currencies. In addition, we consider only exchanges that satisfy a set of requirements in terms of “quality” which we also discuss in this Online Appendix.

13The ranking as of July 2020 is slightly diﬀerent, and it changes frequently. Besides, https://coinmarketcap.com/ reports rankings in terms of volume, adjusted volume (in order to account for possible incorrect reporting on the actual volume by the exchanges), and liquidity.

53 Table A1: Largest Cryptocurrency Exchanges

# Name Currency Number of Trading volume Launch type currency pairs in bln USD (24h) Date 1 Binance crypto only 247 $2.65 2 Bithumb crypto, KRW 12 $2.32 10-Sep-13 3 Upbit crypto only 214 $2.07 24-Oct-17 4 Okex crypto only 410 $1.99 5 Bitﬁnex crypto, USD, EUR 85 $1.62 8-Jun-13 6 Huobi crypto only 145 $1.27 7 Bittrex crypto only 270 $0.59 8 Kraken crypto, EUR, USD, JPY 45 $0.57 10-Sep-13 9 GDAX crypto, USD, EUR, GBR 12 $0.57 25-Jan-15 10 HitBTC crypto only 409 $0.48

Notes: This Table reports the list of the largest cryptocurrency exchanges by trading volume. Data are collected on January 24, 2018, from https://coinmarketcap.com/. The number of pairs refers to the total number of currency pairs (ﬁat and crypto) traded on each exchange. The trading volume is in U.S. dollar billions and is annualized.

A.II Cryptocurrency Prices Investors can trade fiat currencies for bitcoin in several exchanges scattered across the globe. In this paper, we take bitcoin as representative cryptocurrency as it currently accounts for approximately 30% of trading volume and 65% of the market capitalization of all cryptocurrencies. The bitcoin price and volume increased at an incredible pace since its launch in 2009, reached a peak at the end of 2017, and then lost a large fraction of their maximum value. Figure A1 plots the daily U.S. dollar price on Kraken (left axis), together with the daily volume of transactions (right axis) on the same exchange. The bitcoin price increased from approximately $U.S. 740 at the beginning of the sample, on 5/26/2015, to $U.S. 8,300 at the end of the sample, on 10/31/2019, and reached a maximum value of approximately $U.S. 18,900 on December 18, 2017. We start by collecting daily bitcoin price and volume data for all exchanges and currency pairs available on cryptocompare.com. However, in our baseline analysis, we restrict our sample along several dimensions in order to focus on reputable exchanges and currency pairs that are likely to be available to foreign investors. First, we only consider fiat-to-crypto pairs on fiat-to-crypto exchanges (for example, the U.S. dollar-to-bitcoin pair on Coinbase, but not the ether-to-bitcoin pair on Coinbase), because of the lower reliability of crypto-to-crypto exchanges. In what follows, we treat each fiat-to- crypto currency pair on a given exchange as a different asset. For example, we treat differently the dollar-to-bitcoin pair on Coinbase and the dollar-to-bitcoin pair on Kraken. Similarly, the dollar-to-bitcoin and euro-to-bitcoin pairs, both on Coinbase, are also different assets. Second, we consider only fiat-to-crypto exchanges that satisfy several indicators of “quality”, including ratings and web traffic data produced by Nomics and Cryptocompare, crypto- asset data companies providing market data to institutional crypto investors and exchanges; the new liquidity metric produced by CoinMarketCap, which is specifically designed to detect fake volume; and the list of exchanges providing bitcoin quotes for Bloomberg. In Table A3,

54 Figure A1: Bitcoin Price and Volume

bitcoin-to-dollar on Kraken 20 180 18.83 18 160

16 140

14 120 12 100 10 80 8

60 U.S. dollar millions

U.S. dollar thousands 6

40 4

2 20

0 0 2015 2016 2017 2018 2019 2020

Notes: This Figure plots the U.S. dollar price (in thousands and on the left axis) of 1 bitcoin and the daily trading volume (in millions of U.S. dollar and on the right axis) on the Kraken exchange for the bitcoin-to-dollar pair (i.e., our benchmark pair). Data are daily from cryptocompare.com for the period 5/26/2015–10/31/2019. Trading volume is smoothed with a 5-day moving average. we report, for each exchange in the sample, information on these quality indicators. Third, in order to focus on pairs likely available to foreign investors, we include in the sample only pairs that satisfy both of the two following criteria to minimize the role of restrictions to capital flows. The exchanges must be located in one of the following countries: Australia, Canada, Denmark, the E.U., Hong Kong, Israel, Japan, Poland, Switzerland, the U.K., and the U.S. The fiat currencies in pairs must be one of the following: Australian dollar, Canadian dollar, Danish krone, euro, Hong Kong dollar, Israeli shekel, Japanese yen, Polish zloty, Swiss franc, British pound, the U.S. dollar. We select this set of countries and currencies based on the tightness in capital controls with respect to the U.S. using the capital control index constructed by Fernándezet al.(2016). Fourth, we compute U.S. dollar prices for all fiat-to-bitcoin pairs using daily spot rates from WM/Reuters and available on Datastream. For this reason, we exclude non-business days because of the unavailability of spot rates. Fifth, to focus on the most liquid pairs, we consider the following additional restrictions. We exclude pairs with less than 30 days of observations; pairs traded on peer-to-peer platforms, which typically have very low volume; pairs with an average daily bid/ask spread larger than 2%; and observations corresponding to the 5-day average of daily volume smaller than 10 bitcoins. Our baseline sample starts on May 26, 2015, and finishes on October 31, 2019. The sample’s starting date depends on the availability of a minimum number of pairs to build portfolios and the possibility of shorting the dollar-to-bitcoin pair. Our raw sample contains

55 258 fiat-to-bitcoin pairs, while our baseline sample contains, at most, 47 pairs traded in 20 exchanges. Commonly used data sources in finance do not have the bitcoin data coverage we use in this paper. For example, at the time we write this paper, Bloomberg offers bitcoin-to- dollar quotes only for nine fiat-to-crypto exchanges, and does not provide quotes for all the fiat-crypto pairs in our sample. In addition, at the time we write this paper, Bloomberg does not offer data on volume. For this paper, we download daily data on bitcoin prices and volume for different markets from the cryptomarket aggregator cryptocompare.com using a Python script14. Note that all cryptomarkets give users public access to data on prices, order books, and trading volume. Users can access the data either using an application programming interface (API) or a WebSocket. However, in order to download data directly from individual exchanges, exchange-specific scripts to scrap the data are required. On the contrary, cryptocompare.com offers the opportunity of a bulk download of data from different exchanges. We download data on all the fiat-to-bitcoin pairs that were listed on https://www.cryptocompare.com/exchanges/#/overview at the end of October 2019, and we manually check, using each exchange website, the number of currency pairs that were traded at the time. We denote as bitcoin discounts the price differences between a given pair and the bitcoin-to- dollar pair traded on the Kraken exchange, which we take as numeraire. Specifically, we define bitcoin discounts according to equation (1). In a separate paper, Borri and Shakhnov(2018), we provide a detailed analysis of the properties of these discounts and offer a decomposition with respect to time, location-time, and currency-location-time dimensions. Table A2 reports the descriptive statistics of the bitcoin discounts corresponding to the exchanges where the pairs in the baseline sample are traded. In our baseline sample, the largest discount is equal to approximately 30% (Coincheck), and the smallest to approximately -20% (BitTrex). The overall mean discount, over time and over all the exchanges, is equal to 0.39% with a volatility of 1.42%. Note that discounts are persistent: the first-order autocorrelation coefficient is equal to 0.46 on average but ranges from 0.85 on Bitfinex to approximately 0.13 on Liquid. Figure A2 plots the time-series of the cross-sectional volatility in bitcoin discounts for our baseline sample (blue line), for pairs traded in exchanges located in countries excluded from the baseline sample or against fiat currencies excluded from the baseline sample (red- dashed line). In addition, the figure also plots the cross-sectional volatility of bitcoin discounts for bitcoin-to-crypto pairs (black-dotted line). Note that, while domestic and international regulation can potentially limit the transfer of fiat currency across countries, there is very little it can do to limit the transfer of cryptocurrencies. Further, the time and costs faced by investors to transfer cryptocurrencies are largely smaller than to transfer fiat currencies. First, the Figure shows that discounts are volatile, and the volatility appears to be time-varying, and higher in 2017, which is also the year corresponding to the strong increase in bitcoin prices. Second, the Figure shows that the cross-sectional volatility of discounts of the pairs excluded by the baseline sample is approximately twice as large as those of the sample we use in our main analysis. The fact that discounts are larger for pairs traded in exchanges located

14Speciﬁcally, we use API and send organic, grass-fed HTTP/1.1 requests to cryptocompare.com. Note that this is a standard package in Python distributions.

56 in countries with higher capital controls is a stylized fact also documented in Makarov and Schoar(2019), which attribute it to market segmentation, and Borri and Shakhnov(2018), which attribute it to local demand and supply shocks when the market is segmented. Finally, the cross-sectional volatility of the bitcoin discounts in our sample is comparable to that of the bitcoin-to-crypto pairs. Finally, Figure A3 plots the cross-sectional volatility of bitcoin discounts for the baseline sample, along with the total number of available active pairs. While the number of available pairs increases up to approximately 40 at the end of 2018, and then declines to approximately 35 at the end of the sample, the time-series of the discounts are fairly stable, with a small decline towards the end of the sample.

Table A2: Cryptocurrency Discounts

Exchange Mean (%) Std (%) Max (%) Min (%) AC(1) Pairs T Bitﬁnex 0.82 1.48 9.74 -3.09 0.85 4 1111 Bitstamp 0.00 0.60 4.73 -4.15 0.31 2 1119 Coinbase 0.38 0.87 9.19 -4.38 0.47 3 1120 Kraken 0.19 1.95 12.87 -13.48 0.33 5 1119 Gemini 0.11 0.69 7.24 -1.76 0.40 1 1024 itBit 0.05 0.71 13.03 -2.02 0.21 1 593 BitTrex -0.04 1.57 9.86 -19.86 0.69 1 649 Liquid -0.02 0.60 1.67 -2.91 0.13 4 270 DSX 0.05 1.18 5.39 -2.93 0.73 3 389 Cexio 1.58 1.86 12.49 -3.21 0.66 3 1120 bitFlyer 0.47 1.32 14.66 -7.19 0.51 2 1081 BitBay 1.09 2.26 14.36 -7.52 0.51 3 1025 Bitlish 0.18 1.00 5.08 -4.49 0.18 2 318 Bitpoint 0.08 0.32 1.71 -0.71 0.45 3 116 Coincheck 0.72 2.31 30.74 -14.05 0.49 1 1098 Zaif 0.37 2.12 27.30 -12.11 0.60 1 606 BitBank 0.88 2.53 25.20 -4.93 0.68 1 631 Coinﬂoor -0.05 2.12 9.32 -7.17 0.09 3 610 ACX 0.17 0.61 2.51 -2.49 0.34 1 306 BTCMarkets 1.43 2.59 20.64 -3.14 0.80 1 868 CoinJar 0.07 0.56 1.82 -2.43 0.23 1 302 Coinsetter 0.03 1.95 6.69 -13.50 0.44 1 141 Average 0.39 1.42 11.19 -6.25 0.46 2 710

Notes: The Table reports, for each of the 44 exchanges in our baseline sample, the mean (Mean), standard deviation (Std), maximum (Max), minimum (Min), and first-order autocorrelation (AC(1)) of the bitcoin discounts. All moments in percentages, with the exception of the first-order autocorrelation. The column “Pairs” reports the total number of bitcoin-to-fiat pairs traded on each exchange, and the column T reports the maximum number of daily observations across the different traded pairs. When the number of bitcoin-to-fiat pairs is larger than one, all remaining reported moments are the averages across the different pairs. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

B Transaction Costs

This Section presents additional information and results regarding the transaction costs faced by cryptocurrency investors. Speciﬁcally, Section B.I presents results regarding trading fees, while Section B.II presents results regarding bid/ask spreads.

57 Figure A2: Volatility of Discounts (All Samples)

cross-sectional volatility 18 baseline sample all fiat-to-bitcoin pairs 16 all crypto-to-crypto bitcoin pairs

10 % 8

0 2016 2017 2018 2019

Notes: This Figure plots the cross-sectional volatility of bitcoin discounts for the pairs in the baseline sample (blue line); all the ﬁat-to-bitcoin pairs (dashed-red line); and all the crypto-to-crypto bitcoin pairs (dotted-black line). Volatilities are in percentage. The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019 for all the ﬁat pairs, while starts on 3/31/2016 for all the crypto pairs.

B.I Trading Fees

The last two columns of Table A3 report, for all the exchanges in which the pairs in our baseline sample are traded, information on the fees charged to investors. Speciﬁcally, we manually collect, as of October 2019, information on the trading fees. As the data in the Table is obtained directly from the exchange websites, it is likely to be representative of the costs faced by retail investors. Larger investors are, instead, likely to pay smaller fees. The Table reports trading fees for trades that are taker and maker of liquidity (second and third columns). For example, on Kraken, fees are equal to zero for investors with a total monthly volume of transactions larger than 10 million U.S. dollars. Taker fees are paid when investors remove liquidity from the order book by placing an order that is executed against an order on the order book. On the contrary, maker fees are paid when investors add liquidity to the order book of the exchange by placing a limit order below the ticker price for buy, and above the ticker price for sale. Typically, taker trading fees are equal or larger than marker trading fees. In our sample of exchanges, the median trading fees are 25 basis points for a liquidity taker and 15 basis points for a maker of liquidity. Although the majority

58 Figure A3: Volatility of Discounts (Baseline Sample)

10 40

9 N N 35 8 30 7

6 25

5 20 # assets 4 15 3 10 2 cross-sectional discount volatility (%) 1 5

0 01/16 01/17 01/18 01/19

Notes: This Figure plots the cross-sectional volatility of bitcoin discounts for the pairs in the baseline sample (blue line), and the number of available pairs (or assets). Volatilities are in percentage and on the left axis. The number of assets is reported on the right axis. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019 for all the ﬁat pairs, while starts on 3/31/2016 for all the crypto pairs.

of transactions takes place on exchanges that function like standard equity markets where investors submit buy and sell orders that are cleared by a centralized order book, several of the exchanges offer only order-matching services and match buyers and sellers orders when they overlap. Figure A4 and Figure A5 provide a description of the timing of the trading fees for cross and within long/short strategies. Figure A4 refers to the cross strategy. The top (bottom) part of the Figure shows the timing of the strategy that goes long in portfolio 7 (short in portfolio 1). For the sake of exposition, we assume that there is only 1 pair in each portfolio: the bitcoin-to-euro pair on Coinbase in portfolio 7 and the British pound-to-bitcoin pair on Bitfinex in portfolio 1. The long strategy requires paying a trading fee twice: first, at time t, to Kraken when investors buy bitcoins, and then at time t + 1 to Coinbase when investors sell bitcoins for euros. Also, the short strategy requires paying a trading fee twice: first, at time t, to Kraken when investors short bitcoins, then at time t + 1 to Bitfinex when investors buy bitcoins for the British pound. In addition, the short strategy also requires paying a margin fee to Kraken for short positions that extend beyond 1 day. To close their positions, investors transfer bitcoins back to Kraken. We abstract from exchange deposit and withdrawal fees, as well as miner fees, as they are lump sums. Similarly, we abstract from transaction costs on the spot exchange market, as they are typically small for the currencies included in our sample. On the contrary, we account for exchange trading fees, bid/ask spreads, as well as margin fees required by short positions. Note that the trading fees of Kraken are negligible. Therefore, de facto, the cross

59 strategy requires paying the trading fee-only once for each trade.

Figure A4: Carry: Net of Transaction Costs

time t time t + 1

Kraken Kraken-Coinbase Coinbase FX -US$ BTC BTC EUR US$ deposit conﬁrmation time withdrawal trading fee trading fee

Kraken FX Bitfinex Bitfinex-Kraken FX -BTC US$ GBP GBP BTC BTC US$ withdrawal margin fee deposit confirmation time withdrawal trading fee trading fee

Notes: This Figure exemplifies the timing of a Carry investment strategy for a U.S. investor. Carry is a zero-cost cross strategy that goes long in portfolio 7 and short in portfolio 1. The top (bottom) part of the Figure shows the timing of the strategy that goes long in portfolio 7 (short portfolio 1). For the sake of exposition, we assume that there is only 1 asset in each portfolio: the bitcoin-to-euro pair on Coinbase in portfolio 7 and the British pound-to-bitcoin pair on Bitfinex in portfolio 1. Both strategies require paying a trading fee twice: first at time t, and then at time t + 1. In addition, the short strategy also requires paying a margin fee to Kraken for short positions that extend beyond 1 day. In Section4 we discuss the size of the trading fees across different exchanges, as well as the possibility of shorting bitcoin on Kraken and the associated margin fees. We abstract from additional miners fees and trading fees on the foreign exchange market.

Figure A5 refers to the within strategy. The timing of the fees is similar to that of the cross strategy. However, in this case, investors do not start each trade on Kraken and, thereby, must pay twice the trading fee for each trade.

B.II Bid and Ask Spreads We obtain daily bid and ask spreads data from Bitcoinity for a subset of 33 exchange-ﬁat currency pairs in our sample at two levels of market depth. The ﬁrst level corresponds to the minimum ask and maximum bid prices. These are the standard bid and ask prices commonly used to compute net returns, under the assumption that investors can purchase/sell nontrivial amounts of the asset at these prices. The second level corresponds to the ask and bid prices with 10 BTC worth of orders removed from bids and asks. The third and last level measures bid and ask prices for larger orders. Figure A6 plots the bid-ask spreads, computed using the minimum ask and maximum bid prices, for our benchmark bitcoin-to-dollar pair, on the Kraken exchange (solid blue line), along with the median and 75th percentile across all the pairs for which we obtained bid and ask prices. While bid-ask spreads are small on the largest exchanges and for the more traded pairs, they are an order of magnitude larger for less traded pairs. Finally, note that median bid-ask spreads have been stable through the sample, while bid/ask spreads for the benchmark pair have been progressively declining. This depends on the entry, over time, of new pairs and exchanges that are less liquid. Figure A7 compares, for the benchmark pair (top panel) and the median pair (bottom panel), the baseline bid/ask spreads to the bid/ask spreads with 10 BTC worth of orders

60 Figure A5: Long/Short within strategy: Net of Transaction Costs

time t time t + 1

FX Coinbase waiting Coinbase FX -US$ EUR BTC BTC EUR US$ deposit time withdrawal trading fee trading fee

Bitﬁnex waiting time Bitﬁnex withdrawal -BTC US$ US$ BTC US$ trading fee margin fee trading fee

Notes: This Figure exemplifies the timing of a long/short within strategy for a U.S. investor. This is a zero-cost strategy that goes long in portfolio 1 and short in portfolio 7. The top (bottom) part of the Figure shows the timing of the strategy that goes long portfolio 1 (short portfolio 7). For the sake of exposition, we assume that there is only 1 asset in each portfolio: the bitcoin-to-euro pair on Coinbase in portfolio 1 and the bitcoin-to-dollar pair on Bitfinex in portfolio 7. Both strategies require paying a trading fee twice: first at time t, and then at time t + 1. In addition, the short strategy also requires paying a margin fee to Bitfinex for short positions that extend beyond 1 day. In Section4 we discuss the size of the trading fees across different exchanges, as well as the possibility of shorting bitcoins and the associated margin fee. We abstract from miners fees and trading fees on the foreign exchange market.

Table A3: Exchange Fees and “Quality” Characteristics

Name Country FiatCurrencies Bloomberg Bitwise CryptoCompare Nomics Liquidity WebTraffic Maker Fees Taker Fees ACX Australia AUD No D C 946048 20 20 Bitbank Japan JPY No B 2,189,016 13617 15 15 Bitfinex Hong Kong EUR, GBP, JPY+1 Yes Yes B A- 66,690,742 8443 10 20 bitFlyer Japan EUR, JPY, USD Yes Yes AA A 2,856,376 5634 15 15 Bitlish UK EUR, GBP, JPY+2 No D 2,821,061 347244 20 30 BITPoint Japan JPY No A 277038 0 0 Bitstamp UK EUR, USD Yes Yes AA A 9,618,854 16564 25 25 Bittrex US USD No Yes B B 12,817,555 7854 25 25 Cex.io UK EUR, GBP, RUB+1 Yes B 13421 15 25 Coinbase Pro US EUR, GBP, USD Yes Yes AA A 16,802,020 1940 50 50 Coincheck Japan JPY No B D 15249 0 0 Coinfloor UK GBP, EUR No C 1205935 30 30 Coinsetter US USD No B 25 25 DSX UK EUR, GBP, RUB+2 Yes C 7,259,070 348332 15 25 Gemini US USD No Yes A A 4,477,484 26173 25 25 Independent Reserve Australia AUD, NZD, USD Yes B D 194901 50 50 itBit US USD, SGD Yes Yes AA D 168922 0 20 Kraken US CAD, CHF, EUR+3 Yes Yes A A 25,195,082 7935 16 26 Liquid Japan AUD, CNY, EUR+7 No AA A 4,378,101 23118 5 5 Zaif Japan JPY No B D 34502 0 0 median 15.5 25

Notes: This Table reports trading fees for the exchanges where the ﬁat-to-bitcoin pairs of our baseline sample are traded. The second column reports the country of exchange registration. We manually collect data from the websites of the exchanges as of October 2019. Trading fees are always in basis points. Taker fees are paid when investors remove liquidity from the order book by placing an order that is executed against an order on the order book. On the contrary, maker fees are paid when investors add liquidity to the order book of the exchange by placing a limit order below the ticker price for buy, and above the ticker price for sale. The last column reports the number of pairs traded on each exchange and present in our sample.

61 removed from bids and asks. For the benchmark pair, the difference between the 10 BTC bid/ask spread, and the baseline bid/ask spread is usually small (the mean difference is 22 basis points), with the exception of a peak of 162 basis points around the end of 2017. On the contrary, for the median pair, the difference between the 10 BTC bid/ask spread and the baseline bid/ask spread is on average larger (88 basis points) ad more volatile (45 basis points against 17 basis points).

Figure A6: Bid and Ask Spreads

2.5 Kraken (benchmark) median p75

1.5 %

0.5

0 2016 2017 2018 2019

Notes: This Figure plots the bid and ask spreads in percentages. The solid blue line corresponds to the benchmark pair bitcoin- to-dollar on the Kraken exchange; the dashed red line to the median; the dashed black line to the 75th percentile (p75). The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

The coverage of the bid and spread data is limited with respect to all the pairs in our baseline sample. For this reason, we also use the following alternative estimates of the bid and ask spreads.

Bid/Ask spreads based on Roll(1984) Roll’s estimate is based on the idea that in an eﬃcient market, the fundamental value of a security ﬂuctuates randomly. However, bid and ask prices induce negative serial dependence in successive observed market price changes. Therefore, he proposes to estimate the bid/ask spread as

Roll p BAt = 2 −covt(∆pt, ∆pt−1)

62 Figure A7: Bid and Ask Spreads (10 BTC)

bid/ask spread (benchmark pair) 3 Kraken (benchmark) 2.5 Kraken 10BTC (benchmark)

% 1.5

0.5

0 2016 2017 2018 2019

bid/ask spread (median pair) 3 median 2.5 median 10BTC

% 1.5

0.5

0 2016 2017 2018 2019

Notes: This Figure plots the baseline bid and ask spreads (solid blue line) and the bid and spreads with 10 BTC worth of orders removed from bids and asks prices (dashed-red line). Bid/ask spreads are in percentages. The top panel corresponds to the benchmark pair traded on the Kraken exchange; the bottom panel to the median pair. The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

where cov is the ﬁrst-order serial covariance of price changes. We estimate this measure using a rolling-window of 30 days to estimate the covariance and set to zero the argument of the square root when negative.

Bid/Ask spreads based on Abdi and Ranaldo(2017) Abdi and Ranaldo recently have proposed an alternative estimate of bid and ask prices, based on a wider information set with respect to Roll(1984). Speciﬁcally, they consider the close, high and low prices and argue that their measure delivers the most accurate estimates for less liquid stocks. Abdi and Ranaldo(2017) propose the following measure (see equation 11 in their paper)

N 1 X BAAbdi−Ranaldo = pmax{4(c − η )(c − η ), 0} t N t t t t+1 t=1 where c is the log of the closing price; and η = 0.5(h + l), where h is the log of the high price and l the log of the low price; N is the length of the rolling window which we take equal to 30 days.

63 Figure A8 plots, for the benchmark pair traded on Kraken, the actual baseline bid/ask spreads (solid blue line) along the estimates (dashed-red line) based on the measures proposed by Abdi and Ranaldo(2017) (top panel) and Roll(1984) (bottom panel). Both estimates overestimate the baseline bid/ask spreads, even though the measure proposed by Abdi and Ranaldo(2017) is close to the bid/ask spreads with 10 BTC worth of orders removed from bid and ask prices. One possible explanation for the differences between the actual bid/ask spread, and those obtained by the two estimators discussed above is the limited depth for the limit order book. In our baseline analysis, we extend the coverage of bid/ask spreads using the predictive values from regressions based on different pair characteristics, including high/low/close prices, volume, exchange location, and fiat currency.

Figure A8: Alternative Estimates of Bid/Ask Spreads (Benchmark Pair)

actual vs. abdi-ranaldo (RFS, 2017) 2

1.5

% 1 data abdi-ranaldo (RFS, 2017) 0.5

0 2016 2017 2018 2019

actual vs. roll (JF, 1984) 10 data roll (JF, 1984) 8

6 % 4

0 2016 2017 2018 2019

Notes: This Figure plots, for the benchmark pair traded on Kraken, the actual baseline bid/ask spreads (solid blue line) along the estimates (dashed-red line) based on the measures proposed by Abdi and Ranaldo(2017) (top panel) and Roll(1984) (bottom panel). The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

Bid/Ask spreads based on panel regressions We generate synthetic bid and ask prices for all the bitcoin-to-ﬁat currency pairs for which we do not have bid and ask prices using the predicted values from the following panel regressions

panel BAt = αj + γt + B(L)vm,j,t + A(L)hlm,j,t + m,j,t

64 where B(L) and A(L) are fifth-order lag polynomial, v is the log trading volume in bitcoins, hl is the lag of the high-low spread, and αj and γt are currency and time fixed effects. Figure A9 plots, for all the bitcoin exchange-currency pairs, the means (top panel) and 75th percentile (bottom panel) of the actual baseline bid/ask spreads (solid black line); the estimates based on panel regressions (dashed-red line), the measure proposed by Abdi and Ranaldo(2017) (line-dot magenta line), and the measure proposed by Roll(1984) (blue dots). We note that our panel estimates closely follow the actual bid/ask spreads, both in the mean and 75th percentile. The measure by Abdi and Ranaldo(2017) is more stable over the sample, and not too far from the actual data. On the contrary, Roll(1984)’s measure is more volatile, often very large in magnitude, and often equal to zero (i.e., the argument of the square root in the formula is negative).

Figure A9: Alternative Estimates of Bid/Ask Spreads (All Pairs)

mean 6 actual 5 panel abdi-ranaldo 4

% 3

0 2016 2017 2018 2019

p75 6 actual 5 panel abdi-ranaldo 4

% 3

0 2016 2017 2018 2019

Notes: This Figure plots, for all the bitcoin exchange-currency pairs, the means (top panel) and 75th percentile (bottom panel) of the actual baseline bid/ask spreads (solid black line); the estimates based on panel regressions (dashed-red line), the measure proposed by Abdi and Ranaldo(2017) (line-dot mangenta line), and the measure proposed by Roll(1984) (blue dots). The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

65 C Measures of Exchanges “Quality”

Table A3 reports additional information on the “quality” of the exchanges in the sample. Specifically, we report the country of registration of each exchange; the fiat currencies pairs traded against bitcoin; whether the exchange is in the list of data providers for the Bloomberg BTC quotes; whether the exchange is in the list of “good” exchanges in the Bitwise(2019)’s report; the letter rating according to CryptoCompare; the letter rating according to Nomics; the new liquidity and web traffic measures by Nomics. A letter rating of “C” corresponds to a “fair” exchange, while “D” corresponds to “Poor”. According to Nomics, “fair” exchanges usually provide historical trade-level data. The contrary to suspected exchanges. Note that the measure of web traffic refers to the world ranking (Google is the first at the time of this writing). Liquidity refers to the ease of being able to trade in and out of an asset. The liquidity metric takes into account a range of variables from the order book: distance of the order from the mid-price, the size of the order, and the relative liquidity of the asset in question. The metric is made by polling the market-pair at random intervals over a 24-hour period and averaging the result. The units are US dollars.

D Execution Risk

Investors buying bitcoin across exchanges around the globe are exposed to different forms of execution risk, i.e., the risk that a transaction is not executed within the range of market prices observed by investors before sending their orders. Investors following a within strategy on a given exchange face the risk associated with the execution time required to execute a trade within the exchange. Investors following a cross strategy face the additional risk associated with the time required to transfer bitcoin across exchanges. The very large bitcoin volatility exacerbates these risks. For example, for the within dollar returns of the benchmark pair, Figure A10 plots the distribution of 1-day and 5-day returns. The corresponding volatilities are extremely large (respectively, 4.56% and 10.67%), and the distribution shows large kurtosis. It is very hard to exactly quantify the execution times, as they depend both on the “type” of investor (e.g., retail or hedge fund) and the state of the network. The time required by the blockchain to transfer bitcoins across exchanges depends on the congestion of the network and has recently ranged anywhere between 10 minutes to 24 hours. On the other hand, trade execution within an exchange is much faster, with only 3.95% of trades taking longer than one second to be executed for a sample of the largest exchanges (Krückeberg and Scholz, 2020). Note that alternative cryptocurrencies differ in terms of their execution times and transaction costs. Table A4 reports the approximate times for transactions in different cryptocurrencies and for the credit card provider Visa as a reference. The time estimates assume that the transaction is confirmed in the first block after being submitted and are approximate because they depend on the network congestion. For Bitcoin, the maximum number of transactions per second is 7, with an estimated execution time of one hour. Both ethereum and Ripple allow a larger number of transactions per second (respectively, 20 and 1,500) and shorter execution times (respectively, 6 minutes and near-instant execution). The credit card provider Visa allows for a significantly larger number of transactions per second, but also has higher

66 transaction costs than cryptocurrencies.

Figure A10: Distribution of Dollar Returns

dollar returns on Kraken

1-day returns ( =0.43% =4.58%) 5-day returns ( =2.21% =10.70%)

-40 -30 -20 -10 0 10 20 30 40 50 60 percent return

Notes: This Figure plots the distribution of the 1-day (blue line) and 5-day (dashed-red line) dollar returns for the bitcoin-to- dollar pair on Kraken (i.e., within returns). Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

Further, investors might not be able to place their orders at time t+1 because of potential unexpected exchange shutdowns. Exchange shutdowns can be temporary, for example, because of a software malfunctioning or denial-of-service attack, or permanent, in case of bankruptcy. Since the diffusion of cryptocurrencies, exchanges have been subject to several critical events. One of the most common is represented by distributed denial-of-service attacks (DDoS attacks). A standard denial-of-service attack (DoS attack) is a cyber-attack where the perpetrator seeks to make a machine or network resource unavailable to its users by temporarily or indefinitely disrupting services of a host connected to the Internet typically by flooding the service with requests. Instead, in a DDoS attack, the traffic flooding originates from many sources. Exchanges are also subject to thefts, jams due to high user traffic, and suspensions of services due to software maintenance. Table A5 reports a list of the main critical events that cryptocurrency exchanges faced since 2014. We manually collected information on these events using various sources. Panel A reports details on events involving attacks by hackers. Panel B reports details on other

67 Table A4: Cryptocurrencies by Execution Times

Max Transactions Estimated Transaction Transaction per second Execution Time fee in crypto fee in $ Bitcoin (BTC) 7 60 Minutes 0.001 11.29 Etherum (ETH) 20 6 Minutes 0.005 5.18 Ripple (XRP) 1,500 Near-instant 0.02 0.03 Visa 24,000 Near-instant 1.4–2.4%

Notes: We report approximate executions times for transactions in different cryptocurrencies. Execution times can vary depend- ing on the conditions of the network. The time estimates assume that the transaction is confirmed in the first block after being submitted. Dollar fees depend on crypto-US% exchange rate and are for January 2018. Authors manually collect data from different sources.

relevant events. Most of the events are associated with thefts of bitcoins. For these events, in the third and fourth columns, we report the number of bitcoins stolen both in units of bitcoins and in millions of U.S. dollars. For each event, we also report the date corresponding to the first day of service suspension as well as the date, when available, in which the exchanges resumed operations. The last column reports the final outcome of the attack. Most exchanges are still active and have resumed operations, with the exceptions of Mt. Gox, Silk Road, BitGrail, Zaif, and Youbit, who had to declare bankruptcy. The total amount of bitcoin stolen during these events, is approximately equal to 2 billions of U.S. dollars (using the relevant exchange rate at the time of each event). In our sample, exchange shutdowns are relatively common events. Figure A11 plots the total number of pairs that are potentially available to investors in a given day (blue line), and those that are active, i.e., which investors can effectively purchase. On a given day, active pairs have non-zero trading volumes. On the contrary, inactive pairs have zero trading volume but had non-zero trading volume in the previous day and non-zero trading volume in at least one day of the remaining part of the sample. Note that typically more than one pair trades on a given exchange. Therefore, when an exchange shuts down, simultaneously, a number of pairs traded on the exchange becomes suddenly inactive. This, along with the relatively frequent exchange shutdowns, explains the large fraction of inactive pairs, which in our baseline sample has a daily mean of approximately 14 percent. Recall that the definition of cross returns requires purchasing bitcoin, at t, using the benchmark pair, and then selling bitcoin for fiat currency on a different exchange at t + 1 (see equation (2)). In our baseline analysis, to be conservative, we assume that, when at t + 1 the desired currency-exchange pair is unexpectedly unvailable, investors can sell their bitcoin balance at the median price across all the available pairs. Therefore, we exclude the possibility that investment is “stuck” till the exchange resumes trading activities. Since, in principle, this is a possibility that investors face, in Table A6 we report information on the exchange shutdowns with the longest duration in our sample. Specifically, the Table considers all the pairs affected by exchange shutdowns for more than 25 consecutive days. We report the daily returns obtained by investors “stuck” on an exchange because of a shutdown, and that sell their balance as soon as the exchange resumes its activities. The longest shutdown lasted 81 days for the bitcoin-to-euro pair on Liquid. In

68 Table A5: Critical Events on Cryptocurrency Exchanges

name country in tokens in mln USD suspended resume outcome recovered Panel A: Exchange Hacks Coincheck Japan 523 mln NEM 534 26-Jan-18 active MtGox Japan 850000 BTC 500 07-Feb-14 24-Feb-14 bankruptcy 23% BitGrail Italy 17 mln Nano 195 01-Feb-18 bankruptcy NiceHash Slovenia 4700 BTC 75 07-Dec-17 20-Dec-17 active Bitﬁnex Hong Kong 120000 BTC 72 02-Aug-16 active 100% Zaif Japan 5966 BTC 60 14-Sep-18 27-Sep-18 bankruptcy CoinBene China 45 27-Mar-19 active Binance Korea 7000 BTC 40 07-May-19 14-May-19 active Coinrail Korea Pundi X, Astoncoin 37 11-Jun-18 active Youbit Korea 4000 BTC 35 28-Apr-17 active BITPoint Japan 32 12-Jul-19 13-Aug-19 active Bithumb Korea 2016 BTC 31 15-Jun-18 active 30% Bithumb Korea 30 19-Jun-17 active Bithumb Korea 3 mln EOS + 20 mln XRP 19 30-Mar-19 active Bitstamp US 19000 BTC 5 05-Jan-15 09-Jan-15 active Youbit Korea 1000 BTC 2 19-Nov-17 bankruptcy Panel B: Other Events SilkRoad 171955 BTC, hack 270 01-Oct-13 shutdown 100% QuadrigaCX Canada owner death 190 28-Dec-18 bankruptcy Parity UK 513,774.16 ETH, locked funds 160 07-Nov-17 active DAO Germany 3.6 mln Ether 55 17-Jun-16 active 100% Parity UK 15000 Eth, hack 32 01-Jul-17 active Coinroom Poland shutdown over night 07-Jun-19 bankruptcy Bitsane Ireland shutdown over night 13-Jul-19 shutdown Gatecoin Hong Kong due to hack 14-Mar-19 bankruptcy Bitstamp US down 12-Feb-14 14-Feb-14 active Kraken US down 10-Jan-18 12-Jan-18 active Total 2419

Notes: The Table reports a selected sample of critical events that aﬀected the major cryptocurrency exchanges. The third and fourth columns report, in units of currency and U.S. dollars, respectively, either the number of bitcoins or other cryptocurrencies stolen, locked, or unavailable due to software maintenance or a DDoS attack. Panel A refers to “exchange hacks”, while Panel B to “other events”. The authors manually collect data from diﬀerent sources in October 2019.

69 this case, investors had a positive mean cumulated return over the period equal to 2.73%. The median episode lasted 42 days, and investors earned 2 basis points per day over the period in which their balance was possibly “stuck” in the exchange. Finally, note that, even though we observe a relatively large fraction of pairs that are not available for trading on a given day, these do not signiﬁcantly aﬀect portfolio returns as investors cannot trade these pairs.

Figure A11: Total and Active Number of Exchanges

mean 20 10 0 0 5 10 15 20 25 30 35 40 45 std 20 10 0 0 5 10 15 20 25 30 35 40 45 max 200 100 0 0 5 10 15 20 25 30 35 40 45 median 20 10 0 0 5 10 15 20 25 30 35 40 45 min 2 1 0 0 5 10 15 20 25 30 35 40 45

Notes: This Figure plots the total (blue line) and active (red line) number of pairs in our sample. A pair is denoted as “active” when the daily volume of transactions is non-zero. The curves are smoothed with a 5-day moving average. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

Cryptocurrency exchanges function in many ways, like brokers or banks. Customers buy and sell bitcoins (or other cryptocurrencies), but typically maintain balances of both ﬁat currencies and bitcoins on the exchange without retaining direct access to the currency (Gandal et al., 2018, for example). Investors’ trades on an exchange are done oﬀ the blockchain. When investors deposit bitcoins on an exchange, these are put in a shared wallet that the exchange controls (i.e., these are like assets for a bank). The exchange keeps track of investors’ balances and of all the transactions. The blockchain only knows that investors send coins to the exchange, and consider these coins as owned by the exchange. When investors withdraw coins, then the blockchain is informed, and bitcoins are transferred to the investors’ personal wallets. In part because of the uncertain regulatory environment around cryptocurrency exchanges,

70 Table A6: Exchange Shutdowns

Asset j, m t T P1,1,t−1 Pm,j,T rm,j,t,T (%) # days btc-to-usd Coinsetter 27-Jan-2016 – 395.95 – – – btc-to-eur BitBay 23-Feb-2016 28-Apr-2016 433.35 450.34 0.09 47 btc-to-usd BitBay 10-Oct-2017 01-Dec-2017 4772.10 10750.00 3.39 38 btc-to-eur BitBay 11-Oct-2017 01-Dec-2017 4748.50 10713.09 3.49 37 btc-to-pln BitBay 11-Oct-2017 05-Dec-2017 4748.50 11663.78 3.83 39 btc-to-jpy BitBank 22-Jan-2018 29-Mar-2018 11598.10 7451.90 -0.76 48 btc-to-pln Coinfloor 22-May-2018 11-Jul-2018 8390.80 6205.20 -0.77 35 btc-to-usd BitTrex 12-Jul-2018 11-Sep-2018 6377.90 6276.98 -0.04 43 btc-to-pln Coinfloor 04-Sep-2018 19-Nov-2018 7015.80 5287.16 -0.46 55 btc-to-gbp DSX 03-Oct-2018 28-Nov-2018 6500.00 4002.56 -0.99 40 btc-to-pln Coinfloor 20-Nov-2018 – 4741.20 – – – btc-to-jpy Liquid 13-Mar-2019 08-Jul-2019 3863.10 12311.03 2.73 81 btc-to-eur Bitpoint 24-Jun-2019 – 10219.20 – – – btc-to-usd Bitpoint 25-Jun-2019 – 11020.60 – – – btc-to-jpy Bitpoint 15-Jul-2019 – 11798.10 – – – Median 0.02 42

Notes: This Table reports information on all the exchange shutdowns affecting pairs in our baseline sample for at least 25 consecutive days. Each row corresponds to different shutdown episodes. The first column reports the name of the affected pair (j, m) with the format BTC-to-fiat and then exchange name. The second and third columns report the first (t) and last (T ) dates of the shutdown episodes. The fourth column reports the bitcoin price of our benchmark pair on the day immediately before the shutdown episode (i.e., for the bitcoin-to-dollar pair on Kraken, P1,1,t−1). The fifth column reports the price of bitcoin price for pair j, m on date T when the pair is again available to investors. Finally, the last two columns report the daily return for an investor who had purchased bitcoin using the benchmark pair at t − 1 and then sold bitcoin on exchange m and against currency j at T ; and the total number of days the pair was inactive. The reported daily return is computed as the total return ratio between t − 1 and T and the total number of days of inactivity. Missing values in the column Pm,j,T denote pairs that are still unavailable to investors as of 10/31/2019.

the size of the shared wallets is often taken as one of the indicators that investors use to evaluate the reliability of different markets and the risk of not being able to withdraw their balances. In some ways, wallets are like deposits for a traditional financial institution. However, differently from traditional deposits, there is no deposit insurance scheme. We collect daily data on exchange wallets from walletexplorer.com and bitinfocharts.com. We cannot cover all the exchanges in our sample. Also note that wallets data are self-reported, and exchanges started to identify their wallets only after the Mt. Gox bankruptcy in 2013 to improve their transparency. Table 7, in the main paper, reports the average wallet size for each within and cross portfolio. The table shows that the last portfolio, i.e., portfolio 7, on average, contains pairs traded in exchanges with the smallest wallets.

E Cross-sectional Asset Pricing

In this Section we discuss the procedures used to estimate the market price of the CarryBtc risk. k We use Rxt+1 to denote the average excess return in levels on portfolio k in period t + 1.

71 All asset pricing tests are run on excess returns in levels, not log excess returns, to avoid having to assume joint log-normality of returns and the pricing kernel. In the absence of arbitrage opportunities, this excess return has a zero price and satisﬁes the following Euler equation:

k Et Mt+1Rxt+1 = 0. We assume that the stochastic factor M is linear in the pricing factors Φ:

Mt+1 = 1 − b (Φt+1 − µΦ)

where b is the vector of factor loadings, and µΦ denotes the factor means. This linear factor model implies a beta pricing model: the expected excess return is equal to the factor prices λ times the beta of each portfolio βk

E Rxj = λ0βj where X λ = b ΦΦ

and ΣΦΦ is the variance-covariance matrix of the factors

0 ΣΦΦ = E (Φt − µφ) (Φt − µΦ) , and βk denotes the regression coefficients of the return Rxk on the factors. To estimate the factor prices λ and the portfolio betas β, we use two different procedures: a Generalized Method of Moments estimation (GMM) applied to linear factor models, following Hansen (1982), and a two-state OLS estimation following Fama and MacBeth(1973), henceforth FMB. In the first step, we run a time series regression of returns on the factors. In the second step, we run a cross-sectional regression of average returns on the betas. We do not include a constant in the second step (λ0 = 0) and therefore assume that assets with a beta equal to zero must offer zero excess returns.

F Additional Robustness Checks

This Section presents the results of additional robustness checks. First, we document the portfolio allocation of pairs and show that there is no evidence of clustering of subsets of the pairs in any portfolio. Second, we present bitcoin portfolios, sorted by average discounts. Also, for this sort, we obtain a positive and signiﬁcant cross-section of cross net returns. Although it is not a tradable strategy, the results from this alternative sort suggest that also average (unconditional) discounts provide information on the idiosyncratic risks that prevent investors to fully absorb the price diﬀerences. Third, we present portfolios at weekly frequency. A lower trading frequency reduces both transaction costs and the risks associated with the time required to complete a transaction. We show that our baseline results are robust to this lower frequency. Fourth, we show that our results are robust to changes in the number of

72 portfolios.

F.I Portfolio Allocation

Figure A12 documents the allocation of pairs, from the baseline sample to the seven portfolios sorted by bitcoin discounts. The Figure reports on the horizontal axis all the pairs from the baseline sample. The vertical axis has one tick for each portfolio, from 1 to 7. The size of the “bubbles” in the scatter plot denotes the frequency with which each given pair is in a given portfolio. For clarity of exposition, in the Figure, we do not plot any bubble for pairs that are in a given portfolio less than 5% of the days. Even though we find that some pairs are more likely to be in the corner portfolios, the Figure shows that, overall, there is no evidence of clustering of subsets of the pairs in any portfolio. We also formally test for clustering using the k-means clustering test and find no evidence of clustering for the corner portfolios at standard significance levels. Therefore, we rule out the possibility that the portfolio results are driven by a few pairs that are always in the corner portfolios.

Figure A12: Portfolio Allocation

4 portfolios

1 5 10 15 20 25 30 35 40 45 pairs

Notes: This Figure documents the allocation of the bitcoin pairs from the baseline sample to the seven portfolios sorted by bitcoin discounts. In the scatter plot, the bubbles’ size denotes the frequency with which a pair is in a given portfolio. For clarity of exposition, in the Figure, we do not plot any bubble for pairs that are in a given portfolio less than 5% of the days. We also formally test for clustering using the k-means clustering test and ﬁnd no evidence of clustering for the corner portfolios. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

73 F.II Alternative Sort In this Section, we consider an alternative sort. Specifically, we form portfolios by sorting pairs by their average discounts, to investigate whether average characteristics of the pairs drive the cross-section of bitcoin returns. Note that this is not a tradable strategy, as it requires investors’ knowledge about future discounts. Also, note that the average discounts are estimated over different time periods because pairs enter and exit the sample. Table A7 presents our results. The first portfolio has an average discount equal to -13 basis points, while the last portfolio has an average discount equal to 115 basis points. Sorting pairs by their average discounts, produces a cross-section of net cross returns, but not of within returns. The average long/short cross return is positive, statistically significant, and equal to 50 basis points per day. This result suggests that average discounts provide information on the quantity of risk of the different exchange and currency pairs. On the contrary, the average long/short within return is large and negative. Note that the cross strategy goes long in portfolio 7, which has an average volume of transactions equal to approximately 1/3 of the other portfolios’ averages. This indicates that the large returns of the cross strategy might be related to investing in the less liquid pairs.

Table A7: Portfolios by Average Discounts

Portfolio 1 2 3 4 5 6 7 long/short Panel A: discounts Mean -0.13 0.05 0.16 0.31 0.50 0.69 1.15 SE 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Panel B: cross returns 7-1 Mean 0.08 0.26 0.33 0.43 0.72 0.62 1.03 0.50 SE 0.13 0.14 0.14 0.14 0.13 0.14 0.13 0.05 Panel C: within returns 1-7 Mean -0.29 -0.02 -0.05 -0.12 0.37 -0.12 -0.01 -2.70 SE 0.13 0.13 0.14 0.14 0.14 0.14 0.15 0.07 Panel D: volume (in millions of U.S. dollars) Mean 20.11 20.80 23.63 18.20 35.12 17.91 8.15 SE 0.90 0.83 1.44 1.24 2.07 1.08 0.44

Notes: This Table reports bitcoin portfolios sorted by average discounts. We report averages and standard errors for portfolio discounts (panel A); portfolio net cross returns (panel B); portfolio net within returns (panel C); portfolio volume (panel D). Net returns are net of bid/ask spreads. Portfolios are sorted by the average discounts of all pairs. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

F.III Weekly Frequency In our baseline analysis, we consider a period of 1 day and form daily bitcoin portfolios. We have documented how transaction costs, and especially bid-ask prices for some pairs, substantially reduce the daily returns. In addition, we have also documented that bid-ask prices increase steeply with the order size. In this Section, we show that our results also hold at a lower frequency of trading.

74 We build end-of-week bitcoin cross and within returns and discounts and form portfolios sorted by bitcoin discounts, as we did for the portfolios at the daily frequency. For both the cross and within returns, the holding period is equal to 5 days, i.e., from Friday (close) to Friday (close), and exclude non-business days, like Saturday and Sunday. Table A8 presents the properties of the bitcoin portfolios at weekly frequency. Panel A reports the average bitcoin discounts. The mean discounts increase monotonically from -109 basis points for the first portfolio to 196 basis points for the last portfolio. To facilitate the comparison with the portfolios√ at daily frequency, only for returns, we divide averages by 5 and standard deviation by 5, so that the numbers reported are per day. We start with the description of cross portfolio returns (panel B). As for the daily frequency sample, excess returns increase monotonically across the seven portfolios. Specifically, excess returns increase from approximately 38 basis points to 68 basis points per day. The Patton and Timmermann (2010)’s monotonicity tests reject both the null of zero spread between the first and last portfolio (p-value is 0.00%), and of the absence of a monotonic relation (MR) between the portfolio sorting variable and excess returns (studentized p-value of 4.7%). The long/short 7-1 average returns, before transaction costs, are equal to 30 basis points per day. Accounting for bid/ask spreads, reduce the average long/short return by approximately 22 basis points. The net average long/short return is equal to 8 basis point per day, and is statistically different from zero. However, note that net returns do not include trading fees and margin fees, and the latter increase with the holding period. Panel C considers within returns. Also, in this case, we replicate the results of the portfolios at daily frequency. Specifically, we obtain a monotonically decreasing cross-section of portfolio gross and net returns. Note that, in this case, portfolio net returns are also positive, except for the first portfolio. However, the average net long/short within return is still negative and equal to -9 basis points. Table A9 reports the results of regressions of factors on a set of explanatory variables using weekly frequency returns (as for Table 6, in the main paper, which refers to daily frequency returns). The main result from these weekly frequency regressions is that CarryBtc is not spanned by the three factors proposed by Liu et al.(2019) (i.e., a market factor ( CMKTRF ), a size factor (CSIZE), and a momentum factor (CMOM)). Also at the weekly frequency, we find that CarryBtc is significantly related to measures of liquidity (i.e., volume and bid/ask spreads). Instead, we find that the coefficient attached to the Google Trend Index is not significant.

F.IV Out of Sample

In Section4 of the main paper, we show that CarryBtc is a risk factor for the cross-section of bitcoin returns also out-of-sample. In order to do so, we randomly split our sample into two groups and form bitcoin portfolios sorted on discounts for both groups. Because of the lower number of pairs per sample, in this case we form only 5 portfolios. Table 11 shows that we obtain a monotonically increasing cross-section of bitcoin returns in both samples. Specifically, the maximum spread between returns of the last and first portfolio is equal to 316 for the first sample (panel A) and 264 for the second sample (panel B). To confirm that our out of sample results do not only depend on luck in the selection of the pairs for the two groups, we repeat the exercise 1,000 times and report averages and standard errors across

75 Table A8: Bitcoin portfolios: US investor (Weekly Frequency)

Portfolio 1 2 3 4 5 6 7 long/short Panel A: discounts Mean -1.09 -0.28 -0.01 0.20 0.46 0.79 1.96 Std 1.20 0.63 0.60 0.66 0.91 1.15 1.81 Panel B: cross returns 7-1 gross returns Mean 0.38 0.44 0.46 0.46 0.52 0.56 0.68 0.30 Std 4.53 4.55 4.51 4.55 4.63 4.60 4.61 0.72 SE 0.13 0.13 0.13 0.13 0.06 0.06 0.06 0.01 SR 0.08 0.10 0.10 0.10 0.11 0.12 0.15 0.42 returns net of bid/ask Mean 0.28 0.38 0.41 0.42 0.46 0.49 0.56 0.08 Std 4.54 4.55 4.52 4.56 4.61 4.62 4.70 0.94 SE 0.13 0.14 0.13 0.13 0.06 0.06 0.06 0.01 SR 0.06 0.08 0.09 0.09 0.10 0.11 0.12 0.08 Panel C: within returns 1-7 gross returns Mean 0.60 0.50 0.46 0.43 0.43 0.40 0.29 0.32 Std 4.60 4.58 4.52 4.56 4.62 4.57 4.54 0.92 SE 0.13 0.13 0.13 0.13 0.06 0.06 0.06 0.01 SR 0.13 0.11 0.10 0.09 0.09 0.09 0.06 0.35 returns net of bid/ask Mean 0.43 0.39 0.38 0.35 0.33 0.27 0.05 -0.09 Std 4.62 4.60 4.52 4.59 4.61 4.62 4.73 0.55 SE 0.14 0.14 0.13 0.13 0.06 0.06 0.06 0.01 SR 0.09 0.09 0.08 0.08 0.07 0.06 0.01 -0.42

Notes: This Table reports, for each portfolio, the mean and standard deviation for the bitcoin discounts; the excess cross returns, and the high-minus-low returns from a zero-cost strategy 7-1; the excess within returns, and the high-minus-low returns from a zero-cost strategy 1-7. For returns, we also report standard errors by bootstrap and gross and net returns. The holding period is equal to 5 days, from Friday (close price) to Friday (close price), and excludes non-business days like Saturday and Sunday. To facilitate√ the comparison with the portfolios at daily frequency, only for returns, we divide averages by 5 and standard deviation by 5, so that the numbers reported are per day. Transaction costs include bid-ask spreads, but do not include trading, margin, and exchange fees. Portfolios are constructed by sorting assets into seven groups at time t by their discounts. The ﬁrst portfolio contains assets with the lowest negative discounts. The last portfolio contains assets with the highest positive discounts. Data are weekly, from cryptocompare.com and Thomson Reuters. The sample period is 5/29/2015–11/1/2019.

the different random samples. Table A10 summurizes our results. Panel A reports, for portfolio discounts and excess returns of the first group of each pair, the average and standard deviation across the 1,000 iterations. For each iteration, we use these portfolio excess returns as test assets and build CarryBtc from the portfolios obtained from the second group. Panel B reports the estimates for the risk prices for the GMM and FMB estimation procedures. Standard errors are computed over the 1,000 iterations. As for the baseline asset pricing estimation, also, in this case, we find that the market price of the CarryBtc risk is positive and statistically different from zero; and the no-arbitrage condition is satisfied. Average pricing errors are small, and the adjusted R-squares are large. However, we reject the null of zero pricing errors.

76 Table A9: Explaining Crypto Factors (Weekly Frequency)

Btc CarryBtc CMKTRFCSIZECMOM α 1.567 0.439 -1.060 -0.018 4.955 [ 0.718 ] [ 0.405 ] [ 0.637 ] [ 1.989 ] [ 2.181 ] Btc – -0.002 0.954 -0.297 -0.140 – [ 0.030 ] [ 0.064 ] [ 0.186 ] [ 0.391 ]

CarryBtc -0.021 – 0.297 1.096 1.372 [ 0.245 ] – [ 0.271 ] [ 0.707 ] [ 0.683 ] T V ol 0.028 0.119 0.083 0.305 -0.626 [ 0.093 ] [ 0.037 ] [ 0.095 ] [ 0.225 ] [ 0.287 ] σ(V ol) -0.226 -1.524 -0.525 0.086 7.636 [ 0.902 ] [ 0.424 ] [ 0.944 ] [ 1.934 ] [ 3.962 ] E(BA) -0.358 -1.388 0.734 -1.182 0.955 [ 1.075 ] [ 0.692 ] [ 1.032 ] [ 2.541 ] [ 2.813 ] E(HL) -0.187 0.134 0.033 0.122 -0.539 [ 0.159 ] [ 0.062 ] [ 0.160 ] [ 0.245 ] [ 0.420 ] ∆Goog 0.060 0.000 -0.035 -0.049 0.068 [ 0.020 ] [ 0.006 ] [ 0.018 ] [ 0.031 ] [ 0.034 ] CMKTRF 0.736 0.027 – 0.322 0.189 [ 0.044 ] [ 0.023 ] – [ 0.195 ] [ 0.351 ] CSIZE -0.045 0.019 0.063 – -0.050 [ 0.035 ] [ 0.012 ] [ 0.044 ] – [ 0.103 ] CMOM -0.014 -0.014 0.024 -0.032 – [ 0.040 ] [ 0.008 ] [ 0.046 ] [ 0.060 ] – R2 adj. (%) 74.358 31.446 72.284 9.833 2.534 F (%) 0.000 0.000 0.000 0.042 11.525 T 212 212 212 212 212

Notes: This Table presents the results of linear regressions, at weekly frequency, where the dependent variables are the CarryBtc and Btc factors, as well as the factors from the three-factor model of Liu et al.(2019). CarryBtc is the excess return of a zero-cost cross strategy that goes long in portfolio 7 and short in portfolio 1. Btc is the return of a strategy that buys and sells bitcoins always using the dollar-to-bitcoin pair on Kraken. The regressors are the same used in the regressions reported in Table 6 and, in addition, we also include the three-factors of Liu et al.(2019). Standard errors in brackets are Newey and West(1986). The last row reports the p-values (in percentages) of an F-test on all coeﬃcients equal to zero (excluding the constant). Data are weekly from cryptocompare.com, Thomson Reuters, Bloomberg, Google, and the Kenneth French data library for the period 5/26/2015–10/31/2019.

F.V Number of Portfolios

In the baseline analysis we consider a number of portfolios equal to 7. Our choice is motivated by having a minimum number of assets in each portfolio, especially at the beginning of the sample and, at the same time, enough assets to, at least partially, diversify idiosyncratic risk. In this Section, we show that the cross-sections of cross and within returns are robust to diﬀerent numbers of portfolios. Speciﬁcally, Figure A13 plots the mean (gross) long-short excess returns for both the cross (top panel) and within (bottom panel) returns in the case of 3, 5, 7 (baseline) and 10 portfolios. We note that the excess returns are positive and

77 Table A10: Out of sample Montecarlo Results

Panel A: portfolios Portfolio 1 2 3 4 5 discounts Mean -1.19 -0.18 0.16 0.54 1.87 SE 0.10 0.03 0.03 0.05 0.10 net returns Mean -0.65 0.03 0.22 0.36 1.02 SE 0.13 0.05 0.03 0.05 0.14 Panel B: risk prices 2 2 λBtc λCarryBtc bBtc bCarryBtc R RMSE χ (%)

GMM2 0.16 0.47 0.62 4.86 77.77 0.49 0.00 SE 0.14 0.23 0.67 1.30 2.60 0.04 0.00 FMB 0.17 0.47 0.71 3.90 -46.26 0.47 0.00 SE 0.14 0.23 0.67 1.30 13.40 0.04 0.00 Mean 0.43 0.34 SE 0.00 0.17

Notes: This Table reports results of the out of sample Montecarlo experiment. We randomly split our baseline sample into two groups 100 times. For each iteration, we obtain a pair of randomly selected samples with no overlap of pairs. Panel A reports the average portfolio discounts and excess returns along the corresponding standard errors. We report averages with respect time and number of iterations; standard errors correspond to the standard deviations of the average portfolio discounts and returns with respect to the number of iteration. For each iteration, we use these portfolio excess returns as test assets and build CarryBtc from the portfolios formed with the second and non-overlapping, group of pairs. Panel B reports the estimates for the risk prices for the GMM and FMB asset pricing procedures on the ﬁve bitcoin portfolios. Standard errors are computed over the 100 iterations. Market prices of risk λ, the adjusted R2, the square-root of the mean-squared errors RMSE and the p-values of χ2 tests on pricing errors are reported in percentage points. b denotes the vector of factor loadings. All excess returns are multiplied by 100. Shanken(1992)-corrected standard errors are reported in parentheses. We do not include a constant in the second step of the FMB procedure. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

signiﬁcant for all four speciﬁcations (the shaded area corresponds to the 95 percent interval), and weakly increasing in the number of portfolios.

F.VI Time-Varying Risk In order to analyze conditional risk premia we report results with managed portfolios. Investors can adjust their position in a given bitcoin portfolio based on the state of economic conditions, proxied by the lag standardized values of the VIX volatility index. We assume that such managed investment strategies capture the cross-section of conditional expected excess returns in addition to the raw bitcoin cross returns (Cochrane, 2009). To construct the managed portfolios, we multiply each portfolio excess returns by the beginning-of-month value of the VIX. We use the same estimation technique and risk factors presented in section3 on the augmented set of test assets. Table A11 reports the results. The estimate of the CarryBtc risk price is unchanged with respect to the baseline estimation and statistically signiﬁcant. Interestingly, the estimate for

λCarryBtc∗V ix is positive and signiﬁcant. This indicates that the market price of risk is time- varying, and in particular higher when the Vix index is high, i.e., in bad times for ﬁnancial markets.

78 Figure A13: Number of Portfolios and the Cross-Section of Portfolio Returns

Notes: This Figure plots the mean (gross) long-short excess returns for both the cross (top panel) and within returns in case of 3, 5, 7 (baseline) and 10 portfolios. Long-short excess returns correspond to last-ﬁrst portfolio in the case of the cross strategy, and to ﬁrst-last in the case of the within strategy. The shaded area corresponds to the 95 percent interval. Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

F.VII Sample Since 2018

In this Section we present the results of the asset pricing estimation on a shorter sample, that starts in January 2018, after the bitcoin frenzy of 2017. Table A12 reports our results. We ﬁnd that the estimate for the market price of the CarryBtc risk is positive and statistically signiﬁcant, and the no arbitrage conditions are not violated. To sum up, the estimation on the shorter sample produces results that are similar to those we obtain in the longer sample.

G Costly Arbitrage Returns

In this paper, we show that investors could potentially earn positive returns, after transaction costs, by trading on the base of the information contained in bitcoin discounts. We propose a risk-based explanation of bitcoin discounts. In this Section, we consider an alternative explanation based on the idea that arbitrageurs do not fully eliminate these discounts because of idiosyncratic risks. Following Pontiff(1996, 2006), we first show that a measure of idiosyncratic risk is positively related to the absolute size of discounts; second, we show that idiosyncratic risk is related to exchange-specific liquidity and execution risks. However, as explained in our paper, pure arbitrage strategies are not possible because of various frictions that impede them.

79 Table A11: Asset Pricing: US Investor (Conditional)

Panel I: Risk Prices 2 2 λBtc λCarryBtc λCarryBtc∗V ix bBtc bCarryBtc bCarryBtc∗V ix R RMSE χ (%)

GMM1 0.27 1.64 5.95 0.87 -4.35 10.80 85.51 0.40 [ 0.19 ] [ 0.20 ] [ 0.95 ] [ 0.89 ] [ 23.65 ] [ 7.69 ] 0.00

GMM2 0.15 1.83 6.63 0.22 -3.12 11.56 98.58 0.50 [ 0.16 ] [ 0.17 ] [ 0.65 ] [ 0.75 ] [ 8.85 ] [ 3.08 ] 0.00 FMB 0.27 1.64 5.95 0.87 -4.39 10.80 84.36 0.40 [ 0.15 ] [ 0.08 ] [ 0.28 ] [ 0.72 ] [ 9.30 ] [ 2.72 ] 0.00 Mean 0.27 0.68 1.80 s.e. [ 0.14 ] [ 0.20 ] [ 0.60 ] Panel II: Factor Betas Portfolio αk βk βj βj R2 χ2(α) p-value (%) 0 Btc CarryBtc CarryBtc∗V ix 1 -0.47 0.94 -0.24 -0.02 90.17 [ 0.07 ] [ 0.01 ] [ 0.11 ] [ 0.04 ] 2 -0.12 0.98 0.05 -0.03 96.92 [ 0.03 ] [ 0.01 ] [ 0.06 ] [ 0.01 ] 3 -0.02 0.98 0.07 -0.01 96.16 [ 0.04 ] [ 0.01 ] [ 0.07 ] [ 0.02 ] 4 0.11 0.98 0.11 -0.04 96.57 [ 0.04 ] [ 0.01 ] [ 0.06 ] [ 0.02 ] 5 0.16 0.97 0.35 -0.08 95.67 [ 0.05 ] [ 0.01 ] [ 0.10 ] [ 0.03 ] 6 0.33 0.97 0.53 -0.11 94.13 [ 0.06 ] [ 0.01 ] [ 0.18 ] [ 0.04 ] 7 0.60 0.95 0.79 -0.07 94.17 [ 0.07 ] [ 0.01 ] [ 0.16 ] [ 0.05 ] 8 -1.98 3.28 0.01 -0.31 78.49 [ 0.27 ] [ 0.17 ] [ 0.47 ] [ 0.15 ] 9 -0.58 3.42 0.03 -0.07 84.13 [ 0.21 ] [ 0.18 ] [ 0.38 ] [ 0.11 ] 10 -0.28 3.42 0.01 0.00 83.90 [ 0.25 ] [ 0.15 ] [ 0.42 ] [ 0.13 ] 11 0.12 3.44 0.18 -0.09 83.82 [ 0.21 ] [ 0.18 ] [ 0.38 ] [ 0.12 ] 12 0.30 3.40 0.44 -0.10 82.79 [ 0.23 ] [ 0.17 ] [ 0.44 ] [ 0.14 ] 13 1.02 3.38 0.59 -0.06 83.49 [ 0.23 ] [ 0.15 ] [ 0.51 ] [ 0.15 ] 14 1.89 3.32 -0.07 0.51 82.47 [ 0.26 ] [ 0.16 ] [ 0.55 ] [ 0.18 ] 15 0.00 -0.00 1.00 -0.00 100.00 [ 0.00 ] [ 0.00 ] [ 0.00 ] [ 0.00 ] All 630.64 0.00

Notes: Panel I reports results from GMM and Fama and MacBeth(1973) asset pricing procedures on the seven bitcoin portfolios and seven managed portfolios sorted with respect to bitcoin discounts. Managed portfolios are obtained by multiplying the original test assets by the conditioning factor (i.e., the lagged standardized VIX index). Market prices of risk λ, the adjusted R2, the square-root of the mean-squared errors RMSE and the p-values of χ2 tests on pricing errors are reported in percentage points. b denotes the vector of factor loadings. All excess returns are multiplied by 100. Shanken(1992)-corrected standard errors are reported in parentheses. We do not include a constant in the second step of the FMB procedure. Panel II reports OLS estimates of the factor betas. R2s and p-values are reported in percentage points. The standard errors in brackets are Newey and West(1986) standard errors computed with the optimal number of lags according to Andrews(1991). The χ2 test 0 −1 statistic α Vα α tests the null that all intercepts are jointly zero. This statistic is constructed from the Newey and West(1986) variance-covariance matrix (1 lag) for the system of equations (see Cochrane(2009)). Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 5/26/2015–10/31/2019.

80 Table A12: Asset Pricing: US investor (Sample since 1/2018)

Panel I: Risk Prices 2 2 λBtc λCarryBtc bBtc bCarryBtc R RMSE χ (%)

GMM1 -0.19 1.11 -0.51 40.60 73.80 0.26 [ 0.28 ] [ 0.20 ] [ 1.19 ] [ 7.19 ] 0.00

GMM2 -0.08 0.75 -0.11 27.53 94.89 0.31 [ 0.28 ] [ 0.17 ] [ 1.18 ] [ 6.10 ] 0.00 FMB -0.19 1.11 -0.51 40.44 69.55 0.26 [ 0.23 ] [ 0.08 ] [ 0.96 ] [ 2.85 ] 0.00 Mean -0.04 0.79 s.e. [ 0.22 ] [ 0.23 ] Panel II: Factor Betas Portfolio αk βk βk R2 χ2(α) p-value (%) 0 Btc CarryBtc 1 -0.78 0.94 -0.23 95.57 [ 0.08 ] [ 0.01 ] [ 0.11 ] 2 -0.22 0.99 -0.03 99.21 [ 0.03 ] [ 0.01 ] [ 0.01 ] 3 -0.11 0.99 -0.02 99.45 [ 0.02 ] [ 0.01 ] [ 0.02 ] 4 -0.05 0.98 0.01 99.31 [ 0.02 ] [ 0.01 ] [ 0.02 ] 5 -0.05 0.98 0.03 99.02 [ 0.02 ] [ 0.01 ] [ 0.02 ] 6 0.10 0.97 0.18 97.99 [ 0.04 ] [ 0.01 ] [ 0.05 ] 7 0.32 0.95 0.66 96.58 [ 0.08 ] [ 0.01 ] [ 0.12 ] 8 -0.00 -0.00 1.00 100.00 [ 0.00 ] [ 0.00 ] [ 0.00 ] All 460.69 0.00

Notes: Panel I reports results from GMM and Fama and MacBeth(1973) asset pricing procedures on the seven bitcoin portfolios sorted with respect to bitcoin discounts. Market prices of risk λ, the adjusted R2, the square-root of the mean-squared errors RMSE and the p-values of χ2 tests on pricing errors are reported in percentage points. b denotes the vector of factor loadings. All excess returns are multiplied by 100. Shanken(1992)-corrected standard errors are reported in parentheses. We do not include a constant in the second step of the FMB procedure. Panel II reports OLS estimates of the factor betas. R2s and p- values are reported in percentage points. The standard errors in brackets are Newey and West(1986) standard errors computed 2 0 −1 with the optimal number of lags according to Andrews(1991). The χ test statistic α Vα α tests the null that all intercepts are jointly zero. This statistic is constructed from the Newey and West(1986) variance-covariance matrix (1 lag) for the system of equations (see Cochrane(2009)). Data are daily, from cryptocompare.com and Thomson Reuters. The sample period is 1/1/2018–10/31/2019 and contains 940 daily observations per test asset.

G.I Idiosyncratic Risk and Bitcoin Discounts

Pontiﬀ(2006) argues that we observe mispricing in equilibrium, in the presence of rational agents, when arbitrage is costly. By accounting for all the transaction costs, we have already taken into account one of the main cost that can possibly prevent rational traders from completely eliminating mispricing. Holding costs are additional costs that are incurred every period that a position is open. For example, holding costs include the opportunity cost of

81 capital and the idiosyncratic risk exposure. We follow the empirical methodology proposed by Pontiff(1996), and first calculate the idiosyncratic risk for each portfolio by controlling for the systematic exposure15. In particular, we consider the factor model proposed by Liu et al. (2019) for cryptocurrencies. Then, we test whether there is a positive relationship between the magnitude of idiosyncratic risk for each portfolio and the magnitude of its price deviation. Liu et al.(2019) consider a large set of cryptocurrrencies (i.e., all the coins with market capitalization above one million dollars), and show that a three-factor model explains the cross-section of returns for various portfolios sorted by, for example, size, value and momentum. The three factors are a crypto market (CMKT ), size (CSIZE), and momentum (CMOM) factors. In the first step of our empirical analysis we estimate portfolio idiosyncratic risk form the residuals of the following time-series regressions16:

j j j j j j Rt,t+1 = α + βCMKT CMKTt,t+1 + βCSIZECSIZEt,t+1 + βCMOM CMOMt,t+1 + t,t+1 (A1)

j where j = 1,..., 7 and Rt,t+1 are either cross or within portfolio returns. Panel A of Table A13 reports the results of these regressions. For both cross and within portfolio returns, the R- squares of the three-factor model of Liu et al.(2019) are similar and approximately equal to 70%. Most of the variation in returns is explained by the market factor, or CMKT , while the coefficients associated to the size and momentum factors are mostly not statistically different from zero. In fact, all portfolios load similarly on the market factor that, thus, explains a large fraction of the time-series variation but cannot explain any of the cross-sectional differences in portfolio returns. This is why the estimates for the intercepts, α, are mostly significant and increasing in the cross-section of portfolio returns. For each portfolio, we then measure idiosyncratic risk as the volatility of the estimated residuals from the factor regressions (i.e., σ(ˆj)). In the second step of our empirical analysis, we relate portfolio idiosyncratic risks to bitcoin discounts by estimating cross-sectional regressions in the spirit of Fama and MacBeth (1973). Specifically, for each period t we estimate:

|Dt| = ασ(),t + βσ(),tσ(ˆ) + σ(),t (A2) where |Dt| is a vector containing the absolute values of the discounts of each portfolio at time t, and then calculate the average of the regression coefficients. We report our results in Panel B of of Table A13. Specifically, we report the average intercept (ασ() = E(ασ(),t)) and slope (βσ() = E(βσ(),t)) coefficients from the cross-sectional regressions, along standard errors corrected for heteroschedasticity and autocorrelation. Our results indicate that, for both cross and within returns, portfolio idiosyncratic risk is positively and significantly related to bitcoin discounts: i.e., portfolios with higher idiosyncratic risk are also those with the largest price deviations. Because Liu et al.(2019)’s factor are constructed starting from a large

15We typically think that investors can diversify idiosyncratic risk by building portfolios. However, in the presence of mispricing, as explained by Pontiﬀ(2006), idiosyncratic risk limits investors’ position in the mispriced asset because investors cannot share the risk exposure even in the presence of multiple investment opportunities. Similarly, Tuckman and Vila(1992) argue that idiosyncratic risks cause even perfectly hedged arbitrage positions to be risky. 16We are grateful to Yukun Liu for sharing the data on the three factors, which are only available at weekly frequency. We refer to Liu et al.(2019) for details on the construction of the three factors and their interpretation.

82 number of coins, while our portfolios contains only bitcoin-to-fiat pairs, CMKT may not accurately capture the market factor. For this reason, we also estimate a factor model where we replace CMKT with the log bitcoin returns (or, BTC factor), while keeping the CSIZE and CMOM factors unchanged. We report results for this alternative specification in Panel C of Table A13. The coefficient βσ() is positive and highly significant, though smaller in magnitude, thus confirming that portfolios with higher idiosyncratic risk are also those with the largest price deviations.

Table A13: Costly Arbitrage

Panel A: Time-Series Regressions cross returns within returns 2 2 Portfolio α (%) βCMKT βCSIZE βCMOM R (%) α (%) βCMKT βCSIZE βCMOM R (%) 1 0.107 0.765 -0.078 -0.004 70.162 1.745 0.766 -0.067 -0.003 69.824 [ 0.349 ] [ 0.049 ] [ 0.040 ] [ 0.046 ] [ 0.353 ] [ 0.046 ] [ 0.038 ] [ 0.047 ] 2 0.788 0.798 -0.090 0.004 71.077 1.357 0.799 -0.094 0.002 71.614 [ 0.332 ] [ 0.057 ] [ 0.048 ] [ 0.047 ] [ 0.327 ] [ 0.055 ] [ 0.047 ] [ 0.045 ] 3 0.787 0.779 -0.077 0.003 72.102 1.046 0.774 -0.085 -0.001 71.553 [ 0.323 ] [ 0.050 ] [ 0.045 ] [ 0.045 ] [ 0.324 ] [ 0.049 ] [ 0.046 ] [ 0.044 ] 4 0.806 0.783 -0.073 0.009 71.161 0.923 0.771 -0.087 0.005 70.425 [ 0.332 ] [ 0.052 ] [ 0.043 ] [ 0.046 ] [ 0.318 ] [ 0.050 ] [ 0.047 ] [ 0.045 ] 5 1.048 0.785 -0.060 0.003 71.450 0.940 0.771 -0.083 -0.002 69.937 [ 0.335 ] [ 0.051 ] [ 0.036 ] [ 0.049 ] [ 0.328 ] [ 0.050 ] [ 0.044 ] [ 0.048 ] 6 1.084 0.797 -0.060 0.019 72.496 0.679 0.774 -0.088 0.015 70.336 [ 0.297 ] [ 0.051 ] [ 0.037 ] [ 0.044 ] [ 0.297 ] [ 0.050 ] [ 0.047 ] [ 0.044 ] 7 1.445 0.793 -0.054 0.007 70.707 0.152 0.760 -0.097 0.007 67.943 [ 0.373 ] [ 0.055 ] [ 0.034 ] [ 0.049 ] [ 0.358 ] [ 0.054 ] [ 0.047 ] [ 0.049 ] Panel B: Cross-Sectional Regressions cross returns within returns 2 2 ασ() βσ() R (%) ασ() βσ() R (%) -0.177 3.162 12.834 -0.258 4.489 67.107 [ 0.017 ] [ 0.298 ] [ 0.030 ] [ 0.514 ] Panel C: Cross-Sectional Regressions (residuals using BTC factor) cross returns within returns 2 2 ασ() βσ() R (%) ασ() βσ() R (%) -0.006 1.024 27.262 -0.006 0.971 32.124 [ 0.001 ] [ 0.092 ] [ 0.001 ] [ 0.087 ]

Notes: Panel A of the Table reports estimates from time-series regressions of portfolio cross and within returns on the three-factor model of Liu et al.(2019) (see equation (A1)). For each portfolio, we report the estimates for the intercept ( α) and the slope coefficients (β) for the three cryptocurrency factors: the market (CMKT ), size (CSIZE), and momentum (CMOM) factors. In brackets we report Newey and West(1986) standard errors computed with the optimal number of lags according to Andrews (1991). Panel B of the Table reports, for both cross and within portfolio returns, the average intercept and slope coefficients of cross-sectional regressions that relate the (absolute) portfolio discounts to the corresponding portfolio idiosyncratic risk (see equation (A2)). We proxy idiosyncratic risk with the standard deviation of the residuals from the factor regressions (σ(ˆ)). Panel C of the Table also reports average intercept and slope coefficients of cross-sectional regressions. In this case, we estimate idiosyncratic risk from the residuals from a factor model where we replace CMKT with the log bitcoin returns (or, BTC factor). In brackets we report Newey and West(1986) standard errors corrected for heteroschedasticity and autocorrelation. R2 denotes the adjusted R-square. Data are weekly for the period May 27, 2015 to November 11, 2019.

83 G.II Idiosyncratic Risk and Portfolio Characteristics In order to understand the determinants of portfolio idiosyncratic risks, for each portfolio independently and at the weekly frequency, we estimate time-series regressions. The dependent variable is the squared deviation, with respect to the mean of the residuals from the estimation of the three-factor model of Liu et al.(2019) ([ ˆj − E(ˆj)]2). The independent variables are the log bitcoin returns (Btc); the (one-period) lag of the dependent variable; and a set of portfolio-specific characteristics. Specifically, we consider liquidity and counterparty risks and capital market factors. We capture liquidity risks by the mean trading volume (V olume); bid/ask spread (Bid/Ask) and high/low spread (High/Low). We capture counterparty risks by wallet size (W allets) and exchange rating (Ratings). Finally, the capital market factors include a capital control index for each exchange location (K − Controls); and exchange rate growth (∆FX). Table A14 presents our results. First, portfolios with higher discounts contain more idiosyncratic risk. For example, for cross (within) portfolios, the idiosyncratic risk volatility is 6.02% (6.62%) for portfolio 7, while it is equal to 5.87% (5.97%) for the first portfolio. Recall that arbitrageurs should go long the cross portfolio 7 and the within portfolio 1. We find that exactly for these portfolios, a sizeable fraction of the time-series variation in idiosyncratic risk is explained by portfolio-specific characteristics. This is exemplified by the comparison between the R˜2 reported at the bottom of the table, which corresponds to the R-square from a regression with only Btc and the lag idiosyncratic risk as covariates, and the R2 for the regressions including all the covariates. While across all portfolios, portfolio- specific characteristics, on average, explain an additional 10-15% of the time-series variation, for the cross portfolio 7 (the within portfolio 1), the additional contribution of portfolio- specific characteristics is larger and approximately equal to 20%. Third, we find that portfolio- specific-liquidity, in the form of the intraday bitcoin volatility (i.e., high/low spreads), is significantly associated with higher idiosyncratic risk for both cross and within portfolios. One interpretation of this result is related to the execution risk faced by arbitrageurs, i.e., the risk that a transaction is not executed at the desired price because of the large intraday volatility coupled with the uncertainty in the timing of the execution of a trade. In contrast, we find that the contribution of the remaining portfolio-specific characteristics is not statistically significant.

84 Table A14: Idiosyncratic Risk and Portfolio Characteristics

Panel A: cross returns Panel B: within returns Portfolio 1234567 1234567 α 0.836 -0.565 0.140 1.060 0.683 0.598 0.222 0.900 -0.731 -0.029 0.612 0.877 1.061 0.169 [ 0.500 ] [ 0.667 ] [ 0.709 ] [ 0.662 ] [ 0.748 ] [ 0.676 ] [ 0.692 ] [ 0.528 ] [ 0.725 ] [ 0.737 ] [ 0.776 ] [ 0.809 ] [ 0.816 ] [ 0.692 ] Btc 0.017 0.027 0.018 0.021 0.011 0.034 0.036 0.019 0.026 0.021 0.020 0.008 0.035 0.029 [ 0.009 ] [ 0.011 ] [ 0.009 ] [ 0.012 ] [ 0.006 ] [ 0.020 ] [ 0.014 ] [ 0.008 ] [ 0.010 ] [ 0.011 ] [ 0.014 ] [ 0.008 ] [ 0.021 ] [ 0.017 ] V olume -0.029 -0.031 -0.017 0.014 -0.076 -0.003 -0.047 -0.029 -0.045 -0.002 0.027 -0.083 0.012 -0.014 [ 0.035 ] [ 0.044 ] [ 0.057 ] [ 0.036 ] [ 0.054 ] [ 0.039 ] [ 0.052 ] [ 0.037 ] [ 0.046 ] [ 0.074 ] [ 0.038 ] [ 0.056 ] [ 0.044 ] [ 0.068 ] Bid/Ask -0.064 0.067 -0.192 0.033 -0.061 -0.036 -0.024 -0.042 0.040 -0.240 0.038 -0.038 -0.027 0.092 [ 0.050 ] [ 0.073 ] [ 0.138 ] [ 0.032 ] [ 0.079 ] [ 0.083 ] [ 0.020 ] [ 0.041 ] [ 0.067 ] [ 0.147 ] [ 0.036 ] [ 0.078 ] [ 0.086 ] [ 0.060 ] High/Low 0.053 0.064 0.054 0.053 0.043 0.074 0.074 0.055 0.065 0.063 0.055 0.041 0.072 0.060 [ 0.021 ] [ 0.023 ] [ 0.020 ] [ 0.021 ] [ 0.018 ] [ 0.032 ] [ 0.024 ] [ 0.020 ] [ 0.024 ] [ 0.022 ] [ 0.026 ] [ 0.019 ] [ 0.036 ] [ 0.027 ] W allets -0.010 -0.014 0.007 0.020 0.021 0.003 0.007 -0.011 -0.015 0.007 0.024 0.030 0.004 0.031 [ 0.012 ] [ 0.013 ] [ 0.015 ] [ 0.013 ] [ 0.015 ] [ 0.018 ] [ 0.014 ] [ 0.013 ] [ 0.014 ] [ 0.016 ] [ 0.015 ] [ 0.018 ] [ 0.019 ] [ 0.024 ] Ratings -0.011 0.014 -0.001 -0.024 -0.003 -0.015 -0.001 -0.012 0.019 -0.001 -0.019 -0.008 -0.025 -0.008 [ 0.008 ] [ 0.011 ] [ 0.012 ] [ 0.010 ] [ 0.010 ] [ 0.012 ] [ 0.014 ] [ 0.009 ] [ 0.013 ] [ 0.013 ] [ 0.011 ] [ 0.011 ] [ 0.015 ] [ 0.016 ] K − Controls -0.406 0.550 -0.511 -1.516 -0.959 -1.108 -0.788 -0.555 0.796 -0.496 -1.278 -0.894 -1.610 -1.112 [ 0.794 ] [ 0.724 ] [ 0.638 ] [ 0.500 ] [ 0.663 ] [ 1.204 ] [ 0.694 ] [ 0.860 ] [ 0.729 ] [ 0.666 ] [ 0.553 ] [ 0.716 ] [ 1.229 ] [ 0.679 ] ∆FX -0.093 0.160 0.254 0.298 0.020 0.255 0.193 -0.133 0.179 0.273 0.277 -0.045 0.134 0.169 [ 0.154 ] [ 0.168 ] [ 0.153 ] [ 0.149 ] [ 0.133 ] [ 0.176 ] [ 0.097 ] [ 0.160 ] [ 0.180 ] [ 0.164 ] [ 0.158 ] [ 0.133 ] [ 0.185 ] [ 0.128 ] Lag 0.390 0.114 0.379 0.325 0.265 0.278 0.242 0.314 0.242 0.313 0.298 0.282 0.285 0.275 [ 0.124 ] [ 0.109 ] [ 0.121 ] [ 0.096 ] [ 0.071 ] [ 0.104 ] [ 0.098 ] [ 0.149 ] [ 0.129 ] [ 0.110 ] [ 0.106 ] [ 0.078 ] [ 0.079 ] [ 0.108 ] R2 adj. (%) 42.765 35.425 46.866 43.687 31.171 48.370 45.820 41.188 35.235 41.831 36.552 30.301 46.577 37.524 E([ˆj − E(ˆj)]2) 0.344 0.359 0.325 0.345 0.341 0.336 0.361 0.356 0.353 0.328 0.358 0.358 0.363 0.437 σ([ˆj − E(ˆj)]2) 5.876 6.009 5.717 5.886 5.855 5.807 6.023 5.978 5.954 5.739 5.997 5.995 6.041 6.627 R˜2 adj. (%) 35.637 24.185 33.836 30.664 32.560 28.773 24.969 23.894 28.087 28.292 26.166 31.067 27.236 25.286

Notes: This Table presents the results of time-series regressions, for each portfolio and at the weekly frequency. The dependent variable is the squared deviation from the mean of the residuals from the estimation of the three-factor model of Liu et al. (2019) ([ˆj − E(ˆj )]2). The independent variables are the log bitcoin returns (Btc); the (one-period) lag of the dependent variable; and a set of portfolio-speciﬁc characteristics: the mean trading volume (V olume); bid/ask spread (Bid/Ask); high/low spread (High/Low); wallet size (W allets); exchange rating (Ratings); capital control index (K − Controls); and exchange rate growth (∆FX). The last three rows of the Table, for both cross and within portfolio returns, report the unconditional mean and standard deviation of the dependent variables (E([ˆj − E(ˆj )]2)) and the adjusted R-squares from regressions that include only Btc and the lag of the dependent variable as covariates. Data are weekly from cryptocompare.com, Thomson Reuters, Bloomberg, Google, and the Kenneth French data library for the period 5/26/2015–10/31/2019.