Econometric Modeling: Limits to Arbitrage as an Explanation of Cryptocurrency Returns
Abstract
The emergence of cryptocurrency as an alternative asset has fueled research on deciphering what factors best explain cryptocurrency returns. Scholars find that the financial intuition behind risk factors in the traditional stock market are, in some cases, transferable to the cryptocurrency market. Earlier work finds evidence for a CAPM and size explanations of cryptocurrency returns, factors similar to Sharpe (1964) and Fama and French (1993) findings in the stock market. That aside, Köchling, Müller, and Posch (2019) find mispriced cryptocurrencies across different markets which inefficiencies are not arbitraged away by investors due to limits to arbitrage that render arbitrage investment positions highly risky. By using Amihud’s illiquidity framework and idiosyncratic volatility to proxy for these limits to arbitrage, this paper finds that assets with higher arbitrage risk outperform those with lower arbitrage risk in a statistically significant manner. As such, this paper contributes to the literature by offering an explanation to part of the stochastic disturbance (error term) witnessed in earlier cryptocurrency multi-linear factor regression models.
Introduction
The concept of Blockchain as introduced by Nakamoto (2008) gave birth to Bitcoin, the pioneer cryptocurrency. Through Bitcoin’s decentralization of finance gospel, right after the 2008 financial crisis when confidence in financial institutions underwent erosion, the digital asset has emerged to the spotlight within finance corridors. Bitcoin aside, there has been an emergence of alternative cryptocurrencies tradeable in a fashion somewhat akin to stocks. This novel asset class, cryptocurrency, has in turn created a labyrinth around what drives its return, a maze many a scholar are committed to deciphering. It is postulated that financial intuition behind asset pricing models in the stock market are transferable to the cryptocurrency market. For example, a cryptocurrency CAPM factor, which is the beta (\(/beta\)) exposure of a coin to the market is widely appreciated, a financial conclusion drawn by Sharpe (1964) in the stock market. Also, Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020) find a cryptocurrency size factor where the small coins tend to outperform big coins, which mirrors earlier work by Fama and French (1993). Despite the success in establishing cryptocurrency models, however, there is still disagreement on incorporating the factors of momentum or reversals. Unlike Liu, Tsyvinski, and Wu (2019), Shen, Urquhart, and Wang (2020) guide that reversals and not momentum best explain cryptocurrency returns. Shen, Urquhart, and Wang (2020)’s work builds on earlier work by Grobys and Sapkota (2019) who find that momentum returns negative payoffs in the short run.
Therefore, the task of identifying cryptocurrency factor models is actively ongoing as the market evolves into a mainstream one. As such, this paper will focus on proposing the ease of arbitrage trading as a factor that determines cryptocurrency returns. Earlier work by Köchling, Müller, and Posch (2019) find price mismatches in the cryptocurrency market thus availing evidence for arbitrage opportunities. However, Makarov and Schoar (2018) find that these arbitrage opportunities tend to be shunned by arbitrageurs in the cryptocurrency market, a fact which aligns to earlier findings in the stock market by Shleifer and Vishny (1997). This is so because of high volatility, which risks pushing prices further away from the arbitrageurs target price; illiquidity, which poses a risk of spillage thus reducing arbitrage profits; and the presence of unsophisticated investors who subject the market to noisy trading which lessens the possibility of prices conforming to a “fundamental value.” These scenarios in the cryptocurrency market create limits to arbitrage trading. Borrowing from Markowitz (1952)’s modern portfolio theory which conceptualizes the relationship between high risk and high return, the high risk associated with arbitrage trading effectively creates a possibility of a high return event in the cryptocurrency market. By using Amihud (2002)’s illiquidity framework and idiosyncratic volatility to proxy for limits to arbitrage,as previously employed by Lam and Wei (2011) in the stock market, this paper finds that cryptocurrencies with a higher limits to arbitrage measure outperform those with a lower limits to arbitrage measure in a statistically significant manner. The paper goes ahead to find that the limits to arbitrage factor is not subsumed by any earlier cryptocurrency factors of CAPM, size, momentum or reversals. Therefore, this paper contributes to the literature by offering an explanation to part of the error term in earlier cryptocurrency linear regression models.
2. LITERATURE REVIEW
In a bid to explain stock returns, Sharpe (1964) introduced the CAPM as an explanatory regression model of a stocks’ return. The CAPM model dictates that the expected return of a stock is explained by its beta relationship with the broader market return, in addition to the risk-free rate. This in effect, aligns with the MSCI Barra Factor Indexes methodology (2008) work which guided that the risk of a given market portfolio is partly related to the systematic risk exposure of the market.
\[E(ri)=Rf+\beta(E(rm)-Rf)\]
While the CAPM linear model was widely adopted by the finance world, researchers after William Sharpe pointed out the CAPM model’s weakness in explaining the stochastic disturbance component (error term) when a return fails to align to the CAPM regression line. The MSCI Barra Factor Indexes methodology (2008), also notes that risk of an asset possesses not only a systematic component but also some idiosyncratic - residual risk components. As a result, analysis to decipher the unexplained variables that contribute to the return of a stock have dominated contemporary financial research. Fama and French (1993) introduced the 3 factor model that builds onto Sharpe’s earlier work by adding 2 more explanatory variables of size (SMB) and value (HML). The size component demonstrates that small firms tend to outperform big firms which is in line with earlier findings of Banz (1981) which demonstrated the existence of a size effect in stock returns. The value component guides that value stocks outperform growth stocks. In later works, Fama and French (2015) build onto the 3 factor model with an additional two factors of profitability (RMW) and investment (CMA). Work by Fama and French (2015) produced the five factor model broadly used in finance today.
\[E(ri) = a + \beta_0(E(rm) - Rf) + \beta_1 smb + \beta HML + \beta_2 RMW + \beta_3 CMA + e\]
That aside, Carhart (1997) also unearthed the momentum effect whereby winners outperform losers (WML). The MSCI Barra Factor Indexes methodology (2008)’s work on decomposition of risks points out that there is existence of other common risk factors and these can be based on specific geographies or industries. Iwatsubo, Watkins, and Hiroki (2021), for example, introduce production as a common risk factor amongst Japanese companies. Factor modeling in traditional finance has in effect induced similar analysis of risk factors associated with the advent of cryptocurrencies.
Research on cryptocurrency factors has been prompted by the fact that existing factors, as presented by Fama and French (2015), fail to explain returns of cryptocurrencies. By regressing top cryptocurrencies against Fama and French (2015) five factor model, Gregoriou (2019) finds that abnormal returns from cryptocurrency persist indicating the inability of Fama’s models in explaining cryptocurrency returns. Additionally, Liu, Tsyvinski, and Wu (2019) also note that Fama and French (1993) 3 Fama and French (2015) and Carhart (1997) 4 factor model do not explain the cross-section of cryptocurrency returns. By borrowing from Fama-French’s theory, Liu, Tsyvinski, and Wu (2019), Shen, Urquhart, and Wang (2020) construct factor models specific to cryptocurrencies by conducting Fama and MacBeth Regression Analysis. Sovbetov and Sovbetov (2018) also employs the Autoregressive Distributed Lag cointegration framework (ARDL) on five top coins to unearth market beta, trading volume, volatility as significant determinants of the top 5 cryptocurrency prices. It is further postulated that the financial intuition and rationale behind the risk factors of traditional stocks is transferable to cryptocurrencies.
a) Size Effect in Cryptocurrency market:Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020) find that there exists a size factor in cryptocurrency. Both Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020) find statistically significant results on the size factor when market capitalisation is used as a proxy for size. While market capitalisation of stocks is determined from shares issued and share price, a cryptocurrency’s market capitalisation stems from the number of coins in supply and the price per coin. \[Coin.Mkt.Cap= Coins. in .Supply * Coin. Price\] The literature guides that small coins (small market capitalisation) tend to outperform bigger coins (large market capitalisation). The rationale behind this is that investors seek higher premiums for holding more risky small coins.
b) Momentum vs Reversal Effect in Cryptocurrency market:The point of divergence between Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020) is the momentum-reversal factor. Shen, Urquhart, and Wang (2020) show statistically significant evidence that momentum returns are negative on a buy winner-sell loser portfolio if the portfolio is held for 1,2,3 or 4 weeks, thus advocating for a reversal factor. Their findings are supported by earlier work from Grobys and Sapkota (2019) who find negative returns on momentum strategies in the short-run for “proof of work” cryptocurrencies. Upon inspiration from Carhart (1997)’s findings of momentum in traditional stocks, Liu, Tsyvinski, and Wu (2019) point to a momentum factor in cryptocurrency. As such, it is evident that there is disagreement on having reversals or momentum as a cryptocurrency factor. Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020) pick data from the same source (coinmarketcap) and with similar evaluation periods. So how is it possible that both of their papers find statistically significant evidence for completely different factors? It is probable that the strategies employed in portfolio selection differ thus causing a divergence in conclusions. For example, while Shen, Urquhart, and Wang (2020) form different portfolios of winners and losers from 4 weeks, 3 weeks, 2 weeks and 1 week ago, Liu, Tsyvinski, and Wu (2019) form portfolios of winners and losers from only 1 week ago. The difference in formulation periods in the two papers is an ingredient for divergent conclusions.
c) Arbitrage
Sharpe & Alexandar (1990) define arbitrage as the simultaneous buying of a security in a given market and selling it in another market offering a higher price. Price anomalies in traditional stock markets, as demonstrated by Titman, Liu, Tsyvinski, and Wu (2019), exist because investors delay in incorporating all available information into a stock price. In theory whenever there is such a mismatch in prices, arbitrageurs exploit these inefficiencies to close the gap between prices among different markets thereby turning an easy profit. However, in practice, arbitrage opportunities are occasionally shunned by investors. Shleifer and Vishny (1997) show that arbitrageurs avoid particularly volatile assets even when there is opportunity for arbitrage because these positions can be risky if prices drift further away from their “fundamental value.” Therefore, limits to arbitrage in financial markets render arbitrage trading risky and prompt pricing anomalies among securities. Lam and Wei (2011) build on Shleifer and Vishny (1997)’s literature of mispriced assets to construct proxies for limits to arbitrage in the stock market. To measure limits to arbitrage in the stock market, Lam and Wei (2011) employ Idiosyncratic volatility, trading costs (Amihud (2002)’s illiquidity framework) and a count of sophisticated shareholders. Idiosyncratic volatility estimates the level of risk associated with an arbitrage position. High idiosyncratic volatility poses higher arbitrage risk which limits arbitrage trading. Amihud (2002)’s illiquidity theory represents the ease of executing a trade without spillage. A high illiquidity measure increases the cost of trading thus disincentivizing arbitrage positions. Shareholder sophistication is an indicator of how safe a security is from noisy traders. A security with few sophisticated traders is more prone to noise thus posing risk of pushing prices further away from the arbitrageurs target price.
Arbitrage in Cryptocurrency market:With this background from traditional finance, a question arises. Are there price mismatches with arbitrage opportunities in the Cryptocurrency market? In alignment with Titman, Liu, Tsyvinski, and Wu (2019)’s findings in traditional stocks, Köchling, Müller, and Posch (2019) find price delays to react to information and market frictions in cryptocurrency as major drivers of price mismatches in cryptocurrencies.
To showcase the price mismatch effect, this paper pulls Bitcoin-USD price quotes from two major Asian-based cryptocurrency exchanges, Binance and FTX. The timestamp on these price quotes is similar and based on the UTC timezone with a close price at 23:59:59 UTC. Maintaining a similar timezone is important to ensure that the price quotes are stamped at the same time to avoid a price mismatch due to difference in time zones.
On Graph 1 below, the absolute price differences from FTX and Binance is calculated over time.
Köchling, Müller, and Posch (2019) findings that price mismatches exist between different exchanges are unearthed in graph 1. Most variation between the price quotes happens in times of high volatility. On Graph 1, the period of March 2020 and September 2021 experienced the highest deviation in prices of Bitcoin on the two exchanges. These two periods are associated with the COVID stock market crash and the Chinese ban on Cryptocurrency trading respectively.
In addition, Makarov and Schoar (2018) ’s study on arbitrage in the cryptocurrency market finds large price mismatches that persist for several days to be more dominant between exchanges in different geographical locations.
To showcase the price mismatch effect based on different geographies, this paper pulls Bitcoin-USD price quotes from two major Asian cryptocurrency exchanges, Binance and FTX, and one USA based exchange called Gemini. The timestamp on price quotes for all these three exchanges is based on the UTC timezone with a close price at 23:59:59 UTC.
Graph 2 shows how the largest variation in prices is between USA and Asia based exchanges.This is so because, according to Makarov and Schoar (2018), capital controls slow down instant bulk cross border arbitrage trading. This, with the presence of high volatility akin to the cryptocurrency market,illiquidity on some exchanges and irrational noise trading activity, makes it hard for arbitrageurs to keep pace. Makarov and Schoar (2018)’s study on cryptocurrency arbitrage illustrate how volatility and liquidity are pivotal in arbitrage trading in the cryptocurrency, a phenomenal identical to Lam and Wei (2011) study on limits to arbitrage in the traditional stock market.
As such, to study limits to arbitrage in cryptocurrency, one ought to take liquidity and volatility into account. This paper sets idiosyncratic volatility and Amihud (2002)’s illiquidity measure as a proxy for limits to arbitrage in the cryptocurrency market. Lam and Wei (2011) uses idiosyncratic volatility and Amihud (2002)’s illiquidity as estimators of limits to arbitrage in the traditional stock market. The same estimators are used for this paper, but on cryptocurrency, because Makarov and Schoar (2018) demonstrates that volatility limits arbitrage trading and Brauneis et al. (2021) finds the Amihud (2002) measure as a good approximate of trading costs/liquidity in the cryptocurrency market. High trading costs (illiquidity) and excess volatility (idiosyncratic volatility) limit arbitrage trading.The presence of limits to arbitrage trading create a high risk event in the cryptocurrency market which ought to be matched with a high return event.This paper seeks to investigate whether limits to Arbitrage, as witnessed in traditional stocks markets, can explain cryptocurrency returns in a statistically significant manner.
3. DATA & VARIABLES
With the help of R programming language, this paper fetches OHLC, volume and Market Capitalisation data for 570 cryptocurrencies from Coingecko, one of the largest cryptocurrency data aggregators. The data is dated between 02 January 2015 to 02 October 2021.
|
Market.Cap
|
Volume
|
||||
|---|---|---|---|---|---|
| Years | N.o.of.Coins | Mean | Median | Mean.1 | Median.1 |
| 2015 | 79 | 57,051,277 | 256,875 | 3,624,888 | 1,851 |
| 2016 | 101 | 106,324,715 | 950,759 | 20,184,867 | 6,930 |
| 2017 | 159 | 913,367,865 | 38,183,133 | 29,335,807 | 1,037,413 |
| 2018 | 240 | 1,221,360,734 | 37,555,877 | 48,864,378 | 1,219,443 |
| 2019 | 331 | 620,038,392 | 7,181,843 | 131,901,220 | 1,093,830 |
| 2020 | 482 | 654,906,335 | 13,194,405 | 150,088,267 | 2,017,829 |
| 2021 (full) | 570 | 2,815,165,949 | 95,015,007 | 302,062,578 | 11,213,200 |
| Returns Data Panel | |||||
| Weekly Returns Data | Mean | Median | SD | Skewness | Kurtosis (excess) |
| Market | 0.045 | 0.022 | 0.23 | 8.3 | 21.4 |
| Bitcoin | 0.02 | 0.015 | 0.11 | 0.09 | 10.5 |
| Ethereum | 0.04 | 0.016 | 0.18 | 1.23 | 18.5 |
| Correlation (ρ) | Market | Bitcoin | Ethereum | ||
| Market | 1 | ||||
| Bitcoin | 0.42 | 1 | |||
| Ethereum | 0.29 | 0.47 | 1 | ||
Alexander and Dakos (2019) points out the limitations of choosing the right data to conduct cryptocurrency research. Given the open source and decentralised nature of the market, cryptocurrency time series and regression analysis is prone to inaccuracy due to the multiplicity of out of sync Cryptocurrency market indexes and prices with varying timestamps. Alexander and Dakos (2019) also points out inefficiency of using cryptocurrency data aggregators like Coingecko and Coinmarketcap. They state that data from these data aggregators fails at capturing real prices traded in the market because the data from these sites is non-traded data averaged from different exchanges thus skewing actual OHLC data in the market.
Despite that, this paper relies on Coingecko’s data which determines price data through weighting the global volume average price from a pool of different major cryptocurrencies exchanges. The timezone is based on UTC. Opening prices are determined at 00:00:00 UTC and close at 23:59:59 UTC.
\[ \sum_{i=exchange1}^{I} Vol_i * P_i \]
While the price quotes are non-traded data but approximations as noted by Alexander and Dakos (2019), using real traded data from a single exchange would further distort analysis because other exchanges that account for a significant portion trading activity in the markets would be excluded. In addition, accessing real traded data is herculean as some cryptocurrencies do not offer or restrict API access to their servers. Therefore, usage of data from a data aggregator like Coingecko or Coinmarketcap just like Liu, Tsyvinski, and Wu (2019), Sovbetov and Sovbetov (2018), Shen, Urquhart, and Wang (2020), and Brauneis et al. (2021) is the closest to a rational way forward.
4. METHEDOLOGY: Constructing the Factors
This section will reconstruct the cryptocurrency factors of the CAPM and size factor (SMB) as presented by Shen, Urquhart, and Wang (2020), and Liu, Tsyvinski, and Wu (2019). While Shen, Urquhart, and Wang (2020) find a reversal effect, Liu, Tsyvinski, and Wu (2019) find a momentum one, which points to a contradiction in the literature on whether reversal effects or momentum explain cryptocurrency returns. This section will mediate between the divergent findings on reversals or momentum by re-running both strategies employed by Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020). Thereafter, limits to arbitrage will be introduced as a potential factor that explains part of the stochastic disturbance witnessed in cryptocurrency linear models.
a) Market Factor Given the fact that there is no widely adopted market index in the cryptocurrency, published literature on cryptocurrency tends to develop a market index from their own data. This is in line with Liu, Tsyvinski, and Wu (2019), Shen, Urquhart, and Wang (2020) and Sovbetov and Sovbetov (2018), who constructed their own indexes. While there are some cryptocurrency indexes for example CCMIX, CRIX Index among others, these tend to be quoted at different time zones which do not align with the data sourced from Coingecko or Coinmarketcap and other cryptocurrency data aggregators. For accuracy, all data ought to be pulled from one source to avoid errors on modelling as pointed out by Alexander and Dakos (2019). For this paper, a market cap weighted index which is constructed from our 570 coins and is rebalanced on a weekly basis is used as the market index.
\[ Mkt.Ret_t = \sum_{i=1}^{n}ret_i* \frac{Mkt.Cap_i}{Total.Mkt.Cap_t} \] where Mkt.Ret is the market return at time (t), ret and Mkt.Cap is the return and market capitalisation of a given coin (i) and Total.Mkt.Cap is the total cryptocurrency market capitalisation for all coins (n)
b) Size Factor
To investigate the existence of a size effect in cryptocurrency, whether small assets outperform big assets, proxy variables to estimate size ought to be established. Liu, Tsyvinski, and Wu (2019) use cryptocurrency market capitalisation, price, and age to proxy for size. In this paper, the proxies of price and market capitalisation will be used as estimates of size. Age is dropped as a proxy given the fact that Liu, Tsyvinski, and Wu (2019) based the age variable on when assets were first listed on their data source (coinmarketcap), a decision which does not reflect the actual age of the assets under study. Assets are typically listed on data aggregator sites like Coingecko and coinmarketcap long after they are born. So this paper drops age as a proxy.
Equal weighted quintile portfolios are formed based on the proxies of market capitalisation and price. These portfolios are rebalanced weekly from January 4th 2015 to October 2nd 2021 to account for changes in market capitalisation and prices of the assets overtime. The lowest quintile (Q1) represents a portfolio with low market cap assets and low price (small). The highest quintile portfolio (Q5) represents high market cap and highly priced assets (big). The average return on the quintile portfolios is then calculated.
The null hypothesis of the size effect is as follows;
ho: There is no statistical difference between return on a small and a big portfolio.
The results produced, as shown in Table 2 below, indicate with 95% confidence that there is a statistical difference between returns on small and big assets when market capitalisation is used as a proxy. Therefore, null hypothesis (ho) is rejected.
| Quintiles | Q1 | Q2 | Q3 | Q4 | Q5 | Q1.Q5 |
|---|---|---|---|---|---|---|
| Mkt.Cap Proxy | Small | Big | SMB | |||
| 0.33** | 0.297** | 0.14*** | 0.05*** | 0.06*** | 0.27** | |
| (2.13) | (2.24) | (2.91) | (5.43) | (3.03) | (1.71) | |
| Price Proxy | Small | Big | SMB | |||
| 2.2 | 0.08 | 0.24 | 0.16*** | 0.081*** | 2.12 | |
| (1.17) | (2.24) | (1.6) | (2.7) | (3.4) | (1.12) | |
|
a t statistics is reported in parenthesis |
||||||
| b p<0.1; p<0.5; p<0.01 |
The small asset outperforms big asset effect (SMB) is in line with other cryptocurrency literature findings of Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020).
c) Momentum Vs Reversal Returns
Carhart (1997)’s findings of momentum in the equity market (Winners outperform Losers - WML) motivate Liu, Tsyvinski, and Wu (2019) to investigate similar characteristics in the cryptocurrency market. Liu, Tsyvinski, and Wu (2019) run four statistically significant strategies of buying winners and selling losers of the past 1,2,3 and 4 weeks (formation period) and then, rebalances these portfolios weekly. They find momentum effects in cryptocurrency over short periods of time.
On the other hand,Grobys and Sapkota (2019) find that while momentum has strong predictive power in the equity market, the reverse is true in the cryptocurrency market. Grobys and Sapkota (2019) guide that momentum strategies in cryptocurrency return significantly negative payoffs in the short run. This motivates Shen, Urquhart, and Wang (2020) to investigate a reversal factor in the cryptocurrency market. Shen, Urquhart, and Wang (2020) runs 16 strategies of holding losers and selling winners of the past 1,2,3 and 4 weeks (formation period) and then holds the portfolio unchanged for 1,2,3 and 4 weeks (holding period) into the future before rebalancing. Shen, Urquhart, and Wang (2020), in agreement with Grobys and Sapkota (2019), find negative returns on momentum. Instead, they find statistically significant returns from the loser portfolios pointing to a reversal effect that manifests especially over short term horizons.
The strategies implemented by Shen, Urquhart, and Wang (2020) and Liu, Tsyvinski, and Wu (2019) are similar on formation periods but differ on holding periods. While Shen, Urquhart, and Wang (2020) rebalances the portfolios after 1,2,3 and 4 weeks, Liu, Tsyvinski, and Wu (2019) rebalances only weekly.
To mediate between these findings on reversals and momentum, a comprehensive approach that incorporates 1,2,3 and 4 weeks formation periods and 1,2,3 and 4 weeks holding periods is executed. Equally weighted decile portfolios are created. The first decile consists of winners and the tenth decile consists of losers. The winners portfolio represents a momentum strategy and the losers portfolio represents the reversal portfolio. For a 1 week formation period, we look at last week’s winners and losers, and then form two portfolio. For a 1 week holding period, we hold the portfolio unchanged and calculate the future return on both these portfolios for the next week. The process is repeated while rebalancing the portfolios weekly keeping formation and holding periods constant from 4th January, 2015 to October 2nd 2021. The average return on the momentum and reversal portfolio over the period is shown in Table 3.
Understanding Table 3;
1 Momentum = Formation period of past 1 week winners
1 Reversal = Formation period of past 1 week losers
1 week = Holding period of 1 week into the future
2 Momentum = Formation period of past 2 week winners
2 Reversal = Formation period of past 2 week losers
2 week = Holding period of 2 weeks into the future.
This paper finds that long term losers with a formation period of 3-4 weeks produce statistically significant positive returns, which positive return starts to diminish after holding the portfolio for at least 2 weeks. On the other hand, short term winners with a formation period of 1 week produce statistically significant positive returns that also start to peak at the second week of holding before leveling off and dropping.
Despite the fact that, the reversal factor, produces the highest number of statistically significant positive returns, there is no statistically significant difference between reversal and momentum.
| Strategy | Week.1 | Week.2 | Week.3 | Week.4 |
|---|---|---|---|---|
| 1 Momentum | 0.19** | 0.37** | 0.37 | 0.30** |
| (-2.01) | (-2.10) | (-1.60) | (-2.50) | |
| 1 Reversal | 4.621 | 4.411 | 4.3 | 4.24 |
| (-1.2) | (-1.14) | (-0.27) | (-0.28) | |
| 1 Momentum - Reversal | -4.43 | -4.04 | -3.93 | -3.8 |
| (-1.14) | (-1.04) | (-1.01) | (-1.01) | |
| 2 Momentum | 0.47* | 4.68 | 4.63 | 4.83 |
| (-1.86) | (-1.09) | (-1.09) | (-0.27) | |
| 2 Reversal | 4.51 | 0.36*** | 0.51*** | 0.30** |
| (-0.28) | (-3.00) | (-2.67) | (-1.96) | |
| 2 Momentum - Reversal | -4.01 | 4.32 | 4.24 | 4.5 |
| (-0.96) | (1.00) | (0.97) | (1.04) | |
| 3 Momentum | 4.38 | 4.41 | 4.61 | 4.51 |
| (1.12) | (1.13) | (1.06) | (1.10) | |
| 3 Reversal | 0.47*** | 0.16*** | 0.23*** | 0.12*** |
| (-2.82) | (-6.21) | (-2.55) | (-6.01) | |
| 3 Momentum - Reversal | 3.91 | 4.25 | 4.38 | 4.39 |
| (1.00) | (1.09) | (1.00) | (1.07) | |
| 4 Momentum | 0.52** | 4.79 | 0.48* | 4.71 |
| (-2.04) | (-1.1) | (-1.94) | (-1.09) | |
| 4 Reversal | 0.44*** | 0.48** | 0.33** | 0.14*** |
| (-2.65) | (-2.56) | (-2.19) | (-6.32) | |
| 4 Momentum - Reversal | 0.07 | 4.3 | 0.14 | 4.68 |
| (0.24) | (0.98) | (0.5) | (1.06) | |
|
a t statistics is reported in parenthesis |
||||
| b p<0.1; p<0.5; p<0.01 |
4. Limits to Arbitrage
Idiosyncratic volatility and Amihud’s illiquidity measure are used to measure limits to arbitrage. The formula below is used to establish a figure that incorporates both idiosyncratic volatility and Amihud’s illiquidity measure of a given cryptocurrency. The result from the formula represents a limit to arbitrage measure for a given week upon which all the 570 coins are sorted into equally weighted decile portfolios re-weighted on a weekly basis from 4th January, 2015 to October 2nd 2021.
\[Limits. to. Arbitrage = Idiosyncratic .Volatility * Amihud.illquidity\]
\[Weekly.Lim.Arb_i = (\sigma_i^2-\sigma_m^2) * \frac{1}{7}\sum_{t=1}^{7} \frac{|R_i|}{Vol_i}\]
The null hypothesis of limits is arbitrage is as follows
ho: There is no statistical difference between return on a decile with high limits to arbitrage and a low limits to Arbitrage decile.
Table 4 below shows the average return on the decile portfolios. The deciles are in ascending order with D1 representing a portfolio of the lowest limits to arbitrage coins and D10 representing the highest.
| Lim.Arb.Deciles | Return | t.Statistic |
|---|---|---|
| D1 (low) | -0.015*** | (-5.19) |
| D2 | -0.018*** | (-4.01) |
| D3 | -0.018*** | (-3.50) |
| D4 | -0.009 | (-1.51) |
| D5 | -0.006 | (-0.96) |
| D6 | 0.008 | (1.26) |
| D7 | 0.03*** | (4.00) |
| D8 | 0.065*** | (7.38) |
| D9 | 0.13*** | (11.59) |
| D10 (high) | 1.62*** | (3.73) |
| D10 — D1 | 1.63*** | (3.76) |
|
a t statistics is reported in parenthesis |
||
| b p<0.1; p<0.5; p<0.01 |
Table 4 indicates, with 99% confidence, that assets with high limits to arbitrage outperform those with low limits to arbitrage. Therefore, the null hypothesis is rejected. It is also found that assets in the lower deciles, with typically low limits to arbitrage, have no excess return. Assets with high limits to arbitrage have a premium over those with low limits to arbitrage in order to compensate arbitrageurs that take on high risk of arbitraging coins that are difficult to arbitrage.
a) Characteristics of Limits to Arbitrage
A study on different characteristics of the 10 decile portfolios is conducted as shown in table 5. There is monotonic variation across all but one characteristic. Market Capitalisation, Volume and price keep decreasing as the limits to arbitrage increase. The reason for this is that coins with a high market capitalisation (big coins), are well established with typically higher prices and healthy trade volumes, and therefore tend to have lesser volatility or illiquidity issues thus having lower limits to arbitrage and vice versa. The age characteristic has no observable pattern across the deciles.This is so because the age quoted herein is not the actual age but an age of when a particular asset was listed on Coingecko.
An animated plot is included herein to showcase how market capitalisation of three selected coins influences their Limits to Arbitrage over time. As a coin’s market capitalisation increases, the size of the ball plot in the animation increases. From the animation, it is shown that as the market capitalisation increases (see red ball expand) overtime, the asset tends to keep below the 3rd Limits to Arbitrage decile except during times of high volatility. The smaller sized coin with smaller market capitalisation (the green ball) tends to have wild variations over time. In other words, the bigger the ball (market capitalisation), the smaller the bounces. The smaller the bounces, the lower the limits to arbitrage. The lower the limits to arbitrage, the lower the risk, the lower the risk, the lower return and vice-versa.
N.B: Viewing the animation is only possible if document is viewed in HTML format. For PDF format, please refer to Graph 3. It is visible that the red shade which represents a coin with a bigger market cap tends to be in lower limits to arbitrage deciles followed by the blue and then the green coins with smaller and smallest market cap coins respectively
Graph 4 is a time series plot of the first decile portfolio (low limits to arbitrage) and 10th decile decile portfolio (high limits to arbitrage).It is visible that return spreads are wider for assets with high limits to arbitrage. Also, high risk and high return opportunities tend to cluster in times of crises. Graph 4 shows wide swings around the 2018 Cryptocurrency crash and the March 2020 crash induced by COVID-19.
b) Are Limits to Arbitrage subsumed by other Cryptocurrency factors in the literature?
In the stock market, size and liquidity tend to subsume each other. However, in the cryptocurrency market, size does not necessary relate to liquidity of a particular cryptocurrency. This is so because, the cryptocurrency market has multiple exchanges unlike the stock market that has a couple of exchanges within a given country. The multiplicity of cryptocurrency exchanges present differences in liquidity depths for a single cryptocurrency. While a big coin might have healthy liquidity on one exchange, it’s liquidity on the other exchanges might be extremely low. Therefore, it is possible for a big coin to average towards illiquidity when all exchanges are taken into account. A big coin can still be illiquid relative to a small coin that is well spread out across the various exchanges.
To investigate whether the Limits to Arbitrage factor in this paper explains part of stochastic error unexplained by other factors found by Liu, Tsyvinski, and Wu (2019) and Shen, Urquhart, and Wang (2020), an Ordinary least square multiple regression analysis is conducted while controlling for Market Capitalization, Size, Reversals and Momentum factors.
Table 6 shows the results of the regression. It is visible that the alpha component on all deciles is statically significant which points to the fact that the limits to Arbitrage factor contributes to the literature.
| FACTORS | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D10….D1 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| MKT | .023* | 0 .033 | 0.04* | 0.04 | 0.042 | .05* | .06* | .10** | .13** | -1.09 | -1.1 |
| -1.76 | -1.56 | -1.66 | -1.4 | -1.44 | -1.67 | -1.75 | -2.51 | -2.49 | (-0.53 ) | (-0.54) | |
| SMB | 0.0007 | 0.0007 | 0.002 | 0.002 | 0.0008 | -0.0001 | 0.0005 | -0.0005 | -0.002 | -0.04 | -0.04 |
| -0.499 | -0.48 | -1.14 | -0.87 | -0.37 | (-0.04) | -0.2 | (-0.16) | (-0.43) | (-0.25) | (-0.26) | |
| Reversal | 0.00003 | 0.00005 | 0.0001 | 0.0001 | -0.00001 | 0.0001 | 0.00006 | 0.00005 | 0.0002 | -0.001 | -0.001 |
| -0.368 | -0.83 | -1.4 | -1.31 | (-0.12) | -0.63 | -0.62 | -0.39 | -1.35 | (-0.21) | (-0.21) | |
| Momentum | -0.003** | -0.005** | -0.004 | -0.0043 | -0.005* | -0.0025 | -0.003 | -0.003 | -0.002 | -0.016 | -0.013 |
| (-2.13 ) | (-2.03 ) | (-1.44 ) | (-1.38) | (-1.66) | (-0.68) | (-0.78 ) | ( -0.59) | (-0.32) | (-0.07) | (-0.05) | |
| Alpha | -0.15*** | -0.018*** | -0.02*** | -0.01* | -0.01 | 0.01 | 0.028*** | 0.06*** | 0.12*** | 1.67*** | 1.69*** |
| (-5.31) | (-4.04 ) | (-3.62) | (-1.65) | (-0.87 ) | -1.08 | -3.68 | -6.92 | -10.93 | -3.73 | -3.76 | |
| R squared | 0.0232 | 0.0198 | 0.02 | 0.016 | 0.014 | 0.01 | 0.011 | 0.02 | 0.024 | 0.001 | 0.001 |
| Obs | 352 | 352 | 352 | 352 | 352 | 352 | 352 | 352 | 352 | 352 | 352 |
|
a t statistics is reported in parenthesis |
|||||||||||
| b p<0.1; p<0.5; p<0.01 |
The Cryptocurrency 3 Factor Model
This section puts everything together to form a 3 factor model that explains cryptocurrency returns. A 3 factor model incorporating CAPM, size, and Limits to Arbitrage is derived. Reversal and momentum factors are temporarily dropped as factors given the fact that there is no statistical difference between the reversal factor and momentum factor in the cryptocurrency market based on the t-tests results shown in table 3.
Momentum and reversal factors manifest based on the holding period and formulation period i.e it depends on how long one holds the asset in the portfolio or which period’s winners or losers one will buy or short. The highly subjective nature of establishing a reversal or momentum strategy based on an investors’ interests makes it herculean to decide whether momentum or reversals are a cryptocurrency factor. There is no standard holding or formulation period for cryptocurrencies and as such any portfolio can be made to reflect momentum or reversals by tweaking the holding or formualation periods. Therefore, this paper suggests 3 factors; CAPM, size, and Limits to Arbitrage as shown below.
\[ E(ri) = \alpha + \beta_1(E(rm) - Rf) + \beta_2 SMB + \beta_3 Lim.Arb(HML) + e \]
5) CONCLUSION
This paper contributes to the literature by incorporating limits to arbitrage as a factor that explains cryptocurrency returns. Arbitrage opportunities exist in the cryptocurrency market because assets tend to have different price quotes on different exchanges as shown in Graphs 1 and 2. The price mismatches in cryptocurrency are persistent due to market frictions that disincentivize arbitrage trading. These frictions include high trading costs which may lead to spillage and diminish the return on an arbitrage strategy and high volatility which may push the prices further away from the arbitrageurs’ target price. These are the risks associated with arbitrage trading in cryptocurrency. From Markowitz’s modern portfolio theory, high risk is counterbalanced by a high return. Therefore, assets that are hard to arbitrage (high limits to arbitrage) offer a premium to compensate investors willing to take on arbitrage risk. This paper finds that assets sorted on a limits to arbitrage measure, formulated in this paper from idiosyncractic volatility and Amihud’s illiquidity measure, possess monotonic variation in returns, with returns typically increasing as the level of limits to arbitrage increases as shown in Table 4.
Nonetheless, there are areas for further research as the cryptocurrency market continues to evolve and gain broader adoption to mainstream finance. A potential area of research is to investigate whether the arbitrage factor will diminish over the years to come when the cryptocurrency market matures. Also,it is noted with paramount significance that literature on cryptocurrency factors, including this paper (table 6), find that the R-Squared measure is normally low and tends to fall between 2% to 10%. This points to the fact that the randomness of returns akin to many cryptocurrency projects hurts the goodness of fit (R-squared) of regressional analysis. It is probable that more research, outside traditional finance factor models, ought to be conducted to better explain the cryptocurrency market.
Appendix
Key Papers on Cryptocurrency Factors and Arbitrage in the Stock Market| Paper | Authors | Year | Journal | Topic | Data | Econometric.Analysis | Hypothesis..Thesis. |
|---|---|---|---|---|---|---|---|
| 1 | Liu, Tsyvinski & Wu | 2019 | SSRN Electronic Journal | A Cryptocurrency Three factor Model: CAPM, Size & Momentum | Weekly OHLC, Volume & Market Cap data of 1,827 coins from the beginning of 2014 to July of 2020 from Coinmarketcap | Fama-MacBeth cross-sectional regressions | Size and Momentum factors explain returns in cryptocurrencies |
| 2 | Shen, Urquhart & Wang | 2020 | Finance Research Letters | A Cryptocurrency Three factor Model: CAPM, Size & Reversal Return | Weekly OHLC, Volume & Market Cap data of 1786 coins from April 28th 2013 to March 31st 2019, with 309 weekly observations in total. T-Bill data is used as the risk-free asset from the US Department of the Treasury. | Regression Analysis | Size and Reversal (not Momentum) factors explain returns in cryptocurrencies |
| 3 | Grobys & Sapkota | 2019 | Economic Letters | Significant Negative Payoff of a Momentum in Cryptocurrencies | Monthly time series data on 143 Proof of Work cryptocurrencies in the 2014–2018 period from Coinmarketcap | Regression Analysis | There are negative payoffs on momentum strategies amongst Proof-of-Work Cryptocurrencies |
| 4 | Sovbetov | 2018 | Journal of Economics & Financial Analysis | Additional Cryptocurrency Risk Factors of CAPM, Trading Volume & Volatility: Evidence from Top 5 Big Cap Cryptocurrencies | 2010-2018 weekly data from BitInfoCharts for Bitcoin, Ethereum, Dash, Litcoin, and Monero. OHLC, Volume and MarketCap data for 50 coins is fetched from Coinmarketcap to create a market index representing 92% of the market | Autoregressive Distributed Lag (ARDL) cointegration technique | Market beta, trading volume, and volatility factors determine prices for the top five cryptocurrencies in short-run. Also, attractiveness of cryptocurrencies matters in their price determination in long-run. |
| 5 | Sheifer & Vishny | 1997 | Journal of Finance | The Limits to Arbitarge observed in the Traditional Stock Market | N/A | Arbitrage trading, especially on highly volatile stocks, may not bring stocks to their fundamental value as arbitrageurs shun taking extremely risky positions in traditional stock market | |
| 6 | Lam & Wei | 2011 | Journal of Financial Economics | The Effect of Limits to arbitrage and Investment frictions on Stock Mispricing | Data includes U.S. domestic firms traded on the NYSE, Amex, and Nasdaq exchanges. Their financial statements are taken from Compustat. Stock market data comes from the Center for Research in Security Prices (CRSP). Analyst data are from the Institutional Brokers’ Estimate System. Institutional holdings records are from CDA/Spectrum Institutional (13f) Holdings | Fama-MacBeth cross-sectional regressions | An asset growth anomaly in traditional stock markets can be explained by mispricing caused by both limits-to-arbitrage and the q-theory with investment frictions suggested by works from Li and Zhang (2010) |
| 7 | Brauneis, Mestel & Riordan | 2021 | Journal of Banking & Finance | How to measure liquidity levels in the Cryptocurrency market | Bitcoin and Ethereum order book data from Coinbase Pro, Bitstamp, Bitifinex exchanges. Data for Bitcoin volume is from Coinmarketcap. All data is timestamped in UTC timezone. | N/A | The Amihud Illiquidity Ratio & The Kyle-Obizhaeva Estimator are proper Measures of liquidity levels in Cryptocurrency Markets |
BIBLIOGRAPHY