What can we learn from the convenience yield of Bitcoin? Evidence from the COVID-19 crisis

Abstract

In this study, we employ the cost of carrying model to estimate the convenience yields of gold and bitcoin. Using the COVID-19 crisis as a demand shock, we show that the convenience yield of bitcoin decreases, but the convenience yield of gold increases during the pandemic. We further document an inverse relationship between bitcoin user adoption and convenience yield. Such relationship intensifies during the COVID-19 crisis and is mainly driven by the unexpected component of bitcoin's unique active addresses. Our overall findings indicate that gold rather than bitcoin attracts flight-to-safety investors in the pandemic. We provide supporting evidence for the decrease in bitcoin convenience yield, which can be explained by the sudden increase in active trading addresses that short-term speculators own most.

1. Introduction

Bitcoin is a digital store of value that functions as electronic cash through a decentralized peer-to-peer network. Since its introduction, bitcoin has become the most successful cryptocurrency, with the largest market capitalization. Bitcoin has grown enormously in popularity among researchers, investors, and mainstream financial media in recent years. Fig. 1 shows the market capitalization of bitcoin and the corresponding worldwide Google trend from 2013 to 2020. The total market value of bitcoin was around $8 billion in 2013, with a massive increase to $240 billion in 2017. By the end of 2020, the market capitalization of bitcoin has reached over $500 billion.

Fig. 1.

The development of the bitcoin market.

The exponential growth in the bitcoin market value has raised many concerns and debates from practitioners and academics. On the one hand, bitcoin pundits believe that the price increase is due to increased demand for the asset as a hedge against inflation risk.1 The World Gold Council described bitcoin as a high-octane tactical asset that sometimes displays safe-haven-like behavior like gold. On the other hand, economists have characterized bitcoin as a speculative bubble (e.g., Böhme, Christin, Edelman, & Moore, 2015; Cheah & Fry, 2015; Gandal, Hamrick, Moore, & Oberman, 2018; Yermack, 2013). Several regulatory agencies like the Securities and Exchange Commission (SEC) and the Commodity Futures Trading Commission (CFTC) have issued alerts that investments involving bitcoin may have a heightened risk of fraud.2 ^, 3 There are also discussions in the academic community about the crimes generated by bitcoin. For example, Griffin and Shams (2020) show that the increase in bitcoin trading volume in 2017 was primarily due to coordinated price manipulation using tether. Foley, Karlsen, and Putniņš (2019) find that approximately one-quarter of bitcoin users are involved in illegal activity.

Although debatable, bitcoin shares several features with gold, prompting some academics to dub it “digital gold.” The intrinsic values of bitcoin and gold can be linked to their scarcity and high extraction costs. Also, there is no government control over the supply of both commodities, and the overall quantity is finite (Bouri, Molnár, Azzi, Roubaud, & Hagfors, 2017; Dyhrberg, 2016). In terms of functionality, they both serve as a store of value and a transaction motive. Since the CFTC classified bitcoin as a commodity and the Chicago Mercantile Exchange (CME) started trading bitcoin futures in 2017, bitcoin is officially regulated as a commodity like gold.

The comparison between bitcoin and gold has received considerable attention regarding whether these two assets can be classified under one asset class. In particular, Baur and Lucey (2010) and Baur and McDermott (2010) define a safe asset as one that “is uncorrelated or negatively correlated with another asset or portfolio in times of market stress.” Based on this definition, the previous studies on bitcoin and gold asset classification have mixed results, possibly due to different periods of analysis, data frequency, investment horizon, and the country's economic situation (e.g., Guesmi, Saadi, Abid, & Ftiti, 2019; Shahzad, Bouri, Roubaud, Kristoufek, & Lucey, 2019; Smales, 2019; Urquhart & Zhang, 2019). 4 An approach that has not yet been considered is to compare the properties of convenience yield between bitcoin and gold. Employing convenience yield to explain the asset difference dates back to Fama and French (1988). Jiang, Krishnamurthy, and Lustig (2021) and He, Nagel, and Song (2022) confirm this approach by demonstrating that the convenience yield of a safe asset increases during financial turmoil.

In this paper, we employ the cost of carrying model to estimate the convenience yields for gold and bitcoin. Our approach highlights that an asset's class is determined not only by its relationship with another asset but also by its own behavior. We investigate the possible features of gold and bitcoin's convenience yield and compare them to determine if they are in a similar asset class. We first show that the convenience yields of bitcoin and gold exhibit a mean-reverting pattern, confirming that gold and bitcoin can be characterized as commodities. Second, we find that the volatility of gold's convenience yield is symmetric. In contrast, the convenience yield of bitcoin in the low inventory period is more volatile than in the high inventory period. The asymmetric volatility of the bitcoin convenience yield implies that bitcoin shares a similar feature with industrial metals rather than precious metals. Third, we employ the COVID-19 pandemic as an exogenous shock for demand and analyze the changes in convenience yields between gold and bitcoin before and during the pandemic. We intend to distinguish which of the two commodities is a safe or speculative asset. Intuitively, the convenience yield of safe assets expects to increase during financial turmoil as these assets attract flight-to-safety investors. Our analysis documents an increase in the convenience yield of gold and a decrease in the bitcoin convenience yield during the pandemic.

In addition, we empirically test the dynamic asset pricing model of cryptocurrencies by Cong, Li, and Wang (2021). Their proposition posits that the convenience yield of cryptocurrency is inversely related to the number of users. We employ bitcoin's unique active addresses as a proxy for bitcoin usage and provide direct support to their model. Specifically, we document a significantly negative relationship between bitcoin usage and convenience yield and show that such relationship intensifies during the COVID-19 pandemic. We further decompose the unique active address into the expected and unexpected components, and find the impact of the COVID pandemic on the unique active addresses is mainly driven by the unexpected component. The above results suggest that the increased network effect on the bitcoin convenience yield during the pandemic can be attributed to increased speculative demand for bitcoin rather than the demand for flight-to-safety investment. Thus, our findings support that bitcoin cannot be classified as a safe asset during financial turmoil.

Our study contributes to the growing literature about the safe-haven properties of cryptocurrencies. Different from prior studies mostly focusing on the spot price of bitcoin,5 this study follows the introduction of bitcoin futures by CME in 2017, where this market provides a centralized system for bitcoin trading. Thus, our sample employs the price of bitcoin futures contracts in a regulated market. Our findings confirm that gold and bitcoin can be characterized as commodities. However, different from the symmetric volatility of gold's convenience yield, the convenience yield of bitcoin is more volatile in the low inventory period, implying that bitcoin shares a similar feature with industrial metals. Our study also provides empirical evidence for the network effect on the cryptocurrency convenience yield (Cong et al., 2021; Liu & Tsyvinski, 2021). We show that the decrease in bitcoin convenience yield during the COVID-19 pandemic can be explained by the sudden increase of active trading addresses, in which most are owned by short-term speculators rather than long-term investors.

The remainder of the paper is organized as follows. Section 2 discusses the theoretical framework of convenience yield and develops hypotheses. Section 3 provides the sample and descriptive statistics. Section 4 describes the main results, and Section 5 concludes.

2. The theoretical framework of convenience yield

2.1. Literature review

There are three main theories explaining the economic function of futures markets. The insurance theory by Keynes (1930) and Hicks (1939) stands from the supplier's perspective and argues that commodity suppliers sell their products in the futures market to reduce their risk exposure. And thus, the futures price must be less than the spot price (backwardation). The hedging pressure theory by Cootner (1960) generalizes the insurance theory by considering both sides of the consumers and suppliers. According to this theory, commodity futures prices are determined by the net positions of hedgers, which are in turn influenced by their level of risk aversion. As such, spot prices are less than futures prices when suppliers are more risk-averse than consumers. In the same way, spot prices are higher than futures prices (contango) when consumers are more risk-averse than suppliers. The cost of carrying theory (Nicholas, 1939; Telser, 1958; Working, 1949) views the difference between futures and spot prices as a risk-free return to the holders of a commodity. This theory argues that commodities will be held if the premium is equal to the sum of interest opportunity and carrying costs, less the benefit of the commodity when held as inventory, which is known as the convenience yield.

Virtual currencies like bitcoin have been determined to be commodities under the Commodity Exchange Act. Similar to commodities, bitcoin provides a convenience yield, and prior studies have employed the convenience yield approach to understand the cryptocurrency market. For example, Cong et al. (2021) develop a dynamic asset pricing model of cryptocurrencies based on transaction costs, essentially capturing a form of convenience yield. Liu, Tsyvinski, and Wu (2022) test the implications of the trade-off theory between capital gains and the convenience yield as proposed by recent cryptocurrency models (e.g., Prat, Danos, & Marcassa, 2021; Sockin & Xiong, 2020). They show that the size factor of cryptocurrency is positively related to the logged transaction amount and argue the size effect becomes larger when the convenience yield is higher. Hilliard and Ngo (2022) find that jumps in bitcoin returns result in a positive convenience yield and suggest that bitcoin behaves more like a commodity than a currency. Biais, Bisière, Bouvard, Casamatta, and Menkveld (2023) examine bitcoin's fundamental value from a convenience yield perspective and demonstrate that the convenience yield represents the net transactional benefits.

In this paper, we implement the cost of carrying model in which commodity holders face the opportunity cost of holding inventory but, in turn, benefit from the convenience yield. We investigate the possible features of gold and bitcoin's convenience yield and compare them to determine if they are in a similar asset class.

2.2. Hypothesis development

This paper investigates the properties of convenience yield as follows.

The mean-reversion of the convenience yield: The mean reversion in the convenience yield indicates that fluctuations in the difference between spot and futures prices resulting from demand and supply shocks adjust to their long-run equilibrium price. Gibson and Schwartz (1990) examine the time-series properties of the forward convenience yields of crude oil and find a mean-reverting pattern in the series. Bessembinder, Coughenour, Seguin, and Smoller (1995) find significant evidence of mean-reverting in agricultural commodities and crude oil but marginally significant for financial assets. Mazaheri (1999) shows that the convenience yield of petroleum products exhibits a non-stationary and mean-reverting long memory process. In other words, commodities tend to be more mean reverting than financial assets. Recently, Wu, Xu, Zheng, and Chen (2021) find that the spot and futures prices of bitcoin exhibit long memory properties and are fractionally cointegrated. Since the CFTC classified bitcoin as a commodity and the Chicago Mercantile Exchange (CME) started trading bitcoin futures in 2017, bitcoin is officially regulated as a commodity like gold. Thus, we expect that the convenience yield of bitcoin should exhibit a mean-reverting pattern similar to that of gold.

The asymmetric volatility of the convenience yield:Fama and French (1988) examine the asymmetric volatility of the convenience yield of industrial metals (aluminum, copper, lead, tin, and zinc) and precious metals (gold, platinum, and silver). They find strong evidence for industrial metals but less evidence for precious metals. Ng and Pirrong (1994) observe that the forward return is less volatile than the spot returns for five industrial metals. Gao and Wang (2005) confirm the existence of asymmetry volatility patterns based on six London metal futures contracts and NYMEX copper futures, but not for gold and silver futures. The above literature suggests that the convenience yield of non-safety assets exhibits more asymmetric volatility patterns than safety assets. In other words, the convenience yield of non-safety assets is more volatile at a low than a high inventory level. Accordingly, if bitcoin can be considered a safe asset like gold, the volatility of bitcoin convenience yield should be symmetric.

The convenience yield during financial turmoil: Recent studies (e.g., Chang, Du, Lou, & Polk, 2021; Ding, Levine, Lin, & Xie, 2021) have shown that COVID-19 decreases asset prices and increases market volatility. Yarovaya, Brzeszczyński, Goodell, Lucey, and Lau (2022) suggest that the COVID-19 crisis can be considered a “black swan” event: a situation that has never previously occurred and which caused existing risk management models to fail to adequately evaluate the risk (Yarovaya, Matkovskyy, & Jalan, 2021).6 Fama and French (1988) suggest that precious metals such as gold provide a large convenience yield during a financial crisis or economic downturn. Jiang et al. (2021) also show that the convenience yield of U.S. treasury bonds increased during the global financial crisis of 2007–2009. The above evidence implies that safe-haven assets become scarce during an economic downturn since they are sought after by flight-to-safety investors. Hence, the convenience yields of safe-haven assets expect to increase during financial turmoil.

Convenience yield and users network effect:Cong et al. (2021) propose that the convenience yield of cryptocurrency is inversely related to the total user base, namely the user network effect. They assume that the conversion between cryptocurrency and other assets can be costly, especially when a transaction is required within a short period. Cryptocurrency holders enjoy a benefit (convenience yield) to avoid such costs. At the same time, holding cryptocurrency incurs carrying cost (the forgone return from investing in financial assets), and such cost is partially offset by the expected transaction surplus.7 As the number of individual users increases, the transaction surplus derived by each user increases, resulting in a lower convenience yield that compensates for the lower carrying costs. Therefore, if the user network effect exists, the increase in daily usage of bitcoin will lead to a decrease in the convenience yield of bitcoin.

The above arguments lead us to explore the following hypotheses.

H1

The convenience yield of bitcoin expects to have a mean-reverting pattern.

H2

The convenience yield of bitcoin expects to exhibit a symmetric volatility pattern.

H3

During financial turmoil, the convenience yields of bitcoin and gold expect to increase.

H4

The convenience yield of bitcoin expects to have an inverse relationship with daily usage.

2.3. Estimation of convenience yield

To implement the carrying cost model to estimate bitcoin convenience yield, we assume that non-holders of bitcoin incur a transaction cost for using bitcoin platforms. Holding bitcoin saves the transaction cost but incurs a carrying cost (the forgone return from investing in financial assets). We define the bitcoin convenience yield as the benefit derived from holding bitcoin in a wallet rather than a futures contract. While convenience yields are not directly observable, they can be synthetically replicated by taking simultaneous positions in the spot and futures markets.

For no-arbitrage profit, the return an investor obtains from purchasing a bitcoin $P_t$ at time $t$ and selling it at the maturity date $t+\tau$ will be $\left(F_t^{\tau }-P_t\right)$, which equals the lost interest generated from investing $P_t$ at $r_t^{\tau }$ plus the holding cost $W_t^{\tau }$ minus convenience yield ${cy}_t^{\tau }$.

$$F_t^{\tau }-P_t=P_t{r}_t^{\tau }+W_t^{\tau }-{cy}_t^{\tau }$$ (1)

where $P_t$ is the spot price of bitcoin at time $t$. $F_t^{\left(\tau \right)}$ is the price of futures at time $t$ that matures at time $t+\tau$. $r_t^{\left(\tau \right)}$ is the nominal interest rate investors can borrow between periods $t$ and $t+\tau$ . $W_t$ and ${cy}_t^{\tau }$ are the holding cost and the convenience yield.8 We assume that $F_t^{\tau }$, $P_t$, and $r_t^{\tau }$ are observable, and $W_t^{\tau }$ and ${cy}_t^{\tau }$ are unobserved. The holding cost $W_t^{\tau }$ is constant and does not change over time. Considering only the varying components, the convenience yield can be simply the interest-adjusted logarithm basis as follows:

$$\begin{array}{c}{F}_t=P_t{e}^{\left(r_t-{cy}_t\right)\left(T-t\right)}\\ {cy}_t^{\tau }=r_t+\mathit{ln}\left[P_t/F_t\right]/\left(T-t\right)\end{array}$$ (2)

where $\left(T-t\right)$ is the time to maturity of the futures contracts.

2.4. Methodology

Our study employs several long memory time series models to investigate the dynamics of bitcoin and gold convenience yield. We implement the $ARFIMA\left(p,d,q\right)$ estimating procedure proposed by Chung and Baillie (1993) and Baillie (1996), which superimposes a stationary ARMA parameter in the estimation. The model of $ARFIMA\left(p,d,q\right)$ can be written in an operator notation as follows:

$$\begin{array}{c}\Phi \left(L\right){\left(1-L\right)}^d\left(y_t-\mu \right)=\Theta \left(L\right){\epsilon }_t\\ {\epsilon }_t\sim i.i.d.\left(0,{\sigma }_{\epsilon }^2\right)\end{array}$$ (3)

where $y_t$ is the underlying process. $L$ is the backward-shift operator. $\Phi \left(L\right)=1-{\phi }_{1}L-...-{\phi }_p{L}^p\text{,}$ $\Theta \left(L\right)=1+{\theta }_{1}L+...+{\theta }_q{L}^q\text{,}$ and ${\left(1-L\right)}^d$ is the fractional differencing operator defined by ${\left(1-L\right)}^d=\sum _{k=0}^{\infty }\frac{\Gamma \left(k-d\right)L^k}{\Gamma \left(-d\right)\Gamma \left(k+1\right)}$ with $\Gamma (\cdot )$ denoting the generalized factorial fraction. $d$ is the estimated differencing parameter; if $\left|d\right|<0.5$, the process is covariance stationary with a finite variance; $\mathrm{i}\mathrm{f}d<1$, the process exhibits mean-reverting with a finite variance.

To further address the time-dependent conditional heteroscedasticity, Baillie, Chung, and Tieslau (1996) extend the $ARFIMA$ estimation for the fractional differencing parameter by augmenting it with $GARCH$ type innovations. The model of $ARFIMA\left(p,d,q\right)-GARCH\left(P,Q\right)$ specification can be estimated as follows:

$$\begin{array}{c}\Phi \left(L\right){\left(1-L\right)}^d\left(y_t-\mu -{\phi }_1{x}_{t,1}\right)=\Theta \left(L\right){\epsilon }_t={\epsilon }_t\\ {\epsilon }_t|{\Omega }_{t-1}\sim D\left(0,{\sigma }_t^2\right)\\ \beta \left(L\right){\sigma }_t^2=\omega +\alpha \left(L\right)+{\phi }_2{x}_{t,2}\end{array}$$ (4)

where $x_{t,1}$ and $x_{t,2}$ are the vector of exogenous regressors. $Omega\left(\omega \right),ARCHterm\left(\alpha \right),GARCHterm\left(\beta \right)$ are the estimation variations considering heteroscedasticity in the error terms ${\epsilon }_t$. ${\sigma }_t^2$ is the time-dependent heteroscedasticity that follows $GARCH\left(P,Q\right)$. $d$ is the estimated differencing parameter.

3. Sample and descriptive statistics

3.1. Sample and key variables

We obtain daily CME bitcoin futures contract prices, COMEX gold futures contract prices, and the closing spot price of gold from barchart.com. We collect bitcoin's daily closing spot price from coinmarketcap.com, the daily U.S. Treasury bill rate from the U.S. Department of Treasury official website, the unique active addresses of bitcoin from blockchain.com, and the daily closing prices of CME CF bitcoin reference rate from the Thomson Reuters DataStream. We define the daily continuous futures prices series using the nearby futures contract where the price is switched to the price of the following delivery month. The full sample is from 2017 to 12–20 to 2020-12-31. We consider the announcement date of coronavirus as a global pandemic by WHO on 2020/03/11 as the breakpoint. Then we split the full sample into the pre-Covid from 2017 to 12–20 to 2020-03-11 and the during-Covid from 2020 to 03–12 to 2020-12-31.

We construct the main variables at the daily frequency.9 Using the continuous futures series, the spot price, and the U.S. Treasury bill, we construct the convenience yield based on equation (2).10 Specifically, the convenience yield (cy) is calculated as the sum of the risk-free rate and the log difference between the spot prices and the nearest futures contract price, scaled by their maturity time difference (in annualized numbers). We employ bitcoin's daily unique active addresses as a proxy for daily usage. More unique active addresses imply that more people are using or buying bitcoin. Following Fung and Patterson (1999), we transform the unique active addresses (ACT) into stationary time series data using 50 days backward moving average as follows:

$${ACT}_t=\mathit{LN}\left(\frac{{act}_t}{\frac{1}{50}\sum _{i=1}^{50}{act}_{t-1}}\right)$$ (5)

where ${act}_t$ is the unique active addresses of bitcoin at day $t$, and the denominator is 50 days backward moving average of $act$.

3.2. Descriptive statistics

Table 1 displays the summary statistics for variables. Panel A shows that, on average, the convenience yields (cy) of gold and bitcoin are negative. The standard deviations of the convenience yield of the two futures contracts indicate that bitcoin is substantially more volatile than gold. Similarly, Fig. 2 shows the time series plot of the convenience yields of gold and bitcoin. The figure demonstrates that the convenience yield of bitcoin experiences more spikes. Panel B reports the summary statistics of the subsamples before and during the COVID-19 pandemic. The convenience yield of bitcoin is positive before the pandemic and becomes negative during the pandemic. The mean value of bitcoin convenience yield decreases by 0.111, which is statistically significant. The average unique active addresses of bitcoin, proxied for the daily bitcoin usage, increase during the pandemic, suggesting that more people are using bitcoin or buying during the pandemic subsample.

Table 1.

Descriptive statistics.

Panel A: Full sample descriptive
	Mean	Std	Median	Min	Max
Bitcoin cy	−0.014	0.184	−0.027	−1.651	0.904
Gold cy	−0.020	0.034	−0.021	−0.262	0.164
Bitcoin spot	8.999	0.399	9.029	8.084	10.275
Bitcoin futures	8.999	0.408	9.035	8.050	10.288
Gold spot	7.285	0.154	7.244	7.068	7.632
Gold Futures	7.286	0.154	7.245	7.071	7.628
Treasury bill	0.014	0.009	0.017	0.000	0.025
ACT	0.045	0.107	0.058	−0.536	0.283

Panel B: Subsample descriptive
	Bitcoin cy			Gold cy			Bitcoin active addresses ACT
	Pre	During	Diff	Pre	During	Diff	Pre	During	Diff
Mean	0.016	−0.095	−0.111	−0.023	−0.014	0.009	0.038	0.066	0.028
Std	0.185	0.153		0.024	0.052		0.114	0.078
Median	0.015	−0.082		−0.023	−0.010		0.056	0.063
Min	−1.033	−1.651		−0.157	−0.262		−0.536	−0.193
Max	0.904	0.374		0.164	0.139		0.283	0.236

The table displays the descriptive statistics for the bitcoin convenience yield (Bitcoin cy) and gold convenience yield (Gold cy), the bitcoin unique active addresses (ACT), and variables used for constructing the convenience yield. The convenience yield is estimated as the sum of the risk-free rate and the log difference between the spot prices and the nearest futures contract price, scaled by their maturity time difference (in annualized numbers). Futures and spot prices are in the natural log, and the T-bill rate is in decimals. The bitcoin's active addresses (ACT) is the natural log of the ratio of the current unique active address and the 50-day backward moving average. The full sample is from 2017 to 12–20 to 2020-12-31. We consider the announcement of coronavirus as a global pandemic by WHO on 2020/03/11 as the breakpoint. The full sample is split into the pre-Covid sample (from 2017 to 12–20 to 2020-03-11) and the during-Covid sample (from 2020 to 03–12 to 2020-12-31). Panels A and B report the descriptive statistics for the full sample and subsamples, respectively.

Fig. 2.

The time series plot of the convenience yields of gold and bitcoin

Note: The figure plots the time series of the gold and bitcoin convenience yields. The red dashed line indicates when the World Health Organization (WHO) announced COVID-19 as a global pandemic on March 11, 2020. The data span from 2017 to 12–20 to 2020-12-31. The left panel is the convenience yield of gold, and the right panel is the convenience yield of bitcoin. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3 plots the spot and futures prices for bitcoin and gold and shows a synchronous movement for the pair prices during the full sample period. The minimum value of gold convenience yield (-0.262) occurred on 2020/03/24. Given that both the gold futures price and the spot price increased on this date, the futures price rose more than the spot price, which can be explained by the promise of unlimited stimulus by the U.S. Federal Reserve announcement on 2020/03/23 to combat the economic toll of the coronavirus pandemic.11 Gold has been traditionally known as a safe-haven asset during crisis periods, in which the flight-to-safety motives would increase investors’ valuation of gold.12 The sudden increase in the gold price might result from uncertainty over the coronavirus pandemic and the fear of future inflation. The minimum value of bitcoin convenience yield (-1.651) occurred on 2020/03/12, on which the CME bitcoin futures price fell by about 23%, and the spot price fell by about 37%. This is the next day after the coronavirus announcement by WHO as a global pandemic. One reason for the decrease in the bitcoin price might be panic selling in the bitcoin market.

Fig. 3.

The spot and futures prices of bitcoin and gold

Note: The plots display the spot and futures prices for bitcoin and gold from 2017 to 12–20 to 2020-12-31. The black dashed line indicates when the World Health Organization (WHO) announced COVID-19 as a global pandemic on March 11, 2020. The left panel shows the gold futures prices (solid blue line) and spot prices (solid red line). The right panel shows the bitcoin futures prices (solid blue line) and spot prices (solid red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

4. Main results

This section presents the estimated results for evaluating our hypotheses. Table 2 displays the estimated results for investigating the mean-reverting pattern of the convenience yields of gold and bitcoin. Table 3 shows the results regarding the asymmetry volatility of gold and bitcoin convenience yields. Table 4 represents the results from estimating the effect of COVID-19 on gold and bitcoin convenience yields. Table 5, Table 6 examine the relationship between bitcoin's unique active addresses and convenience yield.

Table 2.

Mean reverting feature of convenience yield.

	$Gold{cy}_t$		$Bitcoin{cy}_t$
	(1)	(2)	(3)	(4)
$d$	0.286*** (0.0517)	0.278*** (0.0501)	0.348*** (0.0306)	0.372*** (0.0353)
Constant	−0.020*** (0.0063)	−0.022*** (0.0041)	0.0153 (0.0354)	0.002 (0.0444)
${cy}_{t-1}$	−0.024 (0.0633)	−0.128** (0.0626)	−0.310*** (0.0420)	−0.320*** (0.0481)
Omega		0.000** (0.0000)		0.001*** (0.0002)
ARCH term		0.126*** (0.0353)		0.062*** (0.0138)
GARCH term		0.837*** (0.0438)		0.910*** (0.0136)
Sigma	0.031*** (0.0008)		0.157*** (0.0041)
AIC	−4.075	−4.447	−0.842	−0.929
Ljung-Box [5]	0.527	0.969	0.095	0.724
Ljung-Box2 [5]	0.000	0.852	0.003	0.990
ARCH Lag [5]	0.000	0.403	0.030	0.990

The table shows the fractional difference parameter estimate of gold and bitcoin convenience yields. We follow the $ARFIMA\left(1,d,0\right)$ model and $ARFIMA\left(1,d,0\right)-GARCH\left(\mathrm{1,1}\right)$ model. $d$ is the estimated differencing parameter. Sigma is the estimation variations without considering heteroscedasticity in the error terms ${\epsilon }_t$. $Omega\left(\omega \right)$, $ARCHterm\left(\alpha \right)$ and $GARCHterm\left(\beta \right)$ are the estimation variations considering heteroscedasticity in the error terms. The sample period is from 2017 to 12–20 to 2020-12-31. *p < 0.1; **p < 0.05; ***p < 0.01 specify significance at 10%, 5%, 1% respectively. The robust standard errors are reported in parentheses.

Table 3.

Asymmetry volatility of convenience yields.

	$Gold{cy}_t$	$Bitcoin{cy}_t$
	(1)	(2)
Gamma ($\gamma )$	−0.076 (0.1160)	0.280*** (0.0571)
Omega	0.003 (0.0023)	0.000 (0.0000)
Alpha	0.144*** (0.0240)	0.037*** (0.0120)
Beta	0.853*** (0.0260)	0.870*** (0.0205)
Delta	0.796*** (0.2449)	3.500*** (0.5270)
Constant	−0.022*** (0.0017)	−0.017 (0.0370)
cyt-1	−0.122*** (0.0151)	−0.324*** (0.0483)
$d$	0.248*** (0.0155)	0.360*** (0.0290)
AIC	−4.459	−0.968
Ljung-Box [5]	0.972	0.894
Ljung-Box2 [5]	0.822	0.981
ARCH Lag [5]	0.463	0.963

The table shows the estimation of $ARFIMA\left(1,d,0\right)+APARCH\left(\mathrm{1,1}\right)$ for gold and bitcoin convenience yields. The estimated model is.

$\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-constant\right)={\epsilon }_t$.

${\epsilon }_t|{\Omega }_{t-1}\sim D\left(0,{\sigma }_t^2\right)$.

${\sigma }_t^{\delta }=\omega +\alpha {\left(\left|{\epsilon }_{t-1}\right|-\mathit{\gamma }{\epsilon }_{t-1}\right)}^{\delta }+\beta {\sigma }_t^{\delta }$.

where ${cy}_t$ is the convenience yield. $Omega\left(\omega \right),ARCHterm\left(\alpha \right),GARCHterm\left(\beta \right),Delta\left(\delta \right)$ are the estimated coefficients. $Gamma\left(\gamma \right)$ is the asymmetry measure of the volatility. The sample period is from 2017 to 12–20 to 2020-12-31. *p < 0.1; **p < 0.05; ***p < 0.01 specify significance at 10%, 5%, 1% respectively. The robust standard errors are reported in parentheses.

Table 4.

The effect of COVID-19 on gold and bitcoin convenience yields.

	$Gold{cy}_t$		$Bitcoin{cy}_t$
	(1)	(2)	(3)	(4)
${\psi }_1$	0.016*** (0.0008)	0.018*** (0.0011)	−0.035*** (0.0046)	−0.058*** (0.0073)
Constant	−0.025*** (0.0007)	−0.025*** (0.0008)	0.008*** (0.0013)	0.023*** (0.0044)
cyt-1	−0.069*** (0.0055)	−0.076*** (0.0060)	−0.319*** (0.0080)	−0.302*** (0.024)
$d$	0.185*** (0.0040)	0.189*** (0.0071)	0.331*** (0.0064)	0.352*** (0.0151)
Omega		0.000** (0.0000)		0.001** (0.0004)
alpha1		0.142*** (0.0535)		0.102*** (0.0389)
beta1		0.818*** (0.0605)		0.858*** (0.0458)
Sigma	0.029*** (0.0015)		0.147*** (0.0066)
Shape	0.706*** (0.0411)	0.899*** (0.0553)	0.841*** (0.0491)	0.908*** (0.0510)
AIC	−4.562	−4.717	−1.223	−1.302
Ljung-Box [5]	0.000	0.962	0.189	0.856
Ljung-Box2 [5]	0.000	0.831	0.001	0.997
ARCH Lag [5]	0.000	0.420	0.012	0.972

The table shows $ARFIMAX\left(1,d,0\right)-GARCH\left(1,1\right)$ estimates of gold and bitcoin convenience yields. The estimated model is.

$\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-{\psi }_{1}COVID19-constant\right)={\epsilon }_t$.

${\epsilon }_t\sim i.i.d.\left(0,{\sigma }_{\epsilon }^2\right)$.

${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$.

where ${cy}_t$ is the convenience yield. $Omega\left(\omega \right),ARCHterm\left(\alpha \right),GARCHterm\left(\beta \right)$ are the estimated coefficients. $COVID19$ is a dummy variable that equals 1 after WHO announced COVID-19 as a global pandemic on March 11, 2020, and 0 otherwise. ${\psi }_1$ measures the impact of the pandemic on the convenience yield. The sample is from 2017 to 12–20 to 2020-12-31. *p < 0.1; **p < 0.05; ***p < 0.01 specify significance at 10%, 5%, 1% respectively. The robust standard errors are reported in parentheses.

Table 5.

The relationship between bitcoin unique active addresses and convenience yield.

	(1)	(2)	(3)	(4)
ACT (${\psi }_0$)	−0.178*** (0.0112)	−0.167*** (0.0087)	−0.116*** (0.0198)	−0.072*** (0.0087)
COVID19 (${\psi }_1$)		−0.032*** (0.0027)		−0.057*** (0.0073)
ACT*COVID19 (${\psi }_2$)		−0.035*** (0.0055)		−0.028*** (0.0068)
Constant	0.003** (0.0012)	0.009*** (0.0017)	−0.002** (0.0010)	0.027*** (0.0051)
cyt-1	−0.323*** (0.0074)	−0.308*** (0.0111)	−0.305*** (0.0301)	−0.298*** (0.0163)
$d$	0.338*** (0.0066)	0.321*** (0.0080)	0.364*** (0.0123)	0.359*** (0.0155)
omega			0.001** (0.0004)	0.001** (0.0004)
alpha1			0.092** (0.0369)	0.092*** (0.0336)
beta1			0.864*** (0.0455)	0.869*** (0.0393)
Sigma	0.146*** (0.0065)	0.146*** (0.0066)
Shape	0.842*** (0.049)	0.838*** (0.0492)	0.909*** (0.0504)	0.908*** (0.0509)
AIC	−1.244	−1.239	−1.304	−1.303
Ljung-Box [5]	0.824	0.851	0.896	0.891
Ljung-Box2 [5]	0.418	0.313	0.995	0.994
ARCH Lag [5]	0.654	0.546	0.978	0.974

The table investigates the relationship between bitcoin's unique active addresses (ACT) and convenience yield as follows.

$\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-{\psi }_0{ACT}_t-{\psi }_{1}COVID19-{\psi }_2{ACT}_t*COVID19-constant\right)={\epsilon }_t$.

$\sim i.i.d.\left(0,{\sigma }_{\epsilon }^2\right)$.

${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$.

where ${cy}_t$ is the convenience yield of bitcoin. ${ACT}_t$ is the natural log of the ratio of the current bitcoin unique active address and the 50-day backward moving average. $COVID19$ is a dummy variable that equals 1 after WHO announced COVID-19 as a global pandemic on March 11, 2020, and 0 otherwise. The full sample is from 2017 to 12–20 to 2020-12-31. *p < 0.1; **p < 0.05; ***p < 0.01 specify significance at 10%, 5%, 1% respectively. The robust standard errors are reported in parentheses.

Table 6.

The decomposition of unique active addresses.

Panel A: The relationship between the unexpected component and convenience yield
	(1)	(2)	(3)	(4)
ACT (${\psi }_0$)	−0.044*** (0.0128)	0.019*** (0.004)	−0.035*** (0.0085)	0.020*** (0.0043)
COVID19 (${\psi }_1$)		−0.038*** (0.0075)		−0.065*** (0.0119)
ACT*COVID19 (${\psi }_2$)		−0.101*** (0.0213)		−0.119*** (0.0067)
Constant	−0.007*** (0.0019)	0.011*** (0.0012)	−0.003* (0.0017)	0.020*** (0.0012)
cyt-1	−0.299*** (0.0129)	−0.321*** (0.0074)	−0.293*** (0.019)	−0.319*** (0.0264)
$d$	0.334*** (0.0078)	0.332*** (0.006)	0.359*** (0.0155)	0.358*** (0.0155)
omega			0.001** (0.0004)	0.001** (0.0003)
alpha1			0.094*** (0.0359)	0.091*** (0.0311)
beta1			0.866*** (0.0429)	0.875*** (0.0342)
Sigma	0.146*** (0.0065)	0.146*** (0.0066)
shape	0.853*** (0.0496)	0.842*** (0.0498)	0.907*** (0.0508)	0.906*** (0.052)
AIC	−1.235	−1.233	−1.302	−1.302
Ljung-Box [5]	0.927	0.897	0.861	0.989
Ljung-Box2 [5]	0.325	0.267	0.995	0.994
ARCH Lag [5]	0.588	0.520	0.977	0.972

Panel B: The relationship between the expected component and convenience yield
	(1)	(2)	(3)	(4)
ACT (${\psi }_0$)	−0.213*** (0.0099)	−0.210*** (0.0117)	−0.190*** (0.0174)	−0.214*** (0.0272)
COVID19 (${\psi }_1$)		−0.031*** (0.0031)		−0.062*** (0.0026)
ACT*COVID19 (${\psi }_2$)		0.057 (0.0678)		0.220 (0.2706)
Constant	0.028*** (0.0027)	0.022*** (0.0029)	0.018*** (0.0033)	0.035*** (0.0037)
cyt-1	−0.312*** (0.0086)	−0.314*** (0.0071)	−0.319*** (0.0139)	−0.309*** (0.0767)
$d$	0.343*** (0.0114)	0.330*** (0.0073)	0.366*** (0.0100)	0.365*** (0.0108)
omega			0.001** (0.0004)	0.001** (0.0004)
alpha1			0.091*** (0.0349)	0.096*** (0.036)
beta1			0.868*** (0.0423)	0.865*** (0.0419)
Sigma	0.146*** (0.0065)	0.147*** (0.0068)
shape	0.845*** (0.0493)	0.833*** (0.0497)	0.904*** (0.0509)	0.899*** (0.0511)
AIC	−1.243	−1.239	−1.306	−1.306
Ljung-Box [5]	0.866	0.907	0.956	0.936
Ljung-Box2 [5]	0.445	0.350	0.996	0.995
ARCH Lag [5]	0.711	0.620	0.979	0.977

The table decomposes bitcoin's unique active addresses into two components: the unexpected and expected components. We replace the bitcoin's unique active addresses with these two components and re-estimate the equation in Table 5. Panel A (B) examines the relationship between the convenience yield and the unexpected component (expected component) of unique active addresses. The full sample is from 2017 to 12–20 to 2020-12-31. *p < 0.1; **p < 0.05; ***p < 0.01 specify significance at 10%, 5%, 1% respectively. The robust standard errors are reported in parentheses.

4.1. The mean-reversion of the convenience yield

To study whether the convenience yields of gold and bitcoin exhibit the mean-reverting pattern, we examine the fractional differencing parameter $d$ in equation (3). Table 2 presents the results employing both $ARFIMA\left(1,d,0\right)$ and $ARFIMA\left(1,d,0\right)-GARCH\left(1,1\right)$ models for gold and bitcoin convenience yield. The optimum lag length was selected based on the AIC criteria. Columns (1) and (3) show the estimated results employing the $ARFIMA\left(1,d,0\right)\text{,}$ and Columns (2) and (4) present the estimated results based on $ARFIMA\left(1,d,0\right)-GARCH\left(1,1\right)$ model. For all estimates of the two series, the fractional differencing parameters $d$ are consistently in the range $0<d<0.5$ and are statistically significant at the 1% level. The results indicate that the convenience yields of bitcoin and gold exhibit long memory mean-reverting patterns, suggesting that both assets can be classified as commodities.

4.2. The asymmetric volatility of the convenience yield

To study the asymmetric volatility of the convenience yields, we follow Ding, Granger, and Engle (1993) and employ the $ARFIMA\left(1,d,0\right)+APARCH\left(\mathrm{1,1}\right)$ model as follows:

$$\begin{array}{c}\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-constant\right)={\epsilon }_t\\ {\epsilon }_t|{\Omega }_{t-1}\sim D\left(0,{\sigma }_t^2\right)\\ {\sigma }_t^{\delta }=\omega +\alpha {\left(\left|{\epsilon }_{t-1}\right|-\gamma {\epsilon }_{t-1}\right)}^{\delta }+\beta {\sigma }_t^{\delta }\end{array}$$ (6)

where ${cy}_t$ is the convenience yield. $Omega\left(\omega \right),ARCHterm\left(\alpha \right),GARCHterm\left(\beta \right)\text{,}$ and $Delta\left(\delta \right)$ are the estimated parameters. $Gamma\left(\gamma \right)$ is the asymmetry measure of the volatility. Specifically, if $\gamma$ is not significantly different from zero, no asymmetry effect exists in the convenience yield. If $\gamma$ is significantly positive, convenience yield is more volatile in the low inventory period than in the high inventory period (Fama & French, 1988).

Table 3 investigates whether the convenience yields of gold and bitcoin exhibit asymmetry volatility responses to shocks. In column (1), $Gamma\left(\gamma \right)$ of gold is insignificant, indicating no asymmetric volatility in the convenience yield of gold. In column (2), $Gamma\left(\gamma \right)$ of bitcoin is significantly positive at the 1% level, indicating that asymmetry volatility exists in bitcoin convenience yield. In other words, the convenience yield of bitcoin in the low inventory period is more volatile than in the high inventory period. The asymmetry volatility in the convenience yield of bitcoin implies that bitcoin may share a similar feature with industrial metals (e.g., copper) rather than precious metals (e.g., gold). The difference in $Gamma\left(\gamma \right)$ estimates between gold and bitcoin suggests that these two assets may differ in their investors’ objectives and thus may respond differently to external shocks.13

4.3. The convenience yield during the COVID-19 pandemic

Recent studies (e.g., Chang et al., 2021; Ding et al., 2021) have shown that COVID-19 decreases asset prices and increases market volatility. The possible reason is that sentiment related to coronavirus news drives asset price changes. Yarovaya et al. (2022) discuss the unique characteristics of the COVID-19 crisis and consider this shock a “black swan” event. To examine the behavior of gold and bitcoin convenience yields during the pandemic, we employ $ARFIMAX\left(1,d,0\right)-GARCH\left(1,1\right)$ model to address the time-dependent conditional heteroscedasticity as follows:

$$\begin{array}{c}\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-{\psi }_{1}COVID19-constant\right)={\epsilon }_t\\ {\epsilon }_t\sim i.i.d.\left(0,{\sigma }_{\epsilon }^2\right){\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2\end{array}$$ (7)

where ${cy}_t$ is the convenience yield. $Omega\left(\omega \right),ARCHterm\left(\alpha \right),GARCHterm\left(\beta \right)$ are the estimated parameters. $d$ is the fractional differencing parameter $\text{.}$ $COVID19$ is a dummy variable that equals 1 after WHO announced COVID-19 as a global pandemic on March 11, 2020, and 0 otherwise. ${\psi }_1$ evaluates the impact of the pandemic on the convenience yield, Specifically, a positive value of ${\psi }_1$ implies that investors prefer holding the underlying asset during the pandemic than speculating/liquidating.

Table 4 investigates how the convenience yields of gold and bitcoin change during the COVID-19 pandemic, in which we examine the sign and the magnitude of the estimated coefficient ${\psi }_1$ employing both $ARFIMAX\left(1,d,0\right)$ and $ARFIMAX\left(1,d,0\right)-GARCH\left(1,1\right)$ models. In the first two columns, the estimated coefficients ${\mathrm{\psi }}_1$ for gold convenience yield are significantly positive. The positive coefficients indicate that gold, being a safe-haven commodity, may become more valuable during the pandemic as it may attract flight-to-safety investors, resulting in a larger convenience yield. In contrast, the estimated coefficients ${\mathrm{\psi }}_1$ for bitcoin are significantly negative, implying that bitcoin's convenience yield decreases during the pandemic. The decrease in convenience yield might be due to increased speculative activities rather than holding as a hedge or safe asset.

In summary, the convenience yield of bitcoin decreased during the pandemic, but gold increased,14 suggesting that gold rather than bitcoin attracted flight-to-safety investors during the COVID-19 pandemic. Our empirical results on the coefficients ${\mathrm{\psi }}_1$ coupled with the asymmetric volatility of convenience yield indicate that bitcoin may not be a safe-haven commodity during the crisis period.

4.4. Convenience yield and user network effect

Cong et al. (2021) demonstrate that a higher level of cryptocurrency adoption (proxy by unique active addresses) results from lower carrying costs, namely the user network effect. The reason is that the decrease in carrying costs, in turn, delivers a higher transaction surplus for cryptocurrency users, thereby generating future user base growth. As the number of individual users increases, the transaction surplus derived by each user increases, resulting in a lower convenience yield that compensates for the lower carrying costs. Liu and Tsyvinski (2021) also show that cryptocurrency returns are highly correlated with bitcoin's unique active addresses. Accordingly, we use bitcoin's unique active addresses as a proxy for daily usage, as more unique active addresses imply that more people are using bitcoin or buying it up.

To examine the relationship between the convenience yield and the unique active addresses, we adopt $ARMFIAX\left(1,d,0\right)-GARCH\left(\mathrm{1,1}\right)$ specification is as follows:

$$\begin{array}{c}\left(1-{\phi }_{1}L\right){\left(1-L\right)}^d\left({cy}_t-{\psi }_0{ACT}_t-{\psi }_{1}COVID19-{\psi }_2{ACT}_t*COVID19-constant\right)={\epsilon }_t\\ {\epsilon }_t\sim i.i.d.\left(0,{\sigma }_{\epsilon }^2\right)\\ {\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2\end{array}$$ (8)

where ${cy}_t$ is the convenience yield of bitcoin. ${ACT}_t$ is the natural log of the ratio of the current bitcoin unique active address and the 50-day backward moving average. ${\psi }_0$ is the estimated coefficient that measures the relationship between the unique active addresses and convenience yield. Specifically, a negative value of ${\psi }_0$ indicates that as the number of individual users increases, the transaction surplus derived by each user increases, resulting in a lower convenience yield that compensates for the lower carrying costs. A negative value of ${\psi }_2$ indicates that the negative relationship between the unique active addresses and convenience yield (i.e., the user network effect) intensifies during the pandemic.

Table 5 presents the estimated results employing both $ARFIMAX\left(1,d,0\right)$ and $ARFIMAX\left(1,d,0\right)-GARCH\left(1,1\right)$ models. The estimated coefficients ${\psi }_0$ in all columns are negative and statistically significant at the 1% level, even when we control the COVID dummy. The significantly negative coefficients ${\psi }_0$ indicate that the bitcoin convenience yield is inversely related to the level of user adoption, consistent with Cong et al. (2021). When the number of individual users increases, the increased transaction surplus leads to a lower convenience yield compensating for the lower carrying costs. Columns (2) and (4) display the estimated result of ACT*COVID19 to examine how this inverse relationship changes during the pandemic. The estimated coefficients ${\psi }_2$ are significantly negative, indicating that the negative relationship between the bitcoin convenience yield and user level is even stronger during the pandemic. In other words, the network effect of bitcoin user adoption on its convenience yield becomes stronger during the COVID-19 pandemic, implying an increase in the bitcoin user base.

Furthermore, we follow Bessembinder and Seguin (1992) to decompose the unique active address into the expected and unexpected components. The expected component generally reflects the activities of long-term investors (hedgers), while the unexpected component reflects the activities of short-term investors (speculators). We fit an $ARMA\left(p,q\right)$ model for the unique active addresses, where the optimum lag length $\left(p,q\right)$ is selected based on the AIC criteria. In the fitted model, we define the fitted part as the expected component and the residual term as the unexpected component of unique active addresses. Thus, the unexpected and expected components represent the trading activities of short-term speculators and long-term investors, respectively. Then we replace the unique active addresses with these two components and re-estimate equation (8).

Panel A (B) of Table 6 shows the relationship between the convenience yield and the unexpected component (expected component) of unique active addresses. The estimated coefficients ${\psi }_0$ in both panels are negative and statistically significant at the 1% level. These results imply that the convenience yield is inversely related to daily expected and unexpected usage. To further examine how the inverse relationship changes during the pandemic, columns (2) and (4) in both panels display the estimated result of ACT*COVID19. Panel A shows the significantly negative coefficients ${\psi }_2$, indicating that the negative relationship between the unexpected unique active addresses and convenience yield strengthens during the pandemic. In contrast, the regression coefficients ${\psi }_2$ are insignificant in Panel B, suggesting that the negative relationship between the expected unique active addresses and convenience yield remains unchanged. By comparing the above two panels, we find that the impact of the COVID-19 pandemic on the unique active addresses is mainly driven by the unexpected component.

Overall, our results on bitcoin usage and convenience yield indicate that the increase in bitcoin usage during the COVID-19 pandemic is attributed to short-term speculators (unexpected usage component) rather than long-term investors (expected usage component). The findings support the argument in the previous section that bitcoin cannot be classified as a safe asset during financial turmoil.

5. Discussion and conclusion

Bitcoin prices have experienced a dramatic increase over the years, especially during the COVID-19 crisis. The crazy performance of bitcoin is interesting but is still puzzling. In this paper, we employ the cost of carrying model and the dynamic asset pricing model of cryptocurrencies to estimate the convenience yield of gold and bitcoin. First, we find that the convenience yields of both bitcoin and gold exhibit a mean-reverting process. However, the volatility of bitcoin and gold convenience yield responds differently. Second, the convenience yield of bitcoin decreases during the COVID pandemic, but gold's convenience yield increases, suggesting that gold rather than bitcoin attracts flight-to-safety trading. In addition, we document an inverse relationship between bitcoin's unique active addresses and convenience yield. Such negative relationship intensifies during the COVID-19 pandemic and is mainly driven by the unexpected component of unique active addresses.

In conclusion, our results suggest that bitcoin can be considered a commodity, but not a safe-haven commodity yet during financial turmoil. The increase in bitcoin prices observed during the COVID-19 pandemic is largely driven by the demand from speculative investors rather than flight-to-safety investors.

Data availability

Data will be made available on request.