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Review of Quantitative Finance and Accounting

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09-07-2020 | Original Research

Intertemporal asset pricing with bitcoin

Authors: Dimitrios Koutmos, James E. Payne

Published in: Review of Quantitative Finance and Accounting | Issue 2/2021

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Abstract

This paper develops and tests an intertemporal regime-switching asset pricing model characterized by heterogeneous agents that have different expectations about the persistence and volatility of bitcoin prices. The model is estimated using daily bitcoin price data from 2013 until 2020 whereby three types of agents are considered: mean–variance optimizers, speculators and fundamentalists, respectively. While mean–variance optimizers trade on the basis of conditional first and second moments of the return distribution, speculators engage in trend chasing and buy when prices are rising and sell when prices are declining. Fundamentalists trade on the basis of fundamental factors that can impact the value of bitcoin. The fractions of agents engaging in one strategy over another shows statistically substantial variation during high and low bitcoin price volatility regimes. Estimation results reveal the following. First, unlike in traditional asset classes, there is evidence of mean–variance optimizers. Second, there is evidence of speculators who engage in ‘bandwagon behavior’ and buy bitcoins during price appreciations and sell bitcoins during price declines. Finally, there is evidence of fundamentalists who trade bitcoins when fundamental factors deviate from their long-run trends. Remarkably, these fundamentalists exhibit contrarian-type behaviors during low price volatility regimes while behaving more like fundamental traders during high price volatility regimes.

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Empirical test results of the behavioral heterogeneity framework proposed herein excluding weekend price data are not tabulated for the sake of brevity but are available upon request. These test results yield conclusions that are qualitatively identical with those from the findings reported here which include weekend prices.

Böhme et al. (2015) discuss more the microstructure characteristics of bitcoin and the role and incentives for miners in mining bitcoins and verifying transactions. Many publicly available "bitcoin mining profitability calculators" are accessible online. These allow miners (or would-be miners) to compute their expected revenues based on variables such as bitcoin difficulty, bitcoins per block, hashing power, computing power consumption, cost per KW/h and cost of mining hardware, to name but a few variables. Here are just a few URLs of the many popular websites for computing expected revenues: www.whattomine.com, www.nicehash.com/profitability-calculator, www.bitcoinx.com/profit and www.cryptocompare.com/mining/calculator.

The Hodrick-Prescott filter technique is universally applied in macroeconomic and financial time series analysis. Despite criticism, this technique is considered the workhorse in econometrics as a method for separating long-term trends from their transitory fluctuations. In the words of Ravn and Uhlig (2002, p. 371), "(the Hodrick-Prescott filter) has withstood the test of time and the fire of discussion remarkably well…although elegant new bandpass filters are being developed,…it is likely that the HP filter will remain one of the standard methods for de-trending.".

In conventional asset pricing tests, the risk-free rate (the holding period yield on short-term government treasuries) is usually used as a benchmark to compare the returns on risky assets, such as equities or index funds. Subtracting the risk-free from the returns of risky assets is consistent with the economic notion that investors can either invest in risky assets or risk-free assets (or some combination of both). Investors holding risky assets forgo returns from risk-free assets and, therefore, the risk-free rate serves as a benchmark for the opportunity cost of capital. For this paper, the holding period return for the 1-month Treasury bill is used as a proxy for the risk-free rate, \(r_{f}\). Data is obtained from Professor Kenneth French's publicly available online data library: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Weekend data for the risk-free rate is interpolated using a moving average smoothing approach that fits the weekday data.

Rolling unit root tests (not tabulated for brevity) indicate that the spread between each fundamental factor with its growth rate, \(\text{F}_{i} - {\tilde{F}}_{i}\), is consistently stationary across time. Bitcoin returns, as well as their conditional volatility, are also stationary.

From an econometric perspective, inclusion of a constant serves to ensure the model is unbiased, since the mean of the residuals is zero. From a practical application perspective, inclusion of a constant permits the fitted regression line to colloquially "find its own level" and to provide the best fit for the data. Generally speaking, inclusion of a constant provides less inflated test statistics which make for more conservative economic conclusions.

Various symmetric and asymmetric GARCH-type volatility models are entertained (not reported for brevity). In terms of volatility dynamics, bitcoin returns exhibit volatility asymmetry, whereby negative return shocks lead to more volatility than positive return shocks of equal magnitude, as well as strong volatility persistence—an empirical finding observable from the returns of conventional assets such as stocks and index funds (Katsiampa 2017; Baur and Dimpfl 2018; Ardiar et al. 2019, among others). Various robustness tests that are not reported but available upon request show that the main conclusions of this paper are insensitive to (a) the GARCH specification used to model volatility; (b) the GARCH lag structure; (c) the distributional assumptions regarding the GARCH error terms; and (d) the conventional algorithms used to maximize the various typically-assumed log-likelihood functions.

The expected duration for the low volatility regime (state 1) is \(\mathop \sum \nolimits_{k = 1}^{\infty } k \times {\text{p}}_{11}^{k - 1} \left( {1 - {\text{p}}_{11} } \right) = 1/\left( {1 - {\text{p}}_{11} } \right)\) while for the high volatility regime (state 2) it is \(1/\left( {1 - {\text{p}}_{22} } \right)\). More details are provided in Hamilton (1989, p.374).

The Sharpe ratio for bitcoin returns, \(R_{t}\), is computed as \(\left( {R_{t} - r_{f} } \right)/\sigma\) whereby \(r_{f}\) denotes the risk-free rate (see footnote (4) for an explanation of \(r_{f}\)). The denominator for the Sharpe ratio is the standard deviation of bitcoin returns, \(\sigma\). The VaR for bitcoin returns is calculated as follows: \({\text{VaR}} = W\left( {\mu \Delta t - n\sigma \sqrt {\Delta t} } \right)\) whereby \(\mu\) is the mean return for bitcoin; \(W\) is the value of the portfolio invested in bitcoin; \(n\) is the number of standard deviations depending on the confidence level; \(\sigma\) is the standard deviation of bitcoin returns; \(\Delta t\) is the time window. More discussion and derivations for VaR and MVaR can be found in Signer and Favre (2002).

The behavioral heterogeneity framework in (11) collapses to the Merton (1980) intertemporal capital asset pricing model when we constrain all coefficients, with the exception of \(\beta_{1}\), to zero; \(r_{t} = \beta_{1} \sigma_{t}^{2}\). When this asset pricing model is tested with returns, \(r_{t}\), of conventional asset classes, \(\beta_{1}\), the parameter for risk aversion, is typically found to be either negative and significant or statistically zero. This implies a negative relation between \(r_{t}\) with its conditional volatility, \(\sigma_{t}^{2}\). The 'volatility feedback hypothesis' or 'leverage effect,' two postulations often times used synonymously, are usually evoked to reconcile this finding (Black 1976; Campbell and Hentschel 1992; French et al. 1987).

Using a Markov regime-switching framework to regress bitcoin returns solely against its conditional variance, estimated from the EGARCH in (10), shows a positive linkage between returns and volatility in the low volatility state and a weak relation in the high volatility state (results not tabulated for brevity but available upon request).

Quotes from Satoshi Nakamoto are publicly available here: http://satoshi.nakamotoinstitute.org/quotes/bitcoin-economics/.

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Title: Intertemporal asset pricing with bitcoin
Authors: Dimitrios Koutmos
James E. Payne
Publication date: 09-07-2020
Publisher: Springer US
Published in: Review of Quantitative Finance and Accounting / Issue 2/2021
Print ISSN: 0924-865X
Electronic ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-020-00904-x

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