A multivariate approach for the simultaneous modelling of market risk and credit risk for cryptocurrencies

Abstract

This paper proposes a set of models which can be used to estimate the market risk for a portfolio of crypto-currencies, and simultaneously to estimate also their credit risk using the Zero Price Probability (ZPP) model by Fantazzini et al. (Comput Econ 31(2):161–180, 2008), which is a methodology to compute the probabilities of default using only market prices. For this purpose, both univariate and multivariate models with different specifications are employed. Two special cases of the ZPP with closed-form formulas in case of normally distributed errors are also developed using recent results from barrier option theory. A backtesting exercise using two datasets of 5 and 15 coins for market risk forecasting and a dataset of 42 coins for credit risk forecasting was performed. The Value-at-Risk and the Expected Shortfall for single coins and for an equally weighted portfolio were calculated and evaluated with several tests. The ZPP approach was used for the estimation of the probability of default/death of the single coins and compared to classical credit scoring models (logit and probit) and to a machine learning algorithm (Random Forest). Our results reveal the superiority of the t-copula/skewed-t GARCH model for market risk, and the ZPP-based models for credit risk.

Appendices

Appendix 1: A review of the history and the financial literature devoted to cryptocurrencies

1.1 Brief historical overview: Bitcoin & sons

Bitcoin was not the first cryptocurrency proposed in the IT literature, see e.g. the works by Chaum (1983), Chaum and Brands (1997) and Back (2002). The idea of Bitcoin was presented in a paper published in 2008 by a group of anonymous authors under the pseudonym of (Nakamoto 2008). Since then, Bitcoin has become the most popular online decentralized currency, which allows users to make transactions without any third party by using a peer-to-peer protocol. The system is based on cryptography algorithms which provide security for all operations, see Antonopoulos (2014), Narayanan et al. (2016) for more details. The Bitcoin market capitalization was close to $ 70 bn at the end of 2018 and represented almost 50% of all cryptocurrencies total capitalization. The price for a single bitcoin decreased to approximately $ 4000, after reaching a top of almost $ 20000 at the end of 2017. Most cryptocurrencies are hard forks of the Bitcoin protocol, that is they introduced new rules to create blocks which are not considered valid by the older (Bitcoin) software. A description of the history of bitcoin (and cryptocurrencies in general) can be found in Burniske and Tatar (2017).

A common way to raise funds to create a new coin, app, or service with cryptocurrencies is to launch an Initial Coin Offering (ICO). An ICO is a type of crowdfunding, where funds are collected by selling a fixed number of new coins to investors. It is similar to an Initial Public Offering (IPO) of a company, but there are some important differences: ICOs may fall outside current regulations and can be prone to scams and securities law violations. Moreover, holding coins not necessarily gives dividends, but services and goods manufactured by the company. According to the CoinDesk State of Blockchain Q1 2018¹⁸, the number of ICOs is still increasing (number of ICOs in 2017 was twice larger than in 2016) and its funding attained $ 12 bn. Only in the first quarter of 2018, there were 202 ICOs with $ 6.3 bn of total funding. However, investing in cryptocurrencies can be an extremely risky process and the final return depends on the strategy adopted: Kostovetsky and Benedetti (2018) shows that approximately 56 percent of crypto startups that raise money through ICOs die within four months of their initial coin offerings. However, they also show that the representative ICO investor earns 82% and the safest strategy is to acquire coins in an ICO and then sell them on the first day -if individual investors can participate in ICOs-, or to sell them in the first six months, otherwise.

1.2 Literature review

The Bitcoin phenomenon has attracted significant attention in the academic literature with regard to its fundamental value (Woo et al. 2013; Garcia et al. 2014; Hayes 2015, 2017), price dynamics (Buchholz et al. 2012; Kristoufek 2013; Garcia et al. 2014; Garcia and Schweitzer 2015; Glaser et al. 2014; Bouoiyour and Selmi 2015; Bouoiyour et al. 2015; Ciaian et al. 2016; Bouri et al. 2017), bubble modelling (MacDonell 2014; Cheah and Fry 2015; Gerlach et al. 2018), price discovery (Brandvold et al. 2015), and more recently about univariate volatility modelling (Dyhrberg 2016a, b; Balcilar et al. 2017; Chu et al. 2017; Katsiampa 2017; Liu et al. 2017; Pichl and Kaizoji (2017; Naimy and Hayek 2018; Catania et al. (2018). See Fantazzini et al. (2016, 2017) for a large survey of the econometric and mathematical tools which have been proposed so far to model the bitcoin price and several related issues, highlighting advantages and limits.

With regards to risk management for cryptocurrencies, the number of works is much more limited: Chu et al. (2017) compared twelve GARCH models with seven popular cryptocurrencies, and their fits were assessed in terms of five in-sample criteria and out-of-sample Value at Risk performances. Chan et al. (2017) analyzed the (unconditional) statistical properties of seven cryptocurrencies, while Osterrieder and Lorenz (2017) examined the tail behavior of bitcoin returns using extreme value distributions but no backtesting was performed. Stavroyiannis (2018) examined a set of market risk measures for the BTC and compared these measures with the SP500 index, the Brent crude oil spot price and the gold spot price by performing a backtesting analysis. Gkillas and Katsiampa (2018) studied the tail behavior of the returns of five major cryptocurrencies by using again extreme value analysis and computing the Value-at-Risk and Expected Shortfall, but no backtesting analysis was implemented. Trucios (2019) compared the one-step-ahead volatility forecast of Bitcoin using several GARCH-type models and also evaluated the performance of several procedures when estimating the Value-at-Risk. As it is possible to notice, all these studies dealt only with univariate models, focused almost exclusively on the Value-at-Risk and volatility forecasting, while only three works performed backtesting analysis.

In general, standard models for market risk tend to work poorly with cryptocurrencies due to the frequent presence of structural breaks, see Bouri et al. (2016), Fantazzini et al. (2016), Fantazzini et al. (2017), Mensi et al. (2018) and Thies and Molnar (2018). To make matters worse, price manipulations and market frauds caused by the lack of financial oversight (Gandal et al. 2018; Griffin and Shams 2018) and the fact that cryptocurrencies are still mainly used for speculative purposes, make financial bubbles a recurring phenomenon, see Corbet et al. (2018), Cheah and Fry (2015) and Gerlach et al. (2018). A potential solution could be to use complex model specifications able to accommodate structural breaks and extreme price volatility, but this would come at the cost of lower computational tractability and potential model over-fitting. Moreover, this solution would become quickly unfeasible in the multivariate case, due to the well-known curse of dimensionality. Finally, we remark that credit risk modelling and the implications of having invested in dead coins have not been considered so far.

Appendix 2: Multivariate time series models

We employed two multivariate models for simultaneously computing the market and credit risk measures of a portfolio of cryptocurrencies: the VAR-DCC and the VAR-Copula-GARCH models. The DCC model was originally proposed by Engle (2002) and Tse and Tsui (2002), while copula-GARCH models were discussed in Cherubini et al. (2004), Patton (2006a, b), and Fantazzini (2008, 2009b). We provide below a brief review of these two approaches, while we refer the interested reader to Bauwens et al. (2012) for a more detailed treatment at the textbook level.

These two approaches share similar building blocks for the conditional mean and the conditional variance: for the mean, a Vector Auto-Regression model (VAR) is used, while for the variance a set of GARCH models was employed. Let ${\mathbf {Y}}_t$ be a vector stochastic process of dimension $n \times 1$, then a conditional model for ${\mathbf {Y}}_t$ can be expressed as follows:

$$\begin{aligned} {\mathbf {Y}}_t ={\varvec{\mu }}_t +{\mathbf {D}}_t {\mathbf {z}}_t \end{aligned}$$

where ${\varvec{\mu }}_t$ is a vector of conditional means, ${\mathbf {D}}_t = diag(\sigma _{1,t},\ldots ,\sigma _{n,t})$ a diagonal matrix of conditional standard deviations, while ${\mathbf {z}}_t$ is a vector of standardized errors with a conditional multivariate distribution $H_t (z_{1,t},\ldots , z_{n,t}; {\varvec{\theta }})$ and parameter vector ${\varvec{\theta }}$.

The conditional means are modelled with a VAR(p) model,

$$\begin{aligned} {\varvec{\mu }}_t={\mathbf {a}}_0 + \sum _{m=1}^p {\mathbf {A}}_m {\mathbf {Y}}_{t-m} \end{aligned}$$

while the conditional variances with GARCH(p,q) models,

$$\begin{aligned} \sigma _{i,t}^2=\omega _i+ \sum _{m=1}^p \alpha _{i,m} (\sigma _{i,t-m}z_{i,t-m})^2 + \sum _{k=1}^q \beta _{i,k} \sigma _{i,t-k}^2 \end{aligned}$$

Other univariate GARCH models, like the Exponential-GARCH, the Threshold-GARCH, etc. can be used. Where the VAR-DCC and the VAR-Copula-GARCH models differ is how they specify the conditional joint distribution $H_t$.

1.1 Dynamic Conditional Correlation (DCC) models

The DCC model by Engle (2002) assumes that $H_t$ is a multivariate Normal or Student’s t distribution with correlation matrix ${\mathbf {R}}_t$,

$$\begin{aligned} {\mathbf {R}}_t = (diag {\mathbf {Q}}_t)^{-1/2} {\mathbf {Q}}_t (diag {\mathbf {Q}}_t)^{-1/2} \end{aligned}$$

(5)

where the $n \times n$ symmetric positive definite matrix ${\mathbf {Q}}_t$ is given by:

$$\begin{aligned} {\mathbf {Q}}_t = \left( 1 - \sum \limits _{l=1}^L \theta _{1,l} - \sum \limits _{s=1}^S \theta _{2,s} \right) {\bar{\mathbf {Q}}} + \sum \limits _{l=1}^L \theta _{1,l} {\mathbf {z}}_{t-l}{\mathbf {z}}_{t-l}^\prime + \sum \limits _{s=1}^S\theta _{2,s} {\mathbf {Q}}_{t-s} \end{aligned}$$

where ${\bar{\mathbf {Q}}}$ is the $n \times n$ unconditional variance/covariance matrix of ${\mathbf {z}}_t$, $\theta _{1,l} (\ge 0)$ and $\theta _{2,s} (\ge 0)$ are scalar parameters satisfying $\sum _{l=1}^L \theta _{1,l} + \sum _{s=1}^S \theta _{2,s} < 1$ to have ${\mathbf {Q}}_t>0$ and ${\mathbf {R}}_t>0$. ${\mathbf {Q}}_t$ is the covariance matrix of ${\mathbf {z}}_t$, so that $q_{ii,t}$ is not equal to 1 by construction and it is subsequently transformed into a correlation matrix by (5).

In the multivariate normal case, the total likelihood can be decomposed in two parts, so that the models for the conditional means and variances can be estimated in a first stage, while the parameters of the conditional correlation are estimated in a second stage using the parameters from the first step. Instead, in case of a multivariate student’s t distribution, the estimation should be performed either in one step (so that the shape parameter is jointly estimated for all volatility models), or in two stages: in this latter case, the first step is based on a Gaussian quasi-maximum likelihood (QML) estimator for the conditional means and variances, while the shape parameter is estimated in a second step, as suggested by Bauwens and Laurent (2005).

The DCC model is one of the most used models in applied finance, thanks to its computational flexibility that allows this model to be computed also with portfolios consisting of up to 100 assets (Engle and Sheppard 2001; Bauwens et al. 2012). Despite known problems (Aielli 2013; Caporin and McAleer 2013), it still remains an important benchmark when multivariate volatility modelling is of concern, see Bauwens et al. (2006), Satchell and Knight (2011), Caporin and McAleer (2014) and Bali and Zhou (2016).

1.2 Copula-VAR-GARCH models

Copula theory provides an easy way to deal with the (usually) complex multivariate modelling. Using the so-called Sklar (1959) theorem, a joint distribution can be factored into the marginals and a dependence function called a copula. The joint distribution $H_t$ can be expressed as follows:

$$\begin{aligned} {\mathbf {z}}_t \sim H_t(z_{1,t}, \ldots , z_{n,t}; {\varvec{\theta }}) \equiv C_t(F_{1,t}(z_{1}; \delta _1), \ldots , F_{n,t}(z_{n,t}; \delta _n); {\varvec{\gamma }}) \end{aligned}$$

(6)

which means that the joint distribution $H_t$ of a vector of standardized errors ${\mathbf {z}}_t$ is the copula $C_t(\cdot ; {\varvec{\gamma }})$ of the cumulative distribution functions of the innovation marginals $F_{1,t}(z_{1}; \delta _1), \ldots , F_{n,t}(z_{n,t}, ; \delta _n)$, where ${\varvec{\gamma }}, \delta _1, \ldots , \delta _n$ are the copula and marginal parameters, respectively. For more details about copulas, we refer the interested reader to the textbooks by Joe (1997) and Nelsen (1999), while Cherubini et al. (2004) provide a detailed discussion of copula techniques for financial applications. It follows from (6) that the log-likelihood function for the joint conditional distribution $H_t(\cdot , {\varvec{\theta }})$ is given by

$$\begin{aligned} l({\varvec{\theta }}) =\sum _{t=1}^T\log \bigg ( c\big (F_{1,t}(z_{1}; \delta _1), \ldots , F_{n,t}(z_{n,t}; \delta _n); {\varvec{\gamma }}\big ) \bigg )+\sum _{t=1}^T\sum _{i=1}^n \log f_i (z_{i,t}; \delta _i) \end{aligned}$$

where $c$ is the copula density function, whereas $f_i$ are the marginal densities. The maximization of the previous log-likelihood with respect to the parameters (${\varvec{\gamma }}, \delta _1, \ldots , \delta _n$) can be made in 1 step, or in several steps by partitioning of the parameter vector into separate parameters for each margin and the parameters for the copula. This multi-step procedure is known as the method of Inference Functions for Margins (IFM), see Joe (1997) for details.

It is rather straightforward to show that the previous DCC model can be represented as a special case within a more general copula framework,

$$\begin{aligned} {\mathbf {Y}}_t= & {} {\varvec{\mu }}_t +{\mathbf {D}}_t {\mathbf {z}}_t, \quad \text {where}\quad {\mathbf {z}}_t \sim H_t(z_{1,t}, \ldots , z_{n,t}; {\varvec{\theta }}), \quad \text {and}\quad \\ {\mathbf {z}}_t\sim & {} H_t \equiv C_t^{Normal}(F_{1,t}^{Normal}(z_{1}; \delta _1), \ldots , F_{n,t}^{Normal}(z_{n,t}; \delta _n); {\mathbf {R}}_t) \end{aligned}$$

see e.g. Patton (2006a, b) and Fantazzini (2008, 2009b) for more details.

The copula approach allows us to consider more general cases than a multivariate normal DCC model. For example, if we consider skewed-t distributions for the marginals, like those proposed by Hansen (1994) or Fernandez and Steel (1998), then a multivariate model allowing for marginal skewness, kurtosis and normal dependence can be expressed as follows:

$$\begin{aligned} {\mathbf {Y}}_t= & {} {\varvec{\mu }}_t +{\mathbf {D}}_t {\mathbf {z}}_t \\ {\mathbf {z}}_t\sim & {} H_t \equiv C_t^{Normal}\left( F_{1,t}^{Skewed-t}(z_{1,t}; \delta _1), \ldots , F_{n,t}^{Skewed-t}(z_{n,t}; \delta _n); {\mathbf {R}}_t\right) \end{aligned}$$

where $F_{i,t}^{Skewed-t}$ is the cumulative distribution function of the marginal Skewed-t, and ${\mathbf {R}}_t$ can be made constant or time-varying, as in the constant conditional correlation (CCC) model or in the DCC model, respectively.

If we suppose that our assets have symmetric tail dependence, we can use a Student’s t copula, instead,

$$\begin{aligned} {\mathbf {Y}}_t= & {} {\varvec{\mu }}_t +{\mathbf {D}}_t {\mathbf {z}}_t \\ {\mathbf {z}}_t\sim & {} H_t \equiv C_t^{Student's T}\left( F_{1,t}^{Skewed-t}(z_{1}; \delta _1), \ldots , F_{n,t}^{Skewed-t}(z_{n,t}; \delta _n); {\mathbf {R}}_t, \nu \right) \end{aligned}$$

where $\nu$ denotes the degrees of freedom of the t-copula.