# Value-at-risk and related measures for the Bitcoin

Stavros Stavroyiannis (Department of Accounting and Finance, Technological Educational Institute of Peloponnese, Kalamata, Greece)

ISSN: 1526-5943

Publication date: 19 March 2018

## Abstract

### Purpose

The purpose of this paper is to examine the value-at-risk and related measures for the Bitcoin and to compare the findings with Standard and Poor’s SP500 Index, and the gold spot price time series.

### Design/methodology/approach

A GJR-GARCH model has been implemented, in which the residuals follow the standardized Pearson type-IV distribution. A large variety of value-at-risk measures and backtesting criteria are implemented.

### Findings

Bitcoin is a highly volatile currency violating the value-at-risk measures more than the other assets. With respect to the Basel Committee on Banking Supervision Accords, a Bitcoin investor is subjected to higher capital requirements and capital allocation ratio.

### Practical implications

The risk of an investor holding Bitcoins is measured and quantified via the regulatory framework practices.

### Originality/value

This paper is the first comprehensive approach to the risk properties of Bitcoin.

## Keywords

#### Citation

Stavroyiannis, S. (2018), "Value-at-risk and related measures for the Bitcoin", Journal of Risk Finance, Vol. 19 No. 2, pp. 127-136. https://doi.org/10.1108/JRF-07-2017-0115

### Publisher

:

Emerald Publishing Limited

## 1. Introduction

The recent global financial crisis and the European sovereign debt crisis have urged investors and financial institutions to identify assets to protect their investments in case of extreme market conditions, or systemic risk. With respect to the recent literature (Baur and Lucey, 2010; Baur and McDermott, 2010, Baur and McDermott, 2016) which distinguishes the terms diversifier, hedge and safe haven, gold is the most promising example of a safe haven asset to help investors diversify the risk in case of a crisis. However, owing to the “Internet of Things” through the interconnection of physical and smart devices, a large variety of virtual currencies and cryptocurrencies have started to emerge during the past years. The term “virtual currency” initiated in the online gaming communities including the Bitcoin (BTC, or XBT according to the ISO 4217 standard), in the Magic the Gathering Online Exchange (MtGox) and the Linden Dollar in the Second Life game. These include Litecoin (October 2011), Peercoin (August 2012), Primecoin (July 2013, developed by the same person who developed Peercoin), Dogecoin (December 2013), Darkcoin (January 2014, a more secretive version of Litecoin), Ethereum (2015) and many others. The total market capitalization of digital currencies and cryptocurrencies, as of 17 July 2017, is about US$66bn, with BTC domination of 46 per cent, after a deep decrease from 94 per cent in 2013, followed by Ethereum with 23 per cent. During the past years BTC has attracted significant attention to literature with respect to the formation of the price (Buchholz et al., 2012; Kristoufek, 2013; van Wijk, 2013; Bouoiyour et al., 2014; Ciaian et al., 2016; Bouri et al., 2017), demand and supply issues (Buchholz et al., 2012; Bouoiyour et al., 2014) and univariate GARCH modelling (Dyhrberg, 2016a, 2016b). Buchholz et al. (2012) find that BTC price is driven by the interaction between supply and demand, and Kristoufek, (2013) shows that the price formation of BTC cannot be explained by standard economic theories. In a similar manner, Ciaian et al. (2016) argue that macro-financial developments do not drive BTC price in the long run. They find a significant impact of global macro-financial development, captured by the Dow Jones Index, exchange rate and oil price, only in the short run, but they do not determine BTC price in the long run. The sudden change of BTC price is actually the first bubble formation of a virtual currency. Phillips et al. (2013) identify two bubbles for the BTC/US$ rate daily returns during 2013 using the generalized Sup ADF test and another mini bubble during 2012.

The risk associated with an investment has been the subject of a variety of Accords from the Basel Committee on Banking Supervision (BCBS). Basel II (Basel Committee on Banking Supervision, 2004; Basel Committee on Banking Supervision, 2011a) allowed banks to use their internal risk models to quantify the risk-based capital requirement with respect to the 99th percentile of one-tailed confidence interval for a holding period of 10 trading days. Value-at-Risk (VaR) is defined as an amount lost on a financial asset, given a probability and a fixed number of days, and has become a simple standard tool to quantify market risk by a single number. According to De Grauwe (2009), the Basel Accords have failed to provide stability to the banking sector because the risks linked with universal banks are tail risks associated with bubbles and crises. Expected shortfall (ES) is a natural alternative of value-at-risk, fulfilling all four axioms of a coherent risk measure (Artzner et al. 1997, 1999). It belongs to the category of spectral risk measures which are not elicitable unless they reduce to minus the expected value (Acerbi and Tasche), and this has been taken into consideration by the BCBS (Basel Committee on Banking Supervision, 2011b; Basel Committee on Banking Supervision, 2012) to encompass it in future Accords.

The main focus of this paper and its contribution to the existing literature is to examine risk measures for the BTC, and compare these measures with Standard and Poor’s SP500 as a proxy index, the Brent crude oil spot price and the gold spot price. All of these assets have their own intrinsic characteristics and they are all valued in US\$. Firstly, we examine the econometric properties of these assets using a GARCH methodology, and the performance of the model is tested using a variety of backtesting criteria. Secondly, we use VaR, which has been used by the BCBS as a measure of market risk in the trading book for two decades in several Accords. Thirdly, we use several measures for the ES, which has emerged as a natural alternative of VaR and is going to be implemented in the new Accords. The remaining of the paper is organized as follows. Section 2 describes the data and the econometric methodology used, Section 3 presents the results and discusses the findings and Section 4 concludes.

## 2. Econometric methodology

### 2.1 The data

BTC price data are sourced from Coindesk Price Index from the 1 July 2013 to 10 July 2017 with daily observations, consisting of 1,471 data entries, Standard and Poor’s SP500 Index from finance.yahoo.com and the gold spot price from the World Gold Council (www.gold.org). Owing to the fact that BTC trades seven days per week, we follow the procedure in Dyhrberg (2016a; 2016b), and the missing values in the rest of the series are filled via sequential linear interpolation for the weekend data. The returns are defined via the successive logarithmic differences of the close price multiplied by 100, so that risk measures translate directly to percentage.

### 2.2 The model

The dynamics of the GJR(1, 1) model (Glosten et al., 1993) is expressed as follows:

(1) rt=μ+φrt1+εt
(2) σt2=ω+aεt12+γεt12I(εt1<0)+βσt12
where a and β are the ARCH and GARCH coefficients, γ is the leverage effect capturing the asymmetry effect in return volatility and I is an indicator function taking the value 1 when εt−1 < 0 and zero otherwise. The residuals follow the standardized Pearson type-IV distribution (Stavroyiannis et al., 2012, Stavroyiannis et al., 2013, Stavroyiannis and Zarangas, 2013):
(3) P(z)=σ^Γ(m+12)πΓ(m2)|Γ(m+12+iν2)Γ(m+12)|2exp(ν tan1(σ^z+μ^) )(1+(σ^z+μ^)2)m+12
(4) μ^=νm1
(5) σ^=1m2(1+ν2(m1)2)
where Γ(.) is the gamma function and i the imaginary unit. The log-likelihood (LL) of the model is:
(6) LL=NlnCi=1N[12lnσt2+(m+12)ln(1+(σ^z+μ^)2)+ν tan1(σ^z+μ^)]
(7) lnC=ln[Γ(m+12)/Γ(m2)]+12lnσ^12lnπ+12ln|Γ(m+12+iν2)/Γ(m+12)|

All programming has been performed with the Matlab, MathWorks® computing language.

## 3. Results of the econometric methodology

### 3.1 Descriptive statistics and stylized facts

The time series under consideration is shown in Figure 1 for SP500 (black line), gold (red line) and BTC (green line). The striking observation is the volatile price of the BTC.

The descriptive statistics on the logarithmic differences show that all series exhibit statistically significant skeweness and kurtosis, and the Jarque–Bera test indicates deviation from normality. The ARCH test shows that the series exhibit heteroskedasticity, and the Ljung–Box test for both the returns and the squared returns show the presence of autocorrelation. The mean is positive for SP500 and BTC, and negative for gold. A striking difference of BTC with respect to the other series is the fact that it displays an order of magnitude of higher return, along with a large standard deviation (Table I).

### 3.2 Univariate GJR-GARCH results

The results of the univariate GARCH methodology are shown in Table II. The constant in the mean equation is not statistically significant, and the constant in variance is statistically significant for SP500 and BTC, with BTC having a larger value. The parameters a and β reflect the short-run dynamics of the volatility. The ARCH coefficient a is statistically significant only for BTC which means that volatility reacts to quite intensively to market movements. The GARCH coefficient β is statistically significant for all series, an indication that a shock to the conditional variance takes time to die-out. The leverage coefficient γ is statistically significant for SP500 and BTC. The results indicate that the structural parameters for BTC are different from these of the other series owing to the high volatility of the BTC price.

### 3.3 Value-at-Risk and backtesting criteria

Value-at-Risk for a specific confidence level is the quantile that solves the equation VaRθ = −inf{q ∈ R|F(x) ≥ q}, where F is the cumulative distribution of the probability density function of Eq( ). The VaR levels for the long and short position for the θ confidence level are identified as:

(8) VaR(long)=μt+FPIV1(1θ)σt
(9) VaR(short)=μt+FPIV1(θ)σt
where FPIV1 the inverse of the cumulative distribution function at the specific confidence level. The positioning of the VaR levels for the long and short position of the assets is shown in Figure 2.

Each time an observation exceeds the VaR border it is called a VaR violation, or VaR break. For backtesting criteria, we begin by using the success/failure which counts the number of VaR violations, and this should be as close as possible to the number of VaR breaks specified by the confidence level. Because rarely the exact amount suggested by the confidence level is observed, a widely used test based on failure rates is the proportion of failures (POF) by Kupiec (1995). Measuring whether the number of violations is consistent with the confidence level, under null hypothesis that the model is correct, the number of violations follows the binomial distribution. The Kupiec test is conducted as a likelihood-ratio (LR) test, which is asymptotically chi-squared distributed with one degree of freedom. The Christoffersen (1998) interval forecast test (unconditional coverage) is used to test whether the exceptions are spread evenly over time or they form clustering. The relevant statistic is an LR test asymptotically chi-squared distributed with one degree of freedom. If these two criteria are joined, then the Christoffersen and Pelletier (2004) conditional coverage test is achieved via an LR test which is asymptotically chi-squared distributed with two degrees of freedom.

The results of the tests, the success–failure ratio (S/F ratio), the p-value of the Kupiec test (POF p-value), the p-value for the Christoffersen unconditional coverage (Unconditional p-value) and the p-value for the Christoffersen conditional coverage test (Conditional p-value) are shown for the long and short position in Table III, for SP500 (Panel A), gold (Panel B) and BTC (Panel C) for three confidence levels, 0.95, 0.975 and 0.99. All tests indicate that the model used is accurate especially at the 0.99 confidence level which is suggested by the BCBS in the Accords. It is evident that with respect to the unconditional and conditional coverage tests, the p-values for the BTC are lower for both the long and short position indicating more VaR breaks compared with the rest of the assets.

### 3.4 Expected shortfall measures and regulatory loss functions

The ES measures can be calculated using a variety of methods. The simplest approach which does not require GARCH modelling is given by Acerbi and Tasche (2002), where the ES measure is defined as the empirical average of the data points exceeding the empirical quantile of the data:

(10) ESnθ(X)=1wi=1wXi:n=(Average of at least θ% outcomes Xi)

Two measures used in the software are the average of the data points, exceeding the empirical quantile of the data which is calculated via the GARCH filtering (ESF1), and the ratio of the ES to the VaR level (Hendricks, 1996) defined as ESF2. In cases of few data points, the VaR and ES methodology for high confidence levels might not be applicable, and one solution is to use the theoretical ES or theoretical tail conditional expectation (TTCE) emerging from the cumulative distribution function of the probability density of equation (3). This was shown to be (Stavroyiannis, 2016):

(11) x y f ( y ) d y = Γ ( m + 1 2 ) σ ( m 1 ) π Γ ( m 2 ) | Γ ( m + 1 2 + i ν 2 ) Γ ( m + 1 2 ) | 2 exp ( ν   tan 1 ( σ ^ x + μ ^ )   ) ( 1 + ( σ ^ x + μ ^ ) 2 ) ( m + 1 ) / 2

Two more tests not incorporated in the software and rarely seen in the literature are the Lopez and Sarma approaches. Lopez (1999) suggested the development of a loss function for backtesting different models and proposed to measure the accuracy of the VaR forecasts on the basis of the distance between the observed returns rt, and the forecasted VaR values if a violation occurs:

(12) It={1+(rtVaRt)2, if rt<VaRt0               , if rtVaRt}

A VaR model is penalized when an exception takes place. Hence, the model is preferred to other candidate models if it yields a lower total loss value. This is defined as the sum of these penalty scores. This function incorporates both the cumulative number of exceptions and their magnitude. However, a model that does not generate any violation is deemed the most adequate because the sum is zero. Thus, the risk models must be first filtered by using the aforementioned backtesting measures too.

Sarma et al. (2003), combining the advantages of a loss function with those of backtesting measures, suggested a two-stage backtesting procedure. When multiple risk models meet the backtesting statistical criteria of VaR evaluation, a loss function is brought into play to judge statistically the differences among VaR forecasts. In the first stage, the statistical accuracy of the models is tested by examining whether the mean number of violations is not statistically significantly different from that expected and whether these violations are independently distributed. In the second stage, they propose use of what they term the firm’s loss function, i.e. penalizing failures but also imposing a penalty reflecting the cost of capital suffered on other days, the regulatory loss function:

(13) It={(rtVaRt)2, if rt<VaRt0          , if rtVaRt}

The Sarma methodology addresses the different conceptual approach between the regulators and the risk managers, regarding the aiming of the market risk management tool. Regulators are interested in the number of VaR breaks and the size of the non-covered losses, whereas risk managers disagree on safety and profit maximization. The critical argument is that an excessively high VaR forces them to hold too much capital, imposing the opportunity cost of capital upon firms. The results for the ES measures (ES, ESF1, ESF2 and TTCE) and regulatory loss functions (Lopez, Sarma) are shown in Table IV for the SP500 (Panel A), gold (Panel B) and BTC (Panel C).

It is evident from all risk measures that the asset capital allocation for BTC is higher than the other two series, and the regulatory loss functions results of Lopez and Sarma differ by an order of magnitude.

## 4. Conclusion

The post-crisis financial system is bearing changes and a disputed issue is whether or not digital currencies and cryptocurrencies will have a place in the investors’ positions. The Basel requirements are guided towards a strengthening of the financial system against systemic risks, and this will have implications on risk management and the cost of capital and liquidity. BTC has attracted significant attention after the price increase, but lacks regulation and financial instruments. If finally BTC will encounter a hesitant penetration to the investors, the next step is the regulatory risk management framework, as financial operations entail risks. The capital adequacy rules set a minimum requirement on the size of the financial buffer based on the assumed risk. This paper provides a quantification of such measures indicating that BTC is subject to a higher risk, and therefore, to higher sufficient buffer and risk capital to cover potential losses.

## Figures

#### Figure 1.

Time series under consideration, SP500 (top), gold (middle) and BTC (bottom)

#### Figure 2.

VaR levels and returns for SP500 (top), gold (middle) and BTC (bottom) for θ = (0.01,0.99)

## Table I.

Descriptive statistics of the return series

Asset Mean SD Skew. Kurt. J.B. ARCH(12) LB(12) LB(12)-2
SP500 0.028 0.606 −0.192* 8.317* 1740* 21.05* 12.49 361.9*
Gold −0.002 0.749 0.266* 8.232* 1694* 3.341* 22.38* 42.55*
BTC 0.227 4.185 −0.369* 14.32* 7879* 34.75* 38.60* 518.2*

Notes:

J.B. is the statistic for the null of normality; ARCH(12) denotes the test for heteroskedasticity, LB(12) denotes the Ljung–Box test statistic for serial correlation, LB(12)-2 denotes the Ljung–Box test statistic for serial correlation on the squared residuals with 12 lags, respectively.

*

denotes statistical significance at the 5% critical level

## Table II.

Results of the univariate GJR models

Coefficient SP500 Gold BTC
μ 0.0298 −0.0091 0.0940
ω 0.0232* 0.0160 0.3715*
a (Arch) 0.0000 0.0421 0.2711*
β (Beta1) 0.8464* 0.9695* 0.7845*
γ (Gamma) 0.3095* −0.0231 −0.1070*
ξ (Asymmetry) −0.0423 0.0455 0.2882*
ν (Tail) 2.5967* 2.3298* 3.2076*

Note:

*

Denotes statistical significance at the 5% critical level

## Table III.

Backtesting criteria

Quantile S/F ratio POF test
p-value
Unconditional test p-value Conditional
test p-value
Panel A: SP500
Short position
0.9500 0.9333 0.00517 0.84791 0.01969
0.9750 0.9721 0.48563 0.13366 0.25466
0.9900 0.9931 0.19100 1.00000 0.42530
Long position
0.0500 0.0700 0.00084 0.75900 0.00363
0.0250 0.0448 0.00001 0.54845 0.00005
0.0100 0.0129 0.28080 1.00000 0.55900
Panel B: Gold
Short position
0.95000 0.9326 0.00367 0.04019 0.00179
0.97500 0.9646 0.01638 0.41404 0.04018
0.99000 0.9884 0.55634 1.00000 0.84111
Long position
0.0500 0.0591 0.11594 0.21298 0.13383
0.0250 0.0326 0.07245 0.61652 0.17575
0.0100 0.0102 0.93753 1.00000 0.99693
Panel C: BTC
Short position
0.95000 0.9353 0.01363 0.44147 0.03548
0.97500 0.9646 0.01638 0.41404 0.04018
0.99000 0.9877 0.40333 1.00000 0.70527
Long position
0.0500 0.0571 0.21877 0.02344 0.03601
0.0250 0.0367 0.00700 0.05451 0.00415
0.0100 0.0129 0.28080 0.24348 0.28316

## Table IV.

Expected shortfall measures and regulatory loss functions

Quantile ES ESF1 ESF2 TTCE Lopez Sarma
Panel A: SP500
Short position
0.9500 1.5144 1.2467 1.4146 2.1480 120.0424 22.0424
0.9750 1.8304 1.5305 1.2855 2.8816 47.2826 6.2826
0.9900 2.2624 1.4971 1.1336 4.1808 10.5298 0.5298
Long position
0.0500 −1.5292 −1.2481 1.7455 −2.0871 153.3396 50.3396
0.0250 −1.8950 −1.4458 1.4724 −2.7867 92.5487 26.5487
0.0100 −2.4146 −1.7551 1.3928 −4.0237 29.7573 10.7573
Panel B: Gold
Short position
0.95000 1.8712 1.6499 1.6399 1.8403 187.7320 88.7320
0.97500 2.3591 2.0836 1.4576 2.5191 99.6774 47.6774
0.99000 3.0438 2.8596 1.2764 3.7693 32.2350 15.2350
Long position
0.0500 −1.8284 −1.7014 1.6382 −1.9095 152.6745 65.6745
0.0250 −2.2703 −2.0680 1.4081 −2.6290 77.6728 29.6728
0.0100 −2.7774 −2.6614 1.2298 −3.9561 20.7883 5.7883
Panel C: BTC
Short position
0.95000 9.8573 7.3008 1.5059 2.0957 14,59.6 1364.6
0.97500 12.700 8.0557 1.3808 2.6602 811.64 759.64
0.99000 17.314 9.8422 1.3124 3.5852 359.15 341.15
Long position
0.0500 −11.002 −9.2051 1.8878 −2.4060 2844.8 2760.8
0.0250 −14.668 −10.596 1.6699 −3.1298 1753.4 1699.4
0.0100 −20.046 −13.052 1.7398 −4.3309 937.32 918.32

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## Acknowledgements

This paper forms part of a special section “Digital currencies”, guest edited by Paolo Tasca.

## Corresponding author

Stavros Stavroyiannis can be contacted at: computmath@gmail.com