On the Return Distributions of a Basket of Cryptocurrencies and Subsequent Implications
(First Draft: May 14, 2021)

The augmented Dickey-Fuller test (ADF) is performed considering the autoregressive model for the CC return time series, $y_{t}$, of each CC³³3 Note, the returns $r_{t}$ are calculated in this notation according to $r_{t}=y_{t}-y_{t-1}$.:

With drift coefficient $c$, deterministic trend coefficient $\delta$, AR(1) coefficient $\varphi$ and coefficients $\beta_{i}$ for the lag terms $i=1,\ldots,p$ up to the order $p=10$. In Eq. (1) $\epsilon_{t}$ denotes the innovation process. The aim of the test is to check the hypothesis of trend stationarity, i.e. $\delta=0$ is the null hypothesis, in the tables denoted by H0, see e.g. Wooldridge (2020).

The Heatmap in Fig. 1 visualizes the structure of the fitted autoregressive model Eq. (1).

We find the deterministic trend coefficient $\delta$ to be comparable to zero in all CCs considered. The results in Tab. 3 in the first four columns provide a deeper insight. For the vast majority of CCs, the $p$-values are comfortably high, so that a rejection of the null hypothesis of trend stationarity is not indicated here. But as can be seen in the table the ADF-Test rejects the null hypothesis for the CCs FLO (FLO), Quark (QRK) and Worldcoin (WDC) on a 5% confidence level. At this point we take a closer look at the corresponding trend coefficients: $\delta_{\text{FLO}}=1.6$e-03, $\delta_{\text{QRK}}=1.5$e-04 and $\delta_{\text{WDC}}=1.0$e-04. Since all the coefficients $\delta$ are close to zero, the influence of a possible trend is likely to be of significantly less importance. Therefore, in the following, we assume trend stationarity ($\delta=0$) for the time series of CCs.

The third column in the Heatmap in Fig. 1 illustrates the value of $\varphi$. We find the parameter $\varphi$ to be greater than 0.9 for all CCs and, for the most part, clearly close to 1. The latter is an important condition to be fulfilled for the assumption of a random walk ($\varphi=1$).

The augmented Dickey-Fuller test was performed for lags $p$ up to order ten. More complicated dynamics with serial correlation make themselves noticeable in the analysis by the fact that the coefficients of the terms corresponding to the lags are clearly different from zero. We find the absolute values of the coefficients (that means $\left|\beta_{i}\right|$) to be close to zero. In fact, the majority of the absolute coefficients $\left|\beta_{i}\right|$ are clearly smaller than 0.2 as an upper limit⁴⁴4By similar argumentation as in Wooldridge (2020, Chapter 11 therein) a repeated substitution causes the effectiveness of the corresponding terms to fall below the 5% mark in the next time step and thus become largely insignificant. and a basic model $y_{t}=c+y_{t-1}+\epsilon_{t}$ can be assumed (in Tab. 3 third column denoted with "G"). Only a few single coefficients exceed the value 0.2 by a small margin and in these cases a model with a slightly influencing lag structure $y_{t}=c+y_{t-1}+\bar{\beta}\times(\text{Lag-Structure})+\epsilon_{t}$ with an average coefficient $\bar{\beta}=0.03$ could be guessed at. By marking the corresponding CCs with "L", Tab. 3 shows for which CCs this is the case.

Due to the observed insignificance of the lag stucture, we assume a basic model "G" in these cases as well and expect statistical inaccuracies to distort the result due to the limited length of the time series. We therefore postulate a possible serial correlation in the data sets to be of minor importance. Thus, the detailed analysis of the results of the augmented Dickey-Fuller test suggests that the model $r_{t}=c+\epsilon_{t}$ cannot not be rejected for the returns $r_{t}$. Here, $c$ denotes the individual time-constant drift term for each CC and, as above, $\epsilon_{t}$ denotes the innovation process.

The ARCH-Test is performed for time shifts $q$ up to the order of ten. Up to this lag, the null hypothesis that the innovation process of returns is homoscedastic could not be rejected for most CCs (see Tab. 3), i.e. the basic model $\epsilon_{t}=\sigma z_{t}$ with constant volatility $\sigma$ and $z_{t}$ being an independent and identical distributed process with mean 0 and variance 1 could not be rejected for the majority of CCs. Note, that the results found in literature show a heterogeneous picture and we do not find ARCG effects in contrast to other related studies (Peng et al., 2018; Dyhrberg, 2016; Avital et al., 2014). After all, only 13 of 28 time series examined show ARCH effects and only in 8 cases clearly. The differences in studies may derive from different sampling frequencies or the differently chosen time period or its length and show that the design of the data collection may have an influence on the results.

The aforementioned results would justify the following calculation: $\text{E}[r_{t}]=\mu=c$ and $\text{Var}[r_{t}]=\sigma^{2}$ for the corresponding CCs. Estimates for the mean and the variance of the individual returns are noted in Tab. 2. Their calculation is also justified with the combined consideration of the test results described above.

Combining both tests, we have noted a characterization of the return distribution in the last column of Tab. 3. For the majority of CCs, independent and identical distributed (IID) returns can be assumed. Another part is approximately independent and identical distributed ($\approx$ IID), because either the lag structure is less important in the ADF model or the rejection of homoscedasticity based on the $p$-value is only weakly justified. For eight CCs the assumption of IID returns is clearly rejected.

Note, that the test for independently identical ditstributed returns could have been carried out with a turning point test (Bienaymé, 1874; Kendall and Stuart, 1977). However, as we are interested in a deeper analysis of the possible serial correlation in our data sets, we use a combination of the augmented Dickey-Fuller test and the ARCH-Test instead.

All tests carried out so far, do not reject the assumption that the returns obey an essentially symmetrical, unimodal distribution. Furthermore, for the majority of CCs the assumption of IID or nearly IID holds. In the first case (IID), the modeling of the empirical distribution with a distribution function is justified. In the second case ($\approx$ IID), the model represents a coarser approximation. In the latter case, if the assumption of IID was to be rejected, the distribution function can only be used as a rough approximation and must be – just like the case ($\approx$ IID) – examined more precisely and critically in individual cases as we show in the following section.

In the following we use standard distance measures to determine and compare the model qualities of N, GED, GLD0, GLD3 and SDI for the individual CCs. There are several distance measures available which are suitable to measure the "discrepancy" between the empirical distribution function and a modelled distribution function. The distance measures from Cramér (1928) von Mises (1931) ($W^{2}$-Distance), Anderson and Darling (1952, 1954) ($A^{2}$-Distance) and Kolmogorov (1933) Smirnov (1936, 1948) (KS-Distance) are widely used in literature. A brief summary of the used distance measures are given in A.

In Tab. 4 we summarize the results for the Anderson Darling distance ($A^{2}$). The results for the Kolmogorov Smirnov distance (KS) and the Cramér von Mises distance ($W^{2}$) are compiled in Tab. 11 and Tab. 10 in A.

The values of the various distance measures show that the GED is least suitable to model the empirical distribution function. This may derive from the fact that the GED contains a fundamental skewness, which can only be influenced slightly via a parameter selection. Further, as soon as the shape parameter becomes different from zero a fundamental change of the distribution model occurs and the definition interval on the $x$-axis becomes restricted. Additionally, associated with a change of sign of the shape parameter there is a fundamental change of the distribution model as well and an abrupt change of the sign of the upper (or lower) bound on the $x$-axis, see e.g. Embrechts et al. (1997). In the present case, these properties of the GED make it difficult to precisely adapt the distribution to the data set.

On closer inspection of the calculated distances, the N does not appear to be suitable as a model either, since it is neither suitable for the modeling of empirical distribution functions with fat tails nor for those with a slight skew. Overall, the GLD0 shows significantly smaller distances across all CCs, but is also not ideally suited, as it is completely symmetrical and is therefore not able to model any slight skewness. The best results in terms of the smallest distance can be achieved with the GLD3 and with the SDI. When comparing all CCs, the corresponding distances are very close to each other. If only the Cramér von Mises distance and the Kolmogorov Smirnov distance are considered, see A, it turns out that around half of the empirical distributions of CC returns can be modeled with one or the other distribution. But as soon as more attention is paid to the tail, i.e. if deviations in the tail area are to be weighted more heavily in order to account for tail risks, and the Anderson Darling distance $A^{2}$ is considered, the share of CCs, for which the SDI is the most suitable model, predominates.

This result ties in with the results of Majoros and Zempléni (2018); Kakinaka and Umeno (2020). Using intraday and daily time series for a small sample of CCs they show the SDI family to be the best choice for modeling the empirical distribution function of intraday and daily returns of CCs.

For a broader sample of CCs we found that the SDI is, on average, much better suited to model both slight skewness and pronounced tails in the empirical distribution function. Therefore we use the SDI for all CCs to model the distribution function of returns.

Tab. 5 shows the results of the parameter estimation for the SDI $S(\alpha,\beta,\delta,\gamma)$.

For the individual parameters of the SDI, the 95% scatter intervals are provided, as well. The latter can be determined from the covariance matrix of the estimated parameters. The covariance matrix of the parameter estimates is a matrix in which the off-diagonal element $(i,j)$ resembles the covariance between the estimates of the $i$-th parameter and the $j$-th parameter. For the CC FLO, these scatter intervals cannot be determined numerically using the data set at hand. This is due to the fact that the corresponding empirical distribution on the far right shows a very long, pronounced tail and is strongly skewed to the right. The estimated parameter $\hat{\beta}$ of the SDI takes the value of 1 accordingly, see Tab. 5. It may be the case that the single right tail data point, i.e. the return in period 62, represents an outlier, which is difficult to determine and correct afterwards without any further knowledge.

On average, the estimated parameter $\hat{\alpha}$ exceeds 1.5. With parameter $\alpha$ increasing to its limit value of 2.0, it can be seen that the distribution function becomes similar to the N and the skewness parameter ($\beta\neq 0$) becomes increasingly insignificant. The parameters $\beta$ and $\alpha$ of the SDI are mutually dependent and, as described above, the meaning of $\beta$ decreases when $\alpha$ increases. So it is generally difficult to infer the skewness from the value $\beta$ alone. A relative comparison of the distributions with regard to the skewness is only possible if $\alpha$ has the same value. Hence, for a more precise analysis of a distribution’s skewness, other methods are necessary. In this manner, we used the SIG-Test as an example and note the results in Tab. 2.

For some CCs and the EWCI^- index, Fig. 2 shows the empirical densities and the density of the corresponding SDI in comparison.

In addition, the complete goodness of fit test according to Anderson Darling was carried out. The last two columns of Tab. 5 show that for almost all CCs with very high significance (high $p$-values), the null hypothesis that the adjusted SDI models the data set cannot be rejected. Only for the CC Infinitecoin (IFC) this assumption is rejected at the 5% level. This is probably due to the fact that the empirical distribution suggests a slight bimodal distribution. This peculiarity of the CC Infinitecoin is also indicated in the results of the HDS-Test in Tab. 2.

Overall, it can be stated that the SDI family represents a suitable framework for modeling the distribution function of CC returns. We will make use of this finding for the assessment of tail risks and the comparison of our results with other modeling approaches.

Tab. 7 illustrates the results of the risk assessment for the most common confidence levels found in literature and regulatory requirements. The Value at Risk for the observed CCs are shown for the various model.

Overall, it can be seen that in the overwhelming number of individual cases, the assessment of risk with the tail model (GPD) is closer to the empirical Value at Risk values. This applies to the lower confidence levels in particular, but even with high confidence level of 99.9%, better estimates are possible in individual cases compared to the body model. This becomes apparent from the statistical parameters of the deviation analysis shown in Tab. 8 as well. The mean value and standard deviation over the set of CCs are shown. For this purpose, the deviation between the modeled variable and the corresponding empirical Value at Risks was determined. It turns out that, on average, adjusting the GPD as tail model leads to better risk estimates.

When comparing CCs with each other, a heterogeneous picture emerges, see Tab. 7. If the empirical Value at Risk for the 99.9% confidence interval is taken as a measure, two subgroups can be defined for the cut-off value $\text{VaR}_{99.9\%}\approx$ 100%. One group possesses a significantly lower tail risk ($\text{VaR}_{99.9\%}<$ 100%) as the corresponding $\hat{\xi}$ values in Tab. 6 indicate. The other group ($\text{VaR}_{99.9\%}>$ 100 %) partly embodies a significantly higher tail risk. Correspondingly, large $\hat{\xi}$ values can be determined for these CCs.

Fig. 3 shows the empirical distribution function for the EWCI^- and Bitcoin (BTC), the SDI as a body model and the GPD as a tail model in comparison. The graphics on the right portray an enlargement of the loss area. Particularly in this region, the GPD models the empirical distribution function very well. Considering the afore conducted analysis, it can be stated that the GPD is ideally suited to carry out risk assessment at high quantiles. Therefore, we exclusively consider the GPD to estimate the Conditional Value at Risk as a further risk indicator in the following.

The following Tab. 9 shows the Conditional Value at Risk calculated with the tail model (see Tab. 6) for the individual CCs. The calculation of the Conditional Value at Risk can be conducted using the mean excess function of the GPD:

with $v$ greater than the lower bound of the definition interval of the GPD considering the loss tail and the parameters $\sigma,\xi$ of the GPD. The mean excess function is bound to the restrictions: $\xi<1$, and $\sigma+\xi v>0$, see e.g. Embrechts et al. (1997, Theorem 3.4.13). In Tab. 9 we used Eq. (5) to estimate the Conditional Value at Risk for each CCs setting $v=\text{VaR}_{p\%}$.

Once more, the grouping of CCs described above can be seen. A group of CCs with a fat tail and therefore higher tail risk can be distinguished from a group with moderate risk, see Tab. 9.

Furthermore, two peculiarities are noticeable concerning the CCs Feathercoin (FTC) and Nxt (NXT). For both CCs, the tail is modeled on a small number of data points which have been assigned to the tail. This individual property of the data set deriving from the random distribution of the data in the tail area, is assumed to be given and, as already noted, is not corrected. In particular, when estimating the parameter $\xi$, small samples lead to large statistical errors. Regrading the conspicuous CCs, it can be observed that the parameter is very close to 1 in one case (FTC) and even higher in the other case (NXT), see Tab. 6. As a result, the calculation of the Conditional Value at Risk for the CC NXT is not possible and must be discarded, cf. Eq. (5) and the restriction $\xi<1$. On the other hand, the calculation of the FTC with $\xi\approx 1$ has to be questioned critically. Hence, the Conditional Value at Risk may only be a rough estimate in this case.