Methodology
CRIX formula:
The CRIX is a market index and follows for the derivation the Laspeyres Index. The index of Laspeyres is defined as \[INDEX_t^{Laspeyres} = \frac{\sum_i P_{it} Q_{i0}}{\sum_i P_{i0} Q_{i0}}\] with \(P_{it}\) the price of the crypto \(i\) at time point \(t\) and \(Q_{i0}\) the amount of crypto \(i\) at time point \(0\). \(P_{i0}\) is the price at time point \(0\). The final index formula is a modification of the Laspeyres index, given by \[INDEX_t^{CRIX} = \frac{\sum_i MV_{it}}{Divisor},\] where \(MV_{it}\) is the market capitalization of the crypto \(i\) at time point \(t\). The \(Divisor\) ensures that the changes in the amount of coins of a crypto doesn’t affect the value of the CRIX. At the starting point of the CRIX is the \(Divisor\) chosen such that \[Divisor = \frac{\sum_i MV_{i0}}{1000}.\] The starting value of the CRIX is therefore chosen to be \(1000\). Whenever the amount of coins of a crypto changes, the Divisor has to be adjusted. This shall ensure that just price changes cause changes in the value of the CRIX. The applied formula is \[\frac{MV_{i,t-1}}{Divisor_{t-1}} = INDEX_{t-1}^{CRIX} = INDEX_{t}^{CRIX} = \frac{MV_{i,t}}{Divisor_{t}},\] where \(Divisor_{t-1}\) is the \(Divisor\) directly in front of the change in the amount of coins and \(Divisor_t\) directly afterwards. Therefore, it is ensured that the CRIX starts again with the same value than directly before the change in the amount of coins.
Liquidity rule: It can happen, that a crypto has a high market capitalization and therefore shall be part of the CRIX. But if it is not traded frequently, it can’t be easily converted to traditional fiat money or other cryptocurrencies. On top of this drawback, a low relative trading volume to the rest of the asset universe shows that the users care more for other cryptos. In a representative benchmark shall be just included the cryptos which are used often. Two measures will be applied which are modified versions of the liquidity rules from the STOXX Japan 600 and the AEX Family. From the first is taken the approach to look at the Percentiles of the Average Daily Trading Volume (\(ADTV\)) and from the later the concept to take also the number of shares into account. The applied rules are the following:
- 0.25 percentile of \(ADTV\) \[ADTV_i \geq ADTV_{0.25}\]
where \(ADTV_{0.25}\) is the \(0.25\) percentile of the \(ADTV\)s of all cryptos in the last period and \(ADTV_i\) is the \(ADTV\) of a single crypto. The approach ensures that just cryptos which are liquid in relation to the other cryptos are eligible for the CRIX. The \(0.25\) percentile is chosen as rule of thumb border for the tails of the empirical distribution because this value serves as an important lower bound in statistical analysis, e.g. the Box-plot. It ensures that the relatively less traded cryptos are excluded from the eligible ones for CRIX.
- 0.25 percentile of Average Daily Traded Coins (\(ADTC\)) \[ADTC_i \geq ADTC_{0.25}\]
where \(ADTC_{0.25}\) is the \(0.25\) percentile of the \(ADTC\)s of all cryptos in the last period and \(ADTC_i\) is the \(ADTC\) of a single crypto. Since many coins have a very small price, is it likely that they will be excluded by the first rule from the eligible ones even when many coins are traded but volume will be still low due to the low price. But on the other hand, it doesn’t pay out to look just at the \(ADTC\) since cryptos with a high price will be possibly less traded but attract much liquidity, e.g. Bitcoin.
Finally, if a crypto fulfills at least one of the two rules, it is eligible for the CRIX.
Number of Constituents:
To our knowledge are the number of constituents of an index normally fixed. This may be a good approach for relatively stable markets but the crypto market is a very fast and innovative one. To be able to find always the best benchmark, is it better to variate the amount of member such that always the best benchmark is provided. The used procedure for CRIX is based on the AIC and BIC criterion, which are the Akaike Information Criterion and the Schwartz Information Criterion respectively. First will be computed an index representing the total market by applying the CRIX formula with the necessary capping. But the Liquidity rule won’t be used so that really all cryptos are included. Afterwards will several indices with different numbers of constituents be computed. The index representing the total market will be treated as the observed values for the market situation and be compared against the indices to check which one provides the best approximation while using as less index members as possible. The Likelihood function for computing the AIC and BIC is based on stable distributions to ensure that a broad band of distributions can be covered. For more details, see the presentation in the references.
Weights:
Each cryptos in CRIX is weighted with its market capitalization.
Ranking:
After excluding cryptos which doesn’t fulfill the Liquidity rule will the cryptos be ranked by their market capitalization in the last period from highest to lowest. The number of cryptos, obtained from the rule for the Number of Constituents, with the highest market capitalization will be taken.
Reallocation:
The reallocation period of the CRIX is 1 month to ensure that the index is always up to date. This period is taken rather short such that the CRIX can react faster to changes in the crypto market. At this time point will be checked again the liquidity of the cryptos by taking into account the last month. Furthermore will be builded again the Ranking and afterwards are computed the weights of the cryptos by taking into account the rule described in Weights. On top of that will the \(Divisor\) be adjusted.
Every 3 month will be checked if the number of constituents still fit the market well. For the computation will be used the values from the last 3 month to ensure that just the most recent values influence this important decision for the CRIX.
Special Events
If the current price for a crypto which is part of CRIX is not reachable from the exchanges is the CRIX made insensitive to its value as long as the value can’t be reached.
If a crypto vanished from the market while it is part of CRIX while its value be made insensitive until the next reallocation date is reached.
- CRypto IndeX
