Index Mathematics

Formulas for Index Calculation

Index Construction

The DCgen Ethereum Index ($DCE), a CW index, assigns weights to its 30 constituents based on their market capitalization. This approach effectively reflects the relative impact of each constituent on the overall Ethereum market.

The maintenance of the DCgen Ethereum Index includes a reconstitution and rebalancing interval of eight weeks. This interval ensures that the index remains up-to-date and accurately reflects the most recent market trends and developments in the Ethereum ecosystem.

Index Calculation

The CW index calculation methods assign weights to each constituent based on their respective market capitalization. The Laspeyres index serves as the foundation for the DCgen Ethereum Index's calculation, measuring the relative price changes of a fixed basket of goods and services over time.

The Laspeyres equation can be simplified using the "base-weighted aggregative" method to calculate the index level:

$$Index Level = \frac{\sum_{i} P_i Q_i}{Divisor}$$

Here, $Pi​$ is the price of constituent $i$, and $Qi​$ represents the free-float quantity of constituent $i$ at a specific time. The sum of all constituents results in the total market capitalization of the index constituents, which positions the numerator as a measure of the current market value. The denominator is the divisor, which initially refers to the base period of the Laspeyres index until the divisor is adjusted.

Divisor Adjustments

The divisor plays a critical role in the calculation of the index value by ensuring consistency and continuity throughout reconstitution, rebalancing, and other market events. The initial divisor can be mathematically defined at the time of index creation as follows:

$$Divisor_0 = \frac{\sum_{i} P_i Q_i}{Starting Value}$$

Here, $∑_i​P_i​Q_i​$ represents the initial total market capitalization of all constituents. The divisor at period 0 depends on the specific starting value that the index issuer aims to set when creating the index. Our indices always have a starting value of 100.

When an adjustment to the divisor is required (for instance, during a rebalancing and reconstitution), the new divisor at period $i$ ($Divisor_i​$) is calculated as follows:

$$Divisor_i = \frac{\sum_{i} P_i Q_i \times Divisor_{i-1}}{\sum_{i-1} P_i Q_i}$$

In this equation, $Divisor_{i−1}$ represents the divisor from the previous period ($i−1$). The total market capitalisation of all constituents at period $i$($\sum_{i} P_i Q_i$) is multiplied by the divisor from the previous period and this product is then divided by the total market capitalisation of all constituents at the previous period ($\sum_{i-1} P_i Q_i$).

In conclusion, the DCgen Ethereum Index is constructed and calculated using a CW approach. By employing the Laspeyres index as the basis for calculation and making appropriate divisor adjustments, the DCgen Ethereum Index offers a consistent and comprehensible representation of the Ethereum market's movement.