Calculation

Lookback window

When calculating an index, we calculate it for a lookback window specified by the user. We currently allow a user to specify a lookback window of 1 day, 7 days, 30 days, 90 days or 365 days from the current time period.

Filter use

Themelia allows a user to select their preferred combination of filters and these will be applied at the inception of the index, as well as any rebalancing periods.

There are four different functions for filters:

Function	Description
Include	All constituents in the index must include the selected filter
Exclude	All constituents in the index must not include the selected filter
Min	All constituents in the index must be greater than or equal to the value specified for the filter
Max	All constituents in the index must be less than or equal to the value specified for the filter

Rebalancing

Themelia offers 4 rebalancing options for indexes; Never, Quarterly, Monthly, Weekly,

Function	Description
Never	No rebalancing occurs
Quarterly	Rebalancing occurs at 00:00:00 UTC on the first day of the new quarter. Q1: January 1 – March 31 Q2: April 1 – June 3 Q3: July 1 – September 30 Q4: October 1 – December 31
Monthly	Rebalancing occurs at 00:00:00 UTC on the first day of the new month
Weekly	Rebalancing occurs at 00:00:00 UTC on the first day of the new week. Weeks start on a Monday.

Index Type

Themelia indexes are either Market Capitalisation Weighted or Equal Weighted, depending on the preference specified by the user.

Market Capitalisation Weighted Index calculation

Themelia's Market Capitalisation Weighted Indexes allocate weights based on the relative market capitalisation of each constituent in the index at inception and at any rebalancing time periods.

To calculate a Market Capitalisation Weighted Index, we are first required to determine the constituents at the earliest time period within the lookback window by applying the filters specified by the user.

Based on the constituents at this time period, we then need to calculate the $Market\ Capitalisation$:

$$Market\ Capitalisation_{t=0} = \displaystyle\sum_{i} (P_{i,t=0} * Q_{i,t=0})$$

Where:

• $P_{i,t=0}$ represents the price of the $i^{th}$ constituent at time$\ t=0$
• $Q_{i,t=0}$ represents the circulating supply of the $i^{th}$ constituent at time $t=0$

By building our index based on the summation of the individual market capitalisations of the constituents in the index, we are effectively weighting the index by the relative size of the market capitalisation for each constituent. This happens naturally when we incept the index at the earliest time period, but for subsequent time periods, we will need to eliminate any changes to the Market Capitalisation of the index which do not arise from changes in the prices of the constituents in the index i.e. they have arisen from changes in circulating supply. This ensures that any changes in the index value over time only reflect changes in the prices of constituents.

To handle this requirement, at each time period we need to capture the change in the Market Capitalisation due to the change in circulating supply of any constituents:

• being added to the index

• being removed from the index

We call this value $Delta$, which, for the first time period equates to:

$$Delta_{t=0} =\displaystyle\sum_i (PA_{i,t=0}*QA_{i,t=0}) -\displaystyle\sum_i (PR_{i,t=0}*QR_{i,t=0})$$

Where:

• $PA_{i,t=0}$ represents the price of the $i^{th}$ constituent being added to the index at time$\ t=0$
• $QA_{i,t=0}$ represents the circulating supply of the $i^{th}$ constituent being added to the index time$\ t=0$
• $PR_{i,t=0}$ represents the price of the $i^{th}$ constituent being removed from the index at time$\ t=0$
• $QR_{i,t=0}$ represents the circulating supply of the $i^{th}$ constituent being removed from the index at time$\ t=0$

We then set the initial $Index\ Value$ to an arbitrary value:

$$Index\ Value_{t=0} = 100$$

This incepts the index at the earliest time period within the lookback window.

For subsequent periods, we need a broader calculation for Delta which captures changes in Market Capitalisation resulting from:

• tokens being added to circulating supply for a constituent

• tokens being removed from circulating supply for a constituent

So we expand our calculation for Delta to:

$$Delta_{t} =\displaystyle\sum_i (PA_{i,t}*QA_{i,t}) -\displaystyle\sum_i (PR_{i,t}*QR_{i,t}) + \displaystyle\sum (P_{i,t} * (Q_{i,t} - Q_{i,t-1}))$$

Where:

• $P_{i,t}$ represents the price of the $i^{th}$ constituent within the index at time$\ t$
• $Q_{i,t}$ represents the circulating supply of the $i^{th}$ constituent within the index at time$\ t$
• $Q_{i,t-1}$ represents the circulating supply of the $i^{th}$ constituent within the index at time$\ t-1$

We can now calculate the $Divisor$, which will capture any adjustments required to ensure the Index Value reflects price changes only. We calculate the Divisor using:

$$Divisor_t = Divisor_{t-1} + (Delta_{t-1}\div Index\ Value_{t-1})$$

We can then calculate the Index Value for the current period:

$$Index\ Value{_t}=(Market\ Cap_{t}-Delta_{t})\div Divisor_{t}$$

Finally, we calculate the $Index\ Change$ using:

$$Index\ Change_t = (Index\ Value_t\ \div Index_\ Value_{t=0}) -1$$

After iterating through this process for all time periods required, we make this series available for display in our charts for the front end user.

Equal Weighted Indexes

Themelia's Equal Weighted Indexes allocate weights equally to each constituent included within the index.

To calculate an Equal Weighted Index, we are first required to determine the constituents by applying the filters specified by the user for the earliest time period within the lookback window.

We then count the number of constituents at this time period, denoted as $N$.

Next, we set the initial $Index\ Value$ to an arbitrary value.

$$Index\ Value_{t=0} = 100$$

This incepts the index at the earliest time period within the lookback window.

The $Constituent\ Value$ represents the hypothetical value of each constituent in the index. We ensure that each constituent has an equal weight in the index at the outset with the following calculation:

$$Constituent\ Value_{i,t=0} = Index\ Value_{t=0}\div N_{t=0}$$

For each subsequent period, the Constituent Value is adjusted based on the price change between the previous time period and the current time period.

$$Constituent\ Value_{i,t} = Constituent\ Value_{i,t-1}\ * (Price_{i,t}\ \div Price_{i,t-1})$$

We can then calculate the Index Value by:

$$Index\ Value_t = \displaystyle\sum_i(Constituent\ Value_{i,t})$$

If the current time period is also a rebalancing time period, the index undergoes an additional rebalancing process. The filters specified by the user are applied again to determine the new set of constituents.

Following this, the number of constituents, 𝑁, is counted.

The Constituent Value is then reset to ensure each constituent has an equal weight in the index:

$$Constituent\ Value_{i,t} = Index\ Value_{t}\ \div N_t$$

Now we can calculate the $Index\ Change$ using:

$$Index\ Change_t = (Index\ Value_t\ \div Index_\ Value_{t=0}) -1$$

After iterating through this process for all time periods required, we make this series available for display in our charts for the front end user.