- The net-zero index is an extension of the Automated Index Methodology and for this reason this document only address the areas that are specific to the net-zero index type.
- The net-zero specific methodology only relates to the so-called unconstrained as well as constrained weight calculation steps, whereas the proceeding index weight process is unchanged.
- For the common areas, the Automated Index Methodology document serve as the reference.
- In the following “net-zero” refers to having an equal notional long vs short USD exposure within some group of assets, e.g. across the whole portfolio, within a sector or some custom group of contracts.

**Business Day** for an asset refers to a day where the asset is trading and has a settlement Close Price.

**Asset** refers to one of the selected components of the Index. Assets can be in the form of a fixed contract or a rolling contract.

**Price,** P_{(t,i)} refers to the close price on business day t of Asset i.

**Return,** r ^{%} _{(t,i)} refers to percentage returns, (P_{(t,i)} - P_{(t-1,i)})/ P_{(t-1,i)}.

**Product** refers to the AiLA Index, where its output are daily weights for each asset in the index.

**Mid Cap Logic** requires the execution of instructions on the close of the next Business Day. This is typically used for indices with a capacity lower than US$1 billion.

**Large Cap Logic** is typically used for indices with a capacity of US$1bn and above, where a different execution process is applied as described below.

**Weights** are calculated just after close on each business day.

Inline with the other types of AiLA indices, each Asset is assigned Daily Rebalancing Caps across the curve to prevent trading with a significant amount of slippage. These are specified under the following four Durations to the Expiry Date:

- ≤ 1M (≤ 22 Business Days)
- 1 – 3M (23 – 66 Business Days inclusive)
- 3 – 6M (67 to 132 Business Days inclusive)
- > 6M (More than 132 Business Days)

Each Asset is assigned Daily Asset Caps equal to a given multiple of their rebalancing cap value, where the multiple used is 1x (4x) for the Mid (Large) Cap logic. The multiple hence suggests the number of days necessary to trade in/out of the max allowed weight.

- Just like with the traditional Automated Index Methodology, the purpose of the unconstrained weights is to represent the ideal index, based on the user preferences before any practical constraints have been imposed.
- Unconstrained weights are calculated for the set of assets which have an active long/short position signal on a given date.
- For the net-zero index, the unconstrained weights are calculated similar to the traditional volatility weights, however, respecting net-zero property at portfolio level.

- Distribute the allocation equally w.r.t. asset risk among the assets (𝑖) with an active signal on a given business day (𝑡).
- The asset volatility (𝜎) is calculated from historical returns using an exponentially moving average with a lookback window of about 2-3 months.
- Net-zero achieved by normalizing weights such that sum of long (L) equal sum of short (S) weights, and in case of no active long (short) asset signals across the whole portfolio zero weights are assigned to both long and short assets.

- The constrained weights are taking the unconstrained weights as an input and then determine the most similar (here w.r.t. polar angle as well as size) set of weights given a set of linear constraints.
- The calculation is only considering the assets with active (non-zero) weights on any given business day.
- The following constraints are used similar to the traditional auto methodology.
**Asset cap:**upper bound w.r.t the individual asset weights.**Group cap:**upper bound w.r.t. the sum of weights within a sector or customized asset group.**Index allocation:**upper bound w.r.t. the sum of all weights, i.e. less or equal to 100%.

- In addition, the following net-zero related constraints are included.
**Group Net-Zero:**range bound w.r.t. sum of signed weights within a sector or customized asset group.**Index Net-Zero:**equality bound w.r.t. sum of all signed weights.

- The constrained output weights are signed, i.e. weights are (elementwise) multiplied by signal sign.

- The fit is performed on each business day for all assets with non-zero unconstrained weight (𝑤
^{𝑈}), given that at least two assets have non-zero weights. - As described above, the asset caps are determined by the rebalancing cap values and differs for the Mid/Large Cap logic.
- The group caps are user defined (Group Limit) values.
- The sum of weights cannot exceed 100%.
- The group net-zero ranges are user defined (Group Net Limit) values.
- The sum of signed weights across the portfolio requires a long vs short balance.
- The constrained weights are obtained using the minimize function of the scipy library [SciPy], where the set of weights are determined by the weight vector that minimize the angle together with achieving a similar size w.r.t. the 𝑤
^{𝑈}vector, while respecting the linear constraints (1) to (5). - In case the fit is not possible, the 𝑤
^{𝑈}values are used with weights associated to assets/sectors scaled down in case they breach the corresponding caps.

(* in the formulas s refer to asset allocation signals, positive/negative referring to long/short.)

[SciPy]: E. Jones et al., Open Source Scientific Tools for Python (2001), http://www.scipy.org.