# Automated index process

As explained in the modelling documentation, the model production generates daily signals for individual assets, e.g. the WTI June and December contracts are modelled separately.
The automated index process take these signals as input and index weights are then calculated through three sequential steps.

# Definitions

###### General conventions

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.

###### Rebalancing Caps

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)
###### Asset Caps

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.

# Index Description

The AiLA Alpha Index suite of products is designed to deliver absolute returns investment Alpha to investors. The suite of products is asset agnostic. However, it is presently focused on Commodity assets and equity indices as the underlying assets. These index products are designed to be highly liquid and tradable and constructed taking into account market liquidity. In addition, as AiLA uses opportunistic allocation to generate alpha, the strategy might not be fully allocated all the time.

The Index Methodology calculation starts from using the output of AiLA’s systematic upstream process to decide if we should go Long, Short or stay flat on a given asset. As part of the index construction process, these allocation decisions are used in a first step, together with market volatility and correlation estimates, to construct index weights with the desired risk profile, representing an ideal portfolio.

In a second step, the ideal portfolio is used to find similar but realistic index weights, which respect requirements such as Asset and Sector weight caps. These Asset and Sector weight caps are risk constraints that we would like to implement for a given strategy.

We then take into account liquidity constraints as the index seeks into incorporate actual market liquidity. This is done by setting a Daily Rebalancing Cap for each asset under the Large Cap Logic, ensuring that the strategy can be executed. As the Sector constraints and Daily rebalancing constraints are mutually exclusive, we always prioritise Sector risk constraints.

If the Large Cap Logic is in place, we ensure that all positions are closed with ample time before the expiry of a contract. Therefore, we will starting closing all active positions at least seven Business Days before the expiry of a contract.

Finally, to obtain the index values, we apply simple arithmetic calculations to calculate the Units, and Daily PNL for each Index, taking into account the change of asset prices on each trading day.

# Unconstrained weights (wU)

• The ideal weights are determined based on mean/variance portfolio construction principles, however, due to the dynamic nature of the signals, only aspects that are empirically shown relevant for our circumstances are included in order to achieve a robust and transparent portfolio construction process.
• The purpose of the unconstrained weights is to represent the ideal index, based on the user preferences before any practical constraints have been imposed. The following steps will then aim to achieve an index as close as possible to the unconstrained one while respecting the various user/execution constraints.
• Unconstrained weights are calculated for the set of assets which have an active long/short position signal on a given date.
• The unconstrained weights are calculated with the following aspects considered, of which some are optional.
• Volatility: equally weighted w.r.t. asset risk.
• N effective: varying allocation w.r.t. estimated effective number of assets.
• Index correlation: induce a correlation preference to the standard index BCOM or GSCI.
• Risk target: scale weights inline with index risk target.

# Unconstrained weight calculation

###### Volatility:
• Distribute the allocation equally w.r.t. asset risk among the assets (𝑖) with an active signal (𝑠) 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.
• In case both long and short signals are active, it is possible to assign a relative weight between them. The intention of this option (Long/Short Ratio = rLS) is primarily w.r.t. indices with a larger number of assets and to impact a general long/short bias .
###### N effective: (optional)
• For indices where the number of assets vary significantly over time, a Risk Balance option is available.
• If enabled, the index allocation is varied based on the effective number of assets (𝑁𝑒𝑓𝑓) estimated on a given business day, in order to increase (decrease) the allocation when the variance is expected to be reduced (enhanced) by the large (small)(𝑁𝑒𝑓𝑓).
• Further details about the (𝑁𝑒𝑓𝑓) and corresponding calculation is found in the dedicated section below.

# Unconstrained weight calculation

###### Index Correlation: (optional)
• In case of a particular correlation preference w.r.t. the standard index BCOM or GSCI (Correlated Index), an option is available to bias the weights accordingly.
• This is achieved by rotating the weight vector w.r.t. the standard index weights, within a common risk space.
• The option is currently implemented with a positive/negative (Correlation) preference and four different levels (Degrees of Correlation) of angular (correlation) constraints.
• To maximize the effect, all available assets associated with the corresponding standard index are used to achieve the bias.
• Further details about the standard index bias calculation is found in the dedicated section below.
###### Risk Target:
• The weights obtained from the above steps are finally scaled in order to normalize the total allocation of the ideal/unconstrained index inline with the user defined risk target.
• The scale factor is determined based on the historical index PL, where the scaling is either based on a (Risk Target Type) Max Daily Drawdown (𝑅𝑇𝑑𝑑𝑑) or annual Volatility Target (𝑅𝑇𝑣𝑜𝑙) value.
• The index volatility (𝜎) is calculated using an exponentially moving average with a lookback window of about 2-3 months.

# N effective calculation

• The effective number of assets could typically be estimated from the assets with non-zero weights on a given business day, together with their return correlation.
• To estimate asset correlations which are stable out-of-sample is generally difficult, however, in contrast to many scenarios in quantitative finance, our indices are typically comprised of a medium number of assets (e.g. ≤ 100) with either very high (e.g. between Zinc contracts, or WTI vs Brent) or low (e.g. Zinc vs Soybean) correlations.
• For these reasons, the 𝑁𝑒𝑓𝑓 calculation uses a very simple risk model (𝑅), where assets are grouped together with an assumed correlation of 1 (0) within (between) groups.
• The 𝑁𝑒𝑓𝑓 is then calculated using a signal vector (S) where all non-zero long (short) weights are assigned 1 (− 𝑟𝐿𝑆 = Long/Short Ratio).

[Diversification Benefit]: Perspectives, https://ailaindices.com/diversification-benefit.php

# Standard index bias

• The standard index bias is constructed using a fit w.r.t. to the weights associated with standard indexes.
• The current implementation uses the long and short 𝑤𝑈 weights (i.e. here 𝑤𝑣𝑜𝑙 or 𝑤𝑛𝑒𝑓𝑓) for all assets associated to the standard index constituents, through a common risk model [same as used for the 𝑁𝑒𝑓𝑓 calculation].
• The fit is performed on each business day where at least 5 assets have non-zero weights, where the resulting weights are different to 𝑤𝑈 in size, but not sign.
• The index biased 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 w.r.t. the 𝑤𝑈 vector, while being within a given angular distance to the standard index weight vector.
• The fit preserve the sum of weights equal to sum(𝑤𝑈), however, this can not be guaranteed under all circumstances.

Note: further details w.r.t. this subject are discussed in [Controlling correlation]

[SciPy]: E. Jones et al., Open Source Scientific Tools for Python (2001), http://www.scipy.org
[Controlling Correlation]: Perspectives, https://ailaindices.com/controlling-correlation.php

# Constrained weights (𝑤𝐶)

• The constrained weights are taking the unconstrained weights as an input and then determine the most similar 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 constrained weights are calculated with the following constraints.
• Asset cap: upper bound w.r.t the individual asset weights.
• Sector cap: upper bound w.r.t. the sum of weights within a sector.
• Sharpe ratio: lower bound w.r.t. historical index Sharpe ratios.
• Index allocation: equality bound w.r.t. the sum of all weights.
• The constrained output weights are signed, i.e. weights are (elementwise) multiplied by signal sign.
• In case of a maximum allocation preference, an option (Allocation Range - Max) is available to scale down the output weights to the max value on business days where the sum of weights exceeds this value.
• It should be noted that the sector caps are used without the target risk scaling in order to refer to the USD allocation, therefore if sector caps are used an allocation max option of 100% should be specified to avoid distortion from leverage.

# Constrained weight calculation

###### Constrained weight fit:
• The fit is performed on each business day for all assets with non-zero unconstrained weight (𝑤𝑈), given that at least three 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 sector caps are user defined (Allocation Max %) values.
• The Sharpe ratio (SR) constraint provide an option to require that the weights corresponds to an average historical SR above a given threshold (Target Annual Sharpe Ratio). The historical asset Sharpe ratio values used are capped at ±5.
• The last equality constraint make the fit preserve the sum of weights.
• In case the sum(𝑤𝑈) is too large for the weight region defined by the sector and asset caps, the 𝑤𝑈 are scaled down so that the sum(𝑤𝑈) fits inside the allowed region.
• 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 w.r.t. the 𝑤𝑈 vector, while respecting the linear constraints (1) to (4).
• In case the fit is not possible, the 𝑤𝑈 values are used with weights associated to assets/sectors scaled down in case they breach the corresponding caps.

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

# Index weights (𝑤T and 𝑤A)

###### Constrained weight fit:
• The index weights are calculated using the constrained weights (𝑤C) as input.
• In addition, a trade balancing period criteria is applied, where weights are set to zero when the number of business days to expiry or end-of-year is less than 2 (7) using the Mid (Large) Cap logic.
• The 𝑤C weights with the trade balancing period criteria applied is referred to as the target weights, which the actual index weights try to achieve.
• However, the actual index weights are often not able to equal the target weights, due to the rebalancing restrictions implied by the different caps.

# Index weight calculation

• The index weights involve two type of weights,
• Target weights (𝑤T) are the constrained weights (𝑤C) with the additional trade balancing period criteria applied, discussed above.
• The Actual Weight (𝑤𝐴(𝑡,𝑖)) is the weight for asseti on Effective Datet. There will be no weight instruction for an asset during an asset holiday, in such cases, use the Actual Weight for the previous Effective Date for End-of-day Units calculation.
• On each business day (t) the rebalancing values for the next business day are calculated by the following steps,
• 𝑟𝑒𝑏(𝑡,𝑖) = 𝑤𝑇 (𝑡,𝑖) − 𝑤𝐴 (𝑡,𝑖)
• 𝑟𝑒𝑏(𝑡,𝑖) is set to its rebalancing cap value, if the cap is exceeded.
• 𝑟𝑒𝑏(𝑡,𝑖) values are reduced to meet asset and sector caps, if the actual weight values after rebalancing exceed any such caps, i.e. the affected assets are reduced to equal the 𝑟𝑒𝑏(𝑡,𝑖) allowed by the corresponding cap.
• The final 𝑟𝑒𝑏(𝑡,𝑖) values are then used to rebalance the actual weights on the next business day, i.e. 𝑡+1, and the index is calculated using the same methodology as other types of AiLA indices.

# Index wide aspects

###### Currency
• The default currency used for valuation of all asset prices is USD, i.e. including the PNL and index calculation
• In case another index currency is specified through the user option, or an asset’s domestic price is in another currency, all prices are converted into that index currency, using London 4pm Fixed rates on the same business day as the price
• Therefor the index performance should reflect the prevailing value of its assets in the given index currency
###### Partial Index Holidays
• There are Index Business Days where not all but at least one asset is trading (E.g. in an index which has some assets on CME calendars and others on LME calendars, only one of the two is trading)
• On such Index Business Days, the End-Of-Day Units and Profit and Loss of the Assets that are not trading must be calculated. However, given that prices cannot change on an asset Holiday, the Profit and Loss will surely be 0 for these assets. If there is a non-zero Profit and Loss for that Index Business Day which means that there is a change in Index Value, then the Units calculated for the Assets that are not trading will be different.