Similarity to Target – Fabric

How does Fabric compute the Similarity Score between two portfolios?

Fabric calculates the percent of risk coming from each factor for both portfolios. It then takes the difference between those percentages for each factor. In summary, the Similarity Score is a quantitative way to compare the risk factor makeup of two portfolios. It identifies how similar they are in terms of risk exposures, not specific holdings.

Anything above 80% similar to the target is very close to the target, 50-70% you start to get a big drop down and anything below 50% is pretty far off from the target following an 'S curve'.

To calculate the similarity score, we start with the percent contribution to factor risk (PCTR) of both the client and target portfolio. These values are represented in the attribution chart- e.g., 72% of the client risk comes from equity and 78% of the target risk. We then take the difference between those risk contributions on a factor-by-factor basis- e.g., there is 6\% more risk from the equity factor. These differences are aggregated and normalized to produce the similarity score. Mathematically, it is the L2 norm of the difference between the factor risk contributions, exponentiated with a normalizing factor so that the result is between 0 and 100%.

Formally, we first define the risk distance between assets 1 and 2 as:

Using this distance we can define a similarity score as:

where Λ is a normalizing factor. The similarity score is a number between 0 and 100. A score of 100 indicates that the two portfolios are identical in terms of their risk factor contributions.

Some key features of the similarity score:

Factors with larger differences are given much more weight than factors with small differences.
It is based on the percent contribution to volatility rather than the level of volatility.
Thus volatility of a portfolio can be controlled independently from its similarity (such as by liquidating all assets proportionally to de-lever the portfolio).
The normalization is not linear and is designed to ensure sufficient granularity for both portfolios that are very similar and portfolios that are very dissimilar. The similarity is based solely on the factors. Asset-specific risk (idiosyncratic risk/selection risk) is not incorporated.

In contrast, the tracking error is also calculated with the factor covariance but does take into account the level of volatility and the asset-specific risks. As a result:

A low tracking error always implies a high similarity.
A high similarity may not be accompanied by low tracking error depending on the portfolio’s leverage and the selected securities.