Risk Model Explained

A deep dive into our methodology for the Factor Risk model

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Written by Jeremy Mulder
Updated over a week ago

Currently, the factor values used as input to our model are z-scored versions of raw factors. The z-scoring is based on the mean and standard deviation of the factors of the securities in the T2000, computed daily

Additionally, there is one non-zscored factor: Every security gets a factor score of 1.0 for the "market factor" which is used to describe the daily mean return of the T2000

Also note that the sector and region factors are betas of securities to ETFs. These betas are then z-scored to be some sort of market-adjusted betas


We want to find the factor returns which best describe the proportional daily returns, that is

r≈Xβ

with X being the z-scored raw factor matrix of the T2000.


This leads to the solution

calculated independently for each day.


Using this approach, we can describe about 25% of the daily variance in the equal-weighted T2000. Note here that when the matrix X is permuted, only about 1% of the daily variance can be explained.

However, the off-diagonal sample covariance of the factor returns can be quite high using this model, and the factor returns are sometimes fluctuating over time. To help stabilize the model, we add a Tikhonov regularization term and instead compute

Covariance reduction is not the only reason to regularize. Indeed, for highly correlated factors, too much regularization actually increases covariance.

Regularization is a way of saying : if you can get almost as good a fit with much smaller coefficients, prefer the smaller coefficients as they are more likely to represent an underlying truth.

The regularization strength,

λ

is chosen as the largest value that does not reduce our explanation of the daily variance by more than 1%. We have found empirically that this greatly stabilizes the factor returns over time. There have also been studies of the factor covariance to make sure we don't over-regularize.

The covariance of the factor returns is calculated as follows:

  1. the sample correlation matrix is calculated over 90 calendar days

  2. for each factor, the variance over 90 days is calculated after rejecting outliers. The 2 largest returns are removed from the variance calculation to not let one-time events (like Brexit) influence our prediction of future factor variance

  3.  the individual factor variances are applied to the sample correlation matrix to obtain the factor covariance matrix

We denote the factor covariance matrix as:

F

The covariance matrix of returns due to factors all securities in a portfolio is given by

C = YF(Y^T)

where Y is the z-scored factor matrix for the portfolio holdings.

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Now that we have the portfolio's return covariance due to factors, the factor risk (variance) can be given by

(W^T)CW

where W is the vector of portfolio position sizes.

For the breakdown of factor risk, the matrix C is recalculated by zeroing out all entries of F which are not being considered.

The specific risk for a given security is the sample variance of the specific returns ofthat security over a 3-month lookback. The specific returns on each day are calculatedas:

s=r−Xβ

where r, X, and β are defined as in the first equation.

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Let

Δ

be the diagonal matrix where each the i-th entry is the specific risk of security i in the portfolio.

Then the portfolio's idiosyncratic risk is given by

(W^T)ΔW

and the total risk can then be calculated as

(W^T)(C+Δ)W

However, what is actually being done in the risk attribution for specific risk is that the specific risk for the entire portfolio is calculated. That is, on every day for the past 90days the sum of specific returns is calculated for the portfolio currently held, and the specific portfolio risk is calculated for that day. This is because out factor model doesn't capture all correlated returns and thus the specific returns are also somewhat correlated.

The total portfolio risk is calculated as

(W^T)(C)W + portfolio idio variance

For factor return attribution, the same model is used as that for risk attribution. 


Everyday, the factor attribution to a portfolio is

C=Yβ

for the portfolio z-score matrix Y.

Each individual factor contributes to each security as the z-score of the factor for that security multiplied by the factor return. The idiosyncratic return of a security is the daily return of the security minus the sum of the factor contributions.

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