Quant analytics: Projecting a daily risk decomposition to an annual basis

(Last Updated On: December 14, 2011)

Quant analytics: Projecting a daily risk decomposition to an annual basis

Suppose I have a daily risk factor model. The model generates expected daily log security returns based on the security’s exposures to various factors. Factors consist of economic factor exposures as well as cross-sectional z-scores of fundamental factors (i.e. for example, normalized Price/Book) which vary each day.

My goal is to use the daily log-based model to perform an annualized linear (i.e. arithmetic) risk and return decomposition — namely, decomposing the risk of a security into the sum of the systematic risk factor contributions plus an idiosyncratic component.

I have two procedures but both seem to run into a roadblock because exposures vary by date. (I could assume the mean factor exposures but this simplification is not accurate.)

Method 1:
1. From the daily log returns of the factor model, construct a daily log-based covariance matrix parametrized by vector mu and sigma
2. Project mu and sigma to the annualized horizon (annualized log distribution = mu*250, sigma*250)
3. If exposures were constant, I could simply convert the annualized log-return distribution to an arithmetic distribution (see Meucci – Quant Nugget 4) and calculate:

systematic risk for each security = mean factor exposure * diagonal( Annualized Linear/Arithmetic Matrix) * (mean factor exposure)’.

Since exposures are not constant this approach is invalid. So I try another method.

Method 2:
1. From the daily log returns of the factor model, construct a daily log-based covariance matrix parametrized by vector mu and sigma
2. Since the log returns are an additive invariant, for each security we can sum over the daily contributions to systematic risk over the past year:
securityVarianceByDate = diag(exposuresMatrix * dailyLogMatrix * (exposuresMatrix)’
annualizedSecurityRiskVariance = sum of each element of securityVarianceByDate
3. We have now accounted for changing exposures. However, we are left with an annualized log-variance. The reporting must be in arithmetic terms. We cannot apply steps #1 and #2 with an arithmetic covariance matrix because the linear returns are not invariant.

How should I proceed to project an annualized arithmetic systematic risk decomposition from a daily log risk factor model?

==

Projeccting a cross-sectional linear factor model is a challenging task. I will point out a few issues to highlight why.

Let us use some notation.
* R_t is the Nx1 vector of daily log-returns on N stocks on day t
* b_t is the NxK matrix of exposures on day t. Each column is a set of factor exposures for each of the N stocks such as the standardized price/book ratio.

Then one can fit the cross-sectional model R_t=b_t*F_t+e_t, where
* F_t is the is Kx1 vector of implicit factors, extracted by cross-sectional regression.
* e_t is the Nx1 vector of residuals

* First, contrary to common belief, you will notice that the residuals are not idiosyncratic, i.e. they are correlated with each other
* Second, contrary to common belief, you will notice that the implicit factors are not invariants, i.e. they are not i.i.d.
* Third, contrary to common belief, you will notice that the factors and the residuals are correlated

So, the projection/annualization is a much more challenging task than one would think. On the other hand, in practice we do not (arguably) really need to project it.

==

As pointed out, you may be dealing with unpleasant deviation from plain theoretic factor model assumptions. If you haven’t read yet, I can suggest you to read this paper about a orthogonal transformation of your factors in order to get uncorrelated components.
See here:Â http://www.be.wvu.edu/div/econ//work/pdf_files/10-05.pdf

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