Distinguishing features of Quantal Global Risk Model



  • A consistent factor structure across asset classes, down to the constituent level, that incorporates shifting correlations between equities and interest rates and accounts for the influence of factors on corporate asset values and default probabilities;

  • Daily updates so that the factor structure adapts quickly to shifts in regime;

  • Synchronized Global coverage;

  • A fat-tailed unconditional distribution of market returns that arises from shifts over longer horizons in the factor structure and volatilities

  • Reliable and efficient procedures for measuring portfolio exposures to characteristics such as industry/sector, size, and style;

  • Integration of the basic building blocks for the models that apply to cash market equities, bonds, and credit instruments.  We can also accommodate derivatives on these financial instruments.

  • Bottom-up models that enable a user to drill down to the level of individual assets and not just asset classes, while continuing to let users do analysis using top-down characteristics.

  • Covers global equities, exchange traded funds (ETF), fixed income (term structure and credit), currencies; derivatives on equities, bonds, and interest rates can be replicated using the underlying instruments.



Covariance Matrix


A latent factor model is an implicit statistical model that relates a set of variables to a set of latent variables (variables that are not directly observed). Implicit multiple factor model is specified as:

𝑅𝑖𝑡=𝑎𝑖+𝛽𝑖1𝐹1𝑡+ ⋯+𝛽𝑖𝐾𝐹𝐾𝑡+𝜖𝑖𝑡

where 𝑅𝑖𝑡 is the return on security 𝑖,𝑖=1,…𝑁, in period 𝑡, and 𝐹𝑘𝑡, 𝑘=1,…,𝐾, are pervasive factors affecting the security returns. Quantal factor model is a conditional model; the factor loadings 𝛽𝑖𝐾 and return volatilities are computed daily, so in principle they bear a 𝑡-subscript.


The daily risk generation process generates the the factor loadings, 𝛽𝑖1𝛽𝑖2,…𝛽𝑖𝐾 with respect to each of the K factors (currently 30) and the idiosyncratic variance each of the securities in the universe. We can readily construct the N x N covariance matrix (𝛴 ) for the securities in the universe:

  𝛴=𝐵𝐵𝐷            (B1)

where B is a matrix with N rows and K columns containing the factor loadings and D is an NxN diagonal matrix with the idiosyncratic variances.  The covariance matrix and all of the entries in B and D are in daily units.  To annualize, we simply multiply by the number of trading days:

  𝛴𝑎𝑛𝑛𝑢𝑎𝑙=252 𝛴     (B2)

This formula works because our factor analysis produces orthonormal factors—each factor is a vector with length one, the correlation between factors are absorbed into the exposures B with the idiosyncratic variances independently distributed of the factors.



VIX-Incorporated Model


VIX incorporated risk model uses forward-looking volatility information gleaned from VIX levels to improve the response time of our model to changes in market regime. We insert a new matrix into calculation of the daily covariance matrix Σ (equation B1)


𝐵 is an NxK matrix of exposures of the N securities in the universe to the K factors, D is an NxN diagonal matrix with the idiosyncratic variances of the N securities, and 𝑴 is a KxK matrix that spreads the VIX adjustment across all securities.


We generate the matrix 𝑴 using a weighting scheme that places more or less emphasis on the VIX information depending on the nature of the most recent realized volatility for the S&P 500.  Short-lived spikes in the VIX level have a small impact on the covariance matrix, but the VIX information is emphasized if it is accompanied by increased realized volatility.  As a result, the VIX-adjusted model picks up changes in market volatility regimes within a few days.