In order to measure the contributions of risk factor exposures

to the systematic risk of an instrument, we must decompose the volatility using the Euler formula, as set out in Risk Model Applications. Recall that the Euler formula gives

where each term of the sum corresponds to the contribution attributable to the variable

to the total value of the function.

For decomposing risk at each time

, the function

becomes the portfolio volatility,

, and the parameters

become the factor exposures

​and the specific returns

​. In matrix form, the contribution to risk attributable to factor exposures

and

are given by

​where

is the Hadamard product between two matrices. The derivatives

​ and

​​ are obtained by using a number of identities common in matrix derivatives calculus ^[1] shown

​where

​. The derivative of the volatility with respect to the specific risk is simpler:

The total risk is thus finally given by

where the risk contribution

is an

matrix whose elements

​ correspond to the point in time contribution of the

characteristic of the

instrument to the total risk and

is the weighting of the instruments within the portfolio at point in time

.

Note that summing all risk contributions over every dimension gives the exact volatility of a portfolio.

For single funds, the risk decomposition simplifies considerably and is obtained by applying the following modifications to the above formulas:

Cross-Sectional and historical granularity

The ability to map the risk contributions back to the characteristics allows us to separate the exact risk contributions coming from fundamental and sector-based factors. For instance, the contribution of fundamental risk to the total risk simply corresponds to the sum of the contributions associated with fundamental risk exposures, that is

Another dimension that is often overlooked is the historical one. Grouping risk contributions not only by factor types (fundamental or sector-based) but by time periods provides an insight into the realization of risks. Oftentimes, the largest part of the total risk comes from the contributions of a few short periods of time. Risk tends to appear in bursts, which the above representation is capable of identifying.

References

 

1 Kaare Brandt Petersen and Michael Syskind Pedersen. (2012). The Matrix Cookbook. Technical University of Denmark.

 

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