Risk Decomposition Calculations

Variance Column

Intuitively, the Variance column measures the risk of the actively managed fund relative to its benchmark. We will cover the calculations for the Asset Selection row, the Total Active Common Factor row, and the risk factor rows separately, as they are handled differently.

Although in all the calculations in this section we will be using active weights for the weight vector, one could also use the fund weights or benchmark weights to get fund and benchmark risk decompositions.

Asset Selection Variance

Recall that Asset Selection measures the risk that cannot be explained by the risk model. Therefore, the calculations of its risk statistics are inherently different than those for the risk factors. In particular, neither the risk exposures of the securities (the matrix X) nor the covariance of the risk factors (the matrix C) are needed to calculate its risk characteristics. Its variance is calculated as follows, where we are using active weights for the weight vector.

Here, w₁,...,w_N are the active weights and s₁,...,s_N are the specific risks of each security.

Total Active Common Factor Variance

In the Total Active Common Factor Variance calculations, the security-level exposure matrix X and the covariance matrix C are key inputs. We let E = X^TW denote the active exposure vector. Again, we are using the active weights for the weight vector W. The variance in this case is calculated as follows.

where

is the active exposure of the fund to the j^th risk factor.

Risk Factor Variance

For the individual risk factors and risk factor groups, the computation is similar. The idea is to isolate the portion of the total variance using only a submatrix of the entire covariance matrix. For simplicity, we will describe how to compute the decomposition of the variance corresponding to the first k risk factors. The general case is no different, but more cumbersome with regards to notation.

Let C' be the top left k x k-minor of the covariance matrix C and let E' be the top k entries of the exposure matrix E (again, we are using the active weights for the weight vector in this example). That is,

Given these inputs, we then compute the risk factor variance as follows.

Note that in particular, the risk factor variance for the i^th risk factor is given as follows.

Total Active Variance

The final row that appears in the Risk Decomposition section is the Total Active row. As the name implies, this row combines all risks associated with the choice of active weights of the fund. The Total Active Variance is calculated by adding up the Asset Selection variance, Total Active Common Factor variance, and the Covariance * 2.

Covariance * 2

The remaining rows that appear in the Risk Decomposition section contain the term "Covariance * 2." These lines capture the extent to which the variances do not add, and can be thought of as the error term in the risk decomposition. There will be one row for each risk factor group that is broken down into its constituent risk factors as well as one more to display the interaction amongst risk factor groups.

For the notational ease, we will give an example of how to calculate the Covariance * 2 term for a risk factor group consisting of two factors, which we will call F_i and F_j. On the row for F_i, the variance will be cov_iie_i² and for F_j it will be cov_jje_j² (see the description of Risk Factor Variance above). In this case, the Covariance * 2 term consists of the cross-terms:

Here, we have used the fact that cov_ij = cov_ji. The 2 in the above equation is where the terminology Covariance * 2 comes from. The general case is similar, but more cumbersome with regards to notation.

In the example, there are two Covariance * 2 lines, one that captures the interaction amongst the different risk factors in the Risk group, and the other that captures the interaction between the two risk factor groups Risk and Industry.

Standard Deviation Column

The second column that appears in the Risk Decomposition section is labeled SD, which stands for Standard Deviation. Standard Deviation is another risk measurement, which like variance measures the variability of active returns. One benefit of using standard deviation as a measurement of risk instead of variance is that standard deviation is measured in the same units as the random variable. Thus, we can think of the standard deviation column as measured in percentage returns.

Once the corresponding active variance component has been calculated, the standard deviation is simply its square root.

MC-AR Column

The third column in the Risk Decomposition section is labeled MC-AR, which stands for Marginal Contribution to Active Risk. MC-AR is a measurement of the amount that a small change in active exposure to a given risk factor would change the active risk of the fund. Note that values only appear at the individual risk factor level. By default, the change is Δ = 0.01, so MC-AR measures the change to active risk if the exposure to the given risk were increased by 0.01. It is calculated as follows.

Contribution Column

The last column in the Risk Decomposition section is labeled Contribution. The Contribution column gives contribution to total risk of a given factor as a percentage of the total risk, as measured in variance. That is, for each risk factor or group, contribution is calculated as follows: