Appendix

The delta method

Both factor quality and active risk diversification metrics are fundamentally based on characteristics (shown as​), which are subject to uncertainty. Because these characteristics are obtained from a regression estimation, we can quantify the uncertainty of the characteristics from the standard error of the regression. In order to understand how the standard error affects the values of the diversification metrics, we use a technique called the delta method. Simply put, the delta method ^[1] approximates the confidence interval of a function that results from the uncertainty of its parameters. It assumes that the parameters are random variables with a well-defined mean and variance.

Let each metric be denoted as a function gg which takes as parameters the uncertain (random) variables ​Then, the variance of is given by:

This definition of the function variance as a result of its parameter variances is a simplified version of the definition usually used in the delta method. We have made the decision to neglect the effect of the correlations between the uncertainty of the parameters. Therefore, the above formula is a first-order approximation of the function confidence interval. By first-order, we mean that it represents the uncertainty of the function, though it neglects the smaller, second-order terms arising from the correlations between the parameters. This simplification thus underestimates a more precise estimate involving second-order terms. Also, by neglecting second-order terms, we assume that the uncertainty of each parameter is by and large independent from that of other parameters. It is a common approximation that is made in order to simplify the estimation of a function confidence interval.

Once the variance of the function has been calculated, we can then calculate the confidence interval around the metric as:

​where ​is set to one throughout the application. This corresponds to a confidence level of one sigma.

Confidence interval and robustness

One key property of confidence intervals is that they are proportional to the sample dependency of their associated metrics. For instance, assume that we only have four weeks of data to measure the volatility of a portfolio. It is likely that the value of the volatility estimated from such a small sample will not inform us very much on the true, long-term volatility of the portfolio. This is because a few weeks of data are unlikely to really represent the long-term distribution of the returns. The volatility measured from such a small sample will more than likely only represent the few points present in the sample. In this case, we would expect the confidence interval around the volatility to be large. This is quite reasonable as the level of confidence we can give to the volatility measured on only four points is quite low.

A more robust confidence interval can be obtained by using a method known as bootstrapping. In this example, the confidence interval of the volatility is obtained by the following bootstrap procedure: first we choose four weeks at random and measure the volatility associated with this sample. We then choose another four weeks, and again measure the volatility. We continuously repeat this process a large number of times until we have estimates of the volatilities associated with many different samples. The standard deviation of all of these volatilities is then used to compute a confidence interval. Therefore, an interpretation of the confidence interval is that it is proportional to the stability of the volatility estimates across all different samples. It is thus a measure of the robustness of the volatility should a different sample be chosen. For the volatility, which is based on prices, the confidence interval thus provides an indication of whether the value measured on a given sample depends on the most recent realization of prices, or if it is truly a reliable estimate of the portfolio's true volatility.

In some places, we use the confidence interval to adjust the value of the metric to its lower confidence limit. We often call these adjusted metrics robust. This is because regardless of the sample, including for example, the next realization of returns, the true value of the metric is likely to be larger than its robust version. Robust metrics thus provide a conservative figure such that there is a good chance that the actual metric is going to be at least as high as the robust one. A visual depiction is shown in Figure 1. The robust metric hence provides a more reliable estimate for whatever the next realization of the returns might be. Note that when the confidence interval includes zero, the robust factor quality is set to zero. In such cases, a value of zero means that we cannot with confidence say that the metric is not actually zero.

Robust Factor Quality

Factor quality is defined as the combination of factor intensity (FI) and diversification of fundamental risk exposures (DFRE):

In order to compute the robust factor quality using the delta method, we therefore define the function, , as:

​where is the number of fundamental risk factors (six in this case). We must compute ​ in order to evaluate the standard error around factor quality:

Robust Active Risk Diversification

The active risk of the portfolio is defined as:

In what follows, for notational convenience, we drop the and refer to active risk as . Using the notation from the risk model, ​ with . Active Risk Diversification (ARD) is:

To evaluate the uncertainty of ARD with respect to the uncertainty surrounding all of the ​, we must evaluate the following:

The calculation of two derivatives are necessary. The first is:

​where we use the relationship:

​​The second derivative needed is more involved and requires several steps:

Evaluating the partial derivative in the final term gives:

We obtain :

​We then have all of the elements needed to calculate the confidence interval around the metric.

References

1 J. L. Doob. (1935). The Limiting Distributions of Certain Statistics. The Annals of Mathematical Statistics. 6(3): 160-169. 10.1214/aoms/1177732594.