How to see if a set of asset returns corresponds to a known correlation matrix?

Question

Let's say I have an arbitrary set of $n$ period returns for $k$ assets, and a given $k \times k$ correlation matrix (of asset returns), which is known a priori.

Does it makes sense, or is it even possible, to think about constructing some kind of measure of whether the set of $n$ returns is consistent with the known correlation matrix (or if they suggest some sort of outlier set)?

Can we rank one set of $n \times k$ returns as being a better match to the given correlation matrix than another set?

Does it make more sense to ask this if one assumes each asset has the same standard deviation of returns?

Answer 1

See "Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size" by Ledoit and Wolf.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1031689018

Answer 2

One possibility would be to use a similarity based comparison between sets. There are simple geometric distance based measures that work well (euclidean distance is common).

I've seen papers whereby the authors simply run a histogram of all of the correlations and compare to out of sample histograms to determine if the mean of correlations are statistically different or not (can use t-test or ANOVA for instance).

I have been trying to work on generating reliable synthetic multi-asset data (not so easy), so that I can run many such comparison experiments with some kind of confidence intervals.

I would think that comparison measures across different samples and local temporal stability is what ultimately matters.

Answer 3

First, creating a covar matrix from a set of series of unequal lengths is non-trivial and several approaches (eg: one factor or two factor models) will yield different covar matrices. I do see that you said all asset classes have same length (n periods) meaning the above won't apply, but I thought I'd still say this because unless you cherry pick n while working with many assets, most likely you will run into assets with have unequal time series (eg: some company started recently). The point is you need to be consistent about your covar calc technique especially if the covar comes from different sources.

Another warning when fitting returns to covar matrices; I wouldn't advise it since you can have inconsistent results if you use observed returns with a "given" covar matrix for portfolio construction. Returns are observed 'facts' and the covar matrix embodies the relationships, so to avoid inconsistent portfolio results it's best to compute the covar from the raw result info (and always do so with the same technique)

With that out of the way, lets explore measures of "degree of match". Lets define:

• C = computed covar matrix (from your return series) and
• C' = given covar matrix that you got from 'somewhere' that you're trying to see if it "fits".

Different measures:

1. Principle component analysis. Basically determine the eigenvectors and eigenvalues for the COVAR matrix (there will be N of each for an NxN covar matrix). The largest eigenvalues tell you the eigenvectors (or principle components) that capture the strongest 'features' of the COVAR matrix. An introduction to eigenvectors/values. You then compare them in C vs C' to see if the features have changed. If changed, the returns likely do NOT correspond to your 'given COVAR' matrix. Comparison details for C and C':

   • Find the largest eigenvalue of C => pick corresponding eigenvector
   • Find the largest eigenvalue of C' => pick corresponding eigenvector
   • $\theta$ is angle between those two vectors.
   • Absolute value of $\theta$ less than 2-4 degrees => C and C' are similar ("2-4" depends on asset volatility and time period).
   • If "no change" then the returns likely represent the "given" CoVAR (and vice versa)
   • You can tighten the checks by doing the above for not just the largest but also for the smaller components (higher order components) by then checking the 2nd largest, then 3rd largest and so on. If iterate over all N eigenvectors, then you have checked all N dimensions, progressively going from the strongest dimension (at iteration #1) to the weakest dimension (in iteration N).

2. |C - C'| is a decent choice. Although it's simple, there is no direct physical significance to it because it nets too many individual covars; you are essentially collapsing N dimensions into a single scalar quantity with no prioritization of "strong features" (like above)

3. Another decent heuristic would be signs and magnitudes of elements within the C-C' matrix. i.e.

   • best = have C-C' as zero i.e. identical covars (duh! :) )
   • ok-to-good = all elements of C-C' are either slightly positive (incl 0) or all slightly negative (incl 0). The actual sign (+/-) is unimportant as compared to the consistency of the signs.
   • bad-to-worst = large absolute values with 1/2 positives and 1/2 negatives

Note that C-C' is a relative measure, so you will need to define your own min-max bounds and then your own intervals for "almost no change". Easiest to do so is look at historical min-max's and adjust them for forward looking (and like every forward looking projection in quant finance, it's art+science). For no change, look at recent times when you recall things being ok in your portfolio and use those values.