# What weights should be used when adjusting a correlation matrix to be positive definite?

I have a correlation matrix $A$ for an equity market that is not positive definite. Higham (2002) proposes the Alternating Projections Method, minimising the weighted Frobenius norm $||A-X||_W$ where $X$ is the resulting positive definite matrix.

How should one choose the weight matrix $W$?

The easy alternative is to weigh them equally (W is an identity matrix), but if one has exposures to a portfolio, wouldn't it be natural to weigh the correlations according to your weights of exposure in the different assets, in order to alter their historical correlation less than for those assets you have little exposure in? Or is there a more natural choice?

-
Hi Osloguten, welcome to quant.SE and thanks for submitting this very relevant question. –  Tal Fishman Oct 26 '11 at 18:41
Thanks. Well, so far I have not found any solution and are currently running unweighted approximations. I find this about alright, but as I am approximating correlations from some stocks that are somewhat illiquid it would be satisfying knowing that these will be altered more than the main stocks in our portfolios.. –  AdAbsurdum Nov 8 '11 at 8:18

You may want to have a look at a later paper by Borsdorf, Higham, and Raydan (2010). I believe a variant of the same method may apply in your case. That is, you may want to account for some of the factor structure of your correlation matrix before you apply an unweighted Frobenius norm. Otherwise, using unweighted norms has often given me fine results anyhow, and this is often used only as a quick fix to slightly adjust matrices that are just barely not positive definite. A full approach should definitely be applying some factor structure (see a previous question of mine, as well as others on the site).

-
Yes, I agree, but I have received some pretty nasty results while weighting. Also, how should one evaluate correlation pairs? –  AdAbsurdum Feb 7 '12 at 13:05
@AdAbsurdum can you be more specific about the correlation pairs? Perhaps post a new question, if you feel it is worthy of its own question. –  Tal Fishman Feb 7 '12 at 15:26
Well. Thinking within the model, one would believe that some correlation pairs are more trustworthy than other (e.g. high liquidity implies good data implies a better correlation coefficient). Thinking outside a model, looking at how much the correlation have changed in history could also be a parameter that determine the weight of a pair (high variability in time implies lower weight). Not sure if its worth an own question. Rather it is a part of the overall, although the quesiton above consist of technical and a economical considerations. –  AdAbsurdum Feb 7 '12 at 16:15
@AdAbsurdum your thinking sounds pretty solid to me, both liquidity and variability seem like decent weighting functions, but I haven't seen anything objective on that point. I agree, not really a new question, but could be added to the current question. –  Tal Fishman Feb 7 '12 at 16:19