Meaning of an identity matrix for the covariance in portfolio optimization

Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $$I$$. This approach can be interpreted as a method that shrinks the sample covariance matrix toward the identity matrix, pulling the most extreme coefficients toward more central values, systematically reducing estimation error when it matters most.

The identity matrix contains 0's for off-diagonals, and 1's for the diagonal entries. Is the essence of the importance of the identity matrix in portfolio theory due to the fact that $$I$$ represents a noiseless data structure due to its off-diagonals being 0? Or, instead of noise, does its supposed ideal properties come more from the concept of sparsity?

If so, does this mean that any covariance matrix whose off-diagonals are much smaller than its diagonals must therefore be more amenable to invertibility and quadratic optimization with low estimation error? Or what exactly is so great about a symmetric matrix whose off-diagonals are much smaller than the diagonal elements?

• “Since the identity matrix contains 0's for off-diagonals, and 1's for the diagonal entries, this makes it positive semi-definite right? which in turn means the identity matrix is non-invertible.” You’re wrong here, the identity matrix is invertible and it’s inverse is the identity matrix. It also positive definite. Positive semi-definite doesn’t imply non-invertibility but invertibility. Aug 14 '20 at 19:14
• thanks, i've corrected the question Aug 14 '20 at 19:17
• You'll find this interesting: papers.ssrn.com/sol3/papers.cfm?abstract_id=3469964 Aug 14 '20 at 19:22
• yes I think that signal-induced noise was addressed with random matrix theory portfolios Aug 14 '20 at 19:25

OK, so think of it this way...

Your standard (Markowitz) covariance matrix is a sample observation. That may or not be close to the population sigmas and correlations of your sampled markets. Even if close, the sample-vs-population errors will create asset allocation errors.

The identity matrix here is the "complete strategic ignorance" covariance matrix. Imagine a four asset world - A, B, C & D - and you knew NOTHING about them. You would reasonably assume that each had equal volatility (not knowing any better); and each was uncorrelated with any other (maybe it was +100% or -100%, but you don't know any better, so your best guess is 0). So the identity matrix (or a fraction thereof) is the Markowitz correction for not knowing anything about any parameter, as a correction for historical sampling usually over-predicting, and claiming to over-represent reality...

It's not such a bad idea with respect to multi-asset portfolios. But with respect to stock portfolios, it's not obvious to me that the "correcting" portfolio shouldn't assume a 100% not 0% correlation between any two stocks... just saying :-)

best, DEM

• Right the identity matrix has 0s for off-diagonals which are 0% covariances. The corresponding value for 0% covariance could only be a 0% correlation? So the identity ideal is assuming equal unit variances and independent (equally) uncorrelated assets, for whatever reason. Aug 16 '20 at 6:09
• Exactly. As someone else intimated, it's basically a "Bayesian prior" assuming nothing is known about a stock's vol or its correlation to other stocks. My problem is that assuming stock returns to be independent of each other is actually (I think) a very ambitious (even complacent) kind of prior!!! Aug 16 '20 at 13:25
• And yes, if the variances of A and B are both positive, then a zero covariance must by definition represent a zero correlation. Cov(A,B) = Corr(A,B) * StDev(A) * StDev(B) being a re-formulation of the traditional equation for correlations. Aug 16 '20 at 13:32
• I thought about it more, and think the ignorance (complacent ambition as you call it) is going for both variance and covariance, not just one of them, in terms of the identity matrix. Variance of assets being squarely equal to 1 is the ignorance case, because empirically we know assets have different volatilities from one another, while covariance between assets being all equal to 0 is the ignorance case, because empirically we know assets can have a very high co-dependence with one another. Aug 16 '20 at 14:08
• Exactly. My original point about "complete strategic ignorance" in spades! The validity of what assumptions to make in such ignorance, and the adjectives used to bias others' views thereof, are a whole other discussion ;-) Aug 16 '20 at 23:31

You can think of it in Bayesian terms.

To start with, knowing nothing at all about stocks, you might assume that stock returns are i.i.d with unit variance. This would be your prior. It is very simple and is well behaved because the identity is invertible.

Then you would gather some empirical data on stock returns and measure the actual variances and covariances, finding that it not an identity matrix. Being conservative and not trusting the data very much, you would form an updated estimate as a linear compromise between the prior (identity) and the observed but mistrusted empirical covariance.

• You could also view this as a shrinkage estimator -- with the right weighting scheme. Aug 14 '20 at 20:53