# How can one quantify the incremental value of better covariance matrix modeling in portfolio optimization?

Let's say we have two estimators of the covariance matrix, $$\hat{C}_1$$ and $$\hat{C}_2$$, and the latter is an improvement on the former.

Is there any measure of the improvement that can be sensibly translated into gains in portfolio performance?

To be more concrete, let $$\delta(\hat{C})$$ denote a loss function that measures how bad'' an estimator $$\hat{C}$$ is. We know that $$\delta(\hat{C}_1) > \delta(\hat{C}_2)$$. However, it would be ideal if we can translate the incremental improvement $$\delta(\hat{C}_1) - \delta(\hat{C}_2)$$ into the expected improvement on some portfolio performance metric, such as the Sharpe ratio.

• Hi: that's going to be very difficult to do because, even if you have a "better" covariance matrix ( you didn't define better ), it's unlikely that this will result in consistently improved returns. Just by random luck, one covariance matrix, A, may result in greater returns than another , B, even when B is a better estimate of the true covariance matrix. Jul 30 at 11:20
• @markleeds Of course, I'm talking about incremental improvement in expectation. Jul 30 at 22:49
• Oh okay. Expectation of what ? returns, sharpe ratio ? if its returns, you could calculate efficient frontiers and compare them. if it's sharpe ratio, you could calculate expected sharpe ratios and compare those. Jul 31 at 12:57
• @markleeds, "what" is exactly the question in this post. Jul 31 at 14:14
• Mark and Richard - thank you both for the comments. I am trying to see if there is any "standard way" to do what I have in mind. I guess your answers show that there isn't any. Jul 31 at 17:03