Estimating Tracking Error with Unequal Return Histories

Summary

The document considers how to estimate a portfolio’s ex-ante tracking error against the S&P 500 when holdings have return histories of different lengths. Most stocks have 180 weekly observations, but several large relative positions have fewer than 36. It asks whether a shrinkage estimator applied to a covariance matrix built from all available observations is an accepted solution.

The response points to a full-information maximum-likelihood approach for series with histories of unequal lengths. In outline, regress the short-history returns on the longer-history series to estimate their covariance relationships, then account for the residual covariance from that regression. Once a complete covariance matrix is estimated, tracking error can be calculated using the usual portfolio calculation. The document gives no implementation details, comparison with shrinkage, or empirical results, so it does not establish which method is best for a particular portfolio. Its practical value is introducing a method designed to use unequal-length histories rather than simply discarding observations or treating every pair as equally well estimated.

Key ideas

• Return histories of different lengths complicate covariance estimation for portfolio risk.
• A full-information maximum-likelihood method can use the longer and shorter return series together.
• Regress short-history returns on longer-history series to estimate their covariance relationships.
• Include residual covariance when constructing the full covariance matrix.
• Calculate tracking error from the completed covariance matrix using the standard portfolio approach.

Tags

# How to calculate tracking error given mismatches in available data


# How to calculate tracking error given mismatches in available data












Apologies if this is an overly simple question. I have a series of stock returns, and I would like to estimate my portfolio's ex-ante tracking error versus the benchmark (S&P 500) given the current relative weights and historical returns. However, for many of the stocks there are very few returns (<36 weeks).

What is the industry standard for dealing with mismatching data when estimating portfolio volatility or tracking error? Is it commonly accepted practice to use a shrinkage estimator (as in Ledoit and Wolf) on a covariance matrix estimated using all available data points?

I have 180 weekly returns for the vast majority of the stocks. But, the portfolio's largest relative positions are in stocks that have very little data (FB, LNKD, GRPN).

## Answer by John (score 3)

https://quant.stackexchange.com/a/7456

There are many techniques, but I would begin with Stambaugh Analyzing Investments Whose Histories Differ in Lengths. The full information maximum likelihood approach he describes basically involves regressing the short history series against the long history series to obtain the covariance with the longer history securities and adding back the covariance of the residual from the short history regression. I think there's some Matlab code that implements it in this package

Once you have the full covariance matrix, you can calculate the tracking error as normal.

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