Estimating Covariance with Assets That Have Different History Lengths
Summary
The document considers how to estimate a portfolio variance-covariance matrix when assets began trading at different times. One direct method is to use only the period in which every asset has data, which avoids mismatched histories but discards older observations for longer-lived assets. The discussion argues that the appropriate window depends on the intended use and how quickly the estimates are expected to remain relevant. Shorter lookbacks may suit strategies that update estimates frequently, while market shifts can quickly undermine any historical estimate.
It also outlines alternatives: treat unavailable observations as missing and use an expectation-maximization approach to estimate covariance and fill earlier periods with inferred values, or extend a newer asset's record using returns from a strongly correlated underlying or proxy. These approaches rely on assumptions about the statistical model or the proxy relationship. The document does not compare their out-of-sample performance, and stresses that the choice depends on the assets, data frequency, and modeling purpose.
Key ideas
- Using only the common trading period avoids mismatched observations but discards earlier data.
- Covariance estimates should match the horizon over which they will be used and refreshed.
- An expectation-maximization approach can infer missing historical observations from a covariance model.
- A strongly correlated underlying or proxy can extend the history of a newer asset.
- Market rallies or declines can make historical covariance estimates unreliable.
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Full text
# How to estimate variance-covariance matrix of assets with different length of historical data? # How to estimate variance-covariance matrix of assets with different length of historical data? Consider you have 4 assets `A`, `B`, `C` and `D`, where - Asset `A` started trading on 2 Jan 1990 (i.e. data is available since that point in time for every trading day until today) - Asset `B` started trading on 2 Jan 1995 - Asset `C` also started trading on 2 Jan 1995 - Asset `D` started trading on 2 Jan 2010 In the simplest method, you would just use the joint history of all assets beginning on 2 Jan 2010, maybe fill missing data due to different holidays on different exchanges and compute the sample variance-covariance (VCV) matrix. But this way you would throw away the longer history of the assets `A`, `B` and `C` resulting in a less stable VCV matrix. Is there another way to come around the problem of estimating the VCV matrix for differing length financial time series? Can you e.g. construct the VCV matrix from pairwise covariance? ## Answer by Richi Wa (score 2) https://quant.stackexchange.com/a/21668 The short answer: Take all time series starting from 2010 (at most). The covarianc-matrix tells you something about the assets for a certain amount of time. E.g. if I estiamte the covaraince matrix of those 4 assets taking into account data from the last year (!) then I can expect that this matrix remains valid for the coming 1-3 months - if the markets don't change too much. It pretty much depends on the purpose but e.g. in Momentum and Markowitz: A Golden Combination the authors argue that the seconds moments exhibit some momentum if estimate over a year and held for 1 month (quote: "In this paper we apply short lookback periods (maximum of 12 months) to estimate MVO parameters in order to best harvest the momentum factor.") . Note that all those numbers are just estimates - and if there are market slides or rallies they are possibly worth nothing anymore. ## Answer by nbbo2 (score 2) https://quant.stackexchange.com/a/21678 You can consider old prices for Stocks B, C and D to be "missing data" and apply techniques used by Statisticians to deal with such missing data. One approach, the EM algorithm, suggests you estimate the covariance for the common period, use that covariance matrix and the available data to generate pseudo data for the back period for the third stock and re-estimate the covariance matrix using the combined real/faked data. With more than 3 stocks it would take multiple iterations of backfilling. ## Answer by owner (score 1) https://quant.stackexchange.com/a/21716 I agree on Richard. the simpler you choose, the better it is so as to get reliable estimates. What's your data frequency? purpose? For model construction as far as I am concerned, daily data from 2010 is enough. Otherwise, you could use a proxy asset for asset `D` depending on its nature. To clarify, if `D` is an `ETF` let's say `CAC 40 ETF`, concatenate its return series up to the last available data point with the return series of its underlying - here `CAC 40` - since your starting date. Same process if `D` is a derivatives. By, using this approach (concatenating return series) always make sure that the asset and its underlying are strongly correlated. Hope it helps. Cheers.
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