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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?

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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.

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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.

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  • $\begingroup$ How is this an EM algorithm? I can see that the estimation of the covariance matrix is the second step in the M part (i.e. Maximization), but doesn't the first step E should involve the expectation (hence the E) of the likelihood function conditional on the available observations, i.e. $E_{Y|X, \theta^{(t)}}[L(\theta; X,Y)]$? Or is the simulation of data used as a numerical approximation for this expectation? But if it is to serve as an approximation, shouldn't the number of simulated data points be large? And how does this relate to the number of missing observations? $\endgroup$
    – Confounded
    Commented Dec 2, 2020 at 11:08
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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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