Portfolio optimization with changing portfolio constituents

Question

Say I have time series data for $N$ assets, where for the longest existing asset I have data from $t_0=0$ to $T$, but for several other assets I only have data from say $t_0+k$ to $t_0+l$ for some $0<k<l<T$.

Now, if I want to do portfolio optimization with rolling estimation of the covariance matrix using a window of size $m$, what are some good ways to deal with the assets for which I'm missing data points inside the window?

Answer 1

Go back to the definition of the covariance matrix: for $N$ stocks this matrix is made of the covariances $C_{i,j}$ of any $i$ and $j$ stocks from $1$ to $N$.

You can face different situations:

• $i$ and $j$ are both present during your reference period of $m$ dates: you have your empirical estimate of $C_{i,j}$
• you never observed $i$ and $j$ the same day during your period... Well this is a problem but if you really need a correlation you can
  1. compute a covariance matrix for all the stocks that are there for all the days
  2. perform a PCA on it and keep its first $K$ components ($K$ being low) $P_1,\ldots,P_K$
  3. regress the returns of $i$ on $(P_1,\ldots,P_K)$
  4. regress the returns of $j$ on $(P_1,\ldots,P_K)$
  5. compute their covariance thanks to these two regressions (since any $P_{k}$ and $P_{k'}$ are orthogonal, it is not difficult)
• If $i$ and $j$ have few common dates, say $m'<m$:
  • either you decide $m'$ is enough and you use the corresponding empirical covariance
  • either you do not believe $m'$ points is enough: you use the previous approach (as if they had no date in common)
  • alternatively you could mix the estimation on $m'$ points and the interpolation via $(P_1,\ldots,P_K)$