# Portfolio optimization with changing portfolio constituents

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?

• Usually, when data points are missing - for example in the price series - the best way to deal with it is computing an interpolation between the two available prices and fill the blank with it. However, you have to be sure that your data isn't affected by any other problem. – james42 Jun 1 '17 at 14:19
• Be careful with interpolation of time series, this is technically looking into the future. – msitt Jun 1 '17 at 16:41
• If $k <l <T$ are you saying these securities do not trade anymore? Why would you be interested in such instruments? I'd understand if the case was you have data only starting from $k$ or $l$ to $T$ not from $t_0$. – AK88 Jun 2 '17 at 2:21
• I just included $k<l<T$ for generality. It could be that for instance one asset was very influential during the first half of the lookback window at a given point in time, but then got delisted. Would one not want to include this fact somehow? – L1meta Jun 6 '17 at 11:52

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 $(P_1,\ldots,P_K)$