# Portfolio optimization of unequal length back-tests

I have a portfolio of assets. For each asset I have a back-tested time series of daily profits. I'm tying to optimize, using the correlation of daily returns, to minimize the total draw-down of the portfolio. My problem is that I have 5 years of data for 6 of my assets, and 10 years of data for 4 of my assets. I've taken two approaches so far, neither of which I'm satisfied with.

First - simply optimize using the 5 years where I have data for all assets.

Second - optimize the 4 assets for 10 years independently to create 1 asset, then optimize with remaining 6 assets for the 5 year period with complete data.

My major issue with this is if one of the assets with a longer data-set contains large draw-down 6 years ago, my optimization is ignoring this and likely over-weighting that asset moving forward. On the flip side, if that asset's returns are slightly negative the past 5 years, but strongly positive the years prior, the optimization will under-weight this moving forward.

I understand this is probably a classic issue with trivial solutions, but I'm not well versed in portfolio theory when it comes to optimizing using daily returns. I appreciate any thoughts on this.

First of all: The issue is classic, but by no means trivial.

Your "First" option is probably the easiest. You just adjust your dataset by thorwing away the data points that are not present in all time series. After all, the goal is to strike a balance between "statistical significance" and an identically distributed sample, meaning: If you think the information from the long history is not interesting for the distribution going forward, then you shouldnt tamper with conditional expectations and imputing missing returns...

On point 2: I have adressed the problem in a different answer and I am suggesting to do this:

Page, S., 2013, How to Combine Long and Short Return Histories Efficiently, Financial Analysts Journal 69, 45-52

Its basically suggesting an algorithm to backfill the missing returns based on the current correlation. Please also take note of the caveat section of the paper - there are enough of them of course.

I do not know how your optimization process for "minimizing drawdowns" works. If its some mean/expected-DD optimization using the covariance matrix you should be fine (or even better the empirical distribution). However, I would not calculate historical drawdowns based on imputed returns and compare them.

Another word of warning: When optimizing, your investment horizon can be important. Daily returns are distributed differently from monthly or annual ones.

Best of luck for your optimization!