# Portfolio risk-return when assets have limited and inconsistent historical data / time series?

Lets say we have "today's" snapshot of asset allocation and need to determine the 6mo, 1 yr and 5 yr risk and returns of this portfolio. If the time series for every asset is very long, longer than the longest time horizon of interest then it's simple. Just compute the portfolio returns => get mean and standard deviation and call it a day (ignoring higher order arguments for now).

However, when the time-series data is of different lengths as well as of much shorter durations than the time horizons of risk and return - it's not so straightforward. To illustrate, I've sketched this portfolio time series below.

In such a situation, given only the snapshot of the portfolio 'today' and their time series, what would be a statistically consistent way to determine the risk and return? Lets assume we can't know the portfolio composition/asset allocation in the past - just a single snapshot less than a week old / "today". We would like to employ the same principles across the spectrum of portfolios i.e. use the same even if

• portfolio 1 has only A and B today
• portfolio 2 has only FB and G today
• portfolio 3 has A,B,C,D,E,FB, and G today
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Generally I would annualize risk and returns even when an asset's returns/general time series (ts) does not span over the full year So, both, FB and G present risk and return over the past year. For risk and return that is calculated over longer periods I would not include an asset in the portfolio of which you have no ts available to measure risk and returns. So, if you try to asset the 5 year portfolio risk and return I would not include asset FB and G in the portfolio. Already extrapolating to full years is making assumptions some may find pushing the envelope.

I would strongly advise not to go the correlated asset replacement route. Over a long-term horizon, such as your 5 year risk/return calculations, there is no replacement asset that can be correlated too highly with your asset, whose longer return time series are missing, to make up for unaccounted unsystematic/company specific risk.

A simple calculation should make this clear: Lets take portfolio of 3 assets A,B, FB. Let's assume you have the portfolio risk and return over the past 5 years but not the individual asset time series over 5 years for FB. Now, replace the missing time series FB with a highly correlated asset. Re-calculate the portfolio risk and return profiles. Derive the tracking error to the true portfolio risk and return profiles. Now, before plugging in the correlated asset returns into the missing time series spots, introduce a one-time jump of +-20% at an arbitrary location in the time series. Re-calculate your portfolio risk and return, derive your tracking error. Are you happy with the results? Can you live with the induced jump? Because if you say no you should never even start to think to replace asset returns with any correlated asset returns. 20% moves due to corporate actions, or any unsystematic even for that matter is highly conservative, generally you witness a lot higher deviations over such long period of time, plus correlations generally completely break down after such large moves, something we did not even account for in our back-of-the-envelope calculation. This all assumes we are talking about cash equity as an asset class. Other asset classes may witness higher or lower jumps, and the 20% is purely arbitrary, though as pointed out I believe it is at the low end of what can happen in a 5-year time span.

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So if I understand correctly, a portfolio with 25% weights in A,B,FB,G needing the 5 year risk-return statistics would essentially ignore 50% of the portfolio value under that approach? Seems rather extreme ... – DeepSpace101 Mar 28 '13 at 16:40
@Sid, sounds less extreme to me than constructing an arbitrary time series that in no way reflects the true risk/return profile of the asset in question. I edited my answer to comment on correlated assets as well – Matt Wolf Mar 29 '13 at 1:13
Agreed. Seems like a tough one with very limited / partial data ... – DeepSpace101 Mar 29 '13 at 2:07

For a similar case (Introduction of REITs in Germany a few years ago) we looked for a similar asset (in terms of correlation) and used that time series for the portfolio construction. We had only a very short time series for that REIT Index (around 4-5 months iirc).

Specifically, we used Dutch REITs - due to high correllation between German and Dutch lead indices - and adjusted the NL-REIT timeseries to match the moments of the German REITs. By no means a perfect method, but we tried to analyze the potential for REITs in the German market. To give some context, the result was used to construct some simulated mean/variance portfolio, and not as live strategy.

PS I hope I understand your question correctly, not really sure though. If not, please clarify.

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Thanks. And you got the question right. When you say 'looked for a similar asset in terms of correlation', what specific tests did you conduct to conclude that they are "similar enough"? I ask because the time series for the new one will still be limited ... – DeepSpace101 Mar 28 '13 at 20:22
In our case alternatives were obvious - REITs (think stocks on real estate) were available in several European countries for +10 years, but were recently introduced in Germany (as a new legal form). So all we had to do was find the series for countries were REITs were available, then we looked at correlations among multiple lead indices (i.e. DAX for Germany, AEX Netherlands) and picked that country with the high correlations over most indices, and ended up with Netherlands. Honestly, we looked at it, and picked by guts ;-) But again, that was only a client presentation, not real research. – phi Mar 28 '13 at 22:18
...so for our application, length of the original time series didn't matter much, since we compared country indices and then adjusted the Dutch data to match our and stiched the series together. – phi Mar 28 '13 at 22:21
@phi, I see your point though please consider REITs are much more highly correlated with each other than cash equity over a 5 year horizon. I assume Sid is talking about equity portfolios here but in any case replacement by correlated asset generally only works if correlations are extremely stable which they are not for most all cash equity (I commented on that in my post). Have you looked at correlation stability in REITs? I would be curious if you care to share your findings – Matt Wolf Mar 29 '13 at 1:29
Yes, you are absolutely right. This method works only for assets with very similar sources of risk, and if unsystematic factors can be considered negligible. I was hoping that the context made that clear. Also, we did not analyze the stability of the correlations, but compared a few more-or-less arbitrary periods, which were mostly stable for the REIT indices within Europe. – phi Mar 29 '13 at 18:48