I have to use MVO in practice, in addition to other types of portfolio optimizations. Making correlation adjustments based on assumptions about changes in expected relationships in some of the variables is common practice. What you are talking about, doing MVO with strict historical data, is technically not what one is supposed to do as they are supposed to impart some idea about the expectations of the inputs. But I won’t bore you with that philosophy, mainly pushed by the academics who don’t need to do this in the real world, and we’ll keep the assumption that your historical data approach is fine (it also sounds like you don’t want to introduce that kind of sophistication into your optimization, so no problem). Some points…
The fact that correlation is a dimensionless variable is actually not related to what I’m talking about.
Say you have 40 years of data for your S&P500, Barclays Agg and Cash Indices. You calculate the returns, variances and covariances for all at various frequencies – i.e., annual, quarterly, monthly, weekly, daily. You will see different results for all of them when calculated at different frequencies. In other words, your correlation will not be the same when calculated at a monthly frequency as it is when calculated at an annual frequency, and so on. Dimensionality has nothing to do with it. The timeframe over which an input is relevant to the investor, however, has everything to do with it.
If I have a day trader’s timeframe, I am more interested in intra-day or daily scale for my inputs, as that is the correlation and volatility that is relevant to what I’ll be experiencing. If I have a long-term timeframe of 20-30 years, then I am probably more interested in correlations and vols calculated over monthly, or perhaps even quarterly or annual timeframes. In either case, I’ll want my returns to be scaled consistently with the frequency I’m using with my correlations and vols.
Don’t make the assumption that everything scales up nice and neat. If you are not working with time series that are purely a random walk, scaling up, say, weekly volatility by the square root of time is not necessarily going to yield the same volatililty that would be calculated by using monthly or annual returns. This is due to the fact that very few time series in finance follow a true random walk. So if your calculations are different depending on what frequency you use, there should be a clear philosophical reason why you would calculate correlations and vols based on smaller timeframes versus returns calculated based on longer timeframes.
I see your issue in coming up with estimates for annual returns and the frustration you have. I suspect this may be due to you dealing with limited data sources. If the variability of the return calcs are frustrating, why not roll the end-to-end estimates down in Excel (drop one day off the end, add one to the front), and then take the average of the rolling series?
FYI, depending on what you are talking about measuring in terms of TWRR, this may not be the correct approach to use in estimating return within a traditional MVO framework (are you talking about just doing the TWRR of the individual index price series?).