A Platinum member recently asked me if I would Post the results of back-tests on the 10-asset Rutherford Portfolio, so I started to prepare the Post based on back-tests that I had already performed. These back-tests focused on a new allocation/weighting strategy that I had been working on for a while.
The weighting methodology is based on David Veradi’s ideas on combining the important key variables of correlation and volatility – in this case, momentum would be accounted for simply by filtering assets from the top of the Momentum Ranking list using the standard momentum ranking spreadsheet used on this site. I have referred to this methodology in other posts as the Modified Risk Parity (MRP) Model – it takes the standard “naïve” Risk Parity model (proportional Inverse Volatility weighting) and modifies this using a correction based on Correlation relationships. Adding the momentum filter further “refines” the methodology.
The difference in my new model is that I have chosen to use the Sharpe (or Sortino) ratio to replace the “naïve” risk parity, inverse volatility parameter. I chose to do this since volatility is included in the denominator of these (Sharpe or Sortino) ratios, as in “naïve” risk parity – in addition, momentum is also included directly since the asset return (proportional to momentum) is included in the numerator of the ratio. I shall refer to this allocation strategy as the Sharpe (or Sortino) Weighting (SW) strategy.
The methodology is applicable to any number of assets to be included in the portfolio but, since I have chosen only to select the top 2 ranked assets (in the “live” Rutherford Portfolio account), this presents a special case in which correlation relationships are not important and weightings become proportional to the selected (Sharpe or Sortino) ratio.
The above figure shows the results for a 2-asset Rutherford Portfolio using the MRP (amber line) and SW (Sharpe (dark blue) and Sortino (light blue) lines) weighting strategies. As can be seen from the above, there is very little difference between the Sharpe and Sortino lines with the Sharpe line coming out slightly (but not significantly) ahead. However, both outperform (on the basis of total return) the MRP strategy.
At this point I was a little surprised since I had anticipated that the Sortino (SW) strategy might be superior to the Sharpe (SW) strategy since it was not penalizing positive returns and was only concerned with penalizing negative returns – but this was not reflected in the results (albeit with very little difference). I also recognized that I had not run a back-test using my “benchmark” weighting strategy – the simplest strategy of all – Equal Weighting (EW). Adding this to the picture revealed the following:
Although I have often stated: “If in doubt, Equal Weight”, I was a little surprised by this significant difference in (return) performance.
At this point I started to recall my days spent developing “trading systems” – i.e. systems for short-term trading rather than “investing”. These systems would establish “buy” and “sell” rules based, usually, on some combination of price and/or technical indicator signals. Sometimes, if a “system” that I thought might work well turned out to lose consistently, I would simply “flip” the rules i.e. change “buy” to “sell” and “sell” to “buy” and test the “new system” on a different asset to determine whether the “new system” was robust – i.e. produced winning trades.
As a slight modification of this idea I decided to “flip” the Sharpe/Sortino ratio and use the inverse of the ratio to determine the weighting (an inverse SW strategy). When I did this I got the following results:
The dashed blue line shows the performance when using the Inverse Sortino Ratio (for clarity I have not shown the Inverse Sharpe line, but it is virtually identical). The difference between the performance of a 2-asset portfolio using a direct Sharpe/Sortino ratio and an Inverse Sharpe/Sortino ratio is very significant!
Before saying more about this, for the sake of completeness, I’ll just add plots of a 1-asset (top ranked) portfolio and a 3-asset (Sharpe) SW portfolio:
We note here that the performance of the 1-asset portfolio (100% allocated to a single ETF) is similar to other portfolios with the characteristic that it exhibits significantly more volatility – as might be expected. The 3-asset portfolio, using a direct Sharpe ratio proportionality (and correlation correction) shows a similar volatility to other portfolios but suffers in that it generates significantly lower returns.
The performance characteristics of all portfolios is summarized in the table below:
Only one portfolio exhibits performance better than an equal weighted (EW) portfolio and this is the one with weightings proportional to the inverse of the Sortino (or Sharpe) ratio.
This puts us in the dilemma as to whether we just accept this as a “heuristic” observation or whether there is a more scientific reason for the “anomaly” – or deviation from the original “logic” thought process. At present, the only possible explanation that I can think of is that, since the Sharpe/Sortino ratio’s contain a measure of deviation from a “mean” momentum, there may be a tendency to “revert to the mean” and therefore an advantage of using an inverse relationship i.e. lower deviations may tend to move further from the mean (positive acceleration) whereas higher deviations may tend to slow down and (even if still exhibiting positive momentum) show a lower rate of acceleration. Other thoughts would be welcome.
Before finishing, I should mention that the SW strategy includes the use of a maximum % allocation, followed by a normalization calculation. I have used 2 models, one uses an equal weight (EW) maximum (i.e. if we are selecting 2 assets then the maximum allocation to either asset (before normalization) would be 100/2 = 50% – after normalization this would increase slightly), the other uses a Rank Weighted (RW) maximum (i.e. for a 2 asset portfolio the maximum allocation would be 2/(1+2) = 66.67% – again increasing slightly after normalization. The results shown above all use the EW maximum option. The RW maximum method shows slightly better performance (for the inverse relationship), but not significantly so. The most significant observation is that the “flip” (below/above the green EW line) between direct and inverse proportionality remains consistent, irrespective of whether the EW or RW method is used.