CSSA

The Minimum Variance Algorithm (MVA) follows much of the same logic as the Minimum Correlation Algorithm (MCA) and differs primarily in the objective function which is to minimize portfolio variance versus correlations. Both are “heuristic” algorithms that seek to approximate the results of more complex methods that require employing quadratic optimization. In a recent whitepaper, Newfound performed various simulations and came to the same conclusion that I have shared for a long time:   in the case of uncertainty in the parameter inputs such as returns, correlations and volatilities, simple heuristic methods achieve results that are equivalent to more complex optimization methods. It is therefore feasible that good heuristic methods can exceed the performance of their more complex counterparts especially if they are designed to be less sensitive to parameter uncertainty.

The core principle of both MVA and MCA is to use proportional allocations to generate weightings because they are more stable than using discrete selection of both assets and weights. This principle is supported by information theorists, and is used frequently in technological applications. Cover also covers this principle in his work on Universal Portfolio Theory. A good summary article is presented on Ernie Chan’s blog. Another aspect of both MVA/MCA is that they use a gaussian transformation  to normalize the relative average correlations/covariances. MVA is very similar to “mincorr2” (see the whitepaper for more details) and simply finds the average covariance of each asset versus all other assets -including its own variance- and then converts the average value for each asset to a cross-sectional distribution using normalization. This is used to proportionately weight each asset to find an initial set of weights. The final weights are derived by multiplying each initial asset weight by its inverse variance and then releveraging to sum up the weights to a total of 100%.  The result is that weights reflect both the asset’s own relative variance and also average covariance to the universe of assets. However, the weights are less dependent on correlation estimates (which are critical in complex minimum variance but are noisier than volatility estimates) and do a better job of distributing risk since allocations are made to all assets in the universe.

Below is a backtest of the MVA on eight highly liquid ETFs used for the original MCA tests since 2003. The variance-covariance matrix uses a 60-day parameter with weekly rebalancing. The benchmark used is equal weight:

As you can see the MVA achieves a high sharpe ratio (higher than MCA) and achieves slightly superior returns to an equal weight portfolio with less than 50% of the volatility. The benchmark analysis shows that MVA is simply a means to efficiently reduce downside relative to an equal weight portfolio, and this comes at the cost of some upside performance. MVA captures 75% of the upside in bull markets for the equal weight index, and only 50% of the downside in bear markets using a continuous distribution measurement. The actual results of this one test are not meant to be conclusive, but I have done a large range of tests on different universes with both long-term tests on index data and using recent ETF data and have found similar results. While there is nothing magical about MVA, it supports the point that a heuristic method can be very effective-especially with noisy time series data. For the sake of practicality, it can be implemented easily in just about any platform and like MCA can also be computed very quickly for large datasets. There isn’t really a good case to employ quadratic optimization to minimize variance unless you need handle different constraints. While I haven’t done much in the way of comparisons between the two, I would imagine that MVA would perform at least as well across a wide range of universes.

Often readers ask about methods for approximating minimum variance portfolios. In practice the minimum variance portfolio can be calculated in closed form only for long-short portfolios, and requires a quadratic optimizer to solve for long-only portfolios. Source code and examples for  long-only minimum variance can be found at Systematic Investor – a very good blog that also has a toolkit for a lot of standard optimization methods. Michael Kapler (the man behind Systematic Investor) and I wrote a whitepaper about an algorithm for finding minimum correlation called the Minimum Correlation Algorithm (MCA), which was meant to approximate maximum diversification portfolios. The primary benefits of the algorithm versus the conventional optimization were: 1) speed of computation and ease of calculation 2) greater robustness to estimation error and 3) superior risk dispersion.  Testing results across a wide array of universes also demonstrated the superiority of MCA in terms of risk-adjusted returns versus its maximum diversification counterpart. The Minimum Variance Algorithm (MVA) is a close relative to MCA and shares the same benefits versus conventional minimum variance optimization. In testing, MVA showed superior risk-adjusted returns to MCA across most universes. While I have not yet conducted comparisons versus conventional minimum variance, preliminary results are very competitive. This is encouraging considering that MVA is very simple to calculate. Later this week I will present the logic and also post a spreadsheet for calculation along with some test results.

The recent popularity of “tactical” investment strategies has given rise to a dizzying array of new terminology and strategy descriptions. Most investors and investment professionals lack a deeper understanding of the core nature of such strategies. They can hardly be faulted for all of the marketing material floating around that often obfuscates the difference between a separate brand and a truly separate strategy.  In reality, most “tactical” strategies are very similar and have predictable payoff profiles even if their returns are not predictable. The class of tactical strategies are essentially part of the broader class of “dynamic asset allocation”strategies (DAA). The opposite of DAA is to employ “strategic” or policy-based asset allocation that contains a constant mix (like 50/50 or 60/40).The premise of DAA is that through active shifts in portfolio weightings, one can add value versus buy and hold or stategic/constant mix portfolio allocation.  One of the best articles that helps provide a solid grounding in Dynamic Asset Allocation was a classic paper by Perold and Sharpe.

There are essentially three core strategies compared in the paper (excluding those that are option-based): 1) buy and hold–yes this is a strategy 2) constant mix (CM)– this is like a policy weighting that is constantly rebalanced such as 60/40 stocks/bonds  3) constant proportion (CPPI)– this is essentially synthetic portfolio insurance generated by using a dynamic allocation between stocks and t-bills.  In CPPI, you would buy stocks as they are rising with an increasing proportion, and sell stocks when they are falling with a decreasing proportion until you hit a “floor”- which is essentially akin to a stop loss. These three strategies are  demonstrated in the paper to have very different payoff profiles as a function of market conditions. The table below provides a useful “cheat sheet” that also helps clarify the differences. The best performer in a given market regime is ranked #1 while the worst performer is ranked #4:

Here is a table that relates the different  strategies above as closely as possible  to more commonly used investment strategies or products (note that due to the multi-asset composition of these vehicles/strategies the linkages are not quite perfect):

The first takeaway is that there is no uniform winning strategy in all market conditions. Each strategy has a particular regime in which is it likely to shine. Bull Markets tend to favor Buy and Hold (BAH) unless one is able to successfully employ a CPPI strategy on the underlying asset with leverage (this may or may not be possible .Other things being equal, the degree to which CPPI-L will be able to beat a BAH is proportional to the degree of market noise; as the market becomes noisier, the CPPI-L will have more difficulty matching BAH. With more predictable/trending behavior the CPPI-L will easily beat BAH.Without the use of leverage, it is impossible for CPPI or CM to keep up with BAH in rising markets. This is the case for most tactical strategies- especially if they hold assets other than the equity market. This under-performance can also be compounded by market noise. CM also has difficulty in bull markets because it is constantly “taking profits” via re-balancing and inherently reducing the delta to the market.

In Sideways Markets, both BAH and CPPI struggle  due to a lack of return and a greater abundance of noise. This is where CM shines since it is like Shannon’s Demon– the optimal strategy for capitalizing on entropy. Unlike BAH and CPPI, it is possible to make money with CM even if the market does not produce a positive return.  CPPI is most vulnerable in sideways markets because it is the most sensitive to noise and can get “whipsawed.”

In Bear Markets, CPPI strategies shine because they have a fixed maximum total loss defined by the floor that is always set at a % that is greater than zero. The degree of protection that is guaranteed will be proportional to the reciprocal of the slope (1/m). The protection will also be a function of whether the floor is ratcheted as the asset rises, and also whether the floor is periodically reset- like rolling call options.  BAH obviously does the worst, as it is fully exposed to any losses that incur. CM falls in the middle as it has inherently less exposure through re-balancing, and also tends to capture some of the market volatility.

In general, CPPI-type strategies are most related to trend-following and momentum or relative-strength investing. The broad class of “tactical” or “active” strategies are likely to have a payoff profile very similar to CPPI. In contrast CM-type strategies are either more similar to “balanced” type passive funds or ETFs, “strategic” asset allocation methodologies that are static, or represented by strategies that attempt to capture mean-reversion.  BAH-type strategies represent virtually any passive holding- I would include most equity or bond mutual funds/etfs in here simply because they all seek to have low tracking error and make minor bets on individual holdings in an attempt to outperform on a relative basis.

Perhaps the most important takeaway from the paper was defined as a theory for which strategy will be likely to outperform/underperform:

“the fact that convex and concave strategies are mirror images of one another tells us that the more demand there is for one of these strategies, the more costly its implementation will become, and the less healthy it may be for markets generally. If growing numbers of investors switch to convex strategies, then markets will become more volatile, for there will be insufficient buyers in down markets and insufficient sellers in up markets at previously “fair” prices. In this setting, those who follow concave strategies may be handsomely rewarded. Conversely, if growing numbers of investors switch to concave strategies, then the markets may become too stable. Prices may be too slow to adjust to fair economic value. This is the most rewarding environment for those following convex strategies. Generally, whichever strategy is “most popular” will subsidize the performance of the one that is “least popular.” Over time, this will likely swell the ranks of investors following the latter and contain the growth of those following the former, driving the market toward a balance of the two.” Perold and Sharpe, “Dynamic Strategies for Asset Allocation.”

The theory is simple and is supported by recent market history- if everyone wants for example market upside with protection–ie tactical type strategies- then buy and hold and constant mix will probably outperform. The opposite is true if everyone wants  a more passive approach. Remember the 1990’s when everyone was switching to index funds? That massive shift in demand gave rise to one of the most profitable decades for tactical/active management- 2000- 2008. The spectacular success of tactical strategies in 2008 especially gave rise to tremendous demand for tactical which subsequently underperformed in a big way in 2009. Renewed market problems in 2010 and 2011 and the prospect of a sovereign debt crisis produced good performance for tactical strategies and naturally tremendous demand given all of the renewed “fear mongering”. Naturally 2012 was not kind to tactical strategies–especially those that sought to minimize volatility to protect against a crisis. As investors begin to pile into equity markets and passive investments again, it is possible that tactical products will eventually outperform.

The key lesson is that markets do not exist in a vacuum– the relative peformance of a strategy is inversely proportional to the general demand for that strategy:  when it is least popular to be tactical, it is likely that tactical will outperform. In contrast, when it is least popular to be passive or hold a strategic asset allocation it is most likely that these will outperform. Basically, if it hurts to invest in a given strategy and you have a lot of company in feeling that way it is probably a good idea to invest! The problem is that most investors and advisors want to go with what is working now as this is the easiest sell and also the most comfortable to invest in. On the choice of which dynamic strategy to use in the long run, again Perold and Sharpe had some wise words:

“Which dynamic strategy is demonstrably the best? The goal of this article is to emphasize that “best” should be measured by the degree of fit between a strategy’s exposure diagram and the investor’s risk tolerance (expressed as a function of an appropriate cushion). Ultimately, the issue concerns the preferences of the various parties that will bear the risk and/or enjoy the reward from investment. There is no reason to believe that any particular type of dynamic strategy is best for everyone (and, in fact,only buy-and-hold strategies could be followed by everyone). Financial analysts can help those affected by investment results understand the implications of various strategies, but they cannot and should not choose a strategy without substantial knowledge of the investor’s circumstances and desires.”

In the last post, we introduced the “All-Weather” Sector Portfolio which was developed using data from Fidelity Asset Allocation Research. I created a heuristic approach to integrate a variety of factors (length of  stage, sector performance ranking by stage) in order to create the final portfolio allocation. It is obviously very interesting to examine the performance of this static/strategic portfolio allocation over time. For testing comparisons, I used the Fidelity Sector mutual funds total return series and also the S&P500 total return cash index. The time period for testing was chosen to include all active
sector members to ensure a fair comparison.  Three time series were created: 1) “All-Weather” Sector 2) Equal Weight Sector 3) S&P500 Total Return Index. Rebalancing was conducted on a monthly basis.The graph and table below depict the results:

The results are promising– both higher returns and risk-adjusted returns than both an equal weight benchmark and the S&P500.  The All-Weather Sector also has the lowest risk of the three portfolios. Transaction costs and turnover are likely to be negligible in this case–especially if one were to stay within the minimum holding period for the funds. The broad diversification across sectors and limited “tilting” of sector weights makes this version of the All-Weather Sector Portfolio a desirable core equity holding for investors. The All-Weather Sector Portfolio is arguably a superior theoretical index construction than equal or market cap weightings, with  low tracking error,  tax-efficiency,  and good results to back up the concept.  In subsequent posts I will show some alternative formulations and weighting schemes that have superior performance.

The central concept of the “All-Weather” portfolio is balance: having an allocation that will perform equally well across different economic regimes.  The original portfolio balances portfolio risk and performance with broad asset classes to be neutral to changes in  economic growth and inflation.This basic concept can be extended to create an “All-Weather” equity sector portfolio. One of the traditional ways to look at sector rotation  is to use the economic “business-cycle” to determine which sectors are the most favorable. In this view, the economy goes through four distinct phases that progress in sequential order and repeat in a multi-year cycle: 1) early 2) mid 3) late and 4) recession. In this context, economic growth is highest at the earliest stages and the rate of change declines as the business cycle progresses. Inflation is low at the earliest stages and builds over time to the point where there is “over-heating” at the late stage prior to a recession where inflationary pressures cool down.   Fidelity has conducted a study spanning over 50 years to determine the sectors that perform best in each stage. Below is a graphic (Source: Fidelity Asset Allocation Research) that depicts the business cycle and the expected relative sector performance by stage:

The results of the study indicate some very robust and significant differences in sector performance. The problem with applying this approach is that there is considerable uncertainty as to both which stage the economy is at in a given point of time and how long the current business cycle will last (they can vary from 6 months to several years). It therefore makes logical sense  to create a sector portfolio that is effectively neutral to the business cycle– an “All-Weather” Sector portfolio. This can be accomplished by generating four different portfolios that perform the best in each stage of the cycle and then weight them to account for differences in stage length and performance. The resulting portfolio should perform very well over time with greater consistency than a more naaive allocation.

To create the “All-Weather” Sector Portfolio, I used a simple scoring system to capture relative differences in sector and stage performance. The choice of  using a ranking/scoring model in favor of the actual data avoids the noise associated with using past sector returns and also the historical stage of cycle lengths which can be more difficult to extrapolate in terms of raw magnitude. Theoretically, the relative favorability and length of cycle should be more stable. Both the length of stage and relative performance by stage are ranked (highest to lowest). The cumulative weight is generating by multiplying the length ranking by the relative performance ranking. This determines the relative weighting of each of the four portfolios (early, mid, late and recession). Within each portfolio, each sector is assigned a score according to relative favorability. A score of 3 is given to the historically best performing sector for a given stage, a score of 2 is given to sectors that have outperformed significantly, a score of 1 is given to sectors that have neutral performance (match the market) and a score of 0 is given to sectors that have historically under-performed the market. I have compiled a All Weather Sector Worksheet to show the breakdown of the calculations in greater detail. Below is the final composite and allocation breakdown of the “All-Weather” Sector Portfolio:

The All-Weather Portfolio was designed by Ray Dalio (and clearly influenced by Harry Browne of the Permanent Portfolio) as a robust static allocation that can be used by investors to deliver consistent performance over time. The logic of the portfolio construction is to be neutral to risk/uncertainty with respect to inflation or economic growth–the two primary factors considered to explain all asset returns.  The allocations are a function of the long-term expected sensitivity of each asset to the change in these factors- based on whether they are rising or falling substantially in relation to historical norms.

We know that the “Static” All-Weather Portfolio (using the method above) has a very good long-term track record to back up the story. The more interesting question is whether a dynamic risk allocation can outperform using the static method. Theoretically, risk inputs- especially standard deviations- should be easy to model in a dynamic context since they are fairly predictable. Furthermore, we do not necessarily need to pre-specify the relationships between assets because we can observe their changing relationships via clustering. Since the All-Weather approach has often been considered interchangeable with Risk Parity, it is interesting to see if the purely mechanical and dynamic approaches to risk parity perform in comparison using the same assets. Michael Kapler of Systematic Investor, ran the following tests in R using different risk parity variants. We also show for comparison the more sophisticated “Cluster Risk Parity” (Kapler, Varadi, 2012) which removes the universe bias from portfolio allocation and delivers a more precise risk allocation. The assets used below to represent the different asset classes are a combination of funds and ETFs to maximize data history:

The relative risk-adjusted performance of the Static All-Weather Portfolio versus the dynamic variations is presented below.

We can see that all dynamic methods perform better than the static method by a fairly substantial margin in terms of risk-adjusted returns. This suggests that the changing risk and correlations of each asset class already reflect expectations for changes in the economic factor risk to both inflation and economic growth. Furthermore, these changes can be predicted by looking at recent historical data. In addition, we also can see that more complex versions of risk parity (ERC and Clustering variants) slightly underperform the simplest version of risk parity that ignores the correlations between securities and only uses the risk information. This potentially implies either a constant correlation between assets, or that the careful choice of these different assets already reflects an embedded static clustering method (which would make the correlation information much less useful than risk in a dynamic context). Since previous tests demonstrate the superiority of clustering methods (both static and dynamic) to basic risk parity, this implies that the universe chosen is a good static clustering approach. In conclusion, the results at least suggest that dynamic risk allocation is a valid way to create an effective “All-Weather” Portfolio. In practical terms, using cluster risk parity with a diverse and large asset pool is the easiest way to capture this profile while avoiding a lot of pre-specification.

In the last post we looked at the performance of static versus dynamic clusters on Dow 30 stocks. It is also logical to look at the same comparison on multiple asset classes. Michael Kapler of Systematic Investor ran the same set of tests on major market asset class ETFs for comparison.  To avoid distortion in static versus dynamic clustering, the starting point for the test data was set at the point when all ETF data for each asset class was available. We used the “common sense” method for static clustering, which is typically how investors and traders categorize assets:

The ETFs chosen cover a broad range of asset classes. For dynamic clustering, we again used the  principal components clustering method which is referred to as “hcluster” in “R”. Note that Cluster Risk Parity refers to using dynamic clustering with risk parity allocation both within and across clusters–ideally with risk parity-ERC, or equal risk contribution.The test comparisons are presented below:

While this is not a long backtest, we see that the results are consistent with prior results on the Dow 30 tests and also with what we would logically expect: 1) Cluster Risk Parity is the best performer in terms of risk-adjusted returns (and also annualized returns in this case) 2) dynamic clustering outperforms static clustering in terms of both returns and risk-adjusted returns 3) static clustering outperforms non-clustering and all clustering methods outperform non-clustering in terms of returns and risk-adjusted returns. To further break things down, we also see a logical rank progression based on the risk allocation method: 1) All risk parity variants outperform equal weight in terms of returns and more importantly risk-adjusted returns 2) risk parity-ERC outperforms the more basic risk parity methods- which do not make use of the covariance information. In this dataset, all of the rankings show a greater separation in terms of magnitude than on the Dow 30 tests, which can be expected since assets are less homogenous than stocks.

In general, the purpose of these tests is to show the importance of dynamic clustering and also more precise risk allocation methods in portfolio management. The combination of these two methods leads to a superior risk control and risk-adjusted performance than either in isolation. While the performance improvements are somewhat modest, they are fairly consistent and also more importantly make the portfolio allocation process less sensitive to unfavorable variation arising from universe specification. In fact, it is possible (with some refinement in these methods) to avoid having to carefully pre-select a universe in the first place. This leads to backtest performance that is less likely to be inflated in relation to out of sample results. In a perfect world, we would want to input a large universe of liquid tradeables and have a self-assembing optimization and allocation process with multiple layers based on a set of pre-specified constraints. 

A natural comparison for an allocation method that makes use of dynamic clustering is to use a static clustering method. An example of the use of static clustering are the sector classifications made by large index firms. Typically clusters are formed based on the type of business or industry associated with a company (ie utilities, energy etc). The Dow Jones Industrial Average contains 30 large cap stocks that have a very long trading history. Furthermore, each stock can be easily classified by their respective S&P sector  . This static clustering can also form as the basis for incorporating risk parity methods for  portfolio allocation. The following tests were done in R by Michael Kapler at Systematic Investor using Principal Components for dynamic clustering.

As you can see, dynamic clustering holds a small but consistent advantage over static clustering. The dynamic method produces higher returns and risk-adjusted returns over a long backtest period. Once again, Cluster Risk Parity (dynamic clustering with risk parity or risk parity-erc) does better than any other risk parity variant. Furthermore, dynamic clustering also produces better returns and risk adjusted returns than non-clustering methods. Interestingly, static clustering was not as effective as ignoring clusters altogether. This suggests that the changing volatility and correlation contain information that is exploitable on a dynamic basis. This finding is intuitive with respect to volatility —which is highly forecastable–but may be surprising to many critics that claim correlations are not useful. In fact, looking at dynamic clustering with equal weight versus just regular equal weight suggests that they have the greatest contribution to excess performance in this dataset. This makes sense considering that most stocks have similar volatility, but their correlations can be time-varying.

While this test is by no means conclusive- it again supports the logical and theoretical conclusion that clustering is valuable within a dynamic approach to portfolio allocation.  It also suggests that dynamic clustering is a viable alternative to static clustering-which is cumbersome and may not have a precedent for a given universe of stocks or assets. The usefulness of dynamic clustering versus static clustering depends on the predictability of the distance metric- which in this case were the sample correlations.While I agree that correlations can be noisy and need to be stabilized, it is better to attempt to incorporate information that is likely to be useful if there is a reasonable expectation that it can be forecasted than to omit such information altogether. One can always improve the correlation forecast or use a different set of distance metrics.

Here is a backtest that was done using a dynamic clustering method introduced by Michael Kapler at  Systematic Investor combined with multiple allocation schemes: 1) equal weight within and across clusters 2) risk parity within and across clusters and 3) cluster risk parity (CRP):  equal risk contribution (ERC) is used within and across clusters. For comparison purposes, we show equal weight allocation, risk parity, and risk parity ERC without clustering. For testing we used 10 major asset class ETFs.

The performance of the average of dynamic clustering versions versus the average of their non-clustered counterparts is slightly superior on both a return and risk-adjusted basis. All individual clustering methods also outperform their non-clustered counterparts. More significantly, Cluster Risk Parity (CRP)–or Dynamic Clustering with Risk Parity-ERC– was the best performer and outperforms all other allocation methods in terms of risk adjusted return, and has the second-best annualized return. This is only one universe, and the differences are not substantial–but do conform to what we would expect theoretically. There are a lot of moving parts- both the clustering approach, and the inputs (variance/covariance information and returns) can be used to improve performance and reduce turnover. But the most basic methods tend to demonstrate the validity of this sound theoretical approach. In the next post we will look at static versus dynamic clustering on a different universe.

Here is a  cluster representation  of some of the major markets that are traded internationally. The groupings were formed using data over the past year with a  clustering algorithm that is proprietary (correlation is used as a distance metric). What is interesting is that this particular cluster grouping has persisted  without much change over the past 4 years. Notice that oil has behaved more similarly to equities than it has to gold. This divergence coupled with the fact that the US Dollar has behaved as a distinct cluster suggests that the market is pricing fears of currency debasement as being more likely than commodity inflation (in which case commodities and gold would be grouped together). Another explanation might lie in fact that market sentiment is currently dominated by shifts in the perception of economic growth which outweight the perceived risk of inflation. The grouping of all of the equity indices and even real estate suggest that their risk is being dominated by one or more common factors. Regardless of the explanation, I find it interesting to group clusters and then reverse-engineer a story for market expectations. Interestingly enough, using a 20-day lookback Oil has detached and formed a separate cluster apart from gold, equities and the us dollar….. not sure what to make of that!