Dynamic versus Static Clustering: Dow 30 Stocks 1995- Present
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.
A Backtest Using Dynamic Clustering versus Conventional Risk Parity Methods
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.
A Visual of Current Major Market Clusters
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!
An Elegant Measure of Diversification: The Cluster Gini Score (CGS)
In the paper we wrote on The Minimum Correlation Algorithm we introduced the Composite Diversification Score (CDI). The purpose of this measure was to demonstrate how well a set of portfolio weights has minimized the average portfolio correlations and also balanced the risk contributions from each asset as measured using the Gini coefficient of inequality. The CDI can also be extended in a similar manner to CGS when applied to clusters- this is beyond the scope of this post.
An alternative and potentially more intuitive measure of portfolio diversification uses the concepts embedded in Cluster Risk Parity (CRP). The logic is very simple: if CRP using ERC (equal risk contributions) both within and across clusters represents the “optimally diversified portfolio”, then any portfolio that deviates from that is considered to be imbalanced from a risk/diversification standpoint to varying degrees. Essentially a portfolio from a given universe is considered to have “factor” exposures to each cluster, and also “tracking error” (risk of under-performing a benchmark) to those factors. In practice, we would like to have both a balance of risk across factors/clusters and a balance the risk attributed to having exposure to a factor/cluster. The Gini coefficient was used in the CDI to measure inequality in risk contribution exposure. A measure where (1-Gini) of asset risk contributions would mean that a value of 1 would represent perfect equality and 0 would be perfect inequality. The same concept can be extended to across (inter) factor/cluster risk contributions and within (intra) factor/cluster risk contributions. The formula for CGS would be as follows:
CGS= 100x (sqrt(NC)x(inter-cluster RC (1-Gini))+(avg intra-cluster RC (1-Gini)))/(sqrt(NC)+1)
Essentially this is a weighting of inter-factor risk balance versus the average intra-factor risk balance, where the inter-factor (across clusters) is assigned a greater weight as a function of the square root of the number of clusters available. The resulting equation shows a result between 0 and 100, with 100 being a Cluster Risk Parity allocation, and very low scores representing high concentration and poor distribution of risk. This score can be used to determine the degree of diversification/concentration of any set of portfolio weights across a given universe. In other words, if you have for example 5 assets and a set of weights that you believe to be optimal, the CGS can help to determine how imbalanced you are from a diversification standpoint. This can also be extended to optimization where CGS can help to moderate the objective function to ensure more stable results and generate portfolios that are more intuitive. In machine learning algorithms, and data-mining, CGS can help to produce more stable predictions and enhance the capability of handling large datasets when using clustering.
Cluster Risk Parity (CRP) versus Risk Parity (RP) and Equal Risk Contribution (ERC)
Cluster Risk Parity (Varadi, Kapler, 2012) is a method to improve upon the deficiencies of Risk Parity and Equal Risk Contribution: a) the need for manual universe selection (see All-Weather and Permanent Portfolio) and b) imbalanced risk exposure as a function of the universe selected. To highlight the latter issue it is worthwhile to take a look at an example where we use only the S&P500 (SPY) and Treasury Bond (TLT) time series to create universes for portfolio creation. This example is not that extreme because it is conventional to include multiple ETFs or mutual funds that represent multiple sectors or countries that are included along with multiple bond ETFs/funds. This example below using data over the past year is very important:
The point of this analysis is that the number or composition of bond versus equity will dramatically affect the risk contribution of the portfolio from each category with conventional risk parity methods. In the example above, even when considering only the 5 asset case, the distortion in risk is significant- more than 100% of risk is coming from equities for risk parity, while a whopping 80% of risk is coming from equities for ERC. In contrast, Cluster Risk Parity (CRP) is balanced regardless of the number or composition of the universe. This helps to ensure consistency in risk management and performance and also ensure that portfolio rebalancing has the desired impact. The use of manual compilation/categorization into say “equity” or “bond” is not sufficient to avoid this problem because there are times when there is crossover (like high yield) and even significant differences within each category when correlations are low. Using CRP is the simplest way to avoid these issues without having to run a lot of ad hoc analysis and having to make continual adjustments.
Cluster Risk Parity– A Visual Representation
The following graphic is borrowed from a static risk parity approach via Salient Capital Advisors: http://www.theriskparityindex.com/static/pdfs/Salient-Risk-Parity-Index-White-Paper.pdf. The visual is useful for readers to understand the nuances and relative merits of a Cluster Risk Parity (CRP) approach. In their approach the individual assets and clusters are defined in advance, and thus there is no dynamic clustering method used. However, the concept that they use is similar: balance risk contributions both within and across “clusters” of assets. In this case it is important to clarify that the size/area of each slice of the pie chart is a function of risk contributions NOT percentage capital allocations.
As you can clearly see from this specific chart, it is very similar in spirit to the “All-Weather” Portfolio or even the simpler Permanent Portfolio . The main difference is that the latter portfolio schemes represent “strategic asset allocation” alternatives, while Cluster Risk Parity (and also the Salient Index) is a dynamic asset allocation framework. GestaltU does a good job describing why it is important to prefer dynamic approaches in a recent post: http://gestaltu.blogspot.com/2013/01/the-full-montier-absolute-vs-relative.html. In reference to CRP the advantage is creating a framework that does not require having to pre-specify the assets and weights in advance on a static basis. Instead, it permits the ability for the portfolio to adapt to changes in the variance/covariance matrix of asset returns — which have proven especially useful in a dynamic framework to normalize risk exposure. This framework is so generic that it can be adapted to any type of risk factor or regime framework with relative ease.
Cluster Risk Parity
One of the concepts that I have developed with Michael Kapler at Systematic Investor : http://systematicinvestor.wordpress.com/ is a method of passive portfolio allocation (omitting expected or historical returns) that captures the true spirit of diversification. It is a more elegant but also more complex than our heuristic algorithm: Minimum Correlation. This new method is called “Cluster Risk Parity” (CRP) and combines the use of cluster algorithms with risk parity and equal risk contribution. The core of this method is ideal for indexation: isolate groups of assets (or stocks) and then efficiently allocate both within and across groups. The purpose is to avoid the need for artificial or manual grouping while simultaneously adding a layer of risk management. By clustering, it is possible to dynamically maximize the diversification benefits without having as much sensitivity to the errors in the correlation matrix. Furthermore, this promotes the use of all of the possible assets in the universe which is a desirable way to distribute risk and minimize tracking error. By using risk parity, we can efficiently normalize risk both within and across groups: in CRP we are making nearly equivalent or exactly equal risk bets across the portfolio. The combination is perhaps the most robust method of passive portfolio allocation, and it also produces the best risk-adjusted returns without relying as much on the low-volatility factor or bond/fixed income performance.
More to follow………….
The “All-Weather” Portfolio Derivation
The All-Weather Portfolio was introduced by Ray Dalio- the founder of Bridgewater -which is arguably the largest and most successful hedge fund in the world. His landmark concept was to create a portfolio that would have roughly equal risk in four different economic regimes: 1) rising growth 2) falling growth 3) rising inflation and 4) falling inflation. His other major concept was to leverage up each asset to have the same risk so that returns could come from multiple different sources, and not rely on an equity-centric environment. Of course, it is more accurate to think of each of these as sub-regimes since the change in growth is often accompanied by some change in inflation. Thus, in this adaptation the four major regimes are exactly the same as in the Permanent Portfolio , the only difference is the type of assets included in each regime. By structuring a portfolio to be balanced across economic regimes, the performance and volatility is more stable over time. The inspiration for this post, and a good explanation of the All-Weather Portfolio can be found here.
The “All-Weather Portfolio”
Permanent Portfolio Derivation and Historical Performance
This graphic is designed to help readers understand the logic and assumptions embedded in the Permanent Portfolio model by Harry Browne. It is also a useful framework for understanding how to construct regime-based portfolios. The results are re-published from an earlier article written by Corey Rittenhouse at Catallactic Analysis. It was a very good post (and good blog) and is worth reading for more background. Some other very good posts on the subject are:
GestaltU: An interesting three-part series on the Permanent Portfolio and tactical applications:
Systematic Investor: An interesting article on the Permanent Portfolio showing risk parity applications and implementation in R:
Stats by 5 Year Period
Adaptive Frequency Weighted Moving Average Resource Links
Here is a good link to summarize how to calculate various moving averages:
http://www.metastock.com/Customer/Resources/TAAZ/?c=3&p=74
Here is an interesting post showing a comparison of performance of different moving average variants across a number of markets by moving average length. It is worth noting that recency-weighting (wma) tends to perform better on average than the less responsive but smoothed triangular wma. Note that the weighted moving average and triangular moving average are not highly correlated in terms of performance across lookbacks which demonstrates their complementary nature for an adaptive framework:
http://etfhq.com/blog/2010/10/19/weighted-moving-averages-tests/
