CSSA

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 (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.

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.

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 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”

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

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/

One of the unique aspects of modelling financial time series  is that recent data is more important than previous data. This is because forecasts need to be made for the next time period, and prices compound in value over time. The other unique aspect of time series data is that there tends to be a combination of random noise and mean-reverting tendencies (especially at shorter frequencies). The researcher often faces a tradeoff between weighting based on recency versus amplifying the noise and mean-reversion that exists at shorter frequencies. This tradeoff is logical, and should be captured explicitly as a function in an adaptive framework (success-weighted) versus back-testing different weighting schemes hindsight.

Recency is easily captured by a weighted moving average (wma). In this scheme, prices are weighted in direct proportion to their recency (the recency rank divided by the sum of recency ranks times the price at time t). This link demonstrates the calculation : http://www.investopedia.com/articles/technical/060401.asp . Alternatively,  the best way to eliminate disturbances induced by shorter frequencies is to middle weight the data. That is, we want to give data closest to the middle higher values than more recent data. This is best accomplished using a triangular moving average, which is essentially a moving average of the moving average of the same length. The smoothing induced by a triangular moving average creates maximum weighting on the middle value. The calculation is described here: http://daytrading.about.com/od/indicators/a/Triangular.htm.

To keep things consistent, we should use a weighted triangular moving average (wtma), which is exactly the same except it uses the weighted moving average versus the simple moving average in the calculation. Thus the Adaptive Frequency Weighted Moving Average can be calculated simply as:

AFWMA=  w*(wma)+ (1-w)*(twma)

where “w” is a constant between 0 and 1

The purpose of the AFWMA is to find the optimal balance between recency and avoiding short-term noise. This can be found by solving for “w” to best minimize out of sample error in a forecast format, or to best maximize out of sample sharpe in a trading strategy format. It is a simple and logical way of compressing the value/information content that differs between the weighted and triangular moving averages without resorting to blind optimization (or worse yet the beliefs/opinions of a technical analysis “guru”).

Here is a spreadsheet that shows the calculation of the Gini Coefficient which is used in calculating the Composite Diversification Indicator (CDI) in the Mincorr Paper. For reference, the Gini is a measure of inequality that is expressed on a scale from 0 to 1.  For a perfectly equal portfolio, the Gini would be zero. For a perfectly concentrated portfolio, the Gini would be 1.  In the CDI, we focus on using the Gini on the asset risk contributions- which is similar to the focus for Equal Risk Contribution (ERC) by Roncalli. ERC is often confused with simpler versions of Risk Parity, it is more sophisticated in that it optimizes to make all risk contributions equal in the portfolio (versus having a fixed equal risk- ie target risk of 10%- for each asset). Below is the spreadsheet to calculate the GINI:

GINI calculation

This spreadsheet shows the implementation of the Minimum Correlation Algorithm (mincorr). The power function is a new parameter that regulates the weighting in the averaging process toward the most anti-correlated assets. Currently the default is “1” which is essentially a rank-weighted moving average of pair-wise correlations.  A value of “0” would imply equal weightings, values lower than 1 would imply a balance between equal and rank weighting, while values higher than 1 would more heavily skew the rank-weighting. More spreadsheets and research will be available soon.

Minimum Correlation (Mincorr) Spreadsheet