Thinking through clustering for a different but clearer perspective

There are a couple of visuals that are always used in the money manager's toolbox. One tool for quick comparison is the scatter plot of return and risk across different asset classes. Some analysts will get more sophisticated with this visual by looking at how asset classes have moved through time in risk and return space. Some with better visual dexterity will look at three dimensions and include correlation. 

The visual information of risk and correlation is computed and collected in the covariance matrix which is a core component of any optimization. Unfortunately, the covariance matrix can be difficult to work with as more correlated assets are added to the matrix or if there is instability in covariance through time. The impact of covariance sensitivity is less intuitive on risk measurement and asset weight selection but is a critical part of asset allocation. Any optimization is sensitive to multi-collinearity and the difficulty of inversion. If there are more assets that have similar covariance, small changes in statistical characteristics will lead to significant changes in optimized weights; the optimized asset allocation is unstable. 

A tool that can be useful and that is easily visual for money managers is cluster analysis through the use of principal component factorization. Principal component analysis is a data dimensionality reduction tool. It looks for similarity across data or common feature extraction. By eliminating common features, we can find uniqueness. By grouping common features, we can find clusters which can be graphically displayed. 

When assets cluster around common factors or have similar covariance characteristics, asset allocation becomes more difficult. One, there is less diversification benefit. A cluster of assets around principal components will not offer any benefit to investors. Portfolio risk is not diminished. Two, the covariance matrix becomes less stable and optimization becomes harder. The asset allocation will become sensitive to small changes in the price behavior of any asset in the portfolio.

 

There are quantitative methods for finding clusters and adjusting the covariance matrix, but a first past is to focus on the intuition of cluster analysis. No different than any good data work, plotting the information is a critical first step for analysis. 

As shown in the graph above, there are some well-defined asset clusters and some assets that are unique. Assets in the clusters are going to add little value to the portfolio. Optimization across those clustered assets will shown unstable weights. Assets outside clusters will have strong diversification benefit. Assets outside clusters are more important to a portfolio and should be a place of focused investment effort. Look for commonality of factors and find uniqueness. This process of combination for commonality and uniqueness will always be rewarded.  

CBOE Eurekahedge Tail Risk Hedge Fund Index - A costly downside hedge

 So, what is the price of tail risk investing? Using the CBOE Eurekahedge Tail Risk Index since 2008 as a benchmark shows the cost. The index of leading tail hedge managers would have generated an annual drain of 2.65 percent per year. It is not clear whether this is high or low until you consider the impact on an equity portfolio. 

We compared the SPY returns versus a simple 95/5 allocation between SPY and the Tail Risk Index for the entire history of Tail Risk Index from the end of 2007 through August 2020. The results show that there is no increase in terminal wealth from holding the index as a hedge. There is a slight decrease in annualized return and volatility with the Sharpe ratio only increasing .01. The "insurance" helped during crisis periods, but then the drag continued. Unfortunately, the support was limited given a 5 percent allocation. 

We then compared the stock index and stock/Tail risk portfolio against a 95/5 allocation between SPY and AGG the Barclay Aggregate Index ETF. In this case, the return drag was significantly less than the SPY/Tail Risk combination. The volatility was higher than the SPY/Tail risk combination but lower than the 100% SPY portfolio. 

The AGG has a zero correlation with the SPY benchmark while the correlation of the Tail Risk index was -.48. The volatility of the AGG was less than a quarter of the Tail Risk index. Holding the Tail Risk index provides better diversification because of the negative correlation but at a significant cost. Tail Risk investing as generated through the managers in the CBOE Eurekahedge index is no free lunch and there may be better ways to attack this problem. 

In the case of house insurance, the value is clear; a house destroyed is replaced. It cannot be replaced on its own without the insurance or a new capital infusion. In markets, the stock portfolio will likely increase in value after a recession as long as the firms invested in do not go bankrupt. Needless to say, this is a critical assumption. Insurance is not needed for the long-run but for protecting or smoothing short-term consumption risk. If you have to consume more wealth during a crisis, the hedge adds value. If you are only worried about terminal wealth, the positive impact of a small hedge is limited.

A tail hedge investment has to be considered against a number of factors like the terminal wealth horizon, the size of the hedge, any return or yield drag, and the diversification alternatives. 

Tail events - Historical, Implied and Subjective Distributions

Macro tail events actually occur more often than what you may think. There will be a big negative shock every 2-3 years. We have had the luxury of an extended period of calm before the 2020 pandemic and March liquidity crisis. Unfortunately, the result of less frequent tail events is complacency.

We have discussed the tale of the tail and how to form expectations for tail events, but there are also some market information and indices that can help with measuring the increased likelihood of tail events.

Clearly, there is historical volatility that can provide scale for tail events. An increase in volatility will increase the likelihood of a tail event. The current volatility can be compared with the past as well as current  forward-looking expectations of volatility in implied values. 

Subjective expectations can be overlaid upon these historical and implied volatilities. We focus on three categories: policy uncertainty embedded in the news, surprise data, and financial stress and risk aversion indices. Higher policy uncertainty can lead to policy surprises.  Deviations in forecast will lead to trends and potential extremes as reality and expectations converge.  More financial stress will lead to greater negative market reactions. Theses indices may not forecast a specific event but may indicate an environment more sensitive to extremes. These indicators may help sway expectations of a tail event versus market expectations.   

Forming a tale about tail risk - The narrative matters

There is a current strong focus on tail risk by many investors, yet by definition, tail risks are extremely hard to measure or even describe. The presidential election, pandemic changes, geopolitics, and general policy changes are all the focus of investors as tail risks, yet to be useful these topics have to be quantified and placed in form that is measurable. By definition, there have been few shock events that have driven markets to the tail of the distribution. Hence, there is only a small sample. The frequency distribution is limited, so there is a clear focus on subjective expectations of magnitude and likelihood. 

There has to be a "tale" that creates or describes the subjective expectations for the tail event. There can be a simple process that can be used as a framework for any tail risk discussion. This is the risk assessment and scenario building of whether a tail risk may happen. This assessment analysis is a separate process from what action should be taken. 

The first question is whether the tail event contemplated has ever been experienced before. If it has been seen in the market, then extreme value theory can be applied with perhaps adjustments for the current unique situation. Extreme value theory can also be used as a prior to calibrate any tail analysis.

The alternative from using a prior similar event will be to formulate a subjective narrative for a unique tail situation. This requires a story of what will happen, when it will happen,  and the likelihood of the event. Of course, a tail event cannot be looked at in isolation. A shock to US equity markets will have an impact on bonds and credit, as well as spill-over to the rest of the world. Additionally, the policy response has to be modeled as a second order effect. Hence, there is a tale with a tail event.

After the tail event tale is developed, a hedge or action plan can be developed. Of course, an action plan can also be developed for a tail event that is not imagined yet expected. 

See: 

The Tale of the Tail - Focus on the where, why, and what is wrong; Use strategy diversification as a solution

All the focus on the left tail but what about a right tail event?

Correlation in the tails - You will be surprised at the limited convexity