# Diversification

– Weighted

Performance

Evaluation

# DIVERSIFICATION– WEIGHTED PERFORMANCE EVALUATION

**A new approach to building a portfolio**

**Overview**

In addition, we show portfolio diversification to provide material performance improvements to both risk and return. Part of the attraction to smart-beta euphoria is an intuition that repeatable investment performance is systematic. Investors prefer a repeatable process and especially a process rooted in things that are rather simple and easily understood.

In essence, diversification increases for every equally weighted, uncorrelated asset added to a portfolio. When we put the portfolio in a geometric space, such additions add a dimension, which may measure the diversification therein. When we weight assets in a manner consistent with maximizing the portfolio dimensionality, perhaps subject to a utility function, the portfolio is said to be diversification weighted.

**Defining Diversification: The Framework**

For Measurement And Optimization

For Measurement And Optimization

For readers interested in diversification or how the results are obtained, we need to set some foundations in the geometric modeling. The ensuing geometric modeling framework provides the background for both diversification measurement and optimization.

Asset correlations are measured and assets are projected to a space that maps the measured correlations to the angles separating one asset from another, so that assets with higher correlations have a more acute angle separating the vectors. Assets sharing zero correlation are mapped to a 90° angle and assets with a -1 correlation are mapped as diametric opposites having a 180° angle.

Imagine a three-asset portfolio; each asset has zero correlation to the other two assets. Each asset literally takes the portfolio in a new direction. Note that all available dimensions are used as degrees of freedom to convey the relativity of the assets. It is correlation in every dimension. We refer to this as having a relativity domain space. Imagine that we introduce a fourth uncorrelated asset. Where would it fit? It cannot fit in 3-D without creating some distortion. There is not enough dimensionality to hold it. We need to introduce an additional dimension, which may accurately represent the fourth uncorrelated asset.

In mathematics, there is no limit to dimensionality. So it is with portfolios. Diversification has no upper boundary; there can always be more. Any portfolio has an ambient dimensionality equal to the number of assets in it. The ambient dimension compacts to the spanning dimension after ringing out all of the redundant information caused by disparate weighting and positive correlations. The spanning dimension thus contains all of the portfolio’s information. It is a bit like a .jpeg at 100 percent quality. The dimension of the portfolio can be further reduced when associated with a confidence interval, such that it captures X percent of the portfolio. That dimensionality is called an “effective dimension.” It is like viewing a .jpeg at 90 percent quality; it captures the essence of the image, but does not require same amount of data. Thus, the lowest dimensions 1,2,3 … are the most pervasive and measure systematic diversification and risk exposure.

This framework explains how portfolios sprawl across multiple dimensions. The primary dimensions give the systematic exposure on the left and it integrates with the idiosyncratic exposures on the right. Each new dimension acts as a source of diversification unto the portfolio. The blue shadow line gives the investor an understanding of what perfect diversification (given the investment quantity) would look like. The portfolio would mimic

this line if each asset were equally weighted and uncorrelated. The more exponential the portfolio graph, the greater the risk, as such exponential graphs are indicative of superficial diversification in asset quantity only.

We apply this diversification framework to weighted as well as unweighted portfolios. When the portfolio is already weighted, the framework gives the measurements of diversification as well as the visualization.

When the framework is applied to a weighted portfolio, the given weights for each asset are modeled as the vector lengths for each asset. Intuitively, the weight extends the assets in its given direction (Figure 2).

In essence, it is like a multidimensional pie chart containing relativity data.

The resulting level of symmetry induced by the model structured by the assets’ weights and relationships provides a visual explanation of portfolio diversification. Portfolio balance is manifest as three-dimensional balance in the True Diversification model framework. True Diversification accounts for the diversification directly and captures both the systematic and nonsystematic perspectives (Figure 3). This process can be applied to any asset or any mix of assets, including disparate assets classes, or mixing of various instruments. We recycle this geometric modeling process to optimize diversification.

**Diversification Optimization:**

The Theoretical Foundation

The Theoretical Foundation

The portfolio thus has two elementary inputs: the relationship of each asset to every other asset in the portfolio, as measured by its correlation matrix and modeled by the relative angles; and the utility function for every asset measuring the relative attractiveness of each asset and modeling the magnitude of each vector. These are the only inputted asset variables.

As Figure 5 shows, there are two types of assets: those that combine to create the frame (a) and those trapped inside (b). Efficient assets grow the portfolio in their respective directions. The interior assets are doing an inefficient job of growing the portfolios in their respective direction and are dominated by the assets comprising the diversification frontier. Interior assets are inefficient and normally discarded unless subject to a constraint.

When we model the asset vectors with all-equal lengths, the resulting portfolio is maximizing diversification. Every asset is assured a spot on the efficient frame, and the asset’s weight is exactly proportionate to the asset’s relative uniqueness.

Now that the portfolio has a tangible body, we compute the volume and assign weights to every asset based on the pro rata volume contribution. The equation is simple: Volume = Money (Figure 6).

Highly correlated assets crowd one another out fighting for the space in the portfolio. Such highly correlated assets are also—intuitively— increasingly sensitive to small differences in their utility function, whereas a more uncorrelated asset in an otherwise-correlated portfolio will be rather insensitive to changes in that asset’s utility function. If one could find a historically negatively correlated asset, then it too would be relatively insensitive to the magnitude of its utility, if it had positive utility.

**Diversification Weighted 500:**

Performance Refinements

Performance Refinements

Collect the historical index component data that were in the index (S&P 500) at the start of the observation period. The data source includes delisted assets to eliminate survivorship bias.

**Investment Policies**

For each optimization’s anniversary date on Aug. 1, we would select two sample periods for the historical data. The first sample period was the previous one year; the second sample period was the previous 10-year period. The baseline risk, return and correlation values are calculated for every asset as the average of these two periods. We believe these two periods merge a positive predictive momentum signal arising from the one-year data with the stock’s performance over at least one full market cycle in the long sample. We tested staggered multiple optimization intervals and found no material effect. We applied the sampling procedure uniformly over all sectors and years.

We designate the optimization dimension for each portfolio in an effort to attain a comfortable balancing point between systematic and idiosyncratic diversification. We were also cognizant to set the value high enough that it would more fully allocate to the natural diversification derived from the process and to minimize the application of the constraints. Optimizations taking place in lower dimensions better diversify the portfolio from a perspective of systematic influence, and optimizations taking place in higher-dimensional space more effectively diversify the portfolio from an idiosyncratic perspective.

We apply constraints to all assets. An asset constraint was set at 3 * 1/N (N = Asset Count). For a sector with 20 stocks, N is 5 percent, and the maximum constraint is 15 percent. We applied a similar minimum constraint. While these constraints denied the performance test, the benefit of the diversification/utility efficiency logic, the constraints ensured an allocation to all 500 stocks, creating a more relevant comparison focused on the weighting approach and less on the logic of diversification efficiency.

**Contrast To Other**

Weighting Approaches

Weighting Approaches

Diversification Weighted

Equal Weighted

Fundamental Weighted

Market Cap

Revenue Weighted

Earnings Weighted

Volatility Weighted

Funds of this ilk have seen recent success both in performance and garnering new assets. Now, 16 ETFs collectively manage $16 billion in funds from a strategy that did not exist before 2004. As the nature of indexation becomes more complex, the ETF market is now more accepting of smart-beta strategies. This will likely continue, but when put in context with the $2 trillion of global capital Markowitz estimated as being managed in accord with mean-variance optimization (Markowitz, private conversation, 2010), one can see the ETF space is nascent in its adoption of more complex systematic index products.

Fundamentally, risk and return are not two separate things, but the same. Risk is simply negative returns (Figure 9).

This foundational tenet is always obvious to investors in retrospect. Nevertheless, there are widespread errors in risk estimation, definition and application across the entire industry. Risk cannot be known in advance; it must be estimated. In a risk-based weighting scheme, any such estimation faces an impossible task of measuring the known unknowns and the unknown unknowns. One may liken risk measurement to holding a tape measure around a black hole without succumbing to the event horizon.

One of the traditional methods of portfolio optimization and construction is mean-variance optimization (MVO), which was pioneered by Harry Markowitz in the 1950s (Markowitz, 1952). Markowitz and the variations he has spawned have had significant application in our industry, but they have only recently gained traction as index construction methodologies.

One of the traditional methods of portfolio optimization and construction is mean-variance optimization (MVO), which was pioneered by Harry Markowitz in the 1950s (Markowitz, 1952). Markowitz and the variations he has spawned have had significant application in our industry, but they have only recently gained traction as index construction methodologies.

MVO’s focus on volatility fails to exploit an asymmetric diversification/return relationship (Damschroder, 2009). Furthermore, diversification is shown to have greater stability in a crisis (Damschroder, 2009).

**Performance Analysis**

Diversification looks to have had its worst relative year in 1998, despite putting up a 17.7 percent return. In calendar year 2008, DW was 80 bps under the S&P 500 Index despite having a few percent better drawdown through the crisis. While over the entire crisis the DW strategy did better, it also had a worse losing month, losing 3.43 percent more than the S&P 500 in October 2008.

The full-sample upside- and downside-capture statistics for the DW 500 are:

Up Capture : 105 percent Down Capture : 87 percent

Clearly, the performance looks good against a capitalization- weighted approach. What about other weighting approaches?

(See Figure 12.)

The equal-weighted and fundamentally weighted strategies are reported gross of fees and trading cost. The remaining assets are based on the ETF products, which report net fees. Dividends are reinvested. Before 2010, the DW 500 sectors and sector comparisons have their daily returns adjusted to receive the S&P 500 annual dividend yield (http://www.multpl.com/).

**Risk Analysis**

To the chagrin of efficient market theorists, the Diversification Weighted strategies in most cases produce higher returns while lowering the portfolio variance. This hypothesizes massive market inefficiency, perhaps induced by the trillions of dollars attached to capitalization-weighted approaches.

5 sectors with greater return and less variance

4 sectors with greater returns and more variance

1 sector with smaller returns and less variance

**Diversification Returns &**

Diversification Alpha

Diversification Alpha

The alpha is made possible by exploiting diversification, volatility and some sort of trading activity, such as rebalancing, but it can be other activity such as reconstitution or reoptimization. This trilogy of portfolio variables combines the idiosyncratic asset volatilities to produce alpha.

The diversification-ratchet is very similar to a rebalancing bonus; however, the diversification-ratchet is a more appropriate moniker because the alpha is available to investors without rebalancing. Other forms of portfolio activity produce the alpha, including reconstitution, reoptimization, profit-taking, stop loss, trimming and even dollar cost averaging. In all cases, the alpha requires diversification. In this research, we had annual reconstitution and annual reoptimization, no rebalancing and the strategy was able to benefit from the diversification-ratchet.

As this shows, the incremental return is a function of the amount of variance reduction; it is not a function of the actual level of the portfolio variance. For example, if you start with low-volatility assets, say, having a weighted average standard deviation of 9 percent, the covariances in the portfolio serve to further reduce the portfolio standard deviation to 8 percent. The diversification return is (.09−.08)2 = .857 percent. While the portfolio may claim a low variance, the diversification return of 86 bps is a fraction of the 297 bp diversification return obtained by the DW 500. If your focus is variance and not correlation, it limits the amount of variance reduction, and thus the Diversification Alpha.

**Stacking Diversification**

In Figure 15, the Diversification Alpha is larger at the security level due to the higher volatility and lower correlations of the individual securities.

While beyond the scope of this research, it would seem to be further possible (and likely) for a third alpha stack working at an asset class and/or country level.

**Sector Analysis**

Figure 17 includes a stacked chart to help illustrate the consistency of the incremental performance for each sector over time.

The sector allocation also reveals an interesting and intuitive dynamic: Whenever the market heats up, asset correlations tend to come in, leaving the investor with relatively fewer diversification sources. Telecom and utilities are the largest active weights of the Diversification Weighted strategy over the whole period.

Investors in a Diversification Weighted strategy seem to enjoy a fair amount of consistency in performance. The relative consistency of the sector weights over time are a positive indicator of turnover control. Consistency appears both over time and across the sectors. The Diversification Weighted 500 sector-sleeves beat their benchmark in nine of 10 instances (Figure 19).

**MPT And Factor Analysis**

Value companies by their nature have a greater proportion of idiosyncratic performance, when compared to growth, which often shares homogenizing factors, such as cyclicality (data is from Ken French website (French, 2014)).

**Crucially, the Diversification Weighted 500 produced 652 bps of alpha per year on average for more than 18 years. The alpha for the five-factor model is 414 bps. The statistical significance of the beta-factor alpha is observed having a P value of .00014, or a chance of the alpha being obtained randomly less than 1 in 7,000.**

Remarkably, the alpha was demonstrated purely by the weighting approach, as the product had naive security selection, no timing, leverage, shorting, hedging or other positional risk management, and invested in all the same underlying components of the benchmark.

**Attribution Analysis**

This means that the alpha produced by the strategy did so with a 274 bp head wind at the sector level.

**Sustainability And Capacity**

You can imagine that diverse investor expectations across strategies and platforms may help alleviate matters of moving the market. Whether the DW 500 or another DW strategy—be it equity, asset allocation, currency, fixed income or funds of funds—Diversification Weighted models may likely continue to perform as the massive macro inefficiencies induced by behavioral volatility (overreaction), and passive capitalization approaches appear to provide plenty of ammunition capable of sustaining the performance of Diversification Weighted strategies into the future.

**Conclusion**

The Diversification Weighted 500 demonstrates material and consistent outperformance against other weighting approaches and within the sectors sleeves that we consolidated into the composite.

We explain a framework for diversification measurement and optimization. Diversification Optimization builds on this framework, and the results contrast favorably versus other weighting approaches. We quantify the Diversification Return at 2.97 percent, and we introduce the Diversification Alpha. Total alpha for the strategy is 652 bps annually for over 18 years. We further introduce the ability to stack the Diversification Alpha at the security, sector and asset class level.

The flexibility and efficacy of Diversification Optimization, combined with the demonstrated consistency, attractive risk metrics and total return, presents a strong case for the institutional application of Diversification Weighting as an overlay weighting strategy, product-sleeve, core-satellite model or index.

**References**

Booth, David and Eugene Fama. (May/June 1992). “Diversification Returns and Asset Contributions.” Financial Analysts Journal, vol. 48, No. 3, 26-32.

Damschroder, James. (2010). Internal research report.

Damschroder, James. (2014). “Systematic Risk Exposure of Weighting Approaches,” working paper.

Damschroder, James and Haoyan Sun. (2009). “How True Diversification Preserves Capital.”

French, Kenneth. (2014). Data Library. Retrieved from Darthmouth: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

Ghayur, Khalid (2013). “Low Volatility Investing.” Burridge Center Conference at the University of Colorado, Boulder.

Markowitz, H. (March 1952). “Portfolio Selection.” The Journal of Finance, vol. 7, No. 1, 77-91.

Markowitz, H. (July 2010). private conversation. (James Damschroder, interviewer)

Stein, Charles and W. James. (1961). “Estimation with Quadratic Loss.” Fourth Berkeley Symposium. Berkeley, California.

Stutzer, Michael (Spring 2010). “The Paradox of Diversification.” Journal of Investing, vol. 19, No. 1, 32-35.

Swensen, David. (2005). “Unconventional Success.”