Diversification
Diversification is a fundamental concept that plays a critical role in finance, economics and decision theory (see e.g., [1] and references therein). From a financial perspective, its primary purpose is to hedge the uncertainty of outcomes as much as possible. In simple terms, financial diversification should expose assets to different sources of risks rather than risking them all together [2]. In the context of asset management, measuring and managing diversification is a key objective.
In the context of equities, the market itself represents the largest source of risk and is entirely captured by the market beta of a strategy. Apart from a small subset of specialized funds, most strategies and indices are primarily exposed to the market, and gaining exposure to it is the main objective of many funds and indices. Therefore, market risk is not a risk to diversify, but rather to control. As a result, diversification metrics developed for equities often consider the diversification of risk outside of market risk.
It is heavily documented by academic research that there is an association between a small number of risk factors, known as fundamental, or rewarded risk factors, and long-term excess returns. Exposures to these fundamental risks have been shown to be associated with better prospects in terms of performance than exposures to other drivers of risk [3]. However, exposures to these rewarded risks can also lead to periods of underperformance. Therefore, it is advisable to diversify exposures across rewarded risks to mitigate the impact of underperformance in any any one factor. This observation prompted EDHEC to develop diversification metrics that specifically consider rewarded risks.
Furthermore, academic literature places significant emphasis on going beyond holdings-based diversification and diversifying across risk factors as it can provide several benefits, notably an increased stability in overall risk. In the context of active risk, the natural risk measure through which this stability may be observed is the tracking error. A recent Scientific Portfolio publication [4] reinforces this concept and demonstrates that risk-based diversification can facilitate risk budgeting by identifying funds likely to experience a more stable tracking error due to diversification. Directing attention towards portfolios which are diversified in terms of risk increases the likelihood of being able to manage the level of risk being budgeted.
Hence, three diversification metrics are provided on the Scientific Portfolio platform. Factor quality assesses the long-term performance potential of a portfolio by measuring the quality of exposures to fundamental risk factors. Holdings-based and risk-based diversification are also analyzed, offering a holistic perspective of a portfolio's diversification.
Factor Quality
Factor Quality is a metric developed by EDHEC, a respected leader in the field of portfolio diversification. For the purposes of computing factor quality, only exposures to fundamental risk factors, excluding the Market, are taken into account. Indeed, we are primarily targeting long-only equity ETFs which already have a strong exposure to the Market factor, so we focus specifically on the six alternative risk factors. Rewarded factors have a positive expected return because they represent a common source of risk for which investors require compensation; therefore, any portfolio with positive exposure to these fundamental factors should benefit from a source of long-term performance. In other words, high exposure to fundamental risk factors is desirable. At the same time, these exposures should ideally be spread across each of the fundamental factors to create a more diversified portfolio. Indeed, fundamental factors are expected to individually outperform on average; however, their outperformance does not generally occur at the same time, so a diversified basket of factors has a higher probability of outperformance than a single factor, even if the magnitude of outperformance is potentially reduced compared to a single factor.
The factor quality score evaluates the quality of factor exposures by multiplicatively combining two metrics,
allowing to assess the extent to which the portfolio has exposures to academically validated sources of long-term performance and the extent to which such exposures are appropriately diversified. A low value of Factor Quality therefore indicates that the portfolio is either not materially exposed to fundamental factors, or that it is heavily concentrated into a small number of fundamental factors, or both.
It should be noted that factor exposures are simply the characteristics derived from a multivariate regression, as discussed in the Risk Model.
Factor Intensity
Factor intensity (FI) measures the strength of an instrument's exposures to fundamental risk factors. It is calculated as the sum of the exposures to the six fundamental risk factors given as
where .
A fund with a high FI has a high magnitude of exposure to fundamental risk factors. While the incorporation of FI addresses the strength of fundamental exposures, it does not indicate how well these exposures are diversified across fundamental risk factors. Two funds with the same level of FI can have vastly different diversification levels. For example, a fund with a high FI could be concentrated into a single factor. To address this, diversification of fundamental risk exposures is also considered.
Diversification of Fundamental Risk Exposures
Diversification of fundamental risk exposures (DFRE) assesses the level of diversification across fundamental factor exposures. It is calculated as the inverse of the sum of squared relative characteristics, given as
where . We can therefore rewrite DFRE as
When factor exposures are perfectly diversified (i.e., exposure to each risk factor is equal), diversification of fundamental risk exposures is six, representing the total number of fundamental factors. When factor exposures are concentrated into a small number of factors, diversification of fundamental risk exposures is low.
A fund with both a high FI and high DFRE has a high factor quality, indicating that potential long-term performance attributable to factor exposures is robust. On the contrary, a fund with a high FI but a low DFRE has a high magnitude of exposures to fundamental factors but is concentrated around a small number of these factors.
Robust Factor Quality
Factor quality is fundamentally based on characteristics (shown as ) which are subject to uncertainty. As a result, there is an element of uncertainty around factor quality itself. We can quantify the uncertainty of the characteristics using the standard error of the regression from which they were obtained. In order to understand how the standard error affects the values of the diversification metrics, we use a technique called the delta method to approximate the confidence interval of the factor quality. The robust factor quality is then calculated by adjusting the metric downward to the lower confidence limit. This adjustment ensures that the most conservative figure is used and makes it more robust to different realizations of returns. More specifically,
when factor quality is , the confidence interval is subtracted to compute the robust factor quality.
when factor quality is , the confidence interval is subtracted to compute the robust factor quality.
When the confidence interval includes zero, the robust factor quality is set around zero, keeping the lower confidence limit ranking order.
Given that a confidence level of one sigma is used, it can be assumed that the factor quality will be higher than the figure stated on the platform approximately 2 out of 3 times should different realizations of returns occur.
The appendix provides details on the delta method and the calculation of the robust factor quality.
Holdings-Based Diversification
An intuitive way to think of holdings-based diversification is to simply count the number of instruments within a portfolio. However, this approach can be misleading as often times, portfolio construction involves market cap-weighting, resulting in an uneven weighting of the instruments in the portfolio. For instance, a portfolio comprised of 200 instruments where one single instrument makes up 95% of the portfolio while the other 199 cover the other 5% cannot be classified as well-diversified despite its large number of portfolio constituents. A better measure of diversification evaluates how weights within a portfolio are distributed rather than simply counting the nominal number of positions held [5]. Effective number of stocks (ENS) (as well as effective number of other partitions such as sectors, countries and currencies) analyzes the distribution of capital across the constituents of a portfolio, allowing ENS to account for the dominant effect of large holdings compared to small holdings. The ENS measure is defined as the reciprocal of the Herfindahl-Hirschman index (HHI), where HHI is the sum of squared weights. ENS is given as
where represents the number of constituents within the portfolio and represents the weights of the constituents. While ENS provides a significant improvement over the nominal number of constituents approach, it has an important downside in that it does not consider risk in its analysis. Suppose a portfolio is composed of two instruments (with different volatilities), shown in the table below:
Type | Weight | Volatility |
|---|---|---|
Instrument 1 | 50% | 5% |
Instrument 2 | 50% | 20% |
Despite their weights being perfectly spread (resulting in an effective number of stocks of two), the risk of the portfolio is very highly concentrated in instrument 2. If only effective number of stocks were measured, this portfolio would be considered highly diversified. Therefore, while EN measures can provide valuable insights, they should not be used in isolation as a complete indicator of diversification. Accordingly, we introduce a complementary measure that analyzes diversification in terms of risk.
Risk-Based Diversification
There is a substantial body of research that suggests that factor diversification is a more effective means of managing a portfolio than asset class diversification. Studies have shown that portfolios diversified by equity factors (particularly those weighted by risk parity) have lower volatility and increased Sharpe ratios ([6], [7], [8]).
On the Scientific Portfolio platform, we begin by using the risk model to produce a decomposition of systematic active risk (i.e., the systematic volatility of an instrument's relative performance with respect to a cap-weighted benchmark) into individual factor-based active risk contributions. There are a total of 17 systematic active risk contributions, which add up to the systematic (i.e., explained by our model) portion of tracking error. It is important to note that specific risk is not included.
The decomposition of active risk is then used to construct a measure of diversification called active risk diversification (ARD). ARD measures diversification by assessing the spread of systematic active risk contributions. It therefore reflects the effective number of contributing factors to active risk (see Martellini and Milhau [5] for practical examples of the use of the concept of “effective number of”, directly related to the Herfindahl index). We make use of the risk model to generate the risk contributions used in this metric.
Active risk is defined as
where is the risk contribution of the characteristic and is the number of fundamental and sector-based characteristics. Because we are interested in quantifying how risks that result from active bets are distributed, we define ARD as
ARD is low when active risk is concentrated into a small number of factors and high when active risk is relatively well-spread across all factors.
ARD makes it possible to determine whether a portfolio is well "risk-diversified" and therefore associated with a more stable tracking error for risk budgeting purposes.
Robust Active Risk Diversification
As discussed for factor quality, we measure the confidence interval around ARD to compute a robust, conservative metric. The methodology is discussed in detail in the appendix.
Applicability
When an instrument has a large proportion of its risks unexplained by the model, shown by an , the above diversification analytics will not be relevant and therefore, the portfolio will be excluded. Additionally, for the purposes of diversification, it is necessary that an instrument carries material relative risk with respect to a cap-weighted benchmark. Accordingly, instruments which track a broad, cap-weighted benchmark are excluded. More specifically, all instruments that exhibit a systematic TE of less than 0.5% with respect to the Scientific Portfolio Cap-Weighted index for a given investment zone are excluded. These exclusions ensure that the analysis is focused on instruments which carry material active risk.
References
1 Ola Mahmoud. (2017). The Origin of Diversification: An Evolutionary Theory. SSRN pre-print.
2 Ola Mahmoud. (2021). The Willingness to Pay for Diversification. Management Science. 10.1287/mnsc.2021.4122.
3 Noël Amenc and Felix Goltz. (2016). Long-Term Rewarded Equity Factors: What Can Investors Learn from Academic Research?. The Journal of Index Investing. 7(2): 39-56. 10.3905/jii.2016.7.2.039.
4 Benjamin Herzog and Jenna Jones and Shahyar Safaee. (2023). Remember to Diversify Your Active Risk: Evidence from US Equity ETFs. Journal of Beta Investment Strategies. 14(2): 6-16. 10.3905/jbis.2023.1.033.
5 Lionel Martellini and Vincent Milhau. (2017). Proverbial Baskets Are Uncorrelated Risk Factors! A Factor-Based Framework for Measuring and Managing Diversification in Multi-Asset Investment Solutions. Journal of Portfolio Management. 44(2). 10.3905/jpm.2018.44.2.008.
6 Erik Hjalmarsson. (2011). Portfolio Diversification Across Characteristics. The Journal of Investing. 10.3905/joi.2011.20.4.084
7 Clifford S. Asness and Tobias J. Moskowitz and Lasse Heje Pedersen. (2013). Value and Momentum Everywhere. Journal of Finance. 68(3). 10.1111/jofi.12021.
8 Ulrich Carl. (2018). The Power of Equity Factor Diversification. SSRN Electronic Journal. 10.2139/ssrn.2915443.