*By Bill Wynne*

Although minimum volatility may seem to be a simple strategy, the construction methodology of the iShares MSCI USA Minimum Volatility (NYSEARCA:USMV) is complex and requires explanation. To fully comprehend this product, an investor needs to have a basic understanding of factor-based risk models and quadratic optimization. This article will briefly discuss these topics at a high level; however, it will primarily describe and analyze some of the critical rules that determine the ETF holdings. These insights should inform your views regarding the suitability of this product for you or your clients.

**Determining the Relevant Entities**

As the name of the ETF suggests, two different companies determine the holdings of this fund. iShares is the brand name for BlackRock ETFs, and MSCI constructs indices, or hypothetical baskets of stocks. As the prospectus states, the iShares ETF tracks the MSCI USA Minimum Volatility Index. In addition, the MSCI USA Minimum Volatility Index factsheet states that the constituents of this index derive from the MSCI USA Index. As indicated in its fact sheet, the USA Index contains approximately 600 of the largest companies in the U.S. by market capitalization. Therefore, it is similar to the S&P 500 Index, and the constituent stocks of the ETF are liquid.

*Figure 1 shows the relationships between the MSCI indices and the BlackRock iShares ETF. The constituents of the MSCI USA Index are used to construct the MSCI USA Minimum Volatility Index. The holdings of the ETF are selected to match the Minimum Volatility Index.*

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**Sources: BlackRock prospectus, MSCI factsheets and Poppertech*

**Defining Minimum Volatility**

In quantitative finance, the price volatility of a stock is assumed to be the same as its risk. The intuition behind this idea can be shown with the following hypothetical scenarios:

Â· Scenario 1: You and a friend bet on a coin flip. If the coin shows heads, you win $1 from your friend. If it shows tails, he wins $1 from you.

Â· Scenario 2: You and a friend bet on a coin flip. If the coin shows heads, you win $10 from your friend. If it shows tails, he wins $10 from you.

Scenario 2 is clearly riskier than Scenario 1. In terms of the magnitude of gains and losses, Scenario 2 is also more "volatile." Similarly, stock price volatility refers to the magnitude of possible gains and losses.

"Volatility" as a concept is not quantifiable, but probabilistic variance is quantifiable. Quants represent volatility with variance. Variance is the square of standard deviation, and minimum variance and standard deviation portfolios are equivalent. As discussed below, the ETF and its associated index use decision optimization technology to minimize portfolio variance.

**The Goal: Minimizing Variance**

*Figure 2 shows a schematic of the Minimum Variance Portfolio on the Efficient Frontier according to Modern Portfolio Theory. The Efficient Frontier consists of the portfolios that minimize risk for a given level of reward, or maximize reward for a given level of risk. The Minimum Variance Portfolio contains a combination of stocks with the lowest possible expected risk.*

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**Source: Poppertech*

Based upon mathematical laws, three parameters determine portfolio variance: the weights and variances of the holdings and the correlations between them. Factor-based risk models provide variance and correlation estimates to the decision optimizer. According to the MSCI Global Minimum Volatility Indices Methodology document, the Barra Global Equity Model is the underlying risk model. This model also determines portfolio holdings in other ways that will be described later in the article.

The objective of the decision optimizer is to find the weights that minimize the expected variance of the portfolio. In quantitative finance, optimizations typically attempt to maximize some measure of reward instead of minimizing risk. In some sense, the absence of reward as an objective makes minimum variance portfolios passive investments. From another perspective, the reason for investing in a minimum variance portfolio is to achieve a higher risk-adjusted return than you could achieve from a conventional index. Therefore, it is debatable whether this is an active or passive product.

In theory, MSCI could simply solve for the weights, and BlackRock could create an ETF from the results. In reality, this frequently results in suboptimal portfolios. To understand why, you need to know more about risk models.

**Risk Models**

*Figure 3 diagrams the risk calculations required to optimize a 10-stock portfolio. Each "brick" represents one calculation. Each blue brick represents a variance calculation. Each red brick represents a correlation calculation. In aggregate, 55 statistical estimates are required to optimize a 10-stock portfolio.*

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*Source: Poppertech*

Factor risk models involve multiple stages of complex statistical estimations, and these methods are beyond the scope of this article. Instead, you should know why they exist.

You may have learned basic estimators for variance and correlation. If MSCI were to use these to calculate the variances and correlations for 1000 stocks, they would need to estimate 1000 different variances and almost 500,000 correlations. With approximately 500,000 statistical estimates, some will be low simply by chance. The optimizer will attempt to minimize risk through investing in the lowest variance stocks with the lowest correlations to other stocks. As a result, the optimizer will tend to invest heavily in stocks with spuriously low variances and correlations.

Additionally, more recent historical prices tend to result in more accurate variance and correlation estimates; however, fewer historical prices result in estimates that vary significantly both over time and across different stocks. The result from risk estimates changing over time is high portfolio turnover and trading costs for investors.

To both estimate fewer variances and correlations and base estimates upon recent data, MSCI Barra and other risk model vendors build factor-based risk models. These factors are intended to capture drivers of risk and typically include momentum, value, capitalization, industries, countries, and currencies. Instead of calculating the volatilities and correlations directly, Barra can estimate these quantities for the factors instead. This results in far fewer statistical estimates and, consequently, more stable optimized portfolios.

**Constraints**

*Figure 4 represents the sequential effect of constraints on the total number of possible solutions to the optimization problem. Each additional constraint limits the number of possible solutions. If two constraints conflict, the optimization may be infeasible. Note: These constraints are applied simultaneously in the actual optimization.*

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**Source: MSCI Methodology Document and Poppertech*

Although factor-based risk models alleviate some problems, minimum variance portfolios at this stage still develop several undesirable characteristics. The methodology document hints at these with the following optimization constraints.

*Individual Holding Constraints*

"The maximum weight of an index constituent will be restricted to the lower of 1.5% or 20 times the weight of the security in the Parent Index."

In my experience with running unconstrained, minimum volatility optimizations, a single security frequently represents more than 10% of the resulting portfolio. Sometimes this holding corresponds to a stable company, such as Procter and Gamble (NYSE:PG), with low stock volatility and correlations with the other holdings. Other times, it is a data anomaly, such as an acquisition target whose share price stabilizes close to the offer. Although the latter is more of a problem, both types of concentrations are undesirable. The above constraint specifies that the portfolio have at least 66 holdings. Therefore, it is diversified with respect to possible losses from a single stock.

Despite this constraint, the MSCI Minimum Volatility Index fact sheet states that six of its components exceed 1.5% of the index by weight. How does this happen? A line at the end of the document explains, "Each Minimum Volatility Index is rebalanced (or is re-optimized) semi-annually in May and November." In other words, the latest rebalancing occurred in November. Since then, several of the highly weighted stocks have outperformed other index components and, consequently, have violated this constraint.

Why rebalance so infrequently? Each rebalancing increases turnover in the index, transaction costs in the ETF, and expenses for investors. As a result, MSCI minimizes expenses through such infrequent rebalancing. In addition, they apply the following constraint.

*Turnover Constraint*

"The one way turnover of the MSCI Minimum Volatility Index is constrained to a maximum of 10%."

This means total sales during a rebalance can only account for 10% of the index by weight. Since only two rebalances occur each year, the maximum turnover of the portfolio annually is approximately 20% by weight. This helps explain how the low expense ratio of .15% is achieved.

*Sector Constraints*

"The sector weights of the MSCI Minimum Volatility Index will not deviate more than +/-5% from the sector weights of the Parent Index."

Sector constraints are designed to prevent unintended bets on any given economic sector. Unlike the individual holding constraints, these constraints are applied relative to the USA Index. This explains the low tracking error between the Minimum Volatility Index and USA Index as shown on the fact sheet.

*Figure 5 shows the sector weights of the MSCI USA Minimum Volatility Index, the USA Index, and the difference between the two, respectively. It indicates that the sector constraints of +/- 5% have limited the allocations to several sectors.*

*Source: MSCI factsheets and Poppertech*

Figure 5 shows the sector weights for the Minimum Volatility Index, USA Index, and the difference between them. At first glance, some of the Minimum Volatility Index sector weights seem counterintuitive. For example, Information Technology is a relatively risky sector but accounts for almost 15% of the index by weight; whereas, Utilities are considered safer and only contribute 8.5% by weight.

The differences in weights explain these allocations. Information Technology is 20.5% of the USA Index. Even though the optimizer wants to limit exposure to this sector, the above constraint dictates that it must weight this sector at least 15% in the Minimum Volatility Index. This accounts for the 15% allocation.

In contrast, Utilities only account for 3.3% of the USA Index by weight. Even though the optimizer wants to increase exposure to this sector, the above constraints limit the Minimum Volatility Index to hold at most 8.3% utilities. In fact, the Minimum Volatility Index has exceeded its maximum allocation to this sector by .3%.

Although the sector allocation of the Minimum Volatility Index is moderately constrained, the holdings within each sector are likely to differ significantly from the USA Index. For example, the Financials sector in the ETF contains 61% Real Estate Investment Trusts by weight. The stock prices of REITs tend to have low volatility and correlations to other stocks. Therefore, the optimizer will favor them.

*Style Risk Factor Constraints*

"No constraint will be applied on the exposure of the MSCI Minimum Volatility Index to the Barra Volatility risk index factor. Exposure to all other Barra risk index factors will be restricted to +/- 0.25 standard deviations relative to the Parent Index."

In addition to the constraints on individual holdings and sector allocations, Barra has limited the tilts of the Minimum Volatility Index towards the various style risk factors relative to the USA Index. These include momentum, value, and size. The factor for individual stock volatility is the only style factor excluded from this constraint. As discussed earlier, stocks with low volatility in their share prices are typically the primary candidates for minimum variance portfolios. Therefore, these constraints in aggregate tilt the Minimum Volatility Index towards the only intended bet: low volatility stocks.

It should be noted that the calculation of style factors varies between risk-model vendors. For example, "value" can mean price-to-book, price-to-earnings, price-to-cash flow, or a combination of any of these and others. These factor choices can lead to unintended bets as well. For example, if the value constraint only includes price-to-book, then the Minimum Volatility Index may tilt heavily towards high or low price-to-earnings stocks.

*Figure 6 depicts the historical price-to-earnings, forward price-to-earnings, and book-to-price ratios for the MSCI USA Minimum Volatility Index and MSCI USA Index. The significant difference between the two indicates the Value style constraint is not restrictive.*

*Source: MSCI Minimum Volatility factsheet and Poppertech*

As Figure 6 shows, the Minimum Volatility Index tilts moderately towards growth stocks based upon historical price-to-earnings, forward price-to-earnings and price-to-book. Therefore, the value constraint seems to exert little influence on the holdings with respect to these valuation metrics.

*Unintended Consequences*

It is possible for the constraints to have unintended consequences. For example, large-capitalization stocks tend to exhibit lower share price volatility than smaller ones. The above constraints dictate that the Minimum Volatility Index cannot tilt towards larger companies even if these would reduce variance.

Additionally, safer stocks should outperform during periods of market stress. This would mean these stocks would have high momentum exposures during these periods. As a result, the Minimum Volatility Index would have limited ability to include these names as well.

**When the Optimizer Fails to Converge Upon a Solution**

Decision optimizers are not guaranteed to converge upon an optimal solution for constrained problems, such as this one. In Appendix III in the Methodology document, MSCI describes the procedures used when the initial problem is infeasible. These involve relaxing various constraints until the optimizer converges on a solution. In practice, this is more likely to be a problem with international indices. These impose both sector and country constraints. Additional constraints increase the likelihood of infeasibility.

**Conclusion**

This article examines the iShares MSCI USA Minimum Volatility ETF from a portfolio construction perspective. You should now understand some of the details behind how the ETF allocates its investments. Using this knowledge, you can start to answer the following questions: Do the actual holdings align with the goal of minimizing risk? What unintended bets are being taken, if any? Do statistical estimations of minimum variance based exclusively on historical data adequately capture risk? Is semi-annual rebalancing sufficiently frequent? The answers to these questions and others will determine the suitability of this product for you or your clients.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.