Chapter 10
Benchmarks and the Use of Tracking Error in Portfolio Construction
Expected portfolio return maximization under the mean-variance framework introduced in Chapter 8 is an example of an active investment strategy—a strategy that identifies a universe of potentially attractive investments and ignores inferior investments opportunities. As we mentioned in Chapter 1, a different approach to investing, referred to as a passive investment strategy, or indexing, argues that in the absence of any superior forecasting ability, investors might as well resign themselves to the fact that they cannot outperform the market after adjusting for transaction costs and management fees. From a theoretical perspective, the analytics of portfolio theory tell investors to hold a broadly diversified portfolio. Hence, many mutual funds and equity portfolios are managed relative to a particular benchmark, market index, or stock universe, such as the S&P 500 or the Russell 1000. Bond portfolios are also benchmarked against fixed income indexes with particular characteristics regarding credit quality and maturity. For benchmarked portfolios, portfolio risk is evaluated based on the standard deviation of the portfolio deviations from the benchmark, or the tracking error, rather than the portfolio's overall standard deviation.
The selection of a benchmark is very important for effective portfolio management, and we spend a substantial amount of time in this chapter discussing the types of indexes that are used as benchmarks. We cover both traditional market capitalization-based indexes as well as alternative indexes, leading into a discussion of smart beta strategies, which use smart indexing to implement strategies that fall between active and passive strategies. But first, we set up the discussion by describing the fundamentals of the concept of tracking error and explaining its role in predicting and evaluating portfolio performance.
10.1 Tracking Error versus Alpha: Calculation and Interpretation
As we mentioned in the introduction to this chapter, tracking error measures the dispersion of a portfolio's returns relative to the returns of a specified benchmark. Mathematically, tracking error is calculated as the standard deviation of the portfolio's active return, where
If a portfolio such as an index fund created to match the benchmark regularly has zero active returns (that is, always matches the benchmark's actual return), it would have a tracking error of zero. But if a portfolio is actively managed and takes positions substantially different from the benchmark, it would likely have large active returns, both positive and negative, and thus would have an annual tracking error of, say, 2% to 5%.
Note that because the tracking error measures return variability rather than absolute return, a portfolio does not have tracking error simply because of outperformance or underperformance. For instance, consider a portfolio that underperforms its benchmark by exactly 10 basis points every month. This portfolio would have a tracking error of zero because there is no variability in the portfolio's active returns. On the other hand, the portfolio's average active return, also referred to as alpha, will be negative 10 basis points. It will signal that the portfolio is underperforming the benchmark. However, the tracking error itself is not indicative of outperformance or underperformance—it just measures the degree of variability of the portfolio performance relative to the benchmark performance.
In contrast, consider a portfolio that outperforms its benchmark by 10 basis points during half the months and underperforms by 10 basis points during the other months. This portfolio would have a tracking error that is positive because there is variability in the active returns. The alpha of the portfolio in this case will be zero since no active return has been realized.
Exhibit 10.1 presents the information necessary to calculate the tracking error for a hypothetical portfolio and benchmark using 12 monthly observations. The fourth column in the table shows the active return for the month. The mean (average) and the standard deviation of the monthly active returns are 0.21% and 0.96%, respectively.1 These values are then annualized.
Exhibit 10.1 Data and calculation of tracking error.
| Month | Portfolio | Benchmark | Active |
| 1 | 3.19% | 3.82% | −0.63% |
| 2 | −0.89% | 0.00% | −0.89% |
| 3 | 5.84% | 6.68% | −0.84% |
| 4 | 2.15% | 2.34% | −0.19% |
| 5 | 2.97% | 3.47% | −0.50% |
| 6 | 0.17% | 0.01% | 0.16% |
| 7 | 4.50% | 2.90% | 1.59% |
| 8 | −1.28% | −1.13% | −0.15% |
| 9 | −0.85% | −1.68% | 0.83% |
| 10 | −2.14% | −2.00% | −0.14% |
| 11 | −3.69% | −5.50% | 1.81% |
| 12 | −5.53% | −6.94% | 1.41% |
To annualize average returns computed for a time period of length 1/t years, one multiplies the average returns by t. For example, to annualize average monthly returns (computed for a time period of 1/12 years), one multiplies the average monthly returns by the number of months in a year (12).
To annualize the standard deviation of returns computed for a time period of length 1/t years, one multiplies the standard deviation by the square root of t (i.e., by 
). For example, to annualize the standard deviation of monthly returns, one multiplies the monthly standard deviation by the square root of the number of months in a year (
).
To compute the annual average active return (alpha) in our example, one multiplies the monthly active return (0.21%) by the number of months in a year (12). The annualized active return is
To calculate the annualized tracking error, one multiplies the monthly standard deviation (0.96%) by the square root of the number of months in a year. Hence
If the observations were weekly rather than monthly, the weekly tracking error would be annualized by multiplying by the square root of the number of weeks in a year, that is, by 
. Alpha in that case would be annualized by multiplying by 52.
If one could assume that the active returns are normally distributed, one can estimate a range for the possible portfolio active return and a corresponding range for the portfolio return given the tracking error. For example, assume the following:
- Benchmark = S&P 500
- Expected return on S&P 500 = 20%
- Tracking error relative to S&P 500 = 2%
If active normal returns followed a normal distribution, then 68% of them would fall within one standard deviation of the mean return, 95% would fall within approximately two standard deviations from the mean return, and 99% would fall within approximately three standard deviations from the mean return. Given these well-known facts about the normal distribution,2 one would estimate that the portfolio return has
- A 68% chance of being within one standard deviation (2%) of the expected return (20%) of the S&P 500, that is, between 18% and 22%.
- A 95% chance to be within two standard deviations (2 × 2%) of the expected return of the S&P 500, that is, between 16% and 24%.
- A 99% chance to be within three standard deviations of the expected return of the S&P 500, that is, between 14% and 26%.
Various factors affect the portfolio tracking error. The effects of different factors are investigated in Vardharaj, Fabozzi, and Jones (2004) for the case of equity portfolios in particular. Noteworthy findings include:
- Portfolio tracking error generally decreases with the number of securities from the benchmark included in the portfolio.
- Portfolio tracking error increases as the average market capitalization of the portfolio deviates from that of the benchmark.
- Portfolio tracking error increases when the portfolio beta (measure of market risk) deviates from the benchmark beta.
The fact that tracking error increases due to a particular factor is not necessarily a cause for concern. For example, a portfolio manager may pursue an enhanced indexing strategy in which the portfolio holdings deviate from the index holdings in small amounts in the hope that these bets (or views) will allow for the portfolio to outperform the index slightly. A portfolio manager may also pursue an active/passive strategy, that is, a blend of active and passive strategies, in which a part of the portfolio may be actively managed while another part may be indexed to a benchmark index.
10.2 Forward-Looking versus Backward-Looking Tracking Error
In Exhibit 10.1, we showed the calculation of portfolio tracking error based on realized active portfolio returns over 12 months. The realized portfolio performance is the result of the portfolio manager's decisions during those 12 months with respect to portfolio positioning issues such as beta, sector allocations, style tilt (value versus growth),3 and individual security selection. The tracking error calculated in this manner from historical active returns is referred to as backward-looking tracking error, ex post tracking error, and actual tracking error.
The backward-looking tracking error has its use in portfolio performance evaluation but has little predictive value and can be misleading when it comes to assessing portfolio risk. This is because it does not reflect the effect of the portfolio manager's current decisions (e.g., a change in sector allocations) on the future active returns and the tracking error that may be realized in the future.
Portfolio managers need forward-looking estimates of tracking error to assess future portfolio performance. In practice, this is accomplished by using a multifactor model4 from a commercial vendor that has identified and defined the factors driving the return on the benchmark, or by building such a model in-house. Statistical analysis of historical return data for the securities in the benchmark are used to obtain the factors and to quantify the risks. Using the manager's current portfolio holdings, the portfolio's current exposure to the various factors is calculated and compared to the benchmark's exposures to the factors. From the differential factor exposures, a forward-looking tracking error for the portfolio can be computed. This tracking error is also referred to as ex ante tracking error and predicted tracking error. We provide more detail on the calculation of forward-looking tracking error in Section 10.4 of this chapter.
There is no guarantee that the forward-looking tracking error will match exactly the tracking error realized over the future time period of interest. However, this calculation of the tracking error has its use in risk control and portfolio construction. By performing a simulation analysis on the factors that enter the calculation, the manager can evaluate the potential performance of portfolio strategies relative to the benchmark, and eliminate those that result in tracking errors beyond the client-imposed tolerance for risk. As we mentioned, the backward-looking tracking error, in contrast, is useful for assessing actual performance relative to a benchmark, and enters into the calculation of various performance metrics of concern to investors, such as the information ratio (covered in the next section).
10.3 Tracking Error and Information Ratio
The information ratio (IR) is a widely used reward/risk performance metric. It is calculated as follows:
As we illustrated in Section 10.1, the alpha realized by a portfolio manager historically is the average active return over a time period. The backward-looking tracking error is calculated as explained in Section 10.1.
The IR is essentially a reward–risk ratio. The reward is the average of the active return (alpha). The risk is the standard deviation of the active return (the tracking error), and, more specifically, the backward-looking tracking error. The higher the IR, the better the manager performed relative to the risk assumed. The IR also attempts to measure consistency. A high IR ratio indicates that the portfolio manager outperformed the benchmark by a little every month (as opposed to by a lot in just some of the months). Consistency in performance is considered a desirable trait in a portfolio manager.
To illustrate the calculation of the IR, consider the active returns for the hypothetical portfolio shown in Exhibit 10.1. As we showed in Section 10.1, the annualized average active return is 2.47%. The backward-looking tracking error (annualized) is 3.34%. Therefore, the information ratio is 2.47%/3.34% = 0.74.
10.4 Predicted Tracking Error Calculation
Section 10.1 illustrated the calculation of backward-looking tracking error for the purpose of portfolio performance evaluation. In this section, we illustrate the methodology for calculating forward-looking tracking error for the purpose of portfolio optimization or risk estimation. Specifically, we describe the variance-covariance calculation method for the tracking error and illustrate the calculation using a multifactor model.
Mathematically, determining the optimal portfolio selection for a passive investment strategy involves changing the objective function of the portfolio allocation problem from Chapter 8 so that instead of minimizing the portfolio variance, one minimizes the tracking error with respect to the benchmark. Alternatively, a limit on the tracking error may be included as a constraint in the optimization formulations for active investment strategies. The variance-covariance representation of the tracking error reviewed in this section is useful both in forward-looking portfolio optimization schemes and for estimating the portfolio risk by simulating possible values for the factors influencing future portfolio returns.
10.4.1 Variance-Covariance Method for Tracking Error Calculation
Given a benchmark, the forward-looking tracking error can be calculated using the covariance matrix of returns. Specifically, the tracking error is the standard deviation of the difference between the portfolio return, 
, and the return on the benchmark, 
. Here, 
is the vector of portfolio weights and 
is the vector of benchmark weights. The vector 
is the vector of exposures.
The variance-covariance (VcV) method calculates the tracking error as follows:
where Σ is the covariance matrix of the security returns. One can observe that the formula is very similar to the formula for the portfolio variance derived in Chapter 8.1; however, the portfolio weights in the formula from Chapter 8.1 are replaced by the active weights, that is, by the differences between the weights of the securities in the portfolio and the weights of the securities in the benchmark.
10.4.2 Tracking Error Calculation Based on a Multifactor Model
Using factor models in the context of tracking error calculation has important advantages for reducing security pricing errors. As we explained in Chapter 9, a general factor model has the form
where
is the rate of return on security i,
is the sensitivity (factor loading) of security i to factor k,
is the rate of return (factor return) on factor k, and
is the residual error of security i, a random shock with expected value of 
.5
Suppose there are N securities in the portfolio and K factors, and the weight of each security i is 
. The portfolio variance 
can be expressed as
where
is the variance of 
,
is the covariance of factors k and l, and
is the portfolio sensitivity to factor k, calculated as the weighted sum of the individual securities' sensitivities to factor k:
The tracking error based on the multifactor model is
where 
is the net, or active, sensitivity to factor k, calculated as
In the above formulas, 
is the number of securities in the benchmark and 
is the weight of security 
in the benchmark.
10.5 Benchmarks and Indexes
As we mentioned in the introduction to this chapter, selecting an appropriate benchmark for the purposes of portfolio management and performance evaluation is critical. In this section, we discuss the process of selecting a benchmark and the type of benchmarks used by investment managers and financial advisors.
When institutional investors engage the services of professional asset managers, they usually specify the benchmark by which the manager's performance will be evaluated. Typically, asset managers are hired to manage client assets within an asset class such as equities or fixed income. In fact, the client will often specify not just the major asset class but also a sub-class within the asset class. So, for example, a client seeking an equity portfolio manager might specify a manager who specializes in large capitalization companies or growth-oriented companies. In seeking a bond portfolio manager, a client may specify a manager who specializes in investment-grade bonds or in noninvestment-grade (i.e., high-yield) bonds.
Once a client sets forth the risk exposure and the asset class (and possible sub-asset class), a benchmark represents the portfolio that has the highest expected return given the desired risk exposure. The benchmark's construction requires that a set of rules be specified for the purpose of determining which specific securities from a universe of the asset class (or sub-asset class) should be included and excluded from the benchmark.
Benchmarks can be classified as either market indexes or customized indexes. Historically, investors have required their asset managers to use as a benchmark a market index. However, due to certain limitations of market indexes, since the turn of the century more institutional investors have turned to customized indexes.
10.5.1 Market Indexes
Market indexes can be classified into three groups:
- Those produced by exchanges based on all securities traded on the exchanges.
- Those produced by organizations that subjectively select the securities to be included in the benchmark.
- Those for which security selection is based on an objective measure, such as market capitalization or, in the case of bonds, a minimum credit rating and issue size.
The first group, exchange-provided market indexes, are typically only available for equities because bonds are primarily traded over-the-counter. The more popular equity market indexes that fall into this category include the New York Stock Exchange Composite Index. Although the Nasdaq is not an exchange, the Nasdaq Composite Index falls into this category, too, because the index represents all stocks tracked by the Nasdaq system.
The most popular market index that falls into the second group for equities is the Standard & Poor's 500 (S&P 500). The S&P 500 represents stocks that are chosen from the two major national stock exchanges and the over-the-counter market. The stocks in the index at any given time are determined by a committee of the Standard & Poor's Corporation, which periodically adds or deletes individual stocks or the stocks of entire industry groups. The aim of the committee is to capture overall stock market conditions as reflected in a broad range of economic indicators. Two other market indexes that fall into this group but are rarely used as benchmarks for evaluating the performance of an asset manager are the Value Line Composite Index and the Dow Jones Industrial Average (DJIA). The former market index, produced by Value Line Inc., covers a broad range of widely held and actively traded companies selected by Value Line. The DJIA is constructed from 30 of the largest and most widely held U.S. industrial companies selected by the Dow Jones & Company (publisher of the Wall Street Journal).
Representative examples from the third group are the Wilshire indexes (produced by Wilshire Associates and published jointly with Dow Jones) and the Russell indexes (produced by the Frank Russell Company, a firm that consults to pension funds and other institutional investors). The criterion for inclusion in each of these market indexes is solely a firm's market capitalization. The most comprehensive index is the Wilshire 5000. The Wilshire 4500 includes all stocks of companies in the Wilshire 5000 except for those in the S&P 500. Thus, the stocks of companies in the Wilshire 4500 have smaller capitalization than those in the Wilshire 5000. The Russell 3000 encompasses the 3,000 largest companies in terms of their market capitalization. The Russell 1000 is limited to the largest 1,000 companies of those, and the Russell 2000 has the remaining smaller firms.
For bond benchmarks, the most common market indexes are those used by investment banking firms such as Barclays Capital, Merrill Lynch, JP Morgan, and Morgan Stanley. The broad-based U.S. bond market index most commonly used is the Barclays Capital U.S. Aggregate Bond Index. There are more than 6,000 bond issues in this index, which includes only investment-grade securities. The index is computed daily. The pricing of the securities in each index is either trader priced or model priced. Each broad-based bond index is broken into sectors. The sector breakdown for the Barclays Capital U.S. Aggregate Bond Index is Treasury securities, government agency securities, corporate bonds, agency mortgage-backed securities, commercial mortgage-backed securities, and asset-backed securities. There are then market indexes created for each of these sectors.
The securities included in a market index must be combined in certain proportions, and each security must be given a weight. The two main weighting scheme methodologies are
- Weighting by market capitalization.
- Equal weighting for each security.
Market capitalization weighting (also referred to as value weighting) means that the weight assigned to a company's stock is found by dividing that company's market capitalization by the total market capitalization of the universe of stocks included in the index.6 Market capitalization in the case of a bond market index is calculated differently—in terms of individual bond issues rather than market capitalization for the entire company. One deals with individual bond issues rather than companies since a company typically has more than one bond issue outstanding. The market capitalization for a bond issue is equal to the total par value of a bond issue outstanding multiplied by the price per bond (expressed as a percentage of par value). Market indexes created using the market capitalization weighting scheme are referred to as market-capitalization market indexes.
The equal weighting methodology, the second weighting scheme listed above, is easier. In the case of an equity benchmark, if there are N candidate companies, then the weight assigned to each company's market capitalization is simply 1/N. In the case of a bond benchmark, if there are B bond issues that are candidate securities, then the weight assigned to each bond issue is 1/B.
10.5.2 Noncapitalization Weighted Indexes
Historically, a market index that is market-capitalization weighted has been the benchmark of choice for clients. The theoretical justification for using a market index that is market-capitalization weighted is provided by capital market theory. If a market is price efficient and satisfies the restrictive conditions of the capital asset pricing model (CAPM),7 then the best way to capture the efficiency of the market is to hold the market portfolio that is a market-capitalization-weighted portfolio such an index purports to represent. However, in recent years, research has questioned whether market-capitalization market indexes offer the highest expected return given a client's desired risk exposure and therefore whether such indexes are suitable benchmarks for evaluating portfolio managers.
For example, constructing a portfolio that matches a cap-weighted index such as the S&P 500 can be viewed as an (inefficient) strategy of buying high and selling low (The Economist, July 6, 2013). This is because companies increase their weight in the index when their stock price rises faster than the rest of the market, and decrease their weight when their stock price falls. This and other shortcomings of cap-weighted indexes have been documented in a number of recent studies.8
The limitations of market indexes that are market-capitalization weighted have led to the development of alternative indexes and customized indexes, more generally referred to as noncapitalization weighted indexes.
Alternative indexes can be classified as heuristic-based weighted indexes and optimization-based-weighted indexes. The former include equally weighted indexes, risk-based indexes, fundamental-based indexes, and diversity-based indexes. These indexes seek to weight candidate securities (stocks in the case of an equity index and bond issues in the case of a bond index) by one or more factors that drive asset class returns. Two examples for equity indexes are the Research Affiliates Fundamental Indexes and the MSCI Factor Indexes. Company fundamentals that are used in fundamental-based indexes include cash flow, book-to-value ratio, dividend per share, sales growth, and the like. The MSCI Factor Indexes are a family of factor indexes that are weighted by value style, momentum, and volatility. These alternative equity indexes are referred to as ad-hoc indexes because of the subjective elements involved in constructing them (i.e., in selecting the weights).
We build on an example from Arnott, Hsu, and Moore (2005) to illustrate how one constructs an ad-hoc index. Suppose we would like to create a fundamental-based index that represents company size.9 Arnott, Hsu, and Moore (2005) use the following metrics of company size:
- Book value
- Trailing five-year average cash flow
- Trailing five-year average revenue
- Trailing five-year average gross sales
- Trailing five-year average gross dividends
- Total employment
After ranking the companies according to each metric, one can calculate the relative weight of each stock within the ranking and select, for example, the largest 1,000 according to each metric. The final result could be a Composite Fundamental Index, in which the ranking of the stocks is determined by an equally weighted sum of the stocks' relative weights in each of the metrics for the size factor.
In contrast to ad-hoc constructed indexes, optimization-based indexes are constructed using optimization methodologies to create a diversified index that takes into account the systematic factors that drive returns and thereby a better benchmark that should be used for achieving client objectives and then to evaluate manager performance. The best known example of optimization-based equity indexes is what is popularly referred to as smart beta indexes, which we will discuss in Section 10.6.
Nonweighted market capitalization indexes also include customized indexes where an asset manager in consultation with clients creates customized indexes that can be used to provide tailor-made benchmarks that better reflect a client's investment objectives and risk tolerance or to remove the concentration risk of large issuers dominating the index. To deal with concentration risk for the purpose of avoiding idiosyncratic risk, an index can be created that imposes a maximum allocation constraint on the weight for a given company in the case of an equity customized index, on the weight of the issuer of a corporate bond index in the case of a bond customized index, or on the weight of a given country in the case of a customized international government bond index. An increasing number of corporate sponsors of defined benefit plans create a customized index to reflect better their objectives and risk tolerance when liabilities must be taken into account (i.e., when a liability-driven strategy is pursued). These customized indexes are referred to as liability indexes.
10.6 Smart Beta Investing
We have encountered the term beta in multiple chapters already. Beta is a measure of the sensitivity of a particular security to the market. For example, a stock with a beta of one moves in line with the market—when the market return increases by 1%, the return on the stock with a beta of 1 would be expected to increase by 1% as well (on average). One can buy beta cheaply by investing in an index fund. For example, the largest exchange-traded fund (ETF), the SPDR S&P 500 ETF Trust (SPY), tracks the S&P 500. Alpha, on the other hand, captures the moves in the return of a security that cannot be explained by the market.
A new term—smart beta—has grown in use tremendously in the last few years. To most finance professionals, smart beta investing is about index-like fund management to keep costs down; however, the indexes themselves are “smarter” than the market-capitalization weighted indexes that have been used historically. In Chapter 9.8, we characterized factor investing as a smart beta strategy, and in Section 10.5, we discussed the link to customized indexes. Providers of smart beta indexes include Research Affiliates, BlackRock, Russell, EDHEC-Risk Institute, and STOXX, among others. Several financial institutions are now offering smart beta funds.
The term smart beta is now an umbrella term that encompasses a wide range of strategies, the investment style for which falls between active and passive. In fact, the term is so imprecise that apparently it makes Nobel Prize-winning economist William Sharpe “definitionally sick.”10 The smart beta opportunity set can include (Thomas 2014):
- Alternatives to cap weighting (such as valuation-based, low-volatility, or equal-weighted portfolios).
- Investments in nontraditional asset classes such as commodities, breakeven inflation, volatility, currency carry.
- Alternative asset class payoffs such as collars and leveraged strategies.
- Specialized rules-based strategies such as merger arbitrage, trend-following, convertible arbitrage, and active manager emulation.
Many smart beta indexes employ simple heuristics, such as the fundamental weighting scheme described in Section 10.5. Other heuristic weighting methods include equal weighting, risk-cluster equal weighting (where risk clusters, instead of individual securities, are equally weighted), and diversity weighting (which combines equal weighting and cap weighting).
However, there can be added layers of complexity. Smart beta strategies could combine various proxies for optimal portfolios (e.g., the minimum-variance portfolio calculated with optimization methods) and other portfolios that explore factor exposures that do well under certain conditions. Portfolios based on mixes of different strategies often benefit from the effect of diversification, and appear to do better than single smart beta strategies.11 For some, risk-parity-based strategies (strategies that attempt to allocate risk, rather than capital, equally among investments) are also part of the smart beta strategies family.12 Smart beta index providers list a variety of possible methodologies used for constructing smart beta indexes, including equal weight, minimum variance, dividend, risk weighted (equal risk, low beta, etc.), style, environmental, and sustainability.13
Amenc, Goltz, and Lodh (2012) suggest that the problem of creating a smart beta investment portfolio be separated into two stages: (1) a constituent selection, that is, which characteristics of securities are desirable to hold, and (2) a diversification scheme within the chosen universe of securities. The first stage takes into account stand-alone properties of the securities whereas the second stage attempts to achieve a particular objective by taking into account how the securities interact within the portfolio. Some smart beta index providers focus on the first stage, selecting securities with particular characteristics such as the ad-hoc fundamental weighting scheme described in Section 10.5 or ranking securities based on the output from factor models. Others add optimization to come up with the optimal diversification scheme. Amenc, Goltz, and Lodh (2012) claim that there is a “key distinction between the stock selection decision, which helps tilt the portfolio towards the relevant characteristics, and the choice of diversification scheme which is more effective in attaining the relevant diversification objective than a pure stock selection strategy.”
Consider the creation of the Russell 1000 Low Beta Factor Index offered by Frank Russell Company.14 The first stage reduces to the calculation of a “naive” factor index that delivers exposure to stocks that have low predicted betas according to a screening and ranking methodology applied to the output of the Axioma U.S. Equity Medium Horizon Fundamental Factor Risk Model. The betas are the sensitivities of the stocks to a change in the market price level as measured by the Russell 1000 Index.
The Russell 1000 stocks are ranked based on their predicted betas from the Axioma U.S. Equity Medium Horizon Fundamental Factor Risk Model. The naive target index is then created by starting with the lowest beta stock and adding the next lowest beta stocks until the target portfolio has a total capitalization of 35% of the Russell 1000 Index.
The second stage involves selecting a portfolio of up to 200 stocks from the Russell 1000 Index to track optimally the returns of the naive factor index while managing turnover and neutralizing exposure to other factors, such as volatility and momentum. This is accomplished by solving an optimization problem in which the tracking error between the Russell-Axioma Factor Index and the naive factor index is minimized15 and constraints are added to ensure
- Factor neutrality to factors that are not the target of the index.
- Controlled turnover so that the monthly turnover is not too high.
- Limits on the exposure to the target factor (in this case, the beta).
- Limits on the number of stocks in the portfolio.
We explain the mathematical formulation of such constraints in detail in Chapters 11 and 12.
There can be a variety of alternative diversification schemes for the securities included in the naive index. Amenc, Goltz, and Lodh (2012) study three in particular:
- A minimum volatility weighting with norm constraints: The overall portfolio volatility (standard deviation) is minimized, subject to a constraint on portfolio concentration, which is also known as a norm constraint.16
- Efficient maximum Sharpe ratio weighting: The portfolio Sharpe ratio17 is maximized, where the expected returns are estimated indirectly by assuming that they are proportional to the median downside risk of the risk group a stock belongs to.18
- Maximum de-correlation weighting: The overall portfolio volatility is minimized under the assumption that the individual volatilities are identical across stocks.19 The idea is to combine stocks so as to exploit the risk reduction effect stemming from low correlations between the stocks in the portfolio rather than reducing risk by concentrating in stocks with low volatilities (standard deviations).
Each of these diversification schemes performs differently in bull and bear markets and in high- and low-volatility regimes. As Amenc, Goltz, and Lodh (2012) note, both the systematic (constituent selection) component of the smart beta strategies and the strategy-specific (diversification) component display characteristics that can be exploited by investors.
Smart-beta funds have been utilizing ever-more-complex investment rules. ProShares Large Cap Core Plus, for instance, uses leverage (that is, it borrows money) to buy stocks that target 10 different factors, including value, growth, and price momentum.20 The reason why leverage may be helpful is illustrated in Exhibit 10.2. Suppose factor models are used to create a diversified equity portfolio with best risk/return trade-off (Portfolio A). An investor who wants higher returns generally has two options: (1) invest more in equities, taking on more risk in equities (Portfolio C), or (2) apply leverage to the diversified portfolio to achieve specific target return levels (Portfolio B). A leveraged portfolio—Portfolio B—is one of the options to achieve the target return.
Exhibit 10.2 Illustration of the effect of leverage on investment risk and return.
ProShares Core Plus also identifies underperforming stocks. If some stocks miss the targets, ProShares Core Plus sells them short, betting they'll decline. The resulting product is not an ETF but more of an exchange-traded hedge fund, illustrating the variety of smart-beta products available today.
