### Factor Portfolios And Efficient Factor Investing

Analytic Investors, Inc.

Analytic Investors, Inc.

Marriott School of Management, BYU

June 30, 2015

**Abstract: **

Even in the absence of security-specific alphas, constructing a total portfolio from factor sub-portfolios is not generally mean-variance efficient. For example, an optimal combination of four fully-invested factor sub-portfolios, Low Beta, Small Size, Value, and Momentum, captures only about 45 percent of the potential improvement over the market Sharpe ratio. In contrast, a long-only portfolio of individual securities, using the same risk model and return forecasts, captures about 90 percent of the potential improvement. In this paper, we adapt general portfolio theory to the concept of factor-based investing, and investigate optimal combinations of factor portfolios using the largest one thousand common stocks in the U.S. equity market from 1968 to 2014.

### Factor Portfolios And Efficient Factor Investing – Introduction

Investors increasingly view portfolios not only as a collection of securities, but as a bundle of exposures to the factors that drive security returns. The recent growth in factor-based or “smart beta” strategies under a variety of different names suggests that many investors now view managing factor exposures to be similar to traditional asset allocation. For example, within the Exchange Traded Fund (ETF) marketplace, nearly one quarter (24%) of institutional decision makers currently use smart beta products.1 In the equity market, investors follow well-known factors like size, value, and momentum, although the factor framework applies to other asset classes as well. The unifying theme is that employing sub-portfolios allows investors to effectively manage the risk and return tradeoffs in their factor exposures without having to buy or sell individual securities.

We do not weigh in here on the correct set of equity market factors, how they are defined, or what caused them to emerge and be identified in the historical return data on equity markets. The issues in his paper revolve around portfolio construction and the delivery of risk-adjusted alpha to end investors, given that one accepts a small set of factors as the primary driver of securities returns. While implementation concerns such as turnover, transaction costs, and managerial fees, are all important considerations for investment products and services, we address potentially more material structural issues associated with the long-only constraint under multiple levels of portfolio optimization.

The promise of smart beta strategies rests on the assumption that a combination of sub-portfolios can achieve the same optimal risk-return tradeoff as purchasing the underlying securities. In this paper, we evaluate the validity of that assumption from both a theoretical and empirical perspective. The theory contribution starts with the mathematics of mean-variance optimization that originated with Treynor and Black (1973), extended to accommodate unintended factor exposures and correlated factor returns. We introduce the empirical contribution in Figure 1, which plots the cumulative monthly returns to two long-only portfolios constructed from the largest one thousand US common stocks from 1968 to 2014.

Portfolio “P” above contains 200 individual securities selected at the beginning of each month using a time-varying set of expected factor returns and risks and a simple mean-variance optimization.2 Specifically, the expected factor returns are set equal to the trailing 60-month factor return standard deviation times the following parameters: a market factor Sharpe ratio of 0.400, information ratios of 0.300 for the Small Size, Value, and Momentum factors, and an information ratio of 0.000 for the Low Beta factor. The choice of these particular factors, factor specifications, and the assumed parameter values, are based on their historical track record in the U.S. equity market summarized later in Table 2. Alternatively, portfolio “Q” is an optimal combination of four factor sub-portfolios, using the same set of expected factor returns and risks. Specifically, the Low Beta, Small Size, Value, and Momentum factors are represented by long-only market-capitalization weighted portfolios of the 200 stocks with the highest exposure to each of those factors. “M” is the market-capitalization weighted portfolio of all 1000 stocks, similar to the Russell 1000 Index, used in this study as the benchmark.

Figure 1 shows that despite being based on the same expected factor returns and risks, portfolio P has substantially better realized performance than portfolio Q. As reported in Table 1, the average annual return in excess of the risk-free rate is 11.39 percent for portfolio P compared to 7.94 percent for portfolio Q, with only slightly higher risk. The realized Sharpe ratio of portfolio P composed of individual securities is 0.712, compared to 0.517 for portfolio Q composed of factor sub-portfolios. Both portfolios employ factors that are known in retrospect to have positive payoffs over the last half century, so outperforming the market portfolio Sharpe ratio of 0.373 is not surprising. However, portfolio P built from individual securities has about twice the benchmark-relative value added as portfolio Q built from sub-portfolios, while incurring similar turnover.

In fact, the historical performance of portfolios P and Q could have been anticipated based on their expected Sharpe ratios. Figure 2 plots the ex-ante Sharpe ratios for both actively managed portfolios and the market benchmark at the beginning of each month using the previously stated parameter values. The ex-ante Sharpe ratio of portfolio P varies around 0.600, while the ex-ante Sharpe ratio of portfolio Q varies around 0.500. Thus, the portfolio construction process using individual securities instead of factor sub-portfolios leads to about twice the expected performance over the market Sharpe ratio of 0.400, even though the security-specific alphas are assumed to be zero.

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