It wasn't too long ago that the concept of factors in investing was the exclusive province of professors of finance and a few active "quant" managers. Mainstream portfolio construction was focused primarily on asset allocation. Within equities, that meant achieving the right balance in allocation to various segments, such as large cap and small cap, country and sector, and perhaps value and growth styles.

Today, factor allocation has entered the mainstream as a complementary approach to portfolio construction, alongside traditional asset allocation. An important driver of this development has been the creation of a new array of indexes that sharply focus on one factor at a time. This has opened up new possibilities for asset owners and advisors, including investing in index-replicating financial products, both to seek a desired factor exposure at low cost and to benchmark active managers to assess their value added.

One thing that followers of single factor indexes quickly realize is that the payoff for exposure to any one factor is highly variable. Factors typically follow different return patterns: value usually exhibits pro-cyclical performance, while quality is often countercyclical, for example. Market participants who do not employ a factor-timing or factor-rotation strategy are increasingly looking at strategic combinations of factors to gain potential improvements in risk-adjusted outcomes as compared to single-factor outcomes.

A lively debate has emerged regarding what is the best way to combine several factors into a single index. Roughly speaking, there are two camps in the debate: those who advocate a top-down "mixed" composite of individual factors and those who advocate a bottom-up "integrated" approach which results in an index of stocks that have simultaneous factor exposures. Each side argues that their approach produces strong factor exposures with high diversification.

FTSE Russell stands squarely in the bottom-up "integrated" camp. In this paper, we will illustrate the FTSE Russell sequential tilting or "tilt-tilt" methodology, which is very much a bottom-up approach. After a brief overview of alternative methodologies, we will walk through a simple three-stock example of how we build a single factor and multi-factor index. We will contrast it with the most common and straightforward of composite "mixed" methods using the same factors. Then we will illustrate the alternative approaches with a large universe of stocks. We will augment the empirical illustration with some recent theoretical results on the trade-off with diversification which are independent of any particular data set. Finally, we will show how our multi-factor methodology can be extended to encompass environmental, social and governance (ESG) considerations.

### Multi-factor indexes: Combining factors with meaningful levels of factor exposures

The industry discussion concerning factor combinations focuses on delivering targeted factor exposures and the associated factor premia, whilst maintaining adequate diversification. We will focus on how we construct our multi-factor indexes and contrast it with a simple composite index. But before we do, it's worth mentioning a couple of other common approaches.

Optimization has been an important tool in portfolio construction ever since Markowitz introduced its use in 1952. The important characteristic of using an optimizer for constructing factor indexes is that in theory one can maximize the strength of factor exposures, while satisfying targets on risk, diversification, liquidity, etc. Once the objective function and constraints are set, just let the optimizer run and find a solution. The trouble with this approach is that the optimizer appears to be a "black box" without transparency: it knows why certain stocks are selected and weighted a certain way, but humans might find its choices mysterious. The growth in indexing has been driven in part by a desire for increased transparency. This is one of the reasons why FTSE Russell does not use optimizers in constructing its factor indexes.

Another common approach is to create a "characteristic basket" by using percentile cutoffs on stocks ranked by factor characteristics to select stocks for the index. For example, one might select the top 50% of stocks ranked by some factor characteristic and then weight them by some method, such as capitalization weight, equal weight or characteristic strength. A multi-factor version of this approach would be to create an "intersection basket" of stocks that simultaneously rank highly on all factors. The intersection basket is an alternative bottom-up approach that we will discuss further in the paper.

FTSE Russell "factor tilting" starts with a set of weights, most commonly capitalization weights, but it could also be equal weights or some other weighting scheme. The weights are then perturbed or tilted in the direction of increased factor exposure. This is achieved by multiplying the initial weights by a factor score ranging from 0 to 1, with 0 being the weakest, 1 being the strongest, and 0.5 being average exposure. The appendix contains a summary of the construction of the FTSE Global Factor Index Series single factor scores.

In the next section, we will walk through a three-stock, two-factor example of how we construct single factor indexes and compare two versions of multi-factor indexes, a composite and the FTSE Russell "tilt-tilt" approach.

**A multi-factor composite index.** The most common and simplest way to construct a multi-factor index is to take a weighted average of two or more single factor indexes, say 50% value and 50% quality. The advantage of this approach is its top-down simplicity. In principle, this is no different than replicating single factor indexes in the chosen weights. An advantage in having both factors together in one index is that the index provider maintains the fixed weights, relieving the market participant of having to adjust index-replicating products. The main concern is that the averaging process could dilute the factor exposures. We will show this is a valid concern.

**The FTSE Russell "tilt-tilt" multi-factor methodology.** FTSE Russell constructs a multi-factor index as a multiplicative tilt of one factor on another rather than as an arithmetic averaging of the factors. This multiplicative approach, also called sequential tilting, in our view has the best chance of achieving the multi-factor objectives of strong factor exposures with high diversification.

**A three-stock example.** We will illustrate the mechanics of the two approaches to making a quality and value multi-factor index using just three stocks. First we create a hypothetical capitalization-weighted index, plus hypothetical single-factor quality and value indexes for later reference. Then we illustrate the two ways of combining these two factors into a hypothetical multi-factor index. We base the capitalization weights on the actual capitalization levels in the FTSE Developed Index as at March 30, 2017. Likewise, the quality and value factor scores are the actual scores for these stocks as at March 30, 2017.

We chose three well-known company names with roughly the same capitalization levels so as to better illustrate the effects of tilting away from the cap weights. The first column of numbers in Table 1 shows what a three-stock cap-weighted index would look like based on the actual capitalization of these stocks as at March 30, 2017, adjusted for free float. The quality scores are a metric from 0 to 1, with 1 indicating a high-quality stock based on measures of profitability, efficiency, earnings stability and leverage, and 0 a low-quality stock based on the same measures. A score of 0.50 indicates a stock that has exactly average quality characteristics relative to the universe of the FTSE Developed Index.

We can see that all three stocks have below-average quality scores relative to the FTSE Developed Index universe. But what matters for this simple index are the scores relative to each other: Occidental Petroleum (NYSE:OXY) has the highest quality score, while Barclays (NYSE:BCS) has the lowest. We next multiply the cap weight of each stock by its value score to get the unadjusted weights of a single-factor value index. We then divide the unadjusted weights by the sum 25.5% to gross-up final weights to sum to 100%.

In Table 2 we construct a hypothetical single factor value index in exactly the same way. Ford Motor (NYSE:F) is strongly value, i.e., considered "cheap" in terms of valuation metrics, while the other two stocks are relatively "expensive." The value scores have low correlation with the quality scores, which is typical for these two factors. This results in the two single-factor indexes having very different weights.

In Table 3 we show the construction of a hypothetical composite index combining quality and value. We assume equal weighting of the two factor indexes, but in principle one could choose unequal weights as well. The composite index weights are given in the last column.

Table 4 shows the construction of a hypothetical tilt-tilt quality and value multi-factor index. The scores are multiplied rather than averaged. The unadjusted weights are divided by the sum 12.1% to gross-up the final tilt-tilt weights to sum to 100%.

We have now gone through the simple mechanics of constructing these hypothetical indexes, and we now have two sets of quality and low value multi-factor weights. So what difference does it make? Table 5 summarizes the active weights, with the market capitalization weights subtracted from the index weights. In this form, the contrast between the tilt-tilt methodology and the composite approach is brought out more clearly. In this example, both indexes have the same signs and rank ordering of active weights. This isn't always the case. More noteworthy is that the absolute values of the tilt-tilt active weights are all greater than the absolute values of the composite weights. This is not an unusual comparison and drives a lot of the differences in exposure, as we shall see.

The difference the weightings make is evaluated by the "active exposures" of the factors within each index, i.e., exposures to the factors over what naturally comes with a cap-weighted benchmark index. The quality and low volatility scores, which are a 0 to 1 cumulative normal metric, are converted back to their underlying factor Z-scores and weighted by the active weights:

The active factor exposures are thus measured in Z-score units: the number of standard deviations from a mean of zero. Chart 1 displays the active exposures of our hypothetical three-stock indexes. The active exposures of the tilt-tilt index are greater than the active exposures of the composite index for both factors, although just barely for the quality factor. But the active value exposure is substantially larger in the tilt-tilt index compared to the composite. This is just a three-stock example, of course, and as such might not be very meaningful - except that these qualitative results generalize to a whole stock universe, as we shall see.

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