**Introduction**

Factors have been debated in the academic literature for many decades. That risk premia exist for factors such as value, momentum, size, low volatility and quality seems pretty much settled [1 - 7]. Given this, the discussion has progressed to the circumstances under which one should try to access these risk premia [9, 31] and to what portfolio construction techniques one should employ to do this efficiently.

Some researchers advocate a market neutral approach, accessing “pure” factor premia utilizing long-short portfolio techniques. Others take a long only approach, viewing the premia as more efficient way of accessing market exposure. We feel this is a matter of judgment for the individual manager which will be based in part on risk appetite and the limitations imposed by their investment mandates. In this paper we will consider the long only approach although much of the material may be easily extended to create long-short portfolios.

The use of optimization [10, 11, 12] has been an important tool in the selection and weighting of stocks since the work of Markowitz [29]. Put simply one can specify preferences around factor exposures, diversification, risk etc. and then let the “black box” do the work of delivering a portfolio that satisfies these criteria. If the problem is set up correctly, and the black box does its job of efficiently finding the optimal solution, this can deliver the desired outcomes. The one drawback to optimization is around transparency. The black box “knows” why it has chosen stocks and in what proportion, but this may not be so clear to the human who allocated it the task.

Given this issue with optimization, a number of ad hoc methods of portfolio construction have been developed. The simplest and most commonly employed is the construction of a Characteristic Basket, which selects a given proportion of some initial universe by factor value. Stocks within the basket are then weighted according to the factor value itself or on some other criteria concerned with capacity (e.g. Market Cap weighting), diversification (e.g. Equal weighting) or risk (e.g. Risk weighting).

Another simple, but widely used approach is that of “factor tilting” [8, 14, 32]. The idea here is to take a starting universe of stock weightings and to perturb them in a way that increases the exposure to the factor of interest. This is often achieved by multiplying the initial set of weights by a scoring function; high scores for stocks with large factor values and close to zero scores for stocks with the smallest factor values.

One criticism of such tilting techniques, often made by advocates of a selection approach, is that it can only ever provide relatively weak factor exposure. Quite correctly, they highlight that the factor exposure of say, a Characteristic Basket, can be readily increased by further narrowing of the selection universe, for example by taking the top 10% by factor value rather than the top 50%. In this paper we show that criticism regarding factor exposure strength resulting from tilting approaches is quite wrong, and that exposure outcomes depend on the tilting function employed. Indeed, we show that the Characteristic Basket is a special case of a factor tilt, where the scoring function is the step function.

The final, and arguably the most important question is how should one construct multi-factor portfolios. Leaving aside the possibility of optimized solutions, what is the most appropriate mechanism for obtaining multiple factor exposure, whilst maintaining appropriate levels of stock weight diversification? Two ways in which this is done can be characterized as via “top down” (or mixed) portfolios and “bottom up” (or integrated) portfolios.

In the “top down” approach one constructs a composite portfolio from single factor portfolios. Stock weights in this multi-factor portfolio are a weighted average of their weights in the single factor portfolios. Examples of this are given in [13, 17, 19].

Alternatively, in a “bottom up” portfolio, a particular stock is weighted in consideration of all its factor characteristics simultaneously. An example of this is by use of a composite factor, where individual factor values are combined in some way resulting in an overall factor score that is used for stock selection and weighting.

The relative merits and drawbacks of these general approaches have been discussed extensively in the financial literature [15, 25, 26, 27, 28, 35]. Proponents of “top down” suggest it provides the greatest factor exposure consistent with a high degree of diversification [13, 16, 17]. Although it is accepted that the averaging of stock weights results in an averaging of factor exposures, “top down” advocates argue that high multi-factor exposures can be maintained by averaging high exposure single factor portfolios. However, such high exposure single factor portfolios can only be maintained through the application of increasingly aggressive stock selection and weighting, with adverse implications for levels of diversification. On the other hand some practitioners [16, 17] appear unconcerned regarding the potentially relatively weak factor exposures engendered by averaging and choose to highlight the composite portfolio’s stock weight diversification benefits. This seems strange since, whilst it is clear that diversification is important, the primary target of a factor portfolio should surely be intentional factor exposure.

We are firm proponents of the bottom up approach. In this paper we concentrate on an alternative to the composite factor method described above. This is the concept of multiple factor tilting, where we apply sequential factor tilts to a given starting universe of stocks in the expectation that it will achieve both the multiple factor exposures we require and acceptable levels of stock diversification.

The objective of this paper is to assess the relative benefits and drawbacks of the various factor and multi-factor portfolio construction techniques described above, through the lens of factor exposure and portfolio diversification. Academic and empirical evidence tells us that portfolio exposure to certain factors is a good thing [1 - 7], while modern portfolio theory emphasizes the importance of diversification [34].

To avoid reliance on empirical data and criticisms that our results are sample specific, we prefer a more theoretical approach. We make the assumption that our factors can be modelled by a normal distribution and perform our calculations in the continuous limit. We are clear that whilst real portfolios do not contain infinitely many stocks; that correlated factors are not identically normally distributed and stock weights are not infinitesimal, such abstractions are common in finance, and so long as we appreciate its limitations, it can offer useful guidance for the real world.

Section 2 sets out discrete definitions of composite and multiple tilt portfolios and introduces the definitions of exposure and diversification used throughout the remainder of this paper. Section 3 extends the constructs of Section 2 to the continuous limit and derives formulae for use in later sections. In Section 4 we set out two important tilt functions that form the basis for all of our subsequent results, which are set out in Section 5, where we compare outcomes of one, two, three and - factor portfolios employing alternative construction techniques. Section 6 concludes.

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