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Efficient integration of risk premia exposures into equity portfolios

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We present a stock selection methodology that maximizes the expected returns of equity portfolios by efficiently managing their exposures to a given ensemble of risk premia, also known as factors. Our approach is mathematically grounded, robust in its design, and applicable in practice. It addresses several issues specific to factor investing, such as cross-sectional interactions between factors, the mismatch between the factors performance cycles and typical rebalancing periods, or the mitigation of interactions between the capital allocation schemes and factor exposures.

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Fig. 1

Data sources for financial ratios: factset and worldscope

Fig. 2
Fig. 3

Data sources for financial ratios: factset and worldscope

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  1. For example, in order to select 20% of the initial investment universe, c must be set to \(F^{-1}(0.8)\times \sqrt{{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}}\) where F is the standard normal cdf.


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Correspondence to B. Vaucher.

Appendix A: Optimal scores

Appendix A: Optimal scores

The first crucial observation towards solving the optimization program defined in Eq. 2.6 is that both the vector of returns defined in Eq. 2.3 and the vector of total score are normal variables. Hence, both variables can always be related by a regression of the form

$${\varvec{r}} = \beta {\bar{\varvec{z}}} + \varvec{\eta },$$

where \({\bar{\varvec{z}}}\) and \(\varvec{\eta }\) are un-correlated and the coefficient \(\beta\) is defined by

$$\beta = \frac{ cov ({\bar{\varvec{z}}},{\varvec{r}} )}{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}} = \frac{\varvec{\pi }^T \Omega {\varvec{\lambda }}}{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}},$$

where the matrix \(\Omega _{ij} = cov ({\varvec{z}}_i,{\varvec{z}}_j)\) corresponds to the covariances between cross-sectional z-scores. Because we are dealing with z-scores, the matrix \(\Omega\) will have ones on the diagonal and hence corresponds to the cross-sectional correlation matrix between factors. Beware that because the optimization program of Eq. (2.6) uses the probability distribution of the portfolio’s weights, all the expectations, and hence also covariances, in this section are computed with this distribution.

Definition (A.1) has the useful consequence that the problem of Eq. (2.6) can be restated as

$$E (\varvec{r}| {\bar{\varvec{z}}} \ge c) = \beta E ({\bar{\varvec{z}}}| {\bar{\varvec{z}}} \ge c).$$

Another important observation is that the probability distribution of \({\bar{\varvec{z}}}\) is normal, though not standard. This means that if we define c so as to select a fixed percentage of the universe,Footnote 1 then we have that \(E ({\bar{\varvec{z}}}| {\bar{\varvec{z}}} \ge c) \propto \sqrt{{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}}\). Very importantly, this last equality is true only when the scores \({\bar{\varvec{z}}}\) do not interact with the probability distribution of portfolio weights p. The previous observations allow us to reduce the optimization program to

$$\max _{{\varvec{\lambda }}}\min _{\varvec{\pi } \in Q} E (\varvec{r}| {\bar{\varvec{z}}} \ge c) \equiv \max _{{\varvec{\lambda }}}\min _{\varvec{\pi } \in Q}\frac{\varvec{\pi }^T \Omega {\varvec{\lambda }}}{\sqrt{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}}.$$

This form is reminiscent of a Sharpe ratio. It is also scale invariant in the weights, which means that multiplying the weights by a number will not affect the solution.

The solution to Eq. (A.3) is obtained by first inverting the order of optimization:

$$\max _{{\varvec{\lambda }}} \min _{\varvec{\pi } \in Q} \frac{ \varvec{\pi }^T \Omega {\varvec{\lambda }}}{\sqrt{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}} =\min _{\varvec{\pi } \in Q} \left[ \max _{{\varvec{\lambda }}}\frac{\varvec{\pi }^T \Omega {\varvec{\lambda }}}{\sqrt{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}} \right]$$

This is indeed allowed by the convexity properties of the objective function. The maximization program inside the brackets on the right-hand side is then found to be equivalent to:

$$\begin{array}{ll} \underset{\varvec{y}}{\min }&\sqrt{\varvec{y}^T \Omega \varvec{y}}\\ &\varvec{\pi }^T \Omega \varvec{y} = 1 \\ &t {\varvec{\lambda }}= \varvec{y} \end{array}$$

Here the variable t is a dummy variable whose purpose is to fix the first equality. It also translates the fact that the solution will be leverage invariant. The lagrangian of this problem is

$$\mathcal {L}(\varvec{y},\lambda ) = \sqrt{\varvec{y}^T \Omega \varvec{y}} + \lambda \left( \sqrt{\varvec{\pi }^T \Omega \varvec{y}}- 1\right),$$

and therefore, solving for \(\partial _{\varvec{y}}\mathcal {L} = 0\) reveals that the solution to Eq. (A) is \(t {\varvec{\lambda }} = \varvec{\pi }\). This allows us to transform the problem (A.4) into its final expression:

$${\varvec{\lambda }}^*= {\mathop {{{\text{arg max}}}}\limits _{{{\varvec{\lambda }}}}} {\mathop {\min }\limits _{{\varvec{\pi } \in Q}}} \frac{ \varvec{\pi }^T \Omega {\varvec{\lambda }}}{\sqrt{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}} = {\mathop {{{\text{argmin}}}}\limits _{{{\varvec{\lambda }} \in Q}}} \sqrt{{\varvec{\lambda }}^T \Omega {\varvec{\lambda }}}$$

Since both objective function and solution set are convex, this problem has a unique solution. Convexity also insures straightforward numerical solving. For a number of simple definitions of Q, fully analytical solutions are readily available. For instance, with the condition that \(\sum _{i=1}^M \pi _i \ge 0\), the solution corresponds to the minimum correlation weights. As we have mentioned earlier, the above solution may not be optimal should the distribution of weights in the final portfolio interact with the scores \({\bar{\varvec{z}}}\). This is the case, e.g. when the “small-size” factor is used in a market cap-based allocation. Ways to mitigate the impact of such interactions on the optimality of the solution are discussed in the text.

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Vaucher, B., Medvedev, A. Efficient integration of risk premia exposures into equity portfolios. J Asset Manag 18, 538–546 (2017).

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