Integrated alpha modelling

Abstract

Alpha modelling typically refers to the selection and weighting of various information sources, which when combined are used by active portfolio managers to forecast security returns. It is traditionally seen as an exogenous input in the construction of the investment portfolio. Instead, a growing number of authors have recently argued that alpha modelling should be integrated within the portfolio-construction process to account for the active manager’s objective, constraints and transaction costs. Building in particular upon the frameworks of Sneddon (2008) and Qian et al (2007), we present a parsimonious and analytically tractable alpha modelling approach that aims at maximising the typical objective function of an active manager. Our modelling scheme combines several salient features of previous methodologies and explicitly identifies three critical components necessary in achieving the manager’s portfolio objectives: (i) the predictability of alpha factors as captured by their information decay; (ii) the contribution of alpha factors to portfolio risk and (iii) the autocorrelations of alpha factors in their contribution to portfolio turnover. Our methodology is able to provide various prescriptions relevant to the portfolio manager. For instance, as transaction costs increase, allocating more weight to signals with lower information decay is shown to improve portfolio value added. Factors with higher return-to-risk ratios are also given a higher prominence as they help allocate strategy risk more efficiently.

Appendices

Appendix A

The solution to the portfolio-construction problem

To solve the value-added maximisation in (3) we use the method of Lagrange multipliers, where the Lagrangian function is:

It is trivial to show that the solution to this problem satisfies the second-order conditions for a maximum, so that we concentrate below on the first-order conditions only. The first-order conditions for L are as follows:

Solving for h^t in (A.2), we get:

Substituting h^t in (A.3), we have:

Solving for λ₂, we obtain:

Finally, substituting λ₂ in (A.4) we find the following solution for the portfolio holdings:

In this equation I−X·(X′·Σ⁻¹·X)⁻¹·X′·Σ⁻¹ is an idempotent matrix and can be interpreted as the residual of a cross-sectional regression of α^t on X, weighted by the inverse of asset-specific volatility (1/σ _i ). Further, given our assumption of constant X, we have: (I−X·(X′·∑⁻¹·X)⁻¹·X′·∑⁻¹)·∑·h _i ^t−1=∑·h _i ^t−1−X·(X′·∑⁻¹·X)⁻¹)·X′·h _i ^t−1 = ∑·h _i ^t−1. As a result, the expression for the portfolio holding of asset i in (A.7) can be simplified as follows:

For clarity we further introduce , and write:

Appendix B

Simplification of the solution to the portfolio-construction problem

Equation (6) provides the solution to the value-added maximisation in (3):

We define f(p)=η^p/(λ+η)^p+1, and note that the sum of f(p) is a geometric series with common ratio (η/(λ+η)):

Equivalently:

So that when f(p∗+1) is negligible we have:

For ease of interpretation, we introduce δ(p), a scaled version of f(p), such that ∑_p=0^p∗δ(p)=1. The expression for the portfolio holdings in (B.1) may be written in terms of δ(p) as follows:

Appendix C

The impact of factor performance on portfolio return

The derivations in this section build on several insights from Sorensen et al (2004) and Qian et al (2007). The return of the portfolio at time t+1 is given by:

Using Equation (B.5) to replace h, we get:

Recall that the alpha in (C.2) is a transformed version of the original raw alpha in (1) that is orthogonal to X in (3), and scaled by asset-specific variance. Therefore, the alpha in equation (C.2) is a weighted average of m factors, as follows:

where the factors f _j are also transformed versions of the raw factors F _j in (1). The self-financing and market-neutrality conditions in X ensure that f _j has a mean value of zero and is orthogonal to the market beta of the securities. Instead of scaling the raw factors F _j so that f _j has unit standard deviation, we will see that it is convenient to standardise F _j so that stdev(f_{j, i}^t·σ _i )=1 Replacing alpha in (C.2) by expression (C.3), we have:

Given that the factors, f _j , are orthogonal to the market beta of the securities, the performance of the portfolio can be equally expressed in terms of asset-specific return (). Moreover, the self-financing condition implies that a convenient, if somewhat arbitrary, normalisation for can be used, namely: As a result, the portfolio return may be written as:

In (C.5), is the cross-sectional volatility of risk-adjusted asset-specific return (), which we assume constant over the test period. This assumption is not unrealistic since we have standardised the asset-specific return by specific risk. Moreover, as pointed out in the article, for a meaningful range of portfolio turnover penalties the impact of historical data should die out relatively rapidly. Therefore, in practice, the assumption of constant volatility of asset returns should not prove overly constraining. Furthermore, we define , where IC_{j, p}^t+1 is the IC of factor j, lagged by p, at time t+1. These ICs are a multi-period extension of the risk-adjusted ICs of Sorensen et al (2004). The risk-adjusted ICs strip out unwanted systematic risk exposures and accommodate for asset-specific risk. Finally, we introduce and write:

Appendix D

Portfolio expected return and the multi-period IC

The MIC of factor j is defined in (15) as follows:

We recall from Appendix B that δ(p) is a scaled version of f(p) such that:

Hence:

Given that the above is a geometric series, we have:

We define κ=η/(λ+η), with η and λ being the turnover penalty and the risk-aversion parameter of equation (3), and so we can rewrite (D.4) as:

Finally, using (C.6), the expected return of the portfolio can be written as follows:

Appendix E

Portfolio variance and the multi-period variance of IC

We assume that the variance of each factor IC is constant for all lags p (that is, var(IC_{j, p})=σ _j ²; and that the covariance between the ICs of different factors are negligible, so that cov(IC_{k, l}, IC_{j, m})=0 for all factors k≠j, and all possible lags.

Given these working assumptions, the only significant components for the risk of the portfolio are the autocovariance terms of the ICs, expressed in terms of factor lags. This information for factor j can be summarised as follows:

where V^j(l, m) is the autocovariance of IC _j at time l with its lag |l−m|; cov(IC_{j, l}, IC_{j, |l−m|}).

We define the MVIC for factor j as the variance of the weighted sum of lagged ICs across all horizons. We assume, as we do throughout the article, that the terminal horizon p∗ is chosen so that δ(p), and in turn the variance terms, δ(p)·IC _{j, p} , are negligible for values of p>p∗. This implies that:

Using (E.1) and (E.2), we can express in matrix form the MVIC for factor j as:

Making use of (C.6), (E.3) and the definition of κ in Appendix D, we can write the variance of the portfolio return as follows:

To evaluate the last component, D, in (E.4), we can show that the multiplication of the three elements in D will result in the summation of a set of infinite geometric series, with each series having a common ratio associated to powers of ρ _j . There will be one series associated with ρ _j raised to the power of zero:

While there will be two identical series for each ρ _j with powers>0:

and so on. Taking all the sums together, we have a geometric series that can be evaluated:

Therefore, the expression for the portfolio variance becomes:

or

One can use a similar approach to compute in (19).