Objective function

To acknowledge all the diffculties (i)–(iv) reviewed in Section 4.2.1 and con- nected to the two-step approach around (linear) predictive regressions, we introduce a fexible non-parametric and nonlinear approach which is strictly building upon utility maximization also in empirical implementation to obtain weights wt. We are thus integrating asset allocation decision making and ma- chine learning via a customized gradient boosting algorithm with specifc and important modifcations over the mechanical use of existing machine learning algorithms. Section 4.2.3 presents details of the ‘utility boosting’ algorithm, with various favourable properties.

Our utility boosting approach follows the general lines and arguments made by Brandt (1999), Aït-Sahalia and Brandt (2001), Brandt and Santa-Clara (2006) and Brandt et al. (2009) in the early literature, arguing the importance of

3 _{Sentana (2005) considers formal conditions and assumptions under which least squares-}

based predictions and mean-variance analyses are connected in associated market timing strategies.

direct portfolio decisions instead of the two-step ‘plug-in’ statistical approach. Along their studies, we are not explicitly relying on specifc assumptions on the excess stock return data generating mechanism, such as the arbitrage pricing theory (APT) or the capital asset pricing model (CAPM). Instead, our (empirical) view is that the portfolio weight wtis a direct, potentially highly nonlinear, function of state variables xt−1 maximizing the investor’s utility.

This linkage incorporates all the predictive information contained in xt−1to

determine the weights, irrespective of the conclusions on statistical mean return predictability as discussed in (ii) and (iii) in Section 4.2.1, including also the possible impact of higher conditional moments than just mean and variance.

On these past utility-based approaches, the closest to our approach seems Brandt and Santa-Clara (2006) where they parametrize the portfolio weight as a linear function of predictors (state variables) xt−1 and solve the opti-

mal values of the present parameters maximizing expected quadratic utility function similar to (4.2). Their approach is empirically, however, much more restrictive than ours (see details in the Appendix B) and designed more closely on portfolio choice problems with a genuine cross-sectional dimension as well (i.e. multiple risky assets). Moreover, their resulting portfolio weights can be interpreted as being proportional to the standard OLS regression of a vector of ones on the excess returns and, importantly, additional subsequent constraints to maintain the bounds (4.8) are required to address the point (i) in Section 4.2.1.4

To set a tractable empirical counterpart of (4.2), we frst convert the max- imization problem to a minimization problem. We will train our boosting algorithm with the same training (estimation) data {xt−1, re,t}Tt=1(including

also the initial values for the volatility proxy) as in the least squares-optimal statistical two-step approach in equations (4.4)–(4.7) where re,timplicitly con- tains required information on both rm,tand rf,t. Our utility-based empirical

4 _{In addition to the OLS interpretation of the portfolio weights as obtained in Brandt}

and Santa-Clara (2006), Brandt (1999) and Brandt et al. (2009) build upon on the (statistical) method of moments and ‘maximum utility estimator’, respectively. At the end, they also em- phasize statistical hypothesis testing with the aim to obtain ‘statistically signifcant’ results in the evaluation stage, whereas our perspective is different and specifcally in prediction (forecasting) performance after direct utility-based modelling.

objective function (cf. (4.2)) is ( X T ) ( _T ) X 1 1 1 ₂ σ2

argmin −ut =argmin − rp,t− γ wt t , (4.10) wt T _t=1 wt T _t=1 2

where rp,tdenotes the resulting portfolio returns (see (4.1)). The (negative) utility contribution of the tth observation, −ut, is crucially dependent on the weights wtconstructed with the information contained in xt−1. The selected

volatility proxy σ_t2is already introduced in connection to (4.7): It is the same as in (4.10) and in the two-step approaches throughout this study for comparison reasons (cf. a different formulation in this respect in Brandt and Santa-Clara (2006)). The form (4.10) is the one examined in various return predictability studies as an evaluation diagnostic tool measuring portfolio performance (see Marquering and Verbeek, 2004; Campbell and Thompson, 2008; Rapach et al., 2010; Rossi, 2018) and hence it acts as a natural choice for our advancement.

For simplicity and in accordance with past closely related studies, through-

min

out this study, we set the lower bound in (4.8) as w = 0 and hence excluding

max

short selling.5 Moreover, to respect the pre-determined maximum weight w

explicitly as a part of our procedure (cf. the subsequent weight truncation needed in (4.7)), we set

max_λ

t,

wt=w (4.11)

where the portfolio weight is essentially specifed by the logistic function

λt= 1 . (4.12) 1 1+exp −_{γ σ}2F(xt−1) t

Combined with (4.11), the logistic growth curve form (4.12) guarantees that the weights (4.11) are all the time inside the interval wt∈[0, wmax]. In (4.12), the essential ingredient to determine portfolio weights (cf. (4.4)) is the component

F(xt−1), which is possibly a complex function of the predictive information

xt−1that we aim to teach with our training data. To this end, in Section 4.2.3,

we specifcally develop a customized version of the gradient boosting, which is in principle only one but seemingly highly relevant empirical algorithm over

5 _{An extension allowing also for short selling (i.e. negative weights) is possible but means}

alternatives to determine F(xt−1) when aiming to maximize the empirical

utility.

In specifcation (4.12), the impact of F(xt−1) is adjusted by the risk proxy

σt2and the risk aversion coeffcient γ along the expression (4.3).6 We can also strengthen the linkage to (4.7) by the following ‘payoff to stake’ representation

_max λt λt w F(xt−1) log = log = . wmax_(1−λ t) 1−λt γ σt2

This is the log odds ratio of λtwhere higher λtrefects the likelihood of high portfolio weight should take place. In addition to this representation, the inclusion of the volatility proxy σ2t in (4.12) is strongly motivated by the early empirical evidence of Fleming et al. (2001) and Marquering and Verbeek (2004) on the importance of an explicit volatility component to determine portfolio weights.

In practice, the utility boosting algorithm, to be presented more detail in Section 4.2.3, provides the practical method to determine the weights. It can intuitively be interpreted so that an investor, with given risk aversion preferences and constraints (4.8), aim to continuously optimize and update his or her asset allocation decision mechanism with the available past predictive information targeting to fnd optimal portfolio weight for the next period. This emphasizes the training step of our algorithm before predicting the weight for the next period out of sample. This thinking again somewhat diverges from the goals of Brandt and Santa-Clara (2006), and a few closely related studies, concentrating on full sample portfolio policies.