Neural network approach to portfolio optimization with leverage constraints: a case study on high inflation investment

Abstract

Motivated by the current global high inflation scenario, we aim to discover a dynamic multi-period allocation strategy to optimally outperform a passive benchmark while adhering to a bounded leverage limit. We formulate an optimal control problem to outperform a benchmark portfolio throughout the investment horizon. To obtain strategies under the bounded leverage constraint among other realistic constraints, we propose a novel leverage-feasible neural network (LFNN) to represent the control, which converts the original constrained optimization problem into an unconstrained optimization problem that is computationally feasible with gradient descent, without dynamic programming. We establish mathematically that the LFNN approximation can yield a solution that is arbitrarily close to the solution of the original optimal control problem with bounded leverage. We further validate the performance of the LFNN empirically by deriving a closed-form solution under jump-diffusion asset price models and show that a shallow LFNN model achieves comparable results on synthetic data. In the case study, we apply the LFNN approach to a four-asset investment scenario with bootstrap-resampled asset returns from the filtered high inflation regimes. The LFNN strategy is shown to consistently outperform the passive benchmark strategy by about 200 bps (median annualized return), with a greater than 90% probability of outperforming the benchmark at the end of the investment horizon.

Keywords:

• Cumulative tracking difference
• Leveraged portfolio
• Benchmark outperformance
• Asset allocation
• Machine learning

JEL Classifications:

• G11
• G22

AMS Codes:

• 91G
• 35Q93
• 68T07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The Ministry of Finance of Norway sets the allocation fraction between the equity index and the bond index. It gradually raised the weight for equities from 60% to 70% from 2015 to 2018.

2 Technically, at $t=t_0$, the manager makes the initial asset allocation, rather than a ‘rebalancing’ of the portfolio. However, despite the different purposes, a rebalancing of the portfolio is simply a new allocation of the portfolio wealth. Therefore, for notational simplicity, we include $t_0$ in the rebalancing schedule.

3 Intuitively, the dimensionality comes from tracking the allocation in the $N_a$ assets for both the active portfolio and benchmark portfolio when evaluating the changes in wealth of both portfolios over one period in the action-value function.

4 For example, the state space of problem (48(48) $\begin{array}{r}\left(\mathrm{CD}\right(\text{β}\left)\right):\underset{p\in \mathcal{A}}{inf}{\mathbb{E}}_p^{(t_0,w_0)}\left[{\int }_{t_0}^T{\left(W\right(t)-e^{\text{β}t}\hat{W}(t\left)\right)}^2\text{d}t\right],\end{array}$(48) ) with assumptions of a fixed-mix strategy is a vector in ${\mathbb{R}}^3$.

5 The distance is defined in (38(38) $\begin{array}{r}\underset{\mathit{x}\in \mathcal{X}}{sup}\Vert f_{\mathit{\theta }}\left(\mathit{x}\right)-p^{\ast }\left(\mathit{x}\right)\Vert <ϵ.\end{array}$(38) ), i.e. the supremum of the pointwise distance over the extended state space $\mathcal{X}$.

6 Note that the corresponding set of asset prices can be easily inferred from the set of asset returns, or vice versa.

7 See Pytorch Documentation.

8 For any functional $\psi \left(t\right)$, we use the notation $\psi \left(t^{-}\right)$ as shorthand for the left-sided limit $\psi \left(t^{-}\right)=\underset{\mathrm{\Delta t}\downarrow 0}{lim}\psi (t-\mathrm{\Delta t})$.

9 See Forsyth (2020) for the discussion on the empirical evidence for stock-bond jump independence. Also note that the assumption of independent jumps can be relaxed without technical difficulty if needed (Kou 2002), but will significantly increase the complexity of notations.

10 Note that we consider a two-asset scenario here, thus the scalar $p^{\ast }\in \mathbb{R}$ (allocation fraction for the stock index) fully describes the allocation strategy ${\mathit{p}}^{\ast }$, since ${\mathit{p}}^{\ast }=(p^{\ast },1-p^{\ast }{)}^{\mathrm{\top }}$.

11 Recall that we consider only two assets, a stock index and a bond index, with ${\overline{p}}_{\mathrm{\Delta }t}$ being the fraction of wealth in stocks.

12 The date convention is that, for example, 1926:1 refers to January 1, 1926.

13 More specifically, results presented here were calculated based on data from Historical Indexes, © 2022 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of Business. Wharton Research Data Services (WRDS) was used in preparing this article. This service and the data available thereon constitute valuable intellectual property and trade secrets of WRDS and/or its third-party suppliers. We also use the U.S. CPI index, also supplied by CRSP.

14 The 10-year treasury index was generated from monthly returns from CRSP back to 1941 (CRSP designation ‘b10ind’). The data for 1926–1941 are interpolated from annual returns in Homer and Sylla (1996). The 10-year treasury index is constructed by (a) buying a 10-year treasury at the start of each month, (b) collecting interest during the month, and then (c) selling the treasury at the end of the month. We repeat the process at the start of the next month. The gains in the index then reflect both interest and capital gains and losses.

15 The capitalization-weighted total returns have the CRSP designation ‘vwretd’, and the equal-weighted total returns have the CRSP designation ‘ewretd’.

16 see Appendix D.3 in Ni et al. (2023).

17 See Appendix I in Ni et al. (2023) for a more detailed discussion.

18 Alternative assets often employ explicit leverage strategies.

19 In fact, in this two-asset case, this assumption does not cause loss of generality.

20 For reference, based on the calibrated jump-diffusion model (49(49) $\begin{array}{rl}\frac{\text{d}{S}_i\left(t\right)}{S_i\left(t^{-}\right)}& =({\mu }_i-{\lambda }_i{\kappa }_i)\text{d}t\\ & +{\sigma }_i\text{ d}{Z}_i\left(t\right)+d\left(\sum _{k=1}^{{\pi }_i\left(t\right)}\right({\xi }_i^{\left(k\right)}-1\left)\right),i=1,2.\end{array}$(49) ) on historical high-inflation regimes, ${\mu }_1=0.051,{\mu }_2=-0.014,\vartheta =-0.00024$, and thus both inequalities are satisfied.

Additional information

Funding

Li's work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) [grant RGPIN-2020-04331]. Forsyth's work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) [grant RGPIN-2017-03760].