Robust optimization of time series momentum portfolios.
| Autor | Fague, Jeremy |
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Introduction
For as long as investors have sought capital appreciation by purchasing equities, they have developed strategies to capture returns, thereby compensating them for accepting different sources of risk. No matter the strategy, investors seek an optimal method for determining how to allocate their capital among various investment choices.
In his seminal paper, Markowitz (1972) describes the process of utilizing information about the expected returns, variances, and covariances of financial assets to construct a portfolio, making optimal trade-offs between risk and return. With n assets, the basic problem of portfolio optimization is
[mathematical expression not reproducible]
where [alpha] [member of] [R.sup.n] is a vector of expected returns, [SIGMA] [member of] [R.sub.nxn] is the covariance matrix of assets, w are the portfolio weights, and [r.sub.E] is a target expected return.
While the approach is sensible from a theoretical perspective, in practice the optimization is extremely sensitive to the inputs. Slight changes in the expected returns [alpha] or the covariance matrix [SIGMA] can result in drastically different portfolios. Thus any investor, using mean-variance optimization to decide allocations to a large number of assets, will likely experience substantial instability in the optimal solution to the Markowitz problem. Further, if the first and second moments of the data differ in the future from what they have been in the past, the parameters are misspecified. The resulting portfolio may have quite different realized returns and volatility than expected. This issue calls for the application of methods to make the solution robust to small changes in the inputs. In particular, regularization provides interesting possibilities. In this paper, we provide a new methodology based on an [l.sub.1] regularization of portfolio weights towards the weights obtained on a previous day. We show that this method substantially reduces turnover and obtains high out-of-sample Sharpe ratios, even in the presence of transaction costs.
The remainder of the paper is organized as follows. In Section 2 we discuss the related literature. In Section 3, we describe the all the data adopted, while in Section 4 we introduce our portfolio regularization method. Section 5 presents the empirical results, comparing our shrinkage estimator to the traditional mean-variance estimator with short-sale constraints suggested by Brodie et al. (2009). Section 6 offers conclusions and potential topics for future research.
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Literature Review
Black and Litterman (1992) develop one of the earliest methods for stabilizing mean-variance optimization. Their approach relies on CAPM, determining the equilibrium expected returns based on market-clearing holdings of all market participants, and allowing individual investors to add their own views to the model to construct a "tilt" away from the market portfolio. This model effectively shrinks the expected returns used in mean-variance optimization toward a market-based equilibrium view, generating robustness to investors' specifications of expected returns.
As regularization methods have become ubiquitous in statistics, machine learning, and other applications, a rich body of work on the benefits of applying regularization methods to mean-variance optimization has developed. Jagannathan and Ma (2003) demonstrate that imposing a nonnegativity constraint on the portfolio weights (essentially constraining the 1-norm of the portfolio weights to be 1) results in comparable performance to other regularization methods. Similarly, Brodie et al. (2009) impose a 1-norm penalty on the portfolio weights in the objective function--encouraging sparsity and allowing for negative weights, but not large weights--and find that the resulting portfolios outperform the equally-weighted portfolio in terms of the Sharpe ratio. DeMiguel et al. (2009) examine the effects of imposing a number of different regularizations of the portfolio weights, including the 1-norm, 2-norm, and covariance-matrix norm, and find that the norm-constrained portfolios often have impressive Sharpe ratios compared to benchmarks, including the portfolio formed using the methodology of Jagannathan and Ma (2003).
Most similar to our method is the regularization by Kourtis (2015), in which the portfolio return constraint is [[alpha].sup.T] w - [kappa][[parallel]w - [w.sub.0][parallel].sub.1] = [r.sub.E], where [KAPPA] is the transaction cost and [w.sub.0] are the portfolio weights before rebalancing. With a 1% transaction cost, turnover levels were leveled with those of an equal-weighted portfolio; however, in terms of the Sharpe ratio, this regularized portfolio was unable to consistently beat the equal-weighted portfolio.
The key difference between the method suggested in this paper and that of Kourtis (2015) is the treatment of the turnover regularization. Kourtis (2015) incorporates transaction costs directly into the expected-return constraint so that any trading which does not bring the after-transaction-cost return below the constraint is allowed. Thus he explicitly informs the maximum amount of turnover that is allowed each day by considering the economics of trading. In contrast, the method of this paper does not directly constrain transaction costs to be below a certain level. Turnover is penalized in the objective function, with the magnitude of penalization informed by the data via cross-validation. The range of potential values for the parameter that controls penalization ([lambda]) includes levels which are similar to transaction costs for...
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