Neural network copula portfolio optimization for exchange traded funds

Abstract

This paper attempts to investigate if adopting accurate forecasts from Neural Network (NN) models can lead to statistical and economically significant benefits in portfolio management decisions. In order to achieve that, three NNs, namely the Multi-Layer Perceptron, Recurrent Neural Network and the Psi Sigma Network (PSN), are applied to the task of forecasting the daily returns of three Exchange Traded Funds (ETFs). The statistical and trading performance of the NNs is benchmarked with the traditional Autoregressive Moving Average models. Next, a novel dynamic asymmetric copula model (NNC) is introduced in order to capture the dependence structure across ETF returns. Based on the above, weekly re-balanced portfolios are obtained and compared using the traditional mean–variance and the mean–CVaR portfolio optimization approach. In terms of the results, PSN outperforms all models in statistical and trading terms. Additionally, the asymmetric skewed t copula statistically outperforms symmetric copulas when it comes to modelling ETF returns dependence. The proposed NNC model leads to significant improvements in the portfolio optimization process, while forecasting covariance accounting for asymmetric dependence between the ETFs also improves the performance of obtained portfolios.

Keywords:

• Copulas
• Neural networks
• Portfolio optimization
• ETF

JEL Classification:

• G11
• G17

Notes

¹ The transaction costs for the three ETFs tracking their respective benchmarks do not exceed 0.5% per annum for medium size investors (see, for instance, www.interactive-brokers.com). Before the expansion of ETFs, traders had to pay a separate commission for each individual stock of an industry-specific portfolio. Now there are sector-specific ETFs, which allow traders to pay only one commission to buy or sell short an entire group of stocks.

² Similar results are obtained also in the in-sample period. In-sample results are not provided within text for the sake of space and are available upon request.

³ In order to consider a nonlinear benchmark, we also experiment with a Smooth Transition Autoregressive (STAR) model which is a nonlinear extension of autoregressive models. Nonetheless, the out-of-sample statistical and trading performance was found inferior to the ARMA specifications. As such, it is logical to retain for the portfolio optimization the less complex and better performing linear ARMA. Nonetheless, the STAR results are not included for the sake of space and are available upon request.

⁴ Equivalently, the optimal portfolio can be obtained by maximizing portfolio expected return for a given level of risk as measured by portfolio variance.

⁵ CVaR is the abbreviation of the Conditional Value-at-Risk, which is also known as the Expected Shortfall.

⁶ We use a rolling window instead of the full sample period and set a window size at 250 (one trading year) for all the data sets. We conduct rolling forecast by moving forward a day at a time and end with the forecast for 13/04/2015.

⁷ The Sortino ratio is a modification of the Sharpe ratio, but it only penalizes those returns falling below a user-specified target or required rate of return, while the Sharpe ratio penalizes both upside and downside volatility equally. Both ratios measure the risk-adjusted returns, but they frequently lead to differing conclusions as to the true nature of the investment's return.

⁸ For the sake of space, we present results of portfolios based on ARMA forecasts (benchmark) and PSN and RNN forecasts (best and second best NN model, respectively). MLP results are similar and are not presented for the sake of space.