*A paper from Stanford University has set out a model for determining equity prices using deep learning.*

In June 2019, Luyang Chen, Markus Pelger and Jason Zhu of Stanford University published their paper ‘Deep learning in asset pricing’. In the paper, the trio outline a model which they report allows them to understand the key factors that drive asset prices, to identify mispricing of stocks and to generate a mean-variance efficient portfolio, suggesting it has assembled a portfolio of assets of which the expected return is maximised for a given level of risk.

“Our optimal portfolio has an annual Sharpe Ratio of 2.6, we explain 8% of the variation in individual stock returns and explain over 90% of average returns for all anomaly sorted portfolios,” they wrote.

The paper proposes a new way to estimate asset pricing models for individual stock returns that can take advantage of the vast amount of conditioning information, “while keeping a fully flexible form and accounting for time-variation”.

To achieve this it uses a combination of three different deep neural network structures. It uses a feedforward neural network, a model in which information is not cycled back in to capture non-linearities, e.g. where a shift in the output is not tied to a change in the input.

It also uses a long short-term memory (LSTM )neural network architecture, which has feedback connections, to estimate a small set of economic state processes. This is a flexible approach which is well-suited to detect business cycles, based on the data provided.

It uses a generative adversarial network, which can be characterised as neural networks competing with each other in a game-theory model, to identify the portfolio strategies with the most unexplained pricing information.

“Our crucial innovation is the use of the no-arbitrage condition as part of the neural network algorithm,” the authors wrote. “We estimate the stochastic discount factor that explains all stock returns from the conditional moment constraints implied by no-arbitrage. Our stochastic discount factor (SDF) is a portfolio of all traded assets with time-varying portfolio weights which are general functions of the observable and macroeconomic variables.”

The authors report that their model allows them to understand the key factors that drive asset prices, identify mispricing of stocks and generate the mean-variance efficient portfolio.

“Our primary conclusions are four-fold,” they note.

Firstly, they demonstrate the potential of machine-learning methods in asset pricing, by identifying the key factors that drive asset prices and the functional form of this relationship on a level of generality and with an accuracy which they say that was not possible with traditional econometric methods.

Secondly, they claim to show and quantify the importance of including a no-arbitrage condition in the estimation of machine-learning asset pricing models. The arbitrage pricing theory is a model which predicts a return using the relationship between the expected return combined with macroeconomic factors. A no-arbitrage constraint, which is applied to the pricing of derivatives, assumes assets values can only improve by taking on more risk.

“The ‘kitchen-sink’ prediction approach with deep learning does not outperform a linear model with no arbitrage constraints,” wrote the authors. “This illustrates that a successful use of machine learning methods in finance requires both subject specific domain knowledge and a state-of-the-art technical implementation.”

The third aspect they noted was that financial data have a time dimension, which must be taken into account.

“Even the most flexible model cannot compensate for the problem if the data is inputted in the wrong format… We show that macroeconomic conditions matter for asset pricing and can be summarised by a small number of economic state variables, which depend on the whole dynamics of all time-series.”

Their fourth conclusion is that asset pricing is actually “surprisingly linear”. They note that as long as the portfolio manager looks at anomalies in isolation, the linear factor models provide a good approximation.

“However, the multi-dimensional challenge of asset pricing cannot be solved with linear models and requires a different set of tools,” they wrote.

The authors note that practical benefits for asset pricing researchers potentially “go beyond our empirical findings.”

First, they assert that they have provided a new set of benchmark test assets. New asset pricing models can be tested with portfolios sorted according to the risk exposure in the proposed model.

“These test assets incorporate the information of all characteristics and macroeconomic information in a small number of assets. Explaining portfolios sorted on a single characteristic is not a high hurdle to pass.”

Secondly, the authors provide a set of macroeconomic time series of hidden states that encapsulate relevant macroeconomic information for asset pricing which can be used as an input for new asset pricing models.

“Last but not least, our model is directly valuable for investors and portfolio managers,” they observe. “Given our estimates, the user of our model can assign a risk measure and its portfolio weight to an asset even if it does not have a long time series available.”

©TheScienceOfInvestment 2019