In our last post, we introduced neural networks and formulated some of the questions we want to explore over this series. We explained the underlying architecture, the basics of the algorithm, and showed how a simple neural network could approximate the results and parameters of a linear regression. In this post, we’ll show how a neural network can also approximate a logistic regression and extend our toy example.
What’s the motivation behind showing the link with logistic regression?

We start a new series on neural networks and deep learning. Neural networks and their use in finance are not new. But are still only a fraction of the research output. A recent Google scholar search found only 6% of the articles on stock price price forecasting discussed neural networks.1
Artificial neural networks, as they were first called, have been around since the 1940s. But development was slow until at least the 1990s when computing power rapidly increased.

Our last post discussed using the discounted cash flow model (DCF) as a method to set return expectations that one would ultimately employ in building a satisfactory portfolio. We noted that if one were able to have a reasonably good estimate of the cash flow growth rate of an asset, then it would be relatively straightforward to calculate the required return.
The problem, of course, is figuring out what the cash flow growth rate should be.

The CBOE’s SKEW index has attracted some headlines among the press and blogosphere, as readings approach levels not see in the last year. If the index continues to draw attention, doomsayers will likely say this predicts the next correction or bear market. Perma-bulls will catalogue all the reasons not to worry. Our job will be to look at the data and to see what, if anything, the SKEW divines. If you don’t know what the SKEW is, we’ll offer a condensed definition.

When we were testing random correlations and weighthings in our last post on diversification, we discovered that randomizing correlations often increased portfolio risk. Then, when we randomized stock weightings on top of our random correlations, we began to see more cases in which one would have better off not being diversified. In other words, the percentage of portfolios whose risk exceeded the least risky stock began to rise. By chance, the least risky stock (in terms of the lowest volatility), also happened to enjoy the highest risk-adjusted return, so our random selection of stock returns might be a bit anomalous.

In our last post, we took a detour into the wilds of correlation and returned with the following takeaways:
Adding assets that are not perfectly positively correlated to an existing portfolio tends to lower overall risk in many cases.
The decline in risk depends a lot on how correlated the stocks are in the existing portfolio as well as how the additional stocks correlate with all the existing assets.

In our last post, we asked the simple question of whether an investor is better off being diversified if he or she doesn’t know in advance how a stock is likely to perform. We showed some graphs that suggested diversification lowered risk (or, more precisely, volatility), but this came at the expense of accepting less than maximal returns. We then showed that a diversified portfolio was able to produce better risk-adjusted returns on 8 out of 10 of the stocks we had randomly generated.