The last five days! On Day 25, we compared the peformance of the adjusted vs. unadjusted strategy for different prediction scenarios: true and false positives and negatives. For true positives and false negatives, the adjusted strategy performed better than the unadjusted. For true negatives and

On Day 24, we explained in detail how the error correction term led to somewhat unexpected outperformance relative to the original and unadjusted strategies. The reason? We hypothesized that it was due to the the error term adjusting the prediction in a trending direction when or if the current

On Day 23 we dove into the deep end to understand why the error correction we used worked as well as it did. We showed how traditional machine learning uses loss functions and then hypothesized how our use helped improve predictions through its effect on the correlation of the signs of the

On Day 22 we saw a meaningful improvement in our strategy by waiting an additional week to quantify model error and then using that error term to adjust the prediction on the most recently completed week of data. What was even more dramatic was comparing this improved strategy to one that followed

On Day 21, we wrung our hands with frustration over how to proceed. The results of our circular block sampling suggested we shouldn’t expect a whole lot of outperformance in our 12-by-12 model out-of-sample. To deal with this our choices were, back to the drawing board or off to the waterboard to

On Day 20 we completed our analysis of the 12-by-12 strategy using circular block sampling on the 3 and 7 blocks. We found the strategy did not outperform buy-and-hold on average and its frequency of outperformance was modest – in the 28-31% range – insufficient to warrant actually executing the

On Day 19, we introduced circular block sampling and used it to test the likelihood the 200-day SMA strategy would outperform buy-and-hold over a five year period. We found that the 200-day outperformed buy-and-hold a little over 25% of the time across 1,000 simulations. The frequency of the

On Day 18 we started to discuss simulating returns to quantify the probability of success for a strategy out-of-sample. The reason for this was we were unsure whether or how much to merit the 12-by-12’s performance relative to the 200-Day SMA. We discussed various simulation techniques, but we

On Day 17 , we compared the 12-by-12 and 200-day SMA strategies in terms of magnitude and duration of drawdowns, finding in favor of the 200-day. We also noted that most of the contributors to the differences in performance were due to two periods at the beginning and end of the period we were

On Day 16, we showed the adjusted 12-by-12 strategy with full performance metrics against buy-and-hold, the 60-40 SPY-IEF ETF portfolio, and the 200-day SMA strategy. In all cases, it tended to perform better than the benchmarks. However, against the 200-day SMA that performance came primarily at

On Day 15 we adjusted our model to use more recent data to forecast the 12-week look forward return. As before, we used that forecast to generate a trading signal that tells us to go long the SPY if the forecast is positive, and exit (or short for the long-short strategy) if otherwise. We saw this

On Day 14 we showed how the trading model we built was snooping and provided one way to correct it. Essentially, we ensure the time in which we actually have the target variable data aligns with when the trading signals are produced. We then used the value of the next time step to input into the

Guess what? The model we built in our last post actually suffers from snooping. We did this deliberately to show how easy it is to get mixed up when translating forecasting models into trading signals. Let’s explain. Our momentum model uses a 12-week cumulative return lookback to forecast the next

Unlucky 13! Or contrarian indicator? There’s really nothing so heartwarming as magical thinking. Whatever the case, on Day 12 we iterated through the 320 different model and train step iterations to settle on 10 potential candidates. Today, we look at the best performing candidate and discuss the

In Day 11, we presented an initial iteration of train/forecast steps to see if one combination performs better than another. Our metric of choice was root mean-squared error (RMSE)1 which is frequently used to compare model performance in machine learning circles. The advantage of RMSE is that it is

On Day 10, we analyzed the performance of the 12-by-12 model by examining the predicted values and residuals. Our initial takeaway suggested the model did seem not overly biased or misspecified in the -10% to 10% region. But when it gets outside that range, watch out! We suspected that there was

On Day 9 we conducted a walk-forward analysis on the 12-by-12 week lookback-look forward combination. We then presented the canonical the actual vs. predicted value graph with a 45o line overlay to show what a perfect forecast would look like. Here’s the graph again. As noted previously, we

Yesterday we finished up our analysis of the regression models we built using different combinations of lookback and look forward momentum values. Today, we see if we can generate good forecasts using that data. If you’re wondering why we still haven’t tested Fibonacci retracements with

Yesterday, we discussed the size effects, their statistical significance (e.g., p-values), and some other summary statistics for the various momentum combinations – namely, 3, 6, 9, and 12 week lookback and look forward returns. We found that size effects were small, but a few were significant,

Welcome to the last day of the first week of 30 days of backtesting! We hope you’re enjoying the ride. If you have any questions or concerns, you can reach us at the contact details listed at the bottom of this post. On Day 6 we defined momentum rather roughly and ran a bunch of tests to identify

Yesterday we examined the eponymous Fama-French factors to see if we could find something that will help us develop an investment strategy to backtest. It turned out the best performing factor was the market risk premium, which is essentially the return to the market in excess of the risk-free rate.

The day has finally arrived! Time to start backtesting! We’ve always wanted to test how Fibonacci retracements with Bollinger Band breakouts filtered by Chaikin Volatility would perform while implementing rolling stop-loss updates based on the ATR scaled by the 7-day minus 5-day implied volatility

We’re four days in and you’re probably wondering when are we actually going to start backtesting?! The answer is that while it is natural to want to rush to the fun part – the hope and elation of generating outsized returns and Sharpe Ratios greater than 2 – the reality is getting the

Yesterday we investigated the effect of using the 200-day simple moving average (200SMA) as a proxy for a rules-based investing method. The idea was to approximate what a reasonably rational actor/agent might do in addition to the buy-and-hold approach. When folks talk about research, backtesting,

On Day 1, we decided on a few benchmarks to use for our backtest. That is, a 60-40 and 50-50 weighting of the SPY and IEF ETFs. What we want to add in now is the Hello World version of trading strategies – the 200-DAY MOVING AVERAGE! Why are we adding this to our analysis? As we pointed out

Yesterday we set out our plan to backtest a strategy using the SPY ETF, which tracks the S&P 500. Before we commence, we obviously need to establish a baseline. What metrics will we use to assess the strategy? How will we define success? What benchmarks will we use? Typically, for a single asset

Summer has a way of getting away from you. That is as much relevant for blog writing as it is for life. Nonetheless, before summer ends we wanted to dust off our series on regime prediction and close the loop on the remaining techniques we had yet to investigate. That is, in our last post we

Our last post finished up examining the three different methods used to predict market regimes in the Gold Miners ETF, GDX – namely, clustering, Gaussian Mixture Methods (GMMs), and Hidden Markov Models (HMMs). We found GMMs performed the best in terms of proof-of-concept. But there was a lot of

We conclude our discussion of market regime detection by examining Hidden Markov Models (HMMs). Recall this series was inspired by a post from PyQuant News that highlighted a longer article from the London Stock Exchange Group (LSEG). Those who took the CFA exams probably forgot using HMMs in the

Our previous post, used hierarchical clustering to identify market regimes in the gold miners ETF, GDX. This was inspired by a post from PyQuant News that highlighted a longer article from the London Stock Exchange Group (LSEG). In this post, we’ll continue looking at identifying market regimes

We recently saw a post from PyQuant News that piqued our interest, compelling us to dust off the old blog files and get back into the saddle. The post highlights a longer article from the London Stock Exchange Group (LSEG) on how to use different machine learning models to identify and forecast

We’re taking a short break from neural networks to return to portfolio optimization. Our last posts in the portfolio series discussed risk-constrained optimization. Before that we examined satisificing vs. mean-variance optimization (MVO). In our last post on that topic, we simulated 1,000

For fundamental equity investors, the financial statement is the launchpad for the search for value. True, quants use financial statements too. But they spend less time on what the numbers mean, than on what they are. To produce a financial statement that adequately captures the economic (not GAAP

It’s been over a month since our last post and for that we must apologize. We endeavor to be more prolific, but sometimes work and life get in the way. On the work front, let’s just say we won’t have to spend as much time selling encyclopedias door-to-door, which should free up more time to

Our last post examined the correspondence between a logistic regression and a simple neural network using a sigmoid activation function. The downside with such models is that they only produce binary outcomes. While we argued (not very forcefully) that if investing is about assessing the probability

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.

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

Our last post parsed portfolio optimization outputs and examined some of the nuances around the efficient frontier. We noted that when you start building portfolios with a large number of assets, brute force simulation can miss the optimal weighting scheme for a given return or risk profile. While

Our last few posts on risk factor models haven’t discussed how we might use such a model in the portfolio optimization process. Indeed, although we’ve touched on mean-variance optimization, efficient frontiers, and maximum Sharpe ratios in this portfolio series, we haven’t discussed portfolio

Risk factor models are at the core of quantitative investing. We’ve been exploring their application within our portfolio series to see if we could create such a model to quantify risk better than using a simplistic volatility measure. That is, given our four portfolios (Satisfactory, Naive, Max

In our last post, we looked at using a risk factor model to identify potential sources of variance for our 30,000 portfolio simulations. We introduced the process with a view ultimately to construct a model that could help to quantify, and thus mitigate, sources of risk beyond a simplistic

We’re returning to our portfolio discussion after detours into topics on the put-write index and non-linear correlations. We’ll be investigating alternative methods to analyze, quantify, and mitigate risk, including risk-constrained optimization, a topic that figures large in factor research.

In our last post, we took our analysis of rolling average pairwise correlations on the constituents of the XLI ETF one step further by applying kernel regressions to the data and comparing those results with linear regressions. Using a cross-validation approach to analyze prediction error and

In our last post, we looked at a rolling average of pairwise correlations for the constituents of XLI, an ETF that tracks the industrials sector of the S&P 500. We found that spikes in the three-month average coincided with declines in the underlying index. There was some graphical evidence of a

We recently read two blog posts from Robot Wealth and FOSS Trading on calculating rolling pairwise correlations for the constituents of an S&P 500 sector index. Both posts were very interesting and offered informative ways to solve the problem using different packages in R: tidyverse or xts.

We’re taking a break from our portfolio series and million sample simulations to return to a subject that we haven’t discussed of late despite its featured spot in this blog’s name—options. In this post, we’ll look at the buy-write (BXM) and put-write (PUT) indices on the S&P 500, as

In our last post, we ran simulations on our 1,000 randomly generated return scenarios to compare the average and risk-adjusted return for satisfactory, naive, and mean-variance optimized (MVO) maximum return and maximum Sharpe ratio portfolios.1 We found that you can shoot for high returns or high

In our last post, we explored mean-variance optimization (MVO) and finally reached the efficient frontier. In the process, we found that different return estimates yielded different frontiers both retrospectively and prospectively. We also introduced the concept of satsificing, originally developed

In our last post, we ran through a bunch of weighting scenarios using our returns simulation. This resulted in three million portfolios comprised in part, or total, of four assets: stocks, bonds, gold, and real estate. These simulations relaxed the allocation constraints to allow us to exclude

Our last few posts on portfolio construction have simulated various weighting schemes to create a range of possible portfolios. We’ve then chosen portfolios whose average weights yield the type of risk and return we’d like to achieve. However, we’ve noted there is more to portfolio