The main objective of our research was to find a model that could forecast or, at least, acknowledge the presence of a bear market. In order to estimate such a model, the first necessary step is to date past bear and bull markets. This can be done either through a parametric approach (Markov Switching Models) or through a non-parametric one. We chose to focus on the latter.
The non-parametric approach largely revolves around the algorithm developed by Bry and Boschan (1971). It was originally developed for and applied to the detection of business cycles, in particular for quantitatively replicating the contractions and expansions determined by the National Bureau of Economic Research (NBER). This computer program recognizes the patterns in the time series, detaches these patterns according to a sequence of rules, and locates the turning points (peaks and troughs) in the series. Following the business cycle literature, we assume that the duration of a complete cycle from the trough to the next trough (or alternatively peak to peak) must be at least 15 months. In addition, the time spent in a bear market (time from the peak to the next trough) or bull market (trough to peak) must be at least six months. Once identified the turning points we can build a binary time series where the value one signifies a bear market state.
The model we are going to estimate is called Probit Model and is a particular type of regression where the dependent variable can only take binary values. Given as the binary dependent variable and the regressors’ matrix respectively, we assume the model takes the form where P denotes probability, and is the Cumulative Distribution Function (CDF) of the standard normal distribution. The parameters are typically estimated by maximum likelihood. Therefore, after having estimated the parameters, we can make forecasts on the state variable Y for the next periods.
After a review of the current literature we decided to select as regressors the most significant ones: the 1-period lagged value of Y, the previous period log-return of the stock market and a macroeconomic indicator, which is the Term-Spread (i.e. the difference between the 10yr treasury and the 3months T-bill).
We analyzed 50 years of monthly S&P500 returns. As a first step, we divided the dataset in two subsets and then we estimated the model on the first half of the series and dedicated the second half of the series to out-of-sample forecasts. Forecasts are constructed using an expansive window of observations where the data from the start of the dataset through to the present forecast time are used in estimation to obtain a new forecast. This procedure is repeated until the end of the sample.
In the chart (fig.1) you can see, in the blue line, the forecasted probability of having a bear market the next month plotted against the S&P500 index.
Fig.1 Probability forecast of a bear market (blue line) against S&P 500 (red line) (Source: yahoo.finance)
As you can see, besides some strange solitary extreme values, the forecast successfully predicted the 2 major bear markets of the last decades: the dot-com bubble burst and the 2008 financial crisis. Unfortunately, the model forecasts lag a few months behind the start of every bear market and this is due to the nature of the Bry Boschan algorithm, which requires data points before and after the peak (or troughs) in order to identify it and therefore it takes a few months to identify the most recent turning point. Nonetheless, it still turns out to be quite useful in predicting severe market crashes as stated before.
We developed a market timing strategy which is fully invested in the S&P500 each month whose next-month probability forecast of a bear market is below a certain threshold and fully invested in cash otherwise. In order to avoid unnecessary transactions due to single data point spikes in our probability forecasts we added another constraint to the strategy: it can unwind the stock position only if the probability of a bear market has been over the threshold for 3 consecutive months.
In the chart below (fig.2), you can see the growth of a 100$ capital invested in a buy and hold strategy (red line) vs our market timing strategy (blue line).
Fig.2 Growth of 100$ invested in buy and hold strategy (red line) vs market timing strategy (blue line) (Source: yahoo.finance; fred.stlouisfed.org)
As you can see, the main feature of our market timing strategy is the ability to avoid consistent losses during the most severe market crashes. And this is the advantage that on the long run makes it overperform greatly the buy and hold strategy. The Sharpe Ratio of this strategy is 0.646.
Although being a very basic strategy we believe it offers good results also considering that it has very low costs due to a very limited number of transactions (it is simply a buy and hold strategy that divests in anticipation of severe market crashes). Furthermore, there is ample room for improvement by adding some more complex feature to the trading strategy. For instance, a bear-market-probability dependent leverage.
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