### New input: previous day %return

Let’s refresh our thoughts on the day-of-the-week anomaly.

Here is the collection of studies on the weekend effect:

http://calendar-effects.behaviouralfinance.net/weekend-effect/

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The weekend effect (also known as the Monday effect, the day-of-the-week effect or the Monday seasonal) refers to the tendency of stocks to exhibit relatively large returns on Fridays compared to those on Mondays. This is a particularly puzzling anomaly because, as Monday returns span three days, if anything, one would expect returns on a Monday to be higher than returns for other days of the week due to the longer period and the greater risk.

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In our previous studies, we usually shorted the market on Monday, because of the bearish Monday effect. However, it is worth mentioning that it is not a must. In case the expected shorting profit is not greater then a money market fund daily %rate, it is not worth shorting the market. It is better to stay in cash, and collect the interest payment. It is especially true, because if we are out of the market (we are in cash) on the down Mondays, we earn the cash interest for 3 consecutive days (Sat, Sun, Mon). So, shorting the bearish Monday is only worth doing if the expected value is bigger than 3 times the daily interest on the treasuries. Therefore, in real life, we probably wouldn’t play shorting the market on Friday close. We will test this possibility in the future.

So far in our studies, we used only the day of the week as input. However, there are studies that investigate how the previous day %return affects the next day %return on specific days of the week.

1.

One study here from QUANTIFIABLE EDGES shows that in the 1 year from 2009 March to 2010 March, buying down Friday closes was profitable.

http://quantifiableedges.blogspot.com/2010/03/after-down-fridays-over-past-year.html

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It appears the edge has only been on down Fridays.

It is important to understand that this is what I often refer to as an “environmental edge”. In other words, it is something that has worked in the recent past and seems to be a result of the current market environment. It is not an edge that has persisted over a long period of time nor do I expect it to continue to persist for a long period of time from now. That doesn’t mean it isn’t a useful observation, though. In such cases where I believe a setup contains an environmental edge I will look to use it to my advantage until it appears to be losing its effectiveness.

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2.

Note that in the comment section of that article somebody contends that buying any Friday closes was profitable.

I have to agree with the commenter. The year 2009 was a very bullish day and the Bullish Monday effect was alive.

The commenter also note that this effect worked only on that specific period, in that bull regime. I cannot concur more.

3.

In the list from http://calendar-effects.behaviouralfinance.net/weekend-effect/, there is one from Abraham Abraham,

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=5358

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Abstract:

It is well known that stock returns, on average, are negative on Mondays. Yet, it is less well known that this finding is substantially the consequence of returns in prior trading sessions. When Friday’s return is negative, Monday’s return is negative nearly 80 percent of the time with a mean return of -0.61 percent. When Friday’s return is positive, the subsequent Monday’s mean return is positive, 0.11 percent. This relationship is stronger than for any other pair of trading days and is most acute in small- and medium-size companies. The trading behavior of individual investors appears to be at least one factor contributing to this pattern. Individual investors are more active sellers of stock on Mondays, particularly following bad news in the market.

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Consider 2 inputs for our Russell 2000 index (RUT) investigation from 1997 till November 2010: the day of the week on day T, the %return on day T and the output as the %return on day T+1.

Running the %return input from -2% to +2% by 0.1 increments, and averaging the next day return in the bins, we can plot in 3D this chart: (click the images for the full size version)

The separate slices are:

Monday:

Tuesday:

Wednesday:

Thursday:

Friday:

And here is the aggregation of the different slices together:

We know that the Friday has a bearish output (bearish Monday). This bearishness can be seen in the whole Friday spectrum. Note that close to the zero, there are many samples, but as we move away from zero, there are exponentially less samples (almost Gaussian distribution), so the statistics there is less reliable. These plots contain the average %return however, so the aggregated %gain is divided by the number of samples in the bucket. But be warned that the ANN doesn’t learn exactly this average %gains. In these plots, the samples in the middle are divided by a higher value (high number of samples fall there), while samples at the edge are divided by a lower value. So, the samples at the edge are artificially bumped up. It is given more weight than it deserves. That is OK for these plots, but the ANN training algorithm treats all samples equally, when calculating the Error. Nevertheless, it is sensible to plot these charts, because we can visualize these returns more clearly.

What we would like to point from this discussion that it is worth inspecting the samples close to the zero point. Take the Friday slice plot. The first 4 bars on the left to zero are negative bars, so in the grand average, it is very probably that considering all samples under zero, the grand average is negative. On the other hand, take the samples to the right of zero. You can find 3 positive bars. So, it is possible, that the grand average above zero will be positive.

Note also that as the input approaches +2, the output (next day %gain) become very negative. This is some kind of mean reversion (MR) that works here. If Friday is very-very much up, Mondays are extreme bearish.

It is more clear, if we divide the second input not to 21 buckets (from -2 to 2 by 0.1 increments), but only to 2 bins. The division is made on the borderline of zero. This is the plot:

It is clear instantly that the quote from the study is true

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When Friday’s return is negative, Monday’s return is negative nearly 80 percent of the time with a mean return of -0.61 percent. When Friday’s return is positive, the subsequent Monday’s mean return is positive, 0.11 percent.

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It is a surprise, because we use the RUT and the study mentioned used the SPX and we expect a slightly different behavour from different indices.

So Friday express a kind of follow through (FT), not mean reversion (MR) behaviour.

Similarly, conclusions can be deducted for other days, like Thursday. For example, if Thursday is down, the next Friday is usually very bullish, but if Thursday is up, the next Friday is slightly bearish. It is a kind of mean reversion (MR) behaviour.

Just for comparison, here is the %return when we aggregate everything along the day-of-the-week input:

Our conclusion in this post is that after studying the input output statistics, training the ANN on these 2 inputs seems to be promising. In the next posts, we will do exactly that.

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