# As Goes January?

Despite recent instability, some are predicting that 2007 will be an up year for the stock market. They are basing it on the adage “as goes January, so goes the year” – the so-called January Barometer or January Rule. And January 2007 was, after all, an up month for the S&P 500.

But can we really trust this rule of thumb?

I’m always in a bit of a pickle when it comes to doing simulations on historical stock market data. On one hand, it’s nice to go back as far in time as you can, because you get more data points – or a larger sample size – meaning more accurate statistics.

But if I include market data from the 1950’s, can I be confident that it’s even relevant to how the market behaves today?

So I decided to examine the January Rule’s performance over the past 30 years. It’s a tiny sample size, I know, but seemed a good compromise between sample size and relevancy.

I used the S&P 500 index and only considered price gains (not dividends). In other words, I just looked Yahoo Finance’s historical closing prices for ticker ^GSPC.

You can find my Matlab script for this here.

Prob(Up January in last 30 years) = 0.633
Prob(Up Feb-to-Dec in last 30 years) = 0.767

Well, things are not looking good for seriously validating the January Rule already. First we see the famous January Effect. That is, January alone is usually an up month.

Further, more often than not both January and the 11-month period following are both up. So of course we’ll probably see that, on average, a January up correctly predicts the following February through December as also being up.

But if we’re going to use this rule as a market barometer, it should not only predict that February through December will be up given that January was up, it should also reliably predict that February through December will be down when January is down. Does it?

We’re going to need some conditinal probabilities to gain insight. Recall from you college statistics class that:

Prob(A | B) = Prob(A & B) / Prob(B)

Here are our Prob(A & B)’s for the last 30 years:

Prob(Up Jan & Up Feb-Dec for same year) = 0.533
Prob(Dn Jan & Dn Feb-Dec for same year) = 0.133

And now we’ll use the above to get our conditional probabilities:

Prob(Up Feb-Dec | Up Jan) = 0.533 / 0.633 = 0.842
Prob(Dn Feb-Dec | Dn Jan) = 0.133 / 0.367 = 0.362

Hmmm… Given an up January, there is an 84% chance the rest of the year will also be up. But if January is down, the January Rule is more often than not wrong about what will happen in the following 11 months!

In other words, even given that January was a down month, you still would have been better off betting that the remaining 11 months would turn out up.

Just for completeness, we should combine two above statistics to compute how often the January Rule was correct.

Prob(As goes January, so goes Feb-Dec) = 0.533 + 0.133 = 0.667

At first glance this might not appear too bad, but would you really want to use the above rule, which has only been right 2 times out of 3 when we have the even more reliable (cut-and-paste from above):

Prob(Up Feb-to-Dec in last 30 years) = 0.767

So take your pick: 67% of the time the January Rule has delivered. But 77% of the time the “February through December will be up and who cares about January” rule worked even better!

But there could be perhaps one saving grace of the January Rule. So far we’ve focused on “hard decision” results – that is, how often the rule just gets the market direction correct.

But we should also investigate what returns you would have achieved using the January Rule versus just always investing for February through December. Maybe the slightly less accurate Rule keeps you out of really bad years so that you come out ahead from an average rate of return (ROR) point of view?

Sadly the results don’t show that. Below are the geometric average annual returns for February through December for the last 30 years. You can use the January Rule two ways, one would be to sit out for the rest of the year when January closes down. The second is to short the rest of the year when January is down.

As you can see below, neither of these work as well as just always staying in the market from February-to-December, regardless of what January does.

Average ROR for Feb-to-Dec = 7.7%
Average ROR for Feb-to-Dec using Jan. Rule (long only) = 7.0%
Average ROR for Feb-to-Dec using Jan. Rule (long & short) = 6.2%

In conclusion, the market has more often than not increased from Feb-to-Dec [Prob(Up Feb-to-Dec) = 0.767]. It’s just been even more likely to increase if January was up [Prob(Up Feb-Dec | Up Jan) = 0.842]. But the odds have still been against you ever betting Feb-to-Dec would be down, regardless of what January did.

I’ll trade for the January Effect, and skip the January Rule.