# We Get New Highs More Frequently After New Highs

This post may fall into the “Dog bites Man” bucket, but I will see if I can’t shed a little more light on the phenomenon.  Here’s the question: “When do we see new highs in the stock market most often?”  The punchline: “After a recent new high.”

The red squares above show the probability of hitting a new high so many days after a new high.  The black line near it is a best fit power curve.  The blue diamonds above show the probability of hitting a new high so many days after not hitting a new high.  The green triangles above show the ratio of those two probabilities, matching up against the right vertical axis. The black line near it is a best fit power curve.

As time goes to infinity, both probabilities converge to the same number, which is presently estimated to be 6.8%, the odds that we would hit a new high on any day between 1951 and 2015.  Here’s the table that corresponds to the above graph:

 Probability of a new high after Days after no new high Days after new high Probability Ratio 1st day 3.1% 57.3% 18.29 2nd day 4.2% 43.3% 10.39 3rd day 4.6% 36.7% 7.90 4th day 4.8% 33.8% 6.99 5th day 5.1% 30.0% 5.87 6th day 5.2% 28.2% 5.37 7th day 5.5% 24.2% 4.36 8th day 5.7% 22.5% 3.97 9th day 5.6% 23.4% 4.18 10th day 5.6% 22.6% 4.00 11-15 5.9% 19.0% 3.22 16-20 6.0% 17.2% 2.86 21-30 6.1% 16.4% 2.71 31-40 6.2% 14.5% 2.35 41-50 6.2% 15.2% 2.47 51-60 6.3% 14.2% 2.28 61-75 6.3% 13.9% 2.21 76-90 6.3% 13.6% 2.16 91-105 6.3% 12.8% 2.02 106-120 6.4% 12.5% 1.96 121-140 6.4% 12.0% 1.87 141-160 6.5% 11.3% 1.75 161-180 6.4% 11.5% 1.79 181-200 days 6.4% 11.8% 1.84

E.g., as you go down the table the probability 43.3% represents the probability that you get a new high on the second day after a new high.

Here’s an intuitive way to think about it: if you are not at a new high, you are further away from a new high than if you were at a new high recently.  Thus with time the daily probability of hitting a new high gets higher.  If you were at a new high recently, you daily odds of hitting a new high are quite high, but fall over time, because the odds of drifting lower at some point increase.  Valuation is a weak daily force, but a strong ultimate force.

That said, the odds of hitting new highs a long time away from a new high are significantly higher than the odds of hitting a new high where there has been no new high for the same amount of time.

Closing Thoughts

I could segment the data another way, and this could be clearer: If you are x% away from a new high, what is the odds you will hit a new high n days from now?  As x gets bigger, so will the numbers for n.  Be that as it may, when you have had new highs recently, you tend to have more of them.  New highs clump together.

The same is true of periods with no new highs — they tend to clump together and persist even more.

Valuation and momentum are hidden variables here — momentum aids persistence, and valuation is gravity, eventually causing markets that don’t fairly price likely future cash flows to revert to pricing that is more normal.  Valuation is powerful, but takes a long while to act, often waiting for a credit cycle to do its work.  Momentum works in the short-run, propelling markets to heights and depths that we can only reach from human mimickry.

That’s all.