# Perspectives On Impossible Market Streaks

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by: Salil Mehta

We go through and connect the probabilities associated with streaks from a variety of settings: from sports to investments. Looking at streaks in different settings and differentiating among them is the best way to appreciate the record streaks we are currently seeing in the financial markets (e.g., number of record highs, or unusually large portion of "up" days in a row). We will use numbers close to 20 in the sample calculations below, though specific current market moves are fluid and are only approximately 20 right now.

Sports

We start with a simple sports example from the Michael Lewis book Moneyball. The 2002 Oakland A's won 20 straight games, an American League record since 1901. If a team has an even 50% probability of winning any one game, then the probability associated with a 20-game streak would be 50%^20, which is nearly 0%.

Before the streak, what would have been the probability of winning that would be needed so that 2002 Oakland A's streak would occur 50% of the time? Here are the calculations below.

p^20 = 50%

ln(p^20) = ln(50%)

ln(p) = ln(50%)/20

p = e^(-.035)

p = 97%

Keep in mind that this record was set relatively recently so it is more difficult to empirically estimate how rare such a record exists. Streaks occur in both directions however, and the American League losing streak was 21 games, set by the Baltimore Orioles. In 1988, this too is a relatively recent occurrence. The winning and losing streak records in the parallel National League, on the other hand, both occurred nearly a century ago.

Financial markets

Let's turn our attention to the 2013 Tuesday winning streak in the Dow Jones Industrial average index. The streak is currently at 18, while the previous record was 23 straight Wednesdays. How rare is this, and what if anything can you interpret about the markets based on such long streaks? For benchmarking purposes, we'll do this analysis on 20 straight wins, which is about the average of 18 and 23.

We know from the sports calculation above that the probability associated with 20 straight wins, from a fair random walk process, is a near impossible 0%. A twist of this question is to ask what daily gain probability would be associated with a 20% probability of seeing at least one weekday having a streak of 20 wins (e.g., 20 Tuesdays, or 20 Wednesdays, etc.)? Here again are the equations.

[1-(p^20)]^5 = 100%-20%

[1-(p^20)] = 96%

p^20 = 4%

p = 86%

Say that the daily chance for an up-move on the Dow Jones Industrials is, in fact, nearly 86%. Then what is the precise probability associated with one weekday having this streak of 20? This is a more precise calculation versus the calculation above of "at least one" weekday having this streak of 20. In order to determine the likelihood of seeing such a rare streak, we use a popular probability formula of a binomial distribution where "r"=1 weekday, out of "n"=5 weekdays. The binomial distributions solves for the joint probability of events multiplied by the permutation of events. It will tell us the precise probability of seeing such rare streak occurrences, by chance alone. One may notice that ordering matters in the equation, but it is on both sides of the product so the ordering factor cancels.

permutation * joint probability

= n!/[r!*(n-r)!] * p^r * q^(n-r)

= 5!/[1!*(5-1)!] * [86%^20]^1 * [1-86%^20]^4

= 18%

We see that this 18% chance for one weekday is slightly less than the 20% for the chance for "at least one weekday" having this streak of 20 (note from the examples above that when the probability of an "up" move was 50%, the higher probability of seeing at least one streak still quickly converges to 0%). Such a low 18% probability should not be so low as to signal a red flag for an investor! Now assuming nothing else changes, let's change the problem slightly and ask what is the probability of seeing both one weekday having a streak of 20 gains, while simultaneously another weekday having a streak of 20 losses?

There are two ways to solve this (and non-statisticians can just skip ahead to the results further below), one with a nested binomial formula and the other with a trinomial formula. We show both below.

For the nested binomial we solve first for the complement in three workdays:

l = probability of a down-move, which is 100%-86% (or 14%)

l^20 = nearly 0%

complement = 100% - 86% - 0%

= 14%

permutation * joint probability

= n!/[r!*(n-r)!] * p^r * q^(n-r)

= 5!/[2!*(5-2)!] * [86%^20]^2 * [1-86%^20]^3

= 2%

We then solve for the two remaining workdays being a streak of 20 wins and 20 losses:

p = 86%^50 / (86%^20 + 0%^20)

= 0%

permutation * joint probability

= n!/[r!*(n-r)!] * p^r * q^(n-r)

= 2!/[1!*(2-1)!] * 0%^1 * 100%^(1)

= 0%

So 2%*0% implies a 0% chance using nested binomial. Let's now see this is through a trinomial formula:

s =number of weekdays with a streak of 20 losses

permutation * joint probability

= n!/[r!*s!*(n-r-s)!] * p^r * (l^20)^s * q^(n-r-s)

= 5!/[1!*1!*(5-1-1)!] * [86%^20]^1 * [(100%-86%)^20]^1 * q^(n-r-s)

= 20 * 4%^1 * 0%^1 * [(100%-4%-0%)^20]^(5-1-1)

= 0%

So both the nested binomial and the trinomial approaches show a near 0% probability of having both a 20 winning streak and 20 losing streak, at the same time. So while the 18% probability of seeing a plain vanilla streak in a biased market is not that low, in most circumstances there is a near impossibility of seeing such streaks naturally. One should note that a streak by itself does not state anything about the market being suddenly over or under valued. It simply states the likelihood of seeing such a streak if the markets were randomly moving based on efficient and fair markets to begin with.

Conclusion

So a better explanation for the streaks in the financial market data, as opposed to the sports data above, is that the probability associated with an up-move is not stationary and fixed at 50%. Ex-post we can see regimes that have randomly oscillated about, even as the historical de-trended average is balanced about 0%. A market regime that is only part way biased in one direction, say a 60% daily chance for a performance gain, could see a streak in the mid-single digits with the same probability that we see the current financial market streak of 20. And this is likley what we are seeing in the markets right now, while many bears have been muzzled.

Also recall that streaks can and do occur, in both directions.

A final point is that an additional difference with plain-vanilla financial markets streaks versus sports streaks, is that sports probabilities are paired within zero sum games. A streak of 20 wins by Team X reconciles with 20 losses for all the teams that had played against Team X. It is rare however, as we have eluded above in the financial markets example, to have record losing streaks in sports that occur in the same season(s) as record winning streaks. While negative correlation does appear in plain-vanilla financial markets, and the art of multivariate regression can asses the combination of other asset class performance(s) in one direction versus that counterbalance of an isolated asset class (e.g., X) performance in the opposite direction, it is never inherently perfectly (negative).

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.