Fooled by Seasonality

Kacper Brzezniak
Written by

8th March 2018

Fooled by Seasonality

One characteristic of human beings is our rather weak intuition about probability and randomness, and this permeates our involvement in gambling, financial markets and also our personal lives.

An example I have often seen used (although it still surprised my colleagues around me) is commonly known simply as “the birthday problem”. If you have not come across this, perhaps give this a go yourself. I encourage you to use your intuition, and not an excel spreadsheet.

Question: There are 25 people at a cocktail party. What is the probability that at least two of them have the same birthday?
See bottom of the page.

The actual answer surprises many people: our weak intuition is not aided by the limited time spent at school working through such problems. But this is a financial markets blog, so what does this have to do with financial markets? Quite a lot in fact.

When looking at any repeated events, we can often be very surprised (and even impressed) by unlikely results and rare events. But rare events do happen, even if we assume very basic distributions, and they happen quite randomly. If you flip a coin 10 times, you would be extremely unlikely to see 10 heads in a row. But if you have 100 people flipping the coin 10 times, perhaps someone will be lucky enough. Give this coin to 100,000 people, and you are very likely to find a few of these. When you find them, will you conclude that these winners are special?

I hope not, but move into the domain of financial markets, and you may unknowingly change your approach.

Another question: Do you notice anything unusual in the chart below?
The chart shows the month-on-month changes in 30-year breakeven inflation1 rates in the UK for the past 20 years. Naturally, a green cell shows that breakeven rates moved higher in that month, while a red cell shows that breakevens moved lower. I created this using the SEAG2 function in Bloomberg, and you can look at almost any product there.

Month-on-month changes in 30-year breakeven inflation rates in the UK
Month-on-month changes in 30-year breakeven inflation rates in the UK
Source: Bloomberg

In 17 of the last 20 years, breakeven rates have increased over March. Only twice have they fallen. Surely then, we should be long breakeven rates this month? If you assume that in any month there is a 50% probability that breakeven rates will either move up or down, then the probability of having 17 up-months out of 20 is 0.11% – so surely there must be something special about March?

But hang on a second, is it really so special? Take a look at the table below, which I made using the excel function, “=rand() ”, which assigns each cell a value, randomly, between 0 and 1. Hence, each month in the chart I created has an equal chance of being above and below 0.50. I have then used similar formatting to our SEAG chart above; cells with a number above 0.50 are shaded green, while those below 0.50 are shaded red.

Notice anything? Well in this case, May seems be to a special month, but how can it be, this is entirely random? Again, you could try this for yourself. Notice that the two tables are not too different.

Source: AllianzGI

The point is this: when we look for patterns in data, when we try to understand how rare or special something is, we need to go beyond just a simply probability. We need to ascertain if what we have observed is different from a randomly generated output.

So: it is not unusual to meet someone with the same birthday as you at a party, and it is not unusual to see any market move up during March in most years, and move down most Novembers.

This is not ground-breaking research, and I’m certainly not the only person to have pointed this out, but I think you would be surprised by the sheer volume of research I receive that is based solely on such patterns. And you may receive such research too – so look out.

Not everything is random

I highlighted the word solely above because it’s important not to dismiss everything as random, even if similar outcomes can be randomly generated. The important step is to look deeper into the pattern: is there a fundamental reason that can explain the pattern? Even if we cannot find the answer, we can perform further quantitative analysis to attempt to understand whether the pattern can be repeated into the future. Without further steps, the initial analysis is meaningless.

Answer to The Birthday Problem: To calculate the probability that at least two people out of the 25 have the same birthday, we simply calculate the probability that no two people have the same birthday, and subtract the result from 1. So in this case:

P(each of 25 people has a different birthday) = 365P2536525 = 0.43
P(At least two people have the same birthday) = 1-0.43 = 57%

1 Breakeven inflation is simply the difference between the yield on a conventional bond and an inflation-linked bond. It is the implied inflation in the inflation-linked bond, such that, if it materialised in reality, an investor would be indifferent between the inflation-linked and the conventional bonds.
2 Seasonality price graph.

Investing involves risk. The value of an investment and the income from it may fall as well as rise and investors might not get back the full amount invested. Past performance is not a reliable indicator of future results. This is a marketing communication issued by Allianz Global Investors GmbH,, an investment company with limited liability, incorporated in Germany, with its registered office at Bockenheimer Landstrasse 42-44, 60323 Frankfurt/M, registered with the local court Frankfurt/M under HRB 9340, authorised by Bundesanstalt für Finanzdienstleistungsaufsicht ( Allianz Global Investors GmbH has established a branch in the United Kingdom, Allianz Global Investors GmbH, UK branch, 199 Bishopsgate, London, EC2M 3TY,, which is subject to limited regulation by the Financial Conduct Authority ( Details about the extent of our regulation by the Financial Conduct Authority are available from us on request.

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