I include the following financial exercise because it seems strikingly similar to some aspects of the climate.

In particular, people often feel that the "natural" variations of the temperatures are small. In the same way, many people think that the currency exchange rates are reasonably and "naturally" stable, up to relatively small or infrequent swings. They may end up with a completely wrong intuition.

So I will do the exercise - and debunk the intuition - in the case of the exchange rates but I do think that the same lesson should be learned in the case of the temperatures, too.

Fine. So what's the problem? LBBW, a German-owned bank that also operates on the Czech market, offers the following cool product (that is denominated in Czech crowns, CZK). It's called IQ bonus deposit EUR/CZK: a guaranteed structured depositum.

Chances are that you don't speak Czech so I will have to explain you the matters. LBBW is occasionally producing playful products and I totally praise them for that. But let me try to give you an answer whether this particular new product is good for the consumer.

Their ordinary saving account gives you 1.75% per year right now - the interest rates are insanely low in most of the civilized world. It's somewhat more likely than not that this interest rate is the minimum and it will be raised in the following year. However, this IQ deposit will give you 0.35% if a condition is not satisfied and 3.35% if a condition is satisfied. (15% tax is subtracted from all the interests but because this is a correction that equally influences all situations, I will ignore it.)

The condition is a kind of year-on-year EUR/CZK exchange rate stability whose details I will discuss momentarily. So if the EUR/CZK exchange rate stays pretty constant for a year, you will get 3.35%. In other cases, you will get 0.35%.

**Now, what is the condition exactly?**

You should ideally make your deposit before July 1st, 2010.

At the end of June 2010, they will look at the official EUR/CZK exchange rate, as announced by the Czech National Bank. It will be something like R=25.50 CZK per EUR. The condition looks at all the daily official exchange rates between the end of March 2011 and end of June 2011 to determine how much you should get for the year.

If the euro i.e. the exchange rate R never drops below R-1.00 CZK per EUR (e.g. 24.50 CZK per EUR in my example) and if it also never jumps above R+0.80 CZK per EUR (e.g. 26.30 CZK per EUR in my example), and I really mean if both these conditions are satisfied for every day in the 3-month period defined above, the big condition will be satisfied and you will get 3.35%. Otherwise, you only receive 0.35%.

Is it clear?

The crown is only allowed to weaken by at most 3.1% or so and strengthen by 3.9% or so and this must be true for all dates between 9 and 12 months after you make your deposit. How likely is it that the condition is satisfied and what is the expectation value of the interest you will receive on your IQ deposit?

To find a statistical answer to the question, I imported all the official CNB exchange rates to Mathematica.

This particular database starts on January 1st, 1999 - when the rate was 35.226 - and it continues through June 15th, 2010 - when it was 25.67. The crown was strongest - so far - in 2008 when it was at 23.00 per euro. After those 11.4 years, the rate changed by the factor of 1.37 or 0.73, depending on your viewpoint. In average, the crown has strengthened by 2.8% every year. This trend is within the allowed interval and it really "does feel" that the rate currently wants to live between 25 and 26.

So many people could conclude that it's pretty likely that that the LBBW condition will be satisfied. In fact, I believe that most people would believe that the condition was mostly satisfied in the last 11.4 years.

However, the truth is completely different and the wrong intuition doesn't appreciate that it's Mother Nature (or the invisible hand of the markets), rather than the individual investor who determines the central values of the intervals. Let's look at it more closely. The database contains 2890 daily exchange rates - mostly from workdays of those 11.4 years. The 2626th line is the last day of May 2009. I considered the first 2626 rows as hypothetical starting days of a similar product.

In all these 2626 cases, I asked the question whether the exchange rate stayed in the desired interval during the whole period that occurred 9-12 months after the initial dates. How many days out of the 2626 days actually obeyed the condition?

The condition that the crown won't weaken too much - by a more extreme factor than 26.30/25.50 - was satisfied by 79% of the days. Indeed, it's been pretty unusual that the crown would weaken more than by 0.80 CZK per EUR (or by more than 3.1% - I assume that LBBW would offer a nominally wider tolerance window in the years when the nominal value of CZK was weaker).

However, the condition that the crown won't gain too much - by a more extreme ratio than 25.50/24.50 - was only satisfied by 37% of the days. So for a significant majority of days, you could get a crown that is strengthened relatively to the euro by more than 4% (now: 1 CZK per EUR) at least for one day in the 9-12 month future. That's true despite the fact that the trend growth was only 2.8% per year. The noise in the 3-month period simply allows you to probe a wider spectrum of values.

What if you combine both conditions? They have to hold simultaneously for you to get the higher interest rate. If the two sub-conditions were independent, the combined probability would be 0.37*0.79 = 29%.

However, they're not really independent. In fact, they're kind of anticorrelated with one another. If the crown doesn't strengthen too much, it is actually more likely (relatively to having no assumption) that it may weaken too much. So if you actually count the number of possible initial days among the 2626 candidates for which both conditions are simultaneously satisfied, you only get 17% of days. It's unlikely for the LBBW condition to hold.

So the expectation value of the interest rate is

0.17 * 3.35% + 0.83 * 0.35% = 0.85%which is more than twice lower than the standard 1.75% interest rate they offer you now. You may still profit from the IQ deposit accounts but if you do such things systematically, you have to be much luckier than the average man. ;-) Statistics is speaking against you.

Someone could argue that the first decade of the new millennium was more violent than others. Except that it almost certainly wasn't, especially if you look at quantities such as the EUR/CZK exchange rate. In fact, the rate became more stable during the recent euro turmoil. If things get much more peaceful, the crown will return to the growth that will often exceed 4% per year.

Similar comments apply to the temperatures where the noise is even more important than the (real or fictitious) underlying trend. If you make a bet that the climate will be changing according to some linear extrapolated trend, plus minus dozens of percent, you will almost certainly lose.

**Mathematica notebook:**

Directory; PDF preview;

Data to import; NB notebook

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