12 May 2014

Effect on Statistical Risk Measures of Interplay Between Arrow of Time and Power of Zero1

In our newsletter dated 3rd March 2014, we discussed some deep theoretical chasms, which will in due course invalidate the process of allowing banks to themselves determine their own risk capital based on statistical methods.

Today, we discuss another deep conceptual flaw in the design of risk management systems as defined in currently followed financial theory.

Let us consider a game of tossing of fair coin. One starts with $100. If one gets a head, the money is increased to $150, otherwise, in the instance of getting a tail, the money is reduced to $60.

So, standard textbook formula of expectation from this game is: 0.5*150+0.5*60 = 75+30 = 105

150 X 0.5 + 60 X 0.5 = 105

Since 105 is greater than 100, the modern finance theory says that we should play this game. The idea is that if you play this game enough number of times, say 100,000 times, the variability will smooth out and the average reading of payoff will be very near 105.

Let us look at outcome of Monte Carlo Simulation (MCS) of the above single period game. The Chart on the left shows results of first 400 runs, and the chart on the right shows results of all 100,000 runs. The MCS seem to confirm what the textbook formula suggests, with the MCS results converging very close to 105 in first 400 runs itself, and then remaining virtually unchanged over the remaining 99,600 runs.

Let us look at this concept of expected value very closely. The calculation assumes that, you will play the game over and over again, from same starting point of having $100 in your pocket. That is, the central assumption here is that, an individual is able to go back in time, irrespective of the outcome of fate handed over to her in the previous draw and start from the same position.

If you are an insurance company, aggregating similar risks across a given time frame, this concept of arriving at the end of period expected value is more or less valid, except for rare tail events, such as earthquakes, which make individual risks go massively correlated with each other.

However, if you are an entity who is betting the house on the outcome of a risky event, such as market movement, and you are getting up every morning to play the same game with whatever capital you have carried forward from yesterday night, then calculating expected value based on arithmetic mean does not make any sense because your starting point is not $100, but the starting point is outcome of previous draw.

Let is illustrate this concept in more detail below. If you are playing the above coin toss game once a minute for one hour, that is 60 coin-tosses, the expected value of your game, as per textbook formula is

The chart below shows the result of MCS for 60 consecutive coin-tosses with results from previous period serving as the opening capital for current period.

This chart shows that MCS does not converge to the expected value, and even after a large number of runs, say at approximately 83000 runs in this case, it can show very large jumps.

What is even more interesting is to look at the histogram of the individual 100,000 runs as shown in chart below.

The thing to note here is that bins on x-axis are not evenly distributed. The distribution is so much right-tailed, that to display meaningful data, we had to have 10-gap bins till 200, then 100-gap bins till 1000, then 1000-gaps bins till 10,000 and so on.

You can notice that about 74,000 times, the outcome is 30 or less, that is a loss of 70%. However, there are a few extreme observations, for example, payoff of 24,674,902 in one single case. In other words, this game, where one seem to have 5% edge at every toss of coin, will bring ruin to a player in three out of four cases. However, a few extreme pay-offs will take arithmetic average of expected value to mouth watering returns of more than 18x. That is a typical player is ruined, but the expected value is very attractive because of right-tailed payoff.

Before we decide to analyze why this happens, let us see one more example. If you decide to play this game for 25 hours, that is 1500 coin-tosses, the expected value of your game is, as per standard formula is

or 6 billion trillion trillion from the starting point of 100!

However, lets quickly look at the chart of expected payoffs based on MCS for 1500 consecutive coin-tosses

The pay-off as per MCS is less than one from starting point on 100, which is complete ruin. The table below shows the distribution of 100,000 runs of MCS for 1500 consecutive coin-tosses

So, what is happening? Two simple facts of life, which we learn early in childhood, are coming to play here. We learn to accept that time is irreversible. As a child, we fairly quickly give up trying to piece together broken plastic toys. We somehow accept that loss in most cases is permanent and cannot be reversed. Also, we learn in school that any number multiplied by zero is zero, however large that number be. Add to these two fairly simple concepts the fact that in game of chance, it is possible to have a streak of bad luck. If you have, say 15 consecutive bad draws, whatever maybe your starting capital, you go to near zero fairly quickly. And, once you have taken that big hit to your capital, it is nearly impossible to recover.

So, in the game of 60-consecutive coin-tosses, 75% of observations come with a loss of 70% or more, even though each coin-toss is expected to pay you off 105 over 100, mainly because in 60 tosses, it is more likely than not that you will have five or six consecutive tails. Since you are carrying over positions from yesterday, as most real life institutions do, six losses 40% each take you very near zero from which you cannot recover, even if you do not have leverage.

In 1500-toss simulation, nearly in every case, we encounter 10 to 15 consecutive losses, which took the observations for nearly all runs, except one, below original capital. In the 1500-toss simulation, the outliers are so rare, that we did not observe in 100,000 run. Maybe, if we run a one-trillion run simulation, we will get enough outliers.

In 1500-toss simulation, nearly in every case, we encounter 10 to 15 consecutive losses, which took the observations for nearly all runs, except one, below original capital. In the 1500-toss simulation, the outliers are so rare, that we did not observe in 100,000 run. Maybe, if we run a one-trillion run simulation, we will get enough outliers.

In other words, in real world, for traders and bankers, the expected value is not arithmetic mean but geometric mean. More you play, betting your entire capital each time, the more the geometric mean pulls the typical player towards zero while the expected value as calculated by arithmetic mean may keep rising due to a very few outliers.

Modern financial theory has failed to recognize this fact. The above construct allows us to analyze why every few years the modern capitalist structure encounters banking crisis and how those can be prevented in future.

Empirical evidence has shown that during times of stress, most assets become highly correlated, and diversification does not work. Hence, firstly, it is essential that large institutions should be studied and analyzed as single bet entities with near 100% correlation in various areas of their business, unless they hold assets in risk free short-term deposits with central banks, which they cannot access for funding other assets but only to fund proportionate liability outflow. By using Kelly Criteria one can arrive at this number. Secondly, institutions should be required to hold capital for gap between median observation of geometric value of their risky assets - arrived at based on their holding period and trading pattern – and the current market value of the assets. Since median geometric value is always less than the current market value, this will increase capital requirement significantly, however will reduce risk of failure in future.

The risk management practices, as laid out in current international guidelines fail to address the questions that we are raising, and hence we urge investors to remain watchful of the financial sector for build-up of risks.


1The core concepts of this newsletter are based on work of Prof Ole Peters of Santa Fe Institute & London Mathematical Laboratory

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