MODECUBE

MONTE CARLO SIMULATIONS

Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results.
They are especially useful for modeling phenomena with significant uncertainty in inputs.

The goal of this paper is to give a hint to the reader as to why they could prove useful in asset management (they overcome the single point estimate issue of the classical "static" performance and risk forecast methods and they help take account of correlation between variables).
The specific asset class used as an example is real estate.

COVARIANCE

We all have an intuitive comprehension of the correlation concept. But how to put words on it?

Wikipedia gives some definition but it may appear quite theorical.
I've always needed a deeper understanding of the concept, and especially get where the mathmatical calculation formula came from.
The following is a case in point of what helped my understanding process.

I actually believe that understanding correlation to such an extend helps understand Monte Carlo methods and some related issues.

First let's have a closer look at its parent concept, the covariance.

Let's take some example: the height and weight relationship (if any) of a population.

Here is the data for a sample of Australian people (data made up):

The average height is 182.5 and the average weight is 76 kgs.

Looking at the data, we immediately feel that there is a correlation, ie a relationship.
Indeed, in this case, the taller, the heavier.
But our feeling is also that this correlation is not perfect: if you know the size of a fifth Australian, you cannot predict his weight for sure. The relationship is not linear (you don't exactly have weight = a * size + b where a and b are real)

All of that helped us setting up some characteristics:
(a) A perfect correlation is the case of a linear relationship (knowing one variable determines the other variable)
(a) bis A correlation can be less than perfect: when we know one variable, we can only have a idea regarding the other variable. For instance, we feel that a 177 Australian is more likely to weight 70 than 200 kgs. Of course, exceptions are there to defeat the general rule.
(b) Actually, correlations can be more or less strong, and should be possibly ranked (provided that we have some numerical measurement tool)

Based on that, what does it mean when two variables have no correlation?
We can simply try and answer that knowing X doesn't help predicting Y whatsoever.
If there is no correlation between height and weight of Hungarian people, that means that a short Hungarian is as likely to be obese as to be skinny.

Here is an example of no correlated variables in this fictive Hungarian case:

The average weight is 76 kgs.
The similarity of the values and the averages with the Australian case is striking. What really differs is the dispersion of the weight values.

Lastly, we can notice that negative correlation may exist: the higher X, the lower Y, and the other way round.

We now have enough material to suggest a simple definition.
A correlation express the degree (b) of the relationship between two variables. The relationship looked after is a linear one (a).

Can we find a mathematical tool that enable to measure this degree?
To do so, let's start with the following comment: in the Australian case, we said that the strong positive correlation induces that the taller the heavier and the smaller the slimmer.
Fair enough, but, if you say that 175 cm is small to a Vietnamese, he may be upset. Indeed, Australian people are among the taller people in the world, with a 178 cm average.
In the same fashion, what is deem slim for the Australians may not be relevant for another population.

Relativity is of essence.
To take that into account, one has to compare a value to its average. 175 cm is short and 67 kgs is light in our first sample because the averages are 182.5 cm and 76 kgs respectively.
So, for a pair of values (one height and one weight), what matters is their comparison to their means. And to be more specific, what really matters is their distance to the means (how tall is 190 cm? As tall as it is far from the average height.).

We've got:

Now, to prove the Australian positive correlation, we need to show that very often, a very far height form its average would yield a very far weight, and a very close one would yield a very close one.
To prove the low correlation in the Hungarian case, we need to show that, on average, a far height yields both close and far weights.
To avoid any subjectivity, why not having a mathematical value that translates those ideas? For every person, we could multiply their distances to the means:

Here we clearly see that far distances from means produces high values and closer distances produce low value, and that every result is positive.

And here we see that is more complicated. Taller than average people produce negative and positive results, and shorter than average people also produce negative and positive values.

Lastly, we need to have a single result that summarize what is overall happening, that give us the average case (the expected case):

That's it, we've got single results that can indicate a positive correlation and a null correlation.
What we calculated is called a Covariance.
The Covariance formula is then :

Covariance = Average ( (Height - Average (Height)) * (Weight - Average (Weight)) )

Mathematically, the average is usually called Expected value and its symbol is the letter "E":

Covariance = E ( (Height - E (Height)) * (Weight - E (Weight)) )

Thus, for the general case where we want to calculate the covariance for two variables X and Y:

Covariance = E ((X-E(X)) * (Y-E(Y)))

This is a great mathematical tool that fulfil all our requirements.
But it doesn't come without some limitations...

Interesting links