# Maths make things simpler, rather than more complicated, bring clarity.

Mar 21st, 2013

A discussion on LinkedIN entitled “Does anyone use Standard Deviation, Variance or similar measures to prioritise risks” prompted a reply we are now posting on the blog for our interested readers.

It’s a bit technical, but it shows that maths make things simpler, rather than more complicated, bring clarity!

Author: Horas

Talking about clarity, let’s remember that the context of most of our work at Riskope is in downside risk, in various industries. When we refer below to “Consequences”, we are referring to the “sum” of many components including, for example: replacement cost, Business interruption, loss of market share, reputation, legal costs, crisis cost (boycotts, protests…), etc.

## Does anyone use Standard Deviation, Variance or similar measures to prioritise risks

As many pointed out in the LinkedIN discussion, both probability and consequences are actually stochastic (random) within a population of scenarios (careful not to “sample” different populations with the excuse of simplifying the problem). Both vary between a Min and a Max, with a certain expected value Ave, and a standard deviation St.

Oh, by the way, some may even point out that P (probability),C (consequences) are not necessarily independent variables, but let’s not go there this time around for the sake of space.

Thus Risk= P (probability)*C (consequences) holds, with p, C and, as a result, R being stochastic (random) variables. Thus Risk will have a Min and a Max, Ave and a standard deviation St and it will be possible to evaluate the probability that Risk is larger than a given value, or what is the value that has a x% probability of exceedance.
More importantly, as we do on a daily basis in ORE applications (Optimum Risk Estimates), we can compare probabilistically the risk to the client’s tolerance threshold (another chapter we will leave aside for obvious space limitations).

## To do that all of the above we need to calculate Risk.

Now, let’s make sure we do not use a bazooka to shoot a mosquito here, with all its unpleasant side effects!

Monte Carlo simulation would work provided we know with good approximation Min, Max, Ave, St of p and C. We believe that in most cases we could all convince ourselves we have a grasp of Min, Max and Ave. But frankly St is a difficult one.
Furthermore, Monte Carlo will require that you define the distribution (type) of p,C…and there we are in uncharted territory.

Information theory says that if we do not know (very well) what type a distribution belongs to, we minimize the errors on the estimates by keeping it simple! Some authors, even go so far as to say that you are better off selecting a UNIFORM (yes, you read well) distribution if you are not sure of the type.

Now, using Monte Carlo to calculate Risk as a product of two uniform distribution is really using a space ship to go buy your candies at the shop next door.

Below we are showing the super simple, analytic solution for the case where p and C are symmetric distributions. That solution works for any distribution developing respectively between pMin =CMin =nil and a maximum value pMax, CMax.
Of course the case where pMin = a, CMin= b would also be very easy to derive.

Because both distributions are symmetric their averages are respectively

E(p)= pMax /2;
E(C)= CMax /2.

Risk R= p*C, so Ave (R), is immediately found as follows:

Ave(Risk) = Ave(p) * Ave(C)= (pMax * CMax)/4;

It can be seen that the average risk is four times smaller than the maximum risk. That’s true for any shape of symmetric distributions going from nil to Max.

## A slightly more complicated direct formula leads to St(Risk).

Job done!

Now we can calculate any probability of exceedence you’d like. We create an empirical distribution of R with Min(R), Max(R), Ave(R), St(Risk).

Why would one use Monte Carlo and make the collateral damage of increasing the errors?

If this type of maths intimidates you, there is another approach we often use at Riskope. We consider several sub-cases of a scenario with point estimates of p-C. Then we “lump them up” in a “risk bubble”. We have a number of slideshows you can access from our blog that display such “risk-bubbles”.

Furthermore, if you really believe you have a grasp on the dissimetry of the p,C distribution, then you could use other direct approximations as well. Again still leaving the Monte Carlo bazooka to rest.

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