Professor Forman

# Monte Carlo Simulations

#### Non-Linearities and the Need for Monte Carlo Simulations

The risk of an event can be computed as the product of its likelihood and impact, each of which in turn can be computed as the sum products of source likelihoods with event likelihoods given the sources, and event consequences with the objective priorities. However, these computed risks are distorted by non-linearities when there are multiple sources that are not independent, and/or when there are multiple events with consequences to one or more objectives. Monte Carlo simulations are used to correct for these nonlinearities.

#### Computed vs. Simulated Likelihoods, Impacts, and Risks

A Monte Carlo simulation using the computed likelihoods, impacts produces a Loss Exceedance curve showing the probability of incurring a ‘large’ loss in the ‘short term’, as contrasted with the long term expected loss. The loss exceedance can be expressed either as the probability of exceeding any given loss (for example \$500M in the example below) or the loss amount by which there is a less than a given probability — usually 5% — \$987M in the example below.

The Monte Carlo simulations consists of a series of ‘trials’, each trial representing what could take place in the future. For example the figure below is what happened in trial or step 3 of 9000 trials. In this particular trial, two uncertain sources of events took place (inadequately trained staff and system software technology obsolescence). The inadequately trained staff triggered 4 events and the system software technology obsolescence caused a degradation of the intelligent monitoring system. The impacts of each of the events (shown in dollars) are due to losses to objectives (not shown). The total impact of the events is \$991.96 million dollars.

Results of two other trials, steps 4 and 6 are shown below.

#### How Computed Likelihoods and Impacts are used in Monte Carlo simulations to Reduce Exaggerated Computed Likelihoods, Impacts and Risks

Random numbers between 0 and 1 are generated for each source and event. If the random number is less than the computed probability for a source, then the source has occurred in this trial. A random number between 0 and 1 is generated for each event that can be caused by the source and if it is less than the computed conditional probability of the event given the source, then the event takes place. However, once an event takes place due to a source, it cannot take place again due to another source. This is a non-linearity that is not accounted for when computing likelihoods, impacts and risks without simulation and is one of the reasons that computed values are exaggerated. Another reason is that the consequence of an event on an objective is less from an event causing loss to that objective if another event has also caused a loss to that objective. Thus for example, you can see that the impacts to objectives from an event occurring is not the same for different trials. The simulated results for likelihood, impacts, shown below, are less than the computed results that are exaggerated due to non-linearities.