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Monte Carlo Methods - Concept.

If the technique was being named today it would probably be called Las Vegas calculations and the output known as "Vegas Values".  Monte Carlo methods are based on random numbers.  For a long time, Monte Carlo was one of the best known venues for roulette and because a fair roulette wheel is one of the earliest random number generators, a branch of mathematics has been linked to Mediterranean seaside town.

Follow the link for a demonstration of a Monte-Carlo simulation.  If it is not installed on your machine, your browser will attempt to download Adobe's SVG viewer.

The applications of Monte-Carlo methods are many and various, but many will come under these headings:

The object of study can't be manipulated efficiently by analytical maths; a good example is the application of Monte-Carlo methods to the integration of complex functions.

The analyst wants to use data which describes a process,  Examples include the analysis of investment portfolios and the application of oceanographic data to estimate availability of offshore loading facilities for oil tankers.

This page contains:

A basic illustration of the Monte-Carlo method
The application of the method to a real world situation

Other resources on the site include:

Estimation
Integration
Simulation

The Monte-Carlo Method

A trivial example of the method is the estimation of the area of a circle, its trivial because there is a well known formula which is quick and easy to use, but this example has most of the elements of more complex applications.

Step 1

Draw a square on a piece of paper the length of whose sides are the same as the diameter of the circle.

Step 2

Draw a circle in the square such that the centre of the circle and the square are the same.

Step 3

Randomly cover the surface of the square with dots, so it looks like this:

Step 4

Count all the dots, then count the ones which fall inside the circle, the area of the circle is estimated thus:

The larger the number of dots, the greater the accuracy of the estimate.

Alternatively, you can use your browser to do it for you, enter the radius of the circle (values in the range 1 to 10) into the text box, click on "estimate" and the estimated and accurate estimates will appear:

	Radius	Estimate	Accurate

The estimate is generated by simulating the drawing of 10,000 dots.  By increasing the number of simulations, we can increase the accuracy and also the time taken to complete the process.

A real world situation

Moving on from the trivial to an application which is closer to the real world, that of a venture capital bank, albeit in an example presented in an over simplified form.  Such banks invest in high risk projects and need to manage their risk.  At one extreme, the performance of a bank with a single investment would be dependent on that investment, if it failed, the bank would lose money, it was a spectacular success, the bank would be highly profitable.  However, by spreading its funds over several ventures, the probability of failure is reduced but the profits from the successful ones are offset by the cost of the failures.

Monte Carlo methods provide a means of modelling the behaviour of a portfolio.  In the example below, a fictional bank makes between 1 and 20 investments.  Historically, 50% of investments fail to create marketable products.  This can be modelled with the binomial distribution, which for a given probability of success, provides an estimate of the number of success for a given number of investments.  The example below shows the probability of a given number of successes, for 10 investments.

Of those that start trading, the distribution of revenues is shown in the diagram:

It is a perversity of nature, that the distribution of desirable outcomes are left skewed, i.e. the probability of a modest success is greater than that of a spectacular one, hence sales have been modelled as left skewed, whilst costs are right skewed, i.e. the probability of exceeding budgets is great, this is shown in the diagram below:

Using these models we can estimate the bank's ROR (Rate of Return) using the process outlined in the simplified flowchart below:

The value of 1,000 simulations is arbitrary, in practice the number should be appropriate to the application.  For example, if an event within the process occurs infrequently, the overall number of cycles should be large enough to ensure that the results include all likely outcomes.

The results for investments in 1, 5, 10, 15 and 20 projects have been presented in the form of a line graph showing the probability that a given ROR will be exceeded.

In this graph, the red line shows the probability of the ROR exceeding 0 (i.e. not making a loss).  For a single investment, the probability of not making a loss is 50%, by increasing the number of projects to 20, the probability of not making a loss rises to nearly 80%.  However, this reduction in risk, is offset by decreased upside (i.e. making large profits).  For a single project, the probability of exceeding a 40% ROR is 42%, as the number of investments increases to 20, this figure falls to 34%.

The worldly wise amongst you will be saying that this is just a way of using a computer to illustrate the old adage "don't put all your eggs in one basket".  However, this form of analysis does provide some understanding of how many baskets are needed for a given level of security.

The Monte-Carlo process can also provide some insight into the workings or the bank, for example, by analysing the sum of costs for all the investments, it can be seen that for 20 projects, that the probability that $210m of capital will be adequate is 90%.

Hopefully between these two examples, I have shown the basic concept of the Monte-Carlo method and illustrated its application to a real world situation.

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