# Monte-Carlo simulation

Monte-Carlo simulation is a useful technique for financial modelling that uses random inputs to model uncertainty.

When a financial model is used for forecasting there will clearly be a number of inputs into the model that are unknown. One approach is to take a best estimate for each of these inputs. For example suppose we are using a model to forecast a company's sales volumes that is something like this:

market growth = GDP growth ×multiple
market size = current market size ×(market growth + 1)
market share = current market share + gain
sales volumes = market size ×market share

There are three uncertain inputs here: GDP growth, the relationship between GDP growth and market growth and the increase in market share. The obvious approach is to use the best estimate of each.

Using a probability distribution (for example the normal distribution), rather than using the single best estimate, better reflects reality. Using a probability distribution is not easy. One approach would be to mathematically derive the output of the model as a probability distribution. This is usually very difficult, and often simply not possible.

Mote-Carlo modelling provides an alternative. Monte-Carlo methods do use the probability distributions of the inputs. Rather than using the distributions themselves as inputs, the distributions are used to generate random inputs. The methodology is:

1. Draw a number at random from the probability distribution for each input
2. Calculate and record the outputs given these inputs
3. Repeat from step one as many times as necessary

By doing repeatedly, this it is possible to gradually build up the probability distribution of the outputs.

To apply Monte-Carlo methods to the simple model above we would need to estimate distributions for each of the three inputs. So we may end up doing something like this

1. GDP growth is forecast to be 2%, with a standard deviation of the estimate of 1% and normally distributed. So we randomly take a number from a normal distribution with a mean of 2 and a standard deviation of 1. This gives up the GDP growth percentage, call it x.
2. We have a similar estimate for the multiple that relates GDP growth to market size. Here we draw a random number (call it y) from a normal distribution with a mean of 1.5 and a standard deviation of 0.5
3. We multiply x by y which gives us our market growth estimate. We use this to estimate our forecast market size
4. We have a similar estimate for market share growth. Suppose in this case we draw a number from a normal distribution with a mean of 2% and a standard deviation of 2%
5. We can now calculate sales volumes as above
6. We now record the value we get for sales volumes
7. We now repeat from step one, hundreds of times
8. The recorded values form the output of a Monte-Carlo simulation

We end up with a series of estimates. These can be used to calculate a mean and a standard deviation for the sales volumes. This is a far more meaningful number than a single best estimate, as it gives both a better best estimate (the mean) and a measure of its uncertainty (the standard deviation).

Obviously Monte-Carlo methods are very tedious to use with manual calculations. Computers make the use of Monte-Carlo analysis far easier.

In many cases the most significant amount of work comes not from the Monte-Carlo simulation itself, but from the need to make estimates of probability distributions rather than simple point estimates. Not only is it necessary to estimate the parameters of the distribution (e.g. mean and standard deviation for a normal distribution), but it is also important to select the right probability distribution. This page lists some of the most important, and there is a more comprehensive reference here.