Modelling and Simulation

1.1 Why use models?

A model is a simplification of the real world, with only the most important elements included. A good model enables us to work out what is likely to happen if we take certain actions, so we can avoid actions which have undesirable consequences.

Governments have models of the country's economy, where they can try out any changes which they are thinking of making. Life insurance companies use models to predict profitability of insurance products. Banks offer loans at fixed rates of interest, based on their models of the future behaviour of the economy.

1.2 What does a model involve?

A model expresses relationships between quantities of concern to us. A mathematical model expresses relationships in terms of formulae and equations. It is likely to involve parameters: numerical quantities which need to be estimated.

A model needs to be flexible, including a mechanism for updating parameter estimates.

Models must be monitored, to ensure that they remain valid.

1.3 Setting up a model and keeping it running

• Specify what the model is for
• Work out which quantities are essential and which are not
• If essential quantities have not been monitored in the past, plan how to keep track of them
• Express the relationship between quantities in terms of formulae involving parameters
• Implement the model on a computer
• Use observations to estimate parameters
• Look at the predictions made by the computer

Experts should be involved at all times, to check that the mechanism reflected in the model structure is reasonable, and that the model's predictions reflect reality.

1.4 What kinds of models are there?

Discrete or continuous time

The first choice is whether the model is to run in discrete time or continuous time.

The choice must be determined by the real-life process under consideration. A pension scheme is subject to annual valuations, making a discrete-time model appropriate. But if a merchant bank is modelling an exchange rate a continuous-time model is to be preferred.

Small scale or large scale

For an insurance company it is often helpful to model the behaviour of one individual policyholder. This is a small scale model.

For administrators of a pension fund, assets should be grouped into a small number of classes (risk-free, property, stocks, etc) and each class modelled as a whole - this leads to a large-scale model.

Should a random model be used?

There are times when we can predict with a high degree of certainty what will be the outcome of our actions. The relationship may not be simple, but it can be expressed in terms of a set of formulae.

Usually, though, the possibility of random disturbances should be built in to the model, so that we can see whether our plans are able to cope with unpredictable happenings.

1.5 Testing your model

Apart from the expense of setting a model up, there are a number of drawbacks:

• if unreliable data are used to estimate parameters, then unreliable results will be produced (GIGO)
• models designed for one purpose may sometimes be used for another, or beyond the range of values for which they were intended, with unpredictable consequences
• for a random model, a large number of simulations are required for reliability
• users may attach too much significance to the output from a simulation, because "the computer never makes a mistake".

It is important to test a model and the associated simulation and to estimate the accuracy of output values: there is a difference between $2,000,000 ± $10 and $2,000,000 ± $1,000,000.

One method used is sensitivity analysis: try changing one of the parameters by a small amount and see how large an effect this has on the final result. If a small change makes a big difference, the model is unstable and needs to be refined.

Another method involves checking that predictions would not be substantially changed if variables which have been assumed independent are instead modelled as having a non-zero correlation.

1.6 The role of simulation

One of the aims of modelling is to enable a planner to try out decisions on the model before implementing them in practice. The simulation is the method used to test them out.

Once the model has been put together, and any parameters estimated, the structure of the model must be programmed with care. Most simulations involve a random element, so the random number generation method should be assessed.

Truly random numbers can be generated by a number of possible devices. Most of them take time, so it might make sense to generate a long list in advance, then store the list in the computer for use during the simulation. But a simulation may need more values than can conveniently be stored.

When simulations are being used to compare two or more schemes, it is best to ensure that the schemes all receive the same inputs. Where truly random numbers are used this requires a pre-generated list; a repeatable sequence of numbers which are not truly random (but do appear to be random) is usually preferable.

Re-using the same sequence of random numbers is particularly important for sensitivity analysis. The effect of a small change in a parameter value may be swamped by random effects (noise) if a different set of "random" numbers is used for the simulation.

It is necessary to run several simulations of the same process, so that random effects even out in the long run. But how many are needed?
This depends on the degree of accuracy required. When estimating a parameter value or comparing estimates for alternative schemes, keep an updated confidence interval for the parameter value, stopping when it is narrow enough. After a few simulations there may be enough information to predict how many are going to be required.

1.7 Practicalities of simulation

These will be discussed later on in the course.