Random Numbers and Applied Simulation

Nigel Da Costa Lewis

Simulation is central to modern risk management. The two primary simulation approaches implemented for market risk analysis are historical simulation and Monte Carlo simulation. In this chapter, both of these techniques will be described. In addition, a fully worked case study is also presented, which explores the incremental return and risk to long-term retirement wealth accumulation that arises from the strategic inclusion of gold in lifecycle investing. Before discussing these simulation methods and presenting the case study, we need to know a little about generating random numbers and using these numbers to generate suitable probability distributions for risk factors.

RANDOM NUMBER GENERATION

Random numbers lie at the heart of simulation because they form the basis for generating risk factor probability distributions. A sequence of simulated random numbers should have two basic properties: uniformity and independence. Let {z₁, z₂, …, z_N} be a sequence of random variables, where z_max and z_min are the maximum and minimum value in the sequence, respectively. If we divide the interval [z_min, z_max ] into k sub-intervals of equal length, uniformity implies that the expected number