Monte-Carlo Simulation

Fortune tellers, palm readers, the Farmer’s Almanac, financial analysts. What do these things have in common? They all attempt to anticipate the future. While some use a crystal ball, the ridges of their subject’s palm, or sunspots, financial analysts use numeric models and mathematical techniques to generate simulations of their client’s financial future.

There are two kinds of forecasting models: deterministic models and stochastic models. Deterministic models assume a fixed relationship between the inputs and the output, while stochastic models depends on inputs that are influenced by chance. Deterministic models can be solved analytically via mathematical formulas, while stochastic models require numerical solutions. To solve stochastic models numerically, one must try various values for the model’s parameters and variables. When these variables come from a sequence of random numbers, the solution is called Monte-Carlo simulation.

Monte-Carlo simulation was originally introduced by financial analysts John Von Neumann and Stanislaw Ulam while working on the Manhattan Project at the Los Alamos National Laboratory. They invented a procedure of substituting a random sequence of numbers into equations to solve problems regarding the physics of nuclear explosions. The term Monte-Carlo was inspired by the gambling casinos in Monaco.

Monte-Carlo simulation is performed with a sequence of numbers that are distributed uniformly, are independently of each other, and are random. Random sequences can be generated using mathematical techniques such as the mid-square method, but most spreadsheet software includes random-number generators. It is important to note that most financial analysis applications generate random variables that are not distributed uniformly. In order to perform a Monte-Carlo simulation, the sequence of uniformly distributed random numbers must be transformed into a sequence of normally distributed random numbers.

Applying Monte-Carlo Simulation to the Stock Market

Imagine we invest $100,000 to an S&P 500 index fund in which the dividends are then reinvested, and we want to predict the value of that investment 10 years from now. As previously mentioned, we start by generating a series of 10 random numbers that are uniformly distributed. However, we assume that the S&P’s returns are normally distributed and we therefore need to transform our sequence. This transformation can be accomplished easily by applying the Central Limit Theorem.

The following table shows the relative frequencies of two variables, X and Y.

Value	X	Y	(X+Y)/2
1.0	1/6	1/6	1/36
1.5	0	0	1/18
2.0	1/6	1/6	1/12
2.5	0	0	1/9
3.0	1/6	1/6	5/36
3.5	0	0	1/6
4.0	1/6	1/6	5/36
4.5	0	0	1/9
5.0	1/6	1/6	1/12
5.5	0	0	1/18
6.0	1/6	1/6	1/36

It is clear to see that while neither X nor Y are normally distributed on their own, their average beings to approach a normal distribution. Therefore, we can create a sequence of random numbers that is normally distributed by taking the averages of many sequences of random, uniformly distributed numbers.

Uniform Sequence	Average of 30 Uniform Squences
0.6471	0.4965
0.4162	0.4336
0.5691	0.5747
0.2006	0.4477
0.4685	0.5014
0.7442	0.5126
0.9439	0.4930
0.5556	0.6040
0.2480	0.5267
0.3644	0.4721

Figure 1 shows the relative frequency of both sequences, and we can see from this graph that the average of the 30 uniformly distributed sequences approached a normal distribution.

Figure 1: Relative Frequencies

The next step in Monte-Carlo simulation is to scale the normally distributed sequence so it has a standard deviation of one and a mean of zero. By dividing each observation by 0.05 (the theoretical standard deviation), and then subtracting 10 (the theoretical mean of this sequence) from each other its observations, we can achieve this transformation.

Standardizing a Normally Distributed Sequence

Next, we must rescale our sequence to reflect our assumptions about the mean return and standard deviation of the S&P 500. For this exercise, let us believe that the average return of the S&P 500 is 12% and its standard deviation is 20%. We rescale our standardized normal distribution by multiplying each observation by our assumption of 20% for the standard deviation, and adding to this value our assumption of 12% for its average return. Now we have a sequence of returns that we can then use to simulate our investments performance (column E).

Example: Normally Distributed Random Sequence of S&P 500 Returns

To carry out the Monte-Carlo simulation, we then link the sequence of random returns and multiply the result by 100,000 (our investment) to derive an estimate of its value in 10 years. For example, the simulated returns in Table 4 yield a value of $333,810.

We repeat the entire process, beginning with the generation and averaging of 30 random sequences. We proceed until we generate a sufficiently large quantity of estimates. The distribution of these estimates is the solution to our problem. The figure below shows the frequency distribution of the terminal value of $100,000 over 10 years, resulting from 100 simulations.

Figure 2: Frequency Distribution for $100,000 Ten-Years Forward; based on 100 simulations

While many problems can be solved using an analytical solution, Monte-Carlo simulation is immensely more simple for problems that are too complex to be described by equations. However, in order for Monte-Carlo simulation to obtain a reliable result, it must be repeated enough times.