Monte Carlo Simulation

Monte Carlo Simulation: A simple Example

Modeling uncertainty (by probability distributions) gives the opportunity to make risk analysis.

Simulation risk models use as input uncertainties by probability distributions. Then, the variables of this simulations are recorder letting us analyze them and obtain their characteristics.

Taking the next formula is possible to obtain the expected capital in a future. The tilde is for uncertain quantities and random variables.

C̄t+1= Ct + r̃,t,t+1 Ct or C̄t= (1+ r̃t,t+1)Ct

C̄t+1 Amount of capital at the end of the period

r̃t,t+1 Market return over the time period

Ct Initial investment

The return over a period of time can be computed as:

P Value of the investment

Dt Amount of Dividends

This only will give us an idea of the possible values of the investment. If we want a better result, more attached to reality we must generate different scenarios for the return over the year. This can be done by probability distributions that represent the different scenarios and with the outcomes create a histogram. The more scenarios simulated, a better representation of the distribution will be obtained.

Selecting Probability Distributions for the Inputs

What distribution is appropriate for modeling the future returns? Here we have some options:

-Historical distribution of the past results. Assume the future will behave in the same way.

-Particular probability distribution for future returns, using historical data to estimate the parameters of the distribution (expected value: μ; and standard deviation: σ; Poison distribution: λ; Beta distribution: α or β).

-Start out with a particular distribution, that best fit the data (chi-square hypothesis test, Kolmogorov-Smirnov test, Anderson-Darling test, root-mean-squared-error).

-Contruct a probability distribution based on your subjective guess. Ignore the past and look forward.

None of this will provide the answer, but they will be a good tool to modeling uncertainty.

Interpreting Monte Carlo Simulation Output

We need to simulate a certain number of scenarios, by taking the standard deviation and mean of a certain investment. Then, the output will be represented as an histogram. This histogram is based on the sample of scenarios that was created, and this sample will have certain characteristics than can be calculated: Mean, standard deviation, mode, skewness, Kurtosis, a Minimum an a Maximum.

With this calculated characteristics we can calculate confidence intervals based on percentages. This is done with the next formula:

Then we will obtain two values, between a certain confidence level. That means that based on a percentage of security, the real value of the investment will be between this two values.

Why use simulation?

Simulation enables us to evaluate a function of a random variable. Also it allows us to visualize probability the probability distribution resulting from compounding probability distributions for input variables, to incorporate correlations between input variables and to be able to change the strategy at a low-cost.

Multiple Input Variables and Compounding Distributions

Facts to keep in mind when dealing with multiple input probability distributions:

-When adding a constant to a random variable, the distribution is shift to the right by 1.

-Adding

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