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P1.T2.400. Fabozzi on simulations

Nicole Seaman

Chief Admin Officer
Staff member
Describe different ways of choosing probability distributions in creating simulation models. Describe the relationship between the accuracy of a simulation model and the number of scenarios run in the simulation. Understand and interpret the results generated by Monte Carlo simulation.


400.1. Barbara the Analyst has been tasked to chose a univariate probability distribution in order to create a simulation model of future market returns. One of her key steps utilizes the chi-square distribution. Among the following four methods reviewed by Pachamanova and Fabozzi, which approach is she most likely using?

a. Bootstrapping
b. Assume a distribution, then use historical data to estimate parameters
c. Use historical data to find a distribution
d. Ignore the past and look forward with a subjective choice of distribution

400.2. Peter the Analyst has generated 800 independent scenarios of future single-period portfolio values. He observes the mean (average) of his simulated output distribution and determines a 95.0% confidence interval with a length of approximately $300.00; i.e., length is the difference between the upper and lower bound of the confidence interval. Peter's manager wants him to increase the accuracy of his estimate of the population's mean by reducing the length of the confidence interval to about $60.00. How many scenarios should Peter run?

a. 800; no change in trials but increase the confidence level
b. 4,000
c. 7,200
d. 20,000

400.3. According to Pachamanova and Fabozzi each of the following is true about understanding and interpreting the results generated by Monte Carlo simulation, except which is false?

a. A simulation model applies an input probability distribution(s) to a deterministic model in order to generate many scenarios (a.k.a., trials) which produce output variables and the corresponding output probability distribution
b. Despite several advantages, the key weakness (drawback) of simulations is an inability to generate statistical measures of central tendency and volatility for the output probability distribution
c. Simulation is similar to statistical sampling in that we try to represent uncertainty by generating scenarios, that is, “sampling” values for the output parameter of interest from an underlying probability distribution
d. The simulated output's minimum and the maximum are highly sensitive to the number of simulated values and whether the simulated values in the tails of the distribution provide good representation for the tails of the distribution

Answers here:

David Harper CFA FRM

David Harper CFA FRM
Staff member
HI @[email protected]

I hope my 400.1 is fair (I found it difficult to quiz this reading), here is what I have as the answer (I just edited to acknowledge A is a good try):

400.1. C. Use historical data to find a distribution
Choice (A) is also good, but given the chi-square is a test of goodness-of-fit such that (C) is better. Fabozzi (emphasis mine):
"The first question you need to ask yourself when creating the simulation model about the future values of your funds is what distribution is appropriate for modeling the future market returns. One possible starting point is to look at a historical distribution of past returns, and assume that the future will behave in the same way. When creating scenarios for future realizations, then, you can draw randomly from historical scenarios. This is a very simple approach, which is based on the bootstrapping technique described in section 3.11.3.

Another possibility is to assume a particular probability distribution for future returns, and use historical data to estimate the parameters of this distribution, that is, the parameters that determine the specific shape of the distribution, such as the expected value (µ) and standard deviation (σ) for a normal distribution (see section 3.4), λ for a Poisson distribution (see section 3.7.1), or α and β for a beta distribution (see section 3.7.2). For example, if you assume a normal distribution for returns, then you can use the historical variability of returns as a measure of the standard deviation σ of this normal distribution, and the historical average (mean) as the expected return µ of the normal distribution.

A third approach is not to start out with a particular distribution, but to use historical data to find a distribution for returns that provides the best fit to the data. As we mentioned in sections 3.7.2 and 3.11.4, the chi-square hypothesis test is one possible goodness-of-fit test. Other goodness-of-fit tests include the Kolmogorov-Smirnov (K-S) test, the Anderson-Darling (A-D) test, and root-mean-squared-error (RMSE). Most simulation software packages, including MATLAB and @@risk, have commands that can test the goodness of fit for different probability distributions.

Yet a fourth way is to ignore the past and look forward, constructing a probability distribution based on your subjective guess about how the uncertain variable in your model will behave. For example, using the beta distribution from Exhibit 3.18(A) to model the future market return will express a more pessimistic view about the market than using the beta distribution in Exhibit 3.18(B) or a normal distribution because most of the probability mass in the distribution in 3.18(A) is to the left, so low values for return will happen more often when scenarios are generated. It is important to realize that none of these approaches will provide the answer. Simulation is a great tool for modeling uncertainty, but the outcome is only as good as the inputs we provide to our models. We discuss ways for defining input distributions in specific applications in the book. The art of simulation modeling is in providing good inputs and interpreting the results carefully."