Monte Carlo Modeling: Distributions, Correlations, and Output Analysis

Monte Carlo Modeling: Distributions, Correlations, and Output Analysis

Introduction to Monte Carlo Simulation

Monte Carlo simulation is a powerful computational technique that uses random sampling to obtain numerical results. It is particularly useful for modeling systems with uncertainty, where analytical solutions are intractable or unavailable. In the context of real estate investment analysis, Monte Carlo allows us to simulate a range of possible outcomes by incorporating uncertainty in key input variables, such as rental growth❓, expense growth, and cap rates. This provides a more comprehensive risk assessment compared to deterministic approaches.

• Deterministic Analysis: Focuses on single-point estimates and sensitivities. It typically considers only a few values for each variable, ignoring the full distribution of possibilities and potential correlations.

• Monte Carlo Analysis: Employs every possible value of a random variable, weighting each value by its probability of occurrence. This calculates a probability distribution that reflects the combined impact of uncertainty and correlations.

Probability Distributions

The cornerstone of Monte Carlo simulation is the selection of appropriate probability distributions for each input variable. The choice of distribution directly impacts the accuracy and reliability of the simulation results.

Types of Probability Distributions

• Continuous Distributions: Used for variables that can take on any value within a given range.

  • Normal Distribution: Symmetric and defined by its mean (μ) and standard deviation (σ). Suitable for variables where values tend to cluster around the mean. Its probability density function (PDF) is given by:

    f(x) = (1 / (σ * sqrt(2 * π))) * e^(-((x - μ)^2) / (2 * σ^2))

  • Lognormal Distribution❓❓: The logarithm of the variable is normally distributed. It’s useful for modeling variables that cannot be negative, such as stock prices and certain real estate values. It is characterized by a right skewness.
    > Stock prices are typically lognormal and skewed to the right.
    > A lognormal distribution cannot include negative numbers,
    > which is equivalent to saying that a stock price can never be
    > negative.

  • Triangular Distribution: Defined by its minimum (a), maximum (b), and most likely (c) values. Useful when limited data is available, and expert judgment is used to estimate the range and most probable value. Its PDF is:
    f(x) = { (2(x - a)) / ((b - a)(c - a)) for a <= x <= c (2(b - x)) / ((b - a)(b - c)) for c <= x <= b 0 otherwise }

  • Uniform Distribution: All values within a specified range are equally likely. Useful when there is no information to suggest any particular value is more probable than another.

  • Beta Distribution: Defined on the interval [0, 1]. Flexible shape parameters allow it to represent a wide range of distribution shapes. Often used to model probabilities or proportions.

  • Exponential Distribution: Models the time until an event occurs in a Poisson process.

• Discrete Distributions: Used for variables that can only take on a finite number of values.

  • Bernoulli Distribution: Models the probability of success or failure of a single trial.

  • Binomial Distribution: Models the number of successes in a fixed number of independent Bernoulli trials.

  • Poisson Distribution: Models the number of events occurring within a fixed interval of time or space.

Considerations When Selecting Distributions

• Theoretical Consistency: Choose distributions that align with the underlying theory. For instance, use lognormal for variables that cannot be negative.

• Data Fitting: The distribution should fit the historical data as closely as possible. Statistical tests (e.g., Kolmogorov-Smirnov, Chi-squared) can be used to assess the goodness-of-fit.

• Continuous vs. Discrete: Determine whether the data is continuous or discrete. Discrete data can sometimes be treated as continuous, especially with a large number of observations.

• Histogram Approach: If a standard mathematical distribution doesn’t fit the data adequately, use a histogram. Divide the data into bins and calculate the proportion of data in each bin. Careful consideration should be given to bin number and width.

Fitting Distributions to Data

1. Collect Data: Gather historical data for the variable being modeled.
2. Visualize Data: Create histograms or other plots to visualize the data’s distribution.
3. Identify Candidate Distributions: Based on the data’s characteristics and theoretical considerations, select potential distributions.
4. Estimate Parameters: Estimate the parameters of each candidate distribution using statistical methods (e.g., maximum likelihood estimation).
5. Assess Goodness-of-Fit: Use statistical tests to compare the fit of each candidate distribution to the data.
6. Select Best-Fitting Distribution: Choose the distribution that provides the best fit based on the goodness-of-fit tests and other relevant factors.

Correlation

Correlation measures the statistical relationship between two or more variables. In Monte Carlo simulation, accurately accounting for correlations is crucial to avoid misleading results. If two variables are positively correlated, when one increases, the other tends to increase as well. Conversely, if they are negatively correlated, when one increases, the other tends to decrease.

• Correlation Coefficient (ρ): A measure of the linear relationship between two variables. It ranges from -1 to +1.

  • ρ = +1: Perfect positive correlation.
  • ρ = -1: Perfect negative correlation.
  • ρ = 0: No linear correlation.

Methods for Incorporating Correlation

1. Cholesky Decomposition: A method for generating correlated random variables from uncorrelated ones. It involves decomposing the correlation matrix into a lower triangular matrix (L) such that the correlation matrix (C) can be expressed as:

   C = L * L'

   Where L’ is the transpose of L. Uncorrelated random variables are then multiplied by L to obtain correlated random variables.

   • Generate a vector of uncorrelated standard normal random variables, Z = [Z1, Z2, ..., Zn]'.
   • Calculate the correlated random variables X = L * Z, where X = [X1, X2, ..., Xn]'.
   • Transform the correlated standard normal variables to the desired distributions for each input variable.

2. Copulas: Functions that describe the dependence structure between random variables, independent of their marginal distributions. They allow for more flexible modeling of correlations, especially for non-linear dependencies.

Considerations for Correlation

• Spurious Correlation: Be aware of spurious correlations, which may appear statistically significant but are not causally related.
• Non-Linear Dependencies: Linear correlation coefficients may not fully capture the relationship between variables if the dependency is non-linear. Consider using copulas or other techniques to model non-linear dependencies.
• Time-Varying Correlations: Correlations can change over time, especially during periods of market stress.

Output Analysis

Once the Monte Carlo simulation is complete, the results need to be analyzed to draw meaningful conclusions. The simulation generates a distribution of possible outcomes for the variables of interest (e.g., net present value, internal rate of return).

Methods for Analyzing Output

1. Histograms: Visualize the distribution of the output variable. Histograms show the frequency of occurrence of different values.
2. Summary Statistics: Calculate descriptive statistics such as mean, median, standard deviation, skewness, kurtosis, and percentiles.
   • Mean: The average value.
   • Median: The middle value.
   • Standard Deviation: A measure of the dispersion of the data around the mean.
   • Skewness: A measure of the asymmetry of the distribution. A positive skewness indicates a long right tail, while a negative skewness indicates a long left tail.
   • Kurtosis: A measure of the “tailedness” of the distribution. High kurtosis indicates fat tails (more extreme values), while low kurtosis indicates thin tails.
   • Percentiles: The value below which a given percentage of the data falls. For example, the 25th percentile is the value below which 25% of the data lies.
3. Box-Whisker Plots: A graphical representation of the distribution, showing the median, quartiles, and outliers. The box represents the interquartile range (IQR). The band near the middle is the 50th percentile, or the median. The cross is the mean. The lowest whisker represents data within 1.5 IQR of the lower quartile; the highest within 1.5 IQR of the upper quartile. Data beyond the whiskers are plotted as open squares. The solid outliers are the most extreme data points.
   > Box plots are compact and are especially amenable to data
   > that is highly skewed, flat or peaked, or multi-modal. For
   > example, if the mean is located to the right of the median, the
   > distribution is skewed to the right or non-symmetrical.
4. Sensitivity Analysis: Examine how changes in the input variables affect the output. This helps identify the key drivers of risk.
   • Vary input parameters (e.g., standard deviation, correlation) and observe the impact on the output distribution.
5. Scenario Analysis: Create different scenarios by setting specific values for the input variables. This allows for the evaluation of potential outcomes under different conditions.

Interpretation of Results

• Probability of Loss: Determine the probability of the output variable falling below a certain threshold (e.g., probability of a negative net present value).
• Value at Risk (VaR): Estimate the maximum potential loss at a given confidence level. For example, the 95% VaR is the loss that is not expected to be exceeded 95% of the time.
• Expected Shortfall (ES): The expected value of the loss, given that the loss exceeds the VaR.
• Decision Making: Use the simulation results to make informed decisions about real estate investments, considering the potential risks and rewards.

Example: Stochastic Price Calculation

In Monte Carlo analysis, it’s crucial to account for stochastic growth correctly. If prices fluctuate randomly along an exponential trend, the price in period T is given by:

PT = P0 * e^((μ - 0.5 * σ^2) * T + σ * Z * sqrt(T))

Where:

• PT is the random lognormal price at time T.
• P0 is the initial price.
• μ is the expected growth rate.
• σ is the volatility (standard deviation of the log returns).
• Z is a standard normal random variable (mean 0, standard deviation 1).
• T is the time period.

In the authors’ opinion, some analysts incorrectly believe that
the exponential path is defined just by µ, but it is not if the
variable is random. The greater the standard deviation, the
more the distribution spreads out to the right over time.
(Remember that a lognormal distribution means the price can
never be negative.) Therefore, the mean value increases with
greater volatility. This correction, 0.5 • σ2, can be significant
in high volatility markets. If σ is zero, then the above formula
simplifies to the standard exponential growth formula.

Building a Monte Carlo Model

The process of building a Monte Carlo model involves several key steps:

1. Define the Model Structure: Establish the mathematical relationships between the input and output variables. This involves creating a pro forma or other financial model that incorporates the relevant variables.
2. Identify Random Variables: Determine which variables are subject to uncertainty and should be treated as random variables.
3. Select Probability Distributions: Choose appropriate probability distributions for each random variable, based on historical data, theoretical considerations, and expert judgment.
4. Estimate Correlation: Estimate the correlation between the random variables.
5. Run the Simulation: Use a Monte Carlo simulation software package to run the simulation. The software will generate a large number of random samples from the specified distributions and calculate the output variable for each sample.
6. Analyze the Output: Analyze the distribution of the output variable using the methods described above.
7. Validate the Model: Validate the model by comparing the simulation results to historical data or other benchmarks.

Practical Application

The urge to buy and overpay is irrepressible even within a
slowly recovering market. How can investors avoid the
winner’s curse?5 Monte Carlo analysis can provide some
important insights, especially in the presence of hidden
information and volatility, and can reveal risk.

Example: Modeling Bidding Wars and the Winner’s Curse

Monte Carlo simulation can be used to model bidding wars and assess the risk of the winner’s curse. In a bidding war, multiple parties compete to acquire an asset, often driving up the price. The winner’s curse occurs when the winning bidder overestimates the value of the asset and pays too much.

1. Estimate the Distribution of Asset Values: Assume that the true value of the asset is uncertain and can be represented by a probability distribution (e.g., normal, lognormal).
2. Model Bidding Behavior: Model the bidding behavior of the participants. Assume that each bidder has their own estimate of the asset’s value, which may be based on incomplete or noisy information. Bidders may bid strategically, considering the number of bidders and their perceived level of competition.
3. Run the Simulation: Simulate the bidding process multiple times, drawing random samples from the distribution of asset values and modeling the bidding behavior of the participants.
4. Analyze the Results: Analyze the distribution of winning bids and compare them to the distribution of true asset values. This will provide insights into the probability of the winner’s curse and the expected overpayment.
5. Adjust Bidding Strategy: Use the simulation results to adjust the bidding strategy and mitigate the risk of the winner’s curse. For example, a bidder may choose to bid more conservatively or to walk away from the auction if the price exceeds a certain threshold.