Understanding Distributions & Correlations in Monte Carlo Real Estate Analysis

Understanding Distributions & Correlations in Monte Carlo Real Estate Analysis

probability distributions❓: The Foundation of Uncertainty

Monte Carlo simulation thrives on representing uncertain inputs as probability distributions. These distributions quantify the range of possible values for a variable and the likelihood of each value occurring. Selecting the appropriate distribution is crucial for model accuracy and relevance.

• Definition of a Probability Distribution: A mathematical function that describes the probability of different outcomes for a random variable.
• Key Considerations:
  • Theoretical Consistency: The distribution should align with underlying financial or economic theory. For example, stock prices and nominal interest rates cannot be negative, therefore distributions allowing for negative values might be inappropriate.
  • Data Fit: The distribution should adequately fit historical or available data. Various statistical tests (e.g., Chi-squared, Kolmogorov-Smirnov) can assess the goodness-of-fit.
  • Data Type: Is the data continuous or discrete? While discrete data is sometimes treated as continuous (especially with large sample sizes), certain distributions are specifically designed for discrete variables.
  • Interdependencies: Correlations between random variables must be considered when selecting and defining distributions.

Common Probability Distributions in Real Estate Analysis

• Normal Distribution: Defined by its mean ($\mu$) and standard deviation ($\sigma$). Symmetrical around the mean.

  $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$

  • Useful for variables like expense growth rates, where values tend to cluster around an average.
  • Limitation: Allows for negative values, which may be unrealistic for some real estate variables (e.g., rental rates, cap rates).

• Lognormal Distribution: The logarithm of the variable follows a normal distribution. Positively skewed (long tail to the right).

  $f(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(ln(x)-\mu)^2}{2\sigma^2}}$, for $x > 0$

  • Suitable for variables that cannot be negative and may exhibit exponential growth (e.g., property values, rental income).
  • Commonly used for stock prices, as they are typically lognormal and skewed to the right. A lognormal distribution ensures prices never dip below zero.

  • Important Note: In a stochastic world with lognormal price fluctuations along an exponential trend, the price in period T is:

    $P_T = P_0 \cdot e^{[(\mu - 0.5 \cdot \sigma^2) \cdot T + \sigma \cdot Z \cdot \sqrt{T}]}$

    where:
    * $P_T$ is a random lognormal variable.
    * Z is a standard normal random variable with mean 0 and standard deviation 1.
    * $\mu$ is the growth rate
    * $\sigma$ is the standard deviation
    * $0.5 \cdot \sigma^2$ is a correction to account for the random nature of the lognormal variable, especially significant in high-volatility markets.

• Triangular Distribution: Defined by its minimum (a), maximum (b), and most likely (mode, c) values.

  $f(x) = $$\begin{cases} \frac{2(x-a)}{(b-a)(c-a)}, & \text{for } a \le x \le c \\ \frac{2(b-x)}{(b-a)(b-c)}, & \text{for } c \le x \le b \\ 0, & \text{otherwise} \end{cases}$$ $

  • Useful when limited data is available but estimates for the minimum, maximum, and most likely values can be made.
  • Can be symmetrical or skewed depending on the mode’s position relative to the minimum and maximum.
  • Appropriate when not much data exists to estimate.

• Uniform Distribution: All values within a defined range have equal probability. Defined by a minimum and maximum value.

  $f(x) = $$\begin{cases} \frac{1}{b-a}, & \text{for } a \le x \le b \\ 0, & \text{otherwise} \end{cases}$$ $

  • Simplest distribution; useful when little is known about the variable other than its possible range.
  • Can be applied when you have a prior belief that any result in an interval is equally likely.

• Discrete Distributions: Used for variables that can only take on specific, distinct values (e.g., number of tenants, occupancy rates in whole percentages). Examples include:

  • Binomial Distribution: Models the probability of success in a fixed number of independent trials.
  • Poisson Distribution: Models the number of events occurring in a fixed interval of time or space.

Fitting Distributions to Data

1. Collect Data: Gather historical data❓ or expert opinions related to the variable being modeled.
2. Visualize Data: Create histograms, scatter plots, and other visualizations to understand the data’s characteristics (e.g., shape, skewness, kurtosis).
3. Select Candidate Distributions: Based on the data’s characteristics and theoretical considerations, choose potential probability distributions.
4. Estimate Parameters: Estimate the parameters of each candidate distribution using statistical methods (e.g., maximum likelihood estimation, method of moments).
5. Goodness-of-Fit Tests: Perform statistical tests (e.g., Chi-squared, Kolmogorov-Smirnov) to assess how well each candidate distribution fits the data.
6. Select Best-Fitting Distribution: Choose the distribution with the best goodness-of-fit based on the test results and visual inspection.

Non-Parametric Approaches: Histograms

• When a suitable mathematical distribution cannot be found, a histogram can be used.
• The data is divided into bins, and the proportion of data in each bin is calculated.
• The art lies in determining the number and width of each bin, which impacts the histogram’s shape and representation of the underlying distribution.
• Consider a priori knowledge or theoretical constraints when determining the shape of the distribution.

Box-Whisker Plots (Box Plots)

• A convenient way to represent batches of data, especially data that is not normally distributed.
• A compact way to depict groups of numerical data through five-number summaries:
  • Smallest observation (sample minimum).
  • Lower quartile (25th percentile).
  • Median (50th percentile).
  • Upper quartile (75th percentile).
  • Largest observation (sample maximum).
• The width of 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.
• If the mean is located to the right of the median, the distribution is skewed to the right or non-symmetrical.

Correlations: Modeling Interdependencies

In real estate, variables are rarely independent. Correlations capture the relationships between different random variables. Ignoring correlations can lead to unrealistic simulation results and inaccurate risk assessments.

• Definition of Correlation: A statistical measure of the degree to which two or more variables tend to vary together.

• Correlation Coefficient (r): A value between -1 and +1 that indicates the strength and direction of a linear relationship between two variables.

  • r = +1: Perfect positive correlation (as one variable increases, the other increases proportionally).
  • r = -1: Perfect negative correlation (as one variable increases, the other decreases proportionally).
  • r = 0: No linear correlation.
  • Important Note: Correlation does not imply causation. A correlation between two variables may be due to a third, unobserved variable.

Estimating Correlations

1. Historical Data: Calculate correlation coefficients from historical data.
2. Expert Opinion: Elicit expert opinions to estimate correlations when historical data is limited or unreliable. This could involve surveys or structured interviews with real estate professionals.
3. Regression Analysis: Use regression analysis to identify and quantify relationships between variables. However, be mindful of potential issues like simultaneity and two-way causality. If the error term and explanatory variables are correlated, ordinary least squares produces inconsistent and biased results. Other methods, such as two-stage least squares, are required to deal with the two-way causality.

Modeling Correlations in Monte Carlo Simulation

• Copulas: Advanced statistical functions that allow for modeling complex dependencies between variables, even when the marginal distributions are different. Copulas separate the marginal distributions from the dependence structure, providing greater flexibility in modeling correlations. They are particulary relevant when the random variables are non-normal.
• Cholesky Decomposition: A mathematical technique used to generate correlated random numbers from independent random numbers. It involves decomposing the correlation matrix into a lower triangular matrix.
  1. Create the Correlation Matrix: This matrix represents the pairwise correlations between all random variables. For n variables, the matrix is n x n, with 1s on the diagonal (each variable is perfectly correlated with itself).
  2. Perform Cholesky Decomposition: Decompose the correlation matrix (C) into a lower triangular matrix (L) such that C = L * L^T, where L^T* is the transpose of L.
  3. Generate Independent Random Numbers: Generate a set of n independent random numbers (Z) from the specified distributions (e.g., standard normal).
  4. Calculate Correlated Random Numbers: Multiply the lower triangular matrix (L) by the vector of independent random numbers (Z) to obtain a vector of correlated random numbers (X): X = L * Z*. The resulting vector X contains random numbers that are correlated according to the original correlation matrix C.

• Example: Consider two variables, Rental Growth (RG) and Cap Rate (CR), with a correlation of -0.5.

  • Correlation Matrix:
    > $C = $$\begin{bmatrix} 1.0 & -0.5 \\ -0.5 & 1.0 \end{bmatrix}$$ $
  • Cholesky Decomposition:
    > $L = $$\begin{bmatrix} 1.0 & 0.0 \\ -0.5 & \sqrt{0.75} \end{bmatrix}$$ = $$\begin{bmatrix} 1.0 & 0.0 \\ -0.5 & 0.866 \end{bmatrix}$$ $
  • Generate Independent Random Numbers: Let’s say we generate two independent standard normal random numbers: Z₁ = 0.5, Z₂ = -0.2.
  • Calculate Correlated Random Numbers:
    > $ $$\begin{bmatrix} RG \\ CR \end{bmatrix}$$ = $$\begin{bmatrix} 1.0 & 0.0 \\ -0.5 & 0.866 \end{bmatrix}$$ $$\begin{bmatrix} 0.5 \\ -0.2 \end{bmatrix}$$ = $$\begin{bmatrix} 0.5 \\ -0.423 \end{bmatrix}$$ $
    > So, the correlated random numbers are RG = 0.5 and CR = -0.423. These numbers are now correlated with a coefficient of -0.5.

Examples of Correlation in Real Estate

• Employment Growth and Vacancy Rates: Negative correlation. As employment grows, demand for office space increases, leading to lower vacancy rates.
• Rental Growth and Cap Rates: Negative correlation. Higher rental growth expectations often lead to lower cap rates (higher property valuations).
• Inflation and Interest Rates: Positive correlation. Higher inflation typically leads to higher interest rates as central banks try to control inflation.

Practical Applications and Experiments

1. Property Valuation: Simulate property values based on correlated variables like rental income, expense growth, and cap rates. Experiment with different correlation scenarios (e.g., positive vs. negative correlation between rental income and expense growth) to see how they impact the distribution of possible property values.
2. Portfolio Optimization: Analyze a portfolio of real estate investments, considering the correlations between the returns of different property types or geographic locations. Identify diversification strategies that minimize risk by investing in assets with low or negative correlations.
3. Development Project Feasibility: Model the feasibility of a new development project, incorporating correlations between construction costs, rental rates, and occupancy rates. Assess the project’s profitability under different economic scenarios.
4. Lease Analysis: Account for correlated variables such as inflation and operating expenses when evaluating upward-only adjusting leases. Volatility is the critical input, as the price of the stock already reflects the expected rate of growth when evaluating options.
5. Bidding Wars (Winner’s Curse): Monte Carlo analysis can help investors avoid the winner’s curse by revealing the risks associated with bidding wars, especially in the presence of hidden information and volatility.

Building a Monte Carlo Model

1. Structure: Define the model’s structure (e.g., pro forma, discounted cash flow).
2. Variables and Constants: Identify the key variables (random and deterministic) and constants.
3. Structural Relationships: Establish the relationships between the variables (e.g., how rental income affects net operating income).
4. Software: Choose appropriate Monte Carlo simulation software (e.g., Excel add-ins, specialized simulation software).
5. Relevance: Focus on the most relevant events for modeling, keeping in mind that knowing the difference between relevant and irrelevant events is more art than science.

Effect of Variations in Correlation and Standard Deviation on Output

Sensitivity analysis can be used to explore alternative assumptions with regard to the shape of distributions and the correlations among these distributions.

For example, by increasing the standard deviation of the rental growth rate (investing in a riskier market without changing the discount rate), the discounted value is only barely affected, while the promote increases dramatically. Likewise, reducing the correlation between the rental growth rate and the cap rate from zero to -1.0 produces even more dramatic results.