Beyond Standard Deviation: Quantifying Real Estate Risk

Beyond Standard Deviation: Quantifying Real Estate Risk

Limitations of Standard Deviation in Real Estate

While standard deviation is frequently used as a proxy for risk, particularly in finance, its application to real estate requires careful consideration due to the unique characteristics of the asset class.

• Standard Deviation as Risk in a Random Walk: The notion of standard deviation equaling risk stems from the assumption that asset price changes follow a random walk with drift. In this model, the best forecast is the historical mean, and the standard deviation quantifies the uncertainty around that mean.

• Equation for Random Walk with Drift:

  P(t) = P(t-1) + μ + ε(t)

  where:

  • P(t) is the price at time t.
  • μ is the drift (average change in price).
  • ε(t) is a random error term with a mean of zero and standard deviation σ.
  • The standard deviation (σ) of the error term is often used as a measure of risk in this context.

• Real Estate vs. Random Walk: Applying this model directly to real estate can be problematic because real estate forecasting often involves more sophisticated methods that consider market cycles and other factors, rather than simply relying on historical averages.

• Long-Term Investment Horizon: Real estate investments typically have a long-term horizon, which can make the assumptions of a simple random walk less applicable.
• Data Frequency and Availability: Real estate data is often quarterly or annual, spanning a limited historical period (e.g., a decade or two), in contrast to the high-frequency (daily) data available for stocks, which can be tracked over decades. This limited data can affect the accuracy of standard deviation estimates.
• Serial Correlation: Real estate returns often exhibit serial correlation, meaning that returns in one period are correlated with returns in subsequent periods. This can bias standard deviation downward, underestimating the true risk.

Monte Carlo Simulation: A Probabilistic Approach

Monte Carlo simulation offers a more robust approach to risk assessment by generating a range of possible outcomes based on probability distributions.

• Core Principle: The simulation uses the mean and standard deviation of key variables (e.g., value change, rent growth) to create a large number of possible forecasts.

• Process: Period-by-period growth rates are randomly selected from a specified distribution (e.g., normal distribution) and applied to the initial value or other starting point. This process is repeated hundreds or thousands of times, resulting in a distribution of potential outcomes.

• Benefits over Scenario Analysis: Unlike scenario analysis, which focuses on a few discrete scenarios, Monte Carlo simulation provides a comprehensive probability distribution of potential results.

• Value at Risk (VaR): Monte Carlo simulations facilitate the calculation of VaR, which represents the maximum loss expected over a given time horizon at a specified confidence level. For instance, a 2% VaR indicates the level below which only 2% of the simulated outcomes fall.

  • This approach avoids the subjective and often difficult process of defining sufficiently severe scenarios.

• Equation for VaR:

  VaR = - [μ + σ * z(α)] * V

  where:

  • VaR is the Value at Risk
  • μ is the expected return
  • σ is the standard deviation of returns
  • z(α) is the z-score corresponding to the desired confidence level α (e.g., for a 95% confidence level, α = 0.05 and z(α) ≈ -1.645)
  • V is the value of the asset or portfolio

• Limitations in Real Estate:

  • Long-Term Horizon: Applying a random walk approach to multi-year real estate investments can result in a wide variance of outcomes, particularly in later forecast periods.
  • Data Dependence: The accuracy of the Monte Carlo simulation is limited by the quality and quantity of historical data used to estimate the distributions.
  • Lack of Deterministic Description: The simulation does not explicitly identify the events that would lead to specific outcomes (e.g., the 2% VaR level). It doesn’t trace back to specific inputs that caused that output.

Standard Error of the Estimate (SEE): A Forward-Looking Approach

The Standard Error of the Estimate (SEE) offers a forward-looking approach to quantifying risk, particularly when using econometric models.

• Econometric Models: These models aim to capture the cyclical dynamics of real estate, unlike purely stochastic models.
• Distinction from Historic Standard Deviation: Unlike historical standard deviation❓❓, which measures the volatility of past data, the SEE quantifies the uncertainty around the forecast generated by the econometric model.

• Structured Vector-Autoregressive (VAR) Model: To obtain the SEE for a system of equations, the system must be converted into a structured VAR model.

  Y(t) = c + A1*Y(t-1) + A2*Y(t-2) + ... + Ap*Y(t-p) + ε(t)

  where:

  • Y(t) is a vector of variables at time t.
  • c is a vector of constants.
  • A1, A2, ..., Ap are matrices of coefficients.
  • ε(t) is a vector of error terms with a covariance matrix Σ.
  • The square root of the diagonal elements of Σ provides the SEE for each variable.

• Benefits of SEE:

  • Cycle Capture: The SEE accounts for the cyclical dynamics of real estate, resulting in a more accurate risk assessment.
  • Mean Reversion: Unlike standard deviation, the SEE approach exhibits a limit to the width of the error band as the time horizon increases due to the tendency of real estate returns to revert to the mean over longer periods.
  • Incorporation of Current Conditions: The SEE approach considers current market conditions (e.g., low vacancy rates, construction activity) to refine the risk assessment. This allows the model to predict that prices are more predictable that a random walk due to these current conditions.

• Beyond Historical Data: The SEE approach can incorporate information from longer time series of related variables (e.g., employment) even if the data for the specific real estate asset (e.g. CMBS) is limited.

• Limitations of SEE:

  • Complexity: Implementing the SEE approach requires converting complex models into a VAR form, which can be technically challenging.
  • Transparency: The use of SEE may be less transparent to investors, especially if forecasting is outsourced. Trust in the forecaster and the reasonableness of the forecasting equations becomes crucial.

Combining Approaches: Integrating Scenarios and VaR

Scenario analysis and VaR can be combined to enhance risk management.

• Addressing Stress Test Requirements: Regulatory stress tests often require assessing the impact of specific economic scenarios (e.g., GDP decline, unemployment increase) on real estate portfolios.

• Process:

  1. Generate SEE distributions for the relevant market and property-type combinations.
  2. Develop a specific scenario (e.g., a “double-dip” recession).
  3. At the national level, determine the intersection point between the scenario outcome and the VaR distribution for a given time period.
  4. Calculate the number of standard deviations between the base forecast and the scenario outcome.
  5. Convert the standard deviation measure into a VaR level using an inverse normal distribution function.

• Benefits: This approach allows for rapid assessment of the impact of specific scenarios across a large portfolio.

Using Historic Standard Deviation with Forecasts: A Pragmatic Shortcut

While the SEE is the theoretically correct measure of risk for forecasts generated by complex models, using historical standard deviation can be a practical shortcut.

• Rationale: Historical standard deviation can be viewed as an estimate of the error around the forecast.
• Conservatism: Historical standard deviations are often larger than the SEE, which may lead to a more conservative (i.e., adverse) risk assessment.
• Relationship to SEE: Historical standard deviation and SEE are often correlated (“noise” in the data will inflate both measures).

Converting Forecast Distributions to Probability of Equity Loss

Converting risk measures into probabilities that are easier for equity investors to interpet.

• Challenge for Equity Investors: Equity investors may find it difficult to translate standard deviation into investment decisions.

• Probability-Based Metrics: Convert the distribution of potential returns into probabilities of achieving specific goals or avoiding specific catastrophes.

  • Probability of Beating a Hurdle Rate: Calculate the probability that the investment’s return will exceed a pre-defined hurdle rate (e.g., the required rate of return for a pension fund).
  • Probability of Equity Loss: Calculate the probability of a negative result, such as the investment being underwater at any time during the loan term or at loan maturity.
  • Probability of Default: Using distributions around NOI forecasts to calculate the ability of investor to make loan payments throughout the life of a mortgage.

• Impact of Leverage: Examining the probability of beating a hurdle rate allows investors to assess the impact of leverage on risk. While leverage increases expected return, it also increases standard deviation, potentially reducing the probability of achieving the target return.

Appendix A: Procedure for Approximating VaR Level of a Specific Scenario

1. Produce SEE distributions for the relevant market and property-type combinations.
2. Separately, generate a scenario that differs from the base case (e.g., a recessionary scenario).
3. For a specific time period of the forecast, calculate or identify the standard deviations from the base case for the forecasted value from the scenario in that year.
4. Compute the percentage of the distribution falling below that forecasted value. In the case of a normal distribution, use the standard normal distribution function in Microsoft Excel to calculate the percentage based on the standard deviation entered. This percentage represents the approximate VaR level associated with the specified scenario.