Beyond Standard Deviation: VaR and SEE in Real Estate Risk

Beyond Standard Deviation: VaR and SEE in Real Estate Risk

The Limitations of Standard Deviation in Real Estate

While the standard deviation of historical returns is often used as a proxy for risk, particularly in financial contexts, its application to real estate requires careful consideration. This stems from the inherent characteristics of real estate investments, which deviate from the assumptions underlying the use of standard deviation in traditional finance.

• Random Walk vs. Cyclicality: The assumption that asset price changes follow a random walk, where risk directly corresponds to the standard deviation of growth, often holds for stocks. However, real estate exhibits cyclical patterns, driven by factors like supply, demand, and macroeconomic conditions. Using a random walk model ignores these dynamics and can lead to inaccurate risk assessments.
• Long-Term Focus: Real estate investment horizons are typically long-term, often spanning multiple years or decades. A simple random walk approach❓, while useful for short-term stock price fluctuations, can produce highly variable and unrealistic forecasts over such extended periods.
• Data Availability: Real estate data, particularly transaction data, is often available quarterly or annually, covering a limited time frame. This contrasts with the high-frequency data available for stocks (e.g., daily), making the calculation of statistically robust standard deviations challenging.
• Lack of Deterministic Description: Standard deviation, by itself, doesn’t explain the underlying causes of risk. It provides a statistical measure of volatility but offers❓ no insight into the specific events or factors that could lead to losses.

Value at Risk (VaR) in Real Estate

VaR is a risk management technique that quantifies the potential loss in value of an asset or portfolio over a specific time period, given a certain confidence level. It is defined as the maximum loss that is not exceeded with a given probability. VaR provides a single number summarizing the potential downside risk.

• Definition: VaR answers the question: “What is the maximum loss I can expect to experience with, say, a 95% or 99% confidence level over a given period?”
• Calculation: VaR can be calculated using various methods:
  • Historical Simulation: This method uses historical data to simulate future returns and identify the percentile corresponding to the desired confidence level.

  • Parametric Approach (Variance-Covariance): This method assumes that returns follow a normal distribution and uses the mean and standard deviation to calculate VaR. The formula for VaR is:

    VaR = - (μ + σ * Z)

    where:
    * VaR is Value at Risk
    * μ is the expected return
    * σ is the standard deviation of returns
    * Z is the Z-score corresponding to the desired confidence level (e.g., 1.645 for 95% confidence, 2.33 for 99% confidence)
    * Monte Carlo Simulation: This method generates a large number of random scenarios based on specified probability distributions for relevant variables (e.g., rental income, vacancy rates, interest rates) and calculates the VaR from the resulting distribution of portfolio values.
    * Application in Real Estate:
    * A risk officer might be interested in a result so severe that only 2 percent of the distribution falls below that point. This can be identified via Monte Carlo methods, where period-by-period growth is chosen from a distribution hundreds or thousands of times, producing an array of results.
    * Advantages of VaR:
    * Provides a single, easy-to-understand measure of risk.
    * Allows for comparison of risk across different assets or portfolios.
    * Can be used to set risk limits and allocate capital.
    * Limitations of VaR:
    * Relies on assumptions about the distribution of returns, which may not always hold in real estate. Real Estate returns are often not normally distributed and the VaR metric based on normal distributions will underestimate true risk.
    * Does not provide information about the magnitude of losses beyond the VaR level.
    * Can be sensitive to the input parameters used in the calculation.
    * VaR is a static measure of risk and does not capture the dynamic nature of real estate markets.
    * It cannot identify what events would cause values to get to that VaR level, or trace back through inputs corresponding to the identified output.

Standard Error of the Estimate (SEE) in Real Estate

SEE is a measure of the accuracy of a statistical model, specifically reflecting the average difference between predicted and actual values. In the context of real estate forecasting, SEE offers a more nuanced approach to risk assessment compared to using standard deviation alone, especially when incorporating econometric models.

• Definition: SEE quantifies the dispersion of observed data points around the regression line of a statistical model. A lower SEE indicates a better fit of the model to the data.

• Calculation: SEE is calculated as the square root of the average squared difference between the predicted and actual values:

  SEE = sqrt( Σ (Yᵢ - Ŷᵢ)² / (n - p) )

  where:
  * Yᵢ is the actual value
  * Ŷᵢ is the predicted value from the model
  * n is the number of observations
  * p is the number of parameters in the model. Note: (n-p) represents the degrees of freedom.
  * Application in Real Estate:
  * Econometric Models: Given the widespread use of econometric forecasts in commercial real estate, particularly for rent and vacancy inputs in a pro-forma analysis, there is a desire for risk measures to make use of these forecasts.
  * VAR Models: To obtain such errors requires the system to be converted to a structured vector-autoregressive model (VAR). This allows for capturing the interdependencies between different variables in the real estate market.
  * Advantages of SEE:
  * Forward-Looking: SEE incorporates the forecast itself, attempting to capture cyclical dynamics rather than relying solely on historical data. The use of a forecast can create more or less risk than the past simply because of the point in the cycle that creates the autocorrelation in the series. If vacancy is low, most results would show rents and values to be more likely to increase as the balance of pricing power between tenants and landlords in the forecast period begins to be tilted towards landlords.
  * Captures Cyclicality: SEE accounts for the autocorrelation and mean reversion tendencies observed in real estate returns. The interaction of the equations will not lead to uniformly expanding standard deviation over time. SEE exhibits a limit to the width of the errors as the time frame gets longer and the mean reversion characteristic measure starts to kick in.
  * Model-Based Risk: The ‘worst case’ is not dictated by the worst results in the history available. The model at the heart of the SEE approach can avoid this limitation if the inputs to the model have a longer history (for example, employment) than what is being varied and will dictate how bad the downside can get. As a clarifying example, if the time series of commercial mortgage-backed securities (CMBS) prices is quite short, a Monte Carlo draw based on that time series will be limited to the price changes during that period. In the SEE approach, while the behaviour in the model is established using the short time period, the errors can utilise the longer time series of the employment inputs and are not just limited to the period with CMBS prices.
  * Disadvantages of SEE:
  * Complexity: Converting complex models to a structured VAR form can be challenging.
  * Transparency: The use of errors from equations may be less transparent to investors, requiring trust in the forecaster’s methodology.

Combining Scenario Analysis with VaR

Scenario analysis involves evaluating the impact of specific events or economic conditions on real estate investments. Combining scenario analysis with VaR can provide a more comprehensive understanding of risk.

• Procedure:
  1. Produce SEE distributions.
  2. Separately, generate scenario that differs from base case.
  3. For a specific time period of the forecast, calculate or identify 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, the standard normal distribution function in Microsoft Excel can calculate the percentage based on the standard deviation entered.
• Benefits:
  • Allows for assessing the probability of specific scenarios occurring.
  • Provides a more realistic assessment of potential losses under stress conditions.
  • Facilitates stress testing of portfolios.
  • The combination stems from the ability in VaR methodology (whether stochastic or SEE) to identify a specific VaR level. In cases where a scenario is specified, the process of identifying a VaR level can work in reverse. At the national level, the results of the scenario can be used to determine an intersection point with the VaR distribution for a particular time period.
  • For a normal or pseudo-normal distribution, the scenario intersection point can be used and from there the standard deviations from the base forecast can be calculated. The amount of standard deviations can then be converted into a VaR level using an inverse normal distribution function.

Using Historical Standard Deviation with Forecasts

While SEE is generally a more appropriate measure of risk for complex models, using historical standard deviation as a shortcut might be considered.

• Justification:
  • Historical standard deviation can be thought of as an estimate of the error around the forecast.
  • It is often larger than estimates of risk from SEE, making it a conservative estimate in terms of adverse results.
  • Historical standard deviation and SEE are often related, as “noise” drives up both measures.
• Caveats:
  • Combining a forecast with historical standard deviation has no direct basis in economics.
  • SEE is theoretically more sound for complex models.
  • Historic standard deviation can indeed be thought of as a shortcut to the more complicated estimate. The decision to use historic standard deviation is then a matter of methodological purity. That is, if a forecast more complicated than a stochastic approach is used, there is no question that the SEE is a more appropriate measure of risk, but results will still be informative if historic standard deviation is substituted.

Converting Forecast Distributions to Probability of Equity Loss

To make risk assessment more intuitive for equity investors, forecast distributions can be converted into probabilities of achieving goals or avoiding catastrophes.

• Method:
  • Calculate the percentage of the distribution above or below a specific target or threshold.
  • This translates standard deviation into a more actionable metric.
• Examples:
  • Probability of beating a hurdle rate: Provides a single number to help advisors make investment decisions.
  • Probability of an investment being underwater: Useful for assessing the risk of leverage.

The Impact of Leverage

Crucially, this examination of probability also works when leverage is applied. One of the biggest errors of real estate practitioners leading up to the crisis was the lack of understanding of the risk of applying leverage. In a pro-forma analysis without an attempt to understand risk, investments expected to have positive returns will always appear to improve when leverage is applied. However, this will not be the case in examining the probability of beating a hurdle rate as leverage will increase both the expected return and its standard deviation.