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Real Estate Risk: From Standard Deviation to VaR
Real Estate Risk: From Standard Deviation to VaR
The Standard Deviation as a Measure of Risk
The standard deviation is often used as a proxy for the risk of an asset. This simplification stems from the underlying model used to represent the asset’s behavior.
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When asset price changes are modeled as a random walk❓ with drift, the standard deviation of historical growth rates can be considered as the risk inherent in that random walk. In this model, the forecast is the historical mean of the growth.
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This model can be applied to real estate, but it is at odds with the long-term focused forecasting methods often used in real estate, especially those that attempt to understand the real estate cycle.
Mathematical representation
Let $r_t$ be the return in period t. If the returns are modeled as a random walk with drift $\mu$, then $r_t = \mu + \epsilon_t$, where $\epsilon_t$ is a random error term with mean 0 and standard deviation $\sigma$. Here, $\sigma$ represents the risk.
Limitations of standard deviation in Real Estate:
- Real estate returns are often influenced by cyclical patterns and macroeconomic factors, which aren’t fully captured by a simple random walk.
- Real estate data is often quarterly and covers a limited time horizon, making standard deviation estimates less reliable compared to assets with daily data spanning decades.
- Historic standard deviation is the error of a simple autoregressive model, more complex models will have different errors. The use of historic standard deviation on a complex model is in combining approaches.
Monte Carlo Simulation and Risk Assessment
Monte Carlo simulation is a powerful technique that uses the mean and standard deviation of value change (or any choice of variable) to generate a range of possible forecasts.
- Unlike single scenario analysis, Monte Carlo simulation creates an array of results from repeatedly drawing period-by-period growth values from a specified distribution (e.g., normal distribution) hundreds or thousands of times.
- This produces a probability distribution of outcomes for each time period, allowing for a more comprehensive understanding of potential risks and rewards.
Procedure of Monte Carlo Simulation:
- Define the variables of interest (e.g., property value, NOI, rent growth).
- Estimate the mean and standard deviation for each variable. This can be based on historical data, expert opinion, or econometric models.
- Choose a probability distribution that reflects the behavior of each variable (e.g., normal, log-normal, triangular).
- Simulate the model many times (e.g., 1,000 or 10,000 iterations), with each iteration drawing random values from the specified distributions for each variable.
- Analyze the resulting distribution of outcomes to assess the range of possible results and their associated probabilities.
Advantages of Monte Carlo Simulation:
- Provides a probabilistic view of risk, rather than relying on single-point estimates.
- Allows for the modeling of complex interactions between multiple variables.
- Can be used to assess the impact of different assumptions and scenarios.
Disadvantages of Monte Carlo Simulation:
- The quality of the results depends heavily on the accuracy of the input parameters (mean, standard deviation, distribution).
- It may not be easy to identify the specific events that lead to extreme outcomes.
- For long-term real estate investment, the random-walk approach can create wide variance of results further out in the forecast period.
- A forecast produced by Monte Carlo is only as good as the historical distributions that can be calculated.
Value at Risk (VaR)
Value at Risk (VaR) is a risk management metric that quantifies the potential loss in value of an asset or portfolio over a specific time horizon and at a given confidence level.
- VaR focuses on the tail of the distribution of possible outcomes, rather than the average or expected value.
- It answers the question: “What is the maximum loss I could experience with a certain probability over a given time period?”
Example: A 2% VaR of \$1 million means that there is a 2% probability of losing at least \$1 million over the specified time horizon.
Calculation of VaR using Monte Carlo Simulation:
- Perform a Monte Carlo simulation to generate a distribution of possible outcomes.
- Sort the outcomes from best to worst.
- Identify the value at the desired confidence level. For example, to calculate the 5% VaR, find the value that corresponds to the 5th percentile of the distribution.
Advantages of VaR:
- Provides a single, easy-to-understand measure of risk.
- Can be used to compare the risk of different assets or portfolios.
- Focuses on potential losses, which is particularly relevant for risk management.
Disadvantages of VaR:
- VaR is only a quantile and does not describe the distribution of losses beyond the VaR level. It doesn’t tell you how much you could lose if you exceed the VaR threshold.
- VaR relies on the accuracy of the underlying model and assumptions.
- It does not capture all aspects of risk, such as liquidity risk or operational risk.
- It cannot identify what events would cause values to get to that level.
Standard Error of the Estimate (SEE)
When econometric models are used to forecast real estate variables, such as rent and vacancy, the historical standard deviation is no longer technically the risk of the forecast. A proper measure of risk in this case is the standard error of the estimate (SEE).
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The SEE measures the dispersion of actual values around the predicted values from a regression model.
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In the case of a system of equations used to forecast, to obtain such errors requires the system to be converted to a structured vector-autoregressive model (VAR, not to be confused with VaR).
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Unlike historic standard deviation, the forecast itself is an attempt to capture some of the standard deviation (the cycle).
Mathematical representation
If we have a regression model $y = X\beta + \epsilon$, where $y$ is the dependent variable, $X$ is the matrix of independent variables, $\beta$ is the vector of coefficients, and $\epsilon$ is the error term, then the SEE is given by:
$SEE = \sqrt{\frac{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}{n-p}}$
where $y_i$ is the actual value, $\hat{y}_i$ is the predicted value, n is the number of observations, and p is the number of parameters in the model.
Advantages of SEE:
- Forward-looking: The SEE approach can create more risk or less risk than the past simply because of the point in the cycle that creates the autocorrelation in the series.
- The results need not be confined to the range found in the data available.
- The interaction of the equations will not lead to uniformly expanding standard deviation over time.
Disadvantages of SEE:
- Implementation can be difficult, especially for complex models.
- The use of the errors from the equation is less transparent to investors when the forecasting process is outsourced.
Combining Scenario Analysis and VaR
A powerful approach involves combining scenario analysis with VaR methodology.
- Develop a specific economic scenario (e.g., a recession).
- Use the scenario to generate forecasts of real estate variables.
- Compare the scenario results to a pre-existing VaR distribution (calculated using Monte Carlo or SEE).
- Identify the VaR level that corresponds to the scenario outcome.
Benefits:
- Provides a way to translate scenario results into a probabilistic framework.
- Allows for quick assessment of risk across large portfolios, given existing VaR estimates.
Example:
Based on Year 2, the appropriate VaR estimate for a double-dip scenario is 76 percent.
Probability of Equity Loss
Transforming risk metrics into intuitive measures like the probability of achieving a goal or avoiding a catastrophe is particularly useful for equity investors.
- Instead of focusing solely on mean and standard deviation, calculate the probability of beating a hurdle rate or the probability of an investment remaining “above water” on a mortgage.
- This can be done by determining an intersection point in a particular period and then calculating the percentage of the distribution above or below that point.
Benefits:
- More easily understood by equity investors than standard deviation or VaR.
- Facilitates decision-making by providing a clear indication of the likelihood of success or failure.
- Crucially, this examination of probability also works when leverage is applied.
Example
- Distributions around NOI forecasts can be used to determine the ability for investors to make their loan payments throughout the life of a mortgage.
Chapter Summary
Summary
This chapter focuses on quantifying real estate❓ risk, moving from simple measures like standard deviation to more sophisticated techniques such as Value at Risk (VaR) and incorporating econometric forecasts.
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The chapter critiques the common practice of using standard deviation as a direct proxy for risk in real estate, especially when using long-term forecasting methods that attempt to capture real estate cycle dynamics. Standard deviation can be misleading because it assumes a random walk, which may not be accurate for real estate.
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Monte Carlo simulation utilizes mean and standard deviations to create a distribution of possible future values, providing a more robust alternative to single-scenario❓ analyses. It generates thousands of forecasts, enabling the examination of probability distributions.
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Value at Risk (VaR) determines a specific loss threshold within a probability distribution that is not exceeded a specific percentage of the time (e.g., 2% VaR). This approach bypasses the need to define severe input scenarios by focusing on the distribution of outcomes.
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The chapter highlights the Standard Error of the Estimate (see❓) as a more appropriate risk measure for econometric forecasts. Unlike historical standard deviation, SEE reflects the model’s attempt to capture cyclical dynamics and the interaction of equations, leading to potentially❓ limited error expansion over time. The SEE method is forward looking by using forecasts to determine the level of risk associated with a specific investment, as opposed to a random-walk methodology.
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Combining scenario analysis with VaR by finding the VaR level that corresponds to a specific scenario at the national level is an approach to address the challenge of creating stress test results.
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A practical application involves translating the distribution of forecast results into a probability of equity loss or the probability of achieving a hurdle rate, making risk assessment more intuitive for equity investors, particularly when leverage is involved.
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The chapter concludes that each risk quantification method has advantages and disadvantages. The selection and combination of methods should be based on a thorough understanding of the economic factors that drive the need for the choice.
