Quantifying Real Estate Risk: From VaR to Probability of Loss

Quantifying real estate❓ Risk: From VaR to Probability of Loss

Introduction to Risk Quantification

Understanding and quantifying risk is paramount in real estate investment. While standard deviation is often used as a proxy for risk, it’s important to understand its limitations and explore more advanced methods like Value at Risk (VaR) and probability of loss calculations. These tools help investors make more informed❓ decisions by providing a deeper understanding of potential downside scenarios.

• Traditional risk measures in real estate may underestimate true risk, especially due to factors like serial correlation in returns.
• Advanced techniques like Monte Carlo simulation offer a robust way to model uncertainty and assess potential outcomes.
• Quantifying risk allows investors to translate uncertainty into actionable insights and make better-informed decisions.

Value at Risk (VaR)

VaR is a statistical measure used to quantify the potential loss in value of an asset or portfolio over a specific time period for a given confidence level. It answers the question: “What is the maximum loss I can expect to experience with a certain probability over a certain time horizon?”

• Definition: VaR is the maximum expected loss over a specified time horizon at a given confidence level.
• Example: A 95% one-month VaR of \$1 million means there is a 5% chance of losing more than \$1 million in a month.

Calculating VaR

VaR can be calculated using different methods:

1. Historical Simulation:

   • Involves using historical data❓ to simulate future returns.
   • Steps:
     1. Collect historical returns data for the asset or portfolio.
     2. Sort the returns from worst to best.
     3. Identify the return that corresponds to the desired confidence level.
     4. The VaR is the loss associated with that return.

2. Variance-Covariance Method (Parametric VaR):

   • Assumes that returns are normally distributed.
   • Formula:
     VaR = - (μ + σ * Zα) * V
     Where:
     • μ is the expected return.
     • σ is the standard deviation of returns.
     • Zα is the Z-score corresponding to the desired confidence level (α).
     • V is the initial value of the asset or portfolio.

3. Monte Carlo Simulation:

   • Generates numerous random scenarios based on assumed distributions of relevant variables.
   • Steps:
     1. Define the distribution of each risk factor (e.g., rental growth, vacancy rates).
     2. Generate a large number of random samples from these distributions.
     3. Use these samples as inputs to a valuation model to calculate the portfolio value under each scenario.
     4. Sort the resulting portfolio values from worst to best.
     5. Identify the portfolio value that corresponds to the desired confidence level.
     6. The VaR is the difference between the initial portfolio value and this value.

Advantages and Disadvantages of VaR

• Advantages:
  • Provides a single, easy-to-understand number for quantifying risk.
  • Can be applied to different asset classes and portfolios.
  • Allows for risk comparison across different investments.
• Disadvantages:
  • Relies on assumptions about the distribution of returns, which may not always hold true.
  • May not capture extreme events or “tail risk.”
  • Different calculation methods can yield different results.
  • Doesn’t specify the events that lead to the VaR level.

VaR and Monte Carlo Simulation

Monte Carlo simulation is often used in conjunction with VaR to create a distribution of potential outcomes. This distribution can then be used to determine the VaR at a specific confidence level. Unlike single scenarios, Monte Carlo analysis where period-by-period growth is chosen from a distribution hundreds or thousands of times produces an array of results such that the probability distribution can be examined for any particular time period.

Standard Error of the Estimate (SEE)

The SEE provides a measure of the accuracy of a forecast. In the context of real estate, especially when using econometric models, the SEE can be a more appropriate measure of risk than historical standard deviation.

• Definition: The SEE measures the standard deviation of the errors between predicted and actual values in a regression model.
• Application: When using econometric models to forecast real estate variables (e.g., rent, vacancy), the SEE quantifies the uncertainty around those forecasts.

SEE vs. Historical Standard Deviation

• Historical standard deviation reflects the volatility of past returns. It can be considered the risk of a random walk with drift.
• SEE reflects the uncertainty in a forecast that attempts to capture cyclical dynamics.

The SEE differs from historic standard deviation in that, first, the forecast itself is an attempt to capture some of the standard deviation (the cycle) and, second, the interaction of the equations will not lead to uniformly expanding standard deviation over time. In real estate, the second point becomes important given the tendencies for real estate returns to exhibit signs of both autocorrelation and (over longer periods) mean reversion.

• Autocorrelation: The tendency for returns to be correlated with their past values.
• Mean Reversion: The tendency for returns to revert to their long-term average.

Calculating SEE in a System of Equations

To obtain the SEE for a system of equations used to forecast, the system needs to be converted to a structured vector-autoregressive model (VAR). Software used must be sophisticated enough to produce errors (usually decomposing them as well).

Advantages and Disadvantages of SEE

• Advantages:
  • Forward-looking measure of risk. 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.
  • Captures cyclical dynamics and mean reversion in real estate markets.
  • Can incorporate information not captured in historical data.
• Disadvantages:
  • More complex to calculate than historical standard deviation.
  • Requires a well-specified econometric model.
  • The more complex the model, the more difficult the conversion of the model to a structured VAR form.

Probability of Loss

Another way to quantify real estate risk is to calculate the probability of experiencing a specific loss or failing to achieve a desired return. This approach translates statistical measures into more intuitive and actionable insights for investors. Converting forecast distributions to probability of equity loss can be done by examining the distribution around the forecast to take an intersection point in a particular period and then calculating the percentage of the distribution above or below that point.

• Definition: The probability of loss is the likelihood of an investment falling below a certain threshold, such as breaking even, losing a specified amount of capital, or failing to meet a hurdle rate.

Calculating Probability of Loss

The calculation typically involves the following steps:

1. Develop a Distribution of Possible Outcomes: Use Monte Carlo simulation or SEE to generate a probability distribution of possible future returns or values.

2. Define the Loss Threshold: Determine the level of loss that is of concern. This could be a negative return, a specific dollar amount, or a failure to achieve a target return.

3. Calculate the Probability: Determine the area under the probability distribution curve that falls below the loss threshold. This can be done using statistical software or spreadsheet functions.

Applications of Probability of Loss

• Probability of Negative Return: Investors can calculate the probability of experiencing a negative return on their investment.
• Probability of Capital Loss: Determine the likelihood of losing a certain percentage of their initial investment.
• Probability of Failing to Meet Hurdle Rate: Calculate the probability of not achieving a minimum required rate of return.
• Probability of Being Underwater on a Mortgage: Determining the probability of remaining ‘above water’ on mortgage by examining a value probability distribution.
• Ability to Make Loan Payments: Distributions around NOI forecasts can be used to determine the ability for investors to make their loan payments throughout the life of a mortgage.

Incorporating Leverage

The examination of probability also works when leverage is applied. 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.

• Impact of Leverage: Leverage can amplify both returns and losses, increasing the probability of both exceeding a hurdle rate and experiencing a significant loss.
• Risk Assessment: By calculating the probability of loss with and without leverage, investors can better understand the true risk-adjusted return of their investments.

Tying Scenarios to VaR Stress Levels

In cases where either government regulators or even just corporate risk officers are looking at results stemming from a specific GDP or unemployment scenario, the process of creating a result from scratch for commercial real estate presents a challenge.

One way to address the challenge is to utilise pre-existing SEE results on the market and property-type combination level and comparing VaR results to scenario results at the national level. The combination stems from the ability in VaR methodology (whether stochastic or SEE) to identify a specific VaR level.

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. Once this VaR level is designated, existing results across❓ even large portfolios can be examined quickly.

Procedure for Approximating VaR Level of a Specific Scenario

1. Produce SEE distributions.

2. Separately, generate scenario that differs from base case.

3. For 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 percentage of 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.

Conclusion

Quantifying risk in real estate investments is a complex but essential process. VaR, SEE, and probability of loss are valuable tools that provide different perspectives on potential downside scenarios. By understanding the strengths and limitations of each method, investors can make more informed decisions and better manage their risk exposure.