Quantifying and Combining Risk Measures: From VaR to Probability of Loss

Quantifying and Combining Risk Measures: From VaR to Probability of Loss

Introduction to Risk Measurement in real estate❓

• Real estate, unlike some financial assets, often involves long-term investments and unique market dynamics. Therefore, traditional risk measures may need adaptation or supplementation.
• The use of standard deviation alone as a measure of risk can be misleading, particularly in real estate where cycles and autocorrelations are prevalent.
• This section will explore how Value at Risk (VaR) and the Probability of Loss can provide more robust insights into real estate risk management.

Value at Risk (VaR)

• Definition: VaR is a statistical measure that quantifies the potential loss in value of an asset or portfolio over a specific time period for a given confidence level. In essence, it answers the question: “What is the most I can lose over a given period, with a given probability?”

• Formulaic Representation: Let X be a random variable representing the potential loss of an asset or portfolio. The VaR at a confidence level α (e.g., 95%, 99%) is defined as:
  > VaR_α = -inf{x ∈ ℝ : P(X ≤ x) > 1 - α}
  where:
  > * P(X ≤ x) is the probability that the loss X is less than or equal to x.
  > * α is the confidence level (e.g., 0.95 for 95% VaR).

• Interpretation: A 95% VaR of \$1 million means there is a 5% chance of losing more than \$1 million over the specified time horizon.

• Calculation Methods:

  1. Historical Simulation:
     • Rank historical returns from worst to best.
     • The VaR is the return at the 1-α percentile.
     • Example: If you have 100 days of historical returns and want to calculate 95% VaR, you would find the 5th worst return.
     • Advantages: Simple to implement, non-parametric (doesn’t assume a distribution).
     • Disadvantages: Relies on the past being representative of the future; limited to available historical data.
  2. Variance-Covariance (Parametric) Method:
     • Assumes returns follow a normal distribution.
     • Calculates VaR using the portfolio’s mean and standard deviation.
     • Formula: VaR_α = - (μ + σ * z_α)
       where:
       > * μ is the portfolio’s expected return.
       > * σ is the portfolio’s standard deviation.
       > * z_α is the z-score corresponding to the confidence level α (e.g., for 95% VaR, z_0.95 ≈ 1.645).
     • Advantages: Easy to calculate if the distribution is known or assumed; computationally efficient.
     • Disadvantages: Assumes normality, which may not hold true for real estate returns (especially during crises); sensitive to parameter estimates.
  3. Monte Carlo Simulation:
     • Generates a large number of possible future scenarios based on statistical distributions of key variables (e.g., rent growth, vacancy rates, interest rates).
     • Calculates the portfolio’s value under each scenario.
     • The VaR is determined by the percentile of the resulting distribution of portfolio values.
     • Advantages: Flexible, can handle complex dependencies and non-normal distributions; allows for incorporating expert opinions and forward-looking scenarios.
     • Disadvantages: Computationally intensive; requires careful selection of input distributions and parameter estimation. The forecast produced is only as good as the historical distributions that can be calculated.
  4. Using the Standard Error of the Estimate (SEE) Approach
     • A proper measure of risk in this case is not the historic standard deviation, but the standard error of the estimate (SEE).
     • 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.

• Practical Application: A real estate investment trust (REIT) might use VaR to determine the potential loss in value of its property portfolio over the next quarter. They could use historical simulation based on past property value fluctuations or Monte Carlo simulation incorporating macroeconomic forecasts.

• Experiment: A real estate analyst could perform a Monte Carlo simulation using various growth rates for rent and vacancy. The results can be examined to determine the probability distribution for any particular time period. This is useful for VaR calculations.

Combining VaR with Scenario Analysis

• Scenario analysis involves evaluating the impact of specific, pre-defined events (e.g., a recession, a rise in interest rates) on a portfolio.

• VaR and scenario analysis can be combined to assess the severity of a scenario in terms of its percentile within a broader distribution of possible outcomes.

• Procedure for Approximating VaR Level of a Specific Scenario (from included pdf text):

  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.

• Example: If a “double-dip” recession scenario results in a portfolio loss that falls at the 76th percentile of the VaR distribution, it suggests that the scenario is relatively mild (only 24% of outcomes are worse).

Probability of Loss (or Probability of Achieving a Goal)

• Definition: The probability of loss is the likelihood that an investment will result in a financial loss or fail to meet a pre-defined performance target (hurdle rate).

• Calculation: Can be derived from the same distributions used for VaR calculation (e.g., generated by Monte Carlo simulation). Instead of finding a value at a certain percentile, we calculate the percentage of scenarios that fall below a certain threshold.

• Formula: Let T be a target value (e.g., a hurdle rate or breakeven point). The probability of loss is:
  > P(X < T)
  where X represents the random variable representing the investment’s return.

• Applications:

  • Probability of Beating a Hurdle Rate: Useful for pension fund advisors who are interested in insuring that returns are at least their required rate of return.
  • Probability of a Negative Result such as an investment being underwater at any time in the life of its loan or at loan maturity.
  • The ability for investors to make their loan payments throughout the life of a mortgage.

• Example: Using the SEE approach, one can calculate the probability of NOI being high enough to make loan payments throughout the life of the mortgage.

Leverage and Probability of Loss

• Leverage amplifies both expected returns and potential losses. It is crucial to analyze how leverage affects the probability of achieving a target return or avoiding a significant loss.
• Applying leverage will increase both the expected return and its standard deviation.

The Utility of Including Historic Standard Deviations with Forecasts

• The use of historic standard deviation can be thought of as an estimate of the error around the forecast.
• Historic standard deviations are often larger than estimates of risk to the forecast taken from the SEE.
• The decision to use historic standard deviation is a matter of methodological purity. 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.