Quantifying & Combining Risk: VaR, Scenarios, & SEE

Quantifying & Combining Risk: VaR, Scenarios, & SEE

Value at Risk (VaR)

• Definition: VaR is a risk measure that quantifies the potential loss in value of an asset or portfolio over a specific time period and for a given confidence level. It represents a threshold below which a specified percentage of outcomes falls.
• Conceptual Explanation: Instead of describing risk with descriptive scenarios, VaR focuses on the outputs of a potential distribution of results. It answers the question: “What is the worst loss I can expect to experience x% of the time over a specific timeframe?”

• Mathematical Representation: VaR can be formally defined as:

  P(L > VaR) = 1 - c

  Where:

  • P is the probability
  • L is the loss in value
  • VaR is the Value at Risk
  • c is the confidence level (e.g., 95%, 99%)
  • Example: A 2% VaR means that there is a 2% probability that the loss will be greater than the calculated VaR value during the specified period.
  • Calculation Methods: VaR can be calculated using various methods:
  • Historical Simulation: This method uses historical data❓ to simulate future price movements and calculate potential losses.
  • Variance-Covariance Method: This method assumes that asset returns follow a normal distribution and uses the mean and standard deviation of returns to calculate VaR.
  • Monte Carlo Simulation: This method generates a large number of random scenarios based on assumed distributions of asset returns and calculates VaR from the simulated outcomes.
  • Advantages:
  • provides❓ a single number that summarizes the risk of a portfolio.
  • Easy to understand and communicate.
  • Widely used in the financial industry.
  • Disadvantages:
  • Relies on assumptions about the distribution of asset returns, which may not always hold in reality.
  • Can underestimate risk in situations with non-normal distributions or extreme events (tail risk).
  • Doesn’t explain why the values reach this point or correlate to certain inputs.
  • Practical Application: A risk officer might be interested in a result so severe that only 2 percent of the distribution falls below that point. VaR helps identify such a risk level, focusing on the outputs.

Scenario Analysis

• Definition: Scenario analysis involves creating different potential future scenarios, often focusing on adverse conditions, and evaluating the impact of these scenarios on an asset or portfolio’s value.
• Process:
  1. Define the scenarios: This involves identifying key risk factors and developing plausible but stressful scenarios.
  2. Estimate the impact: This involves estimating the impact of each scenario on the asset or portfolio’s value.
  3. Evaluate the results: This involves analyzing the results of the scenario analysis and identifying potential vulnerabilities.
• Example: A scenario might involve a significant increase in interest rates, a decline in economic growth, or a major natural disaster. The impact of each of these scenarios on a real estate❓ portfolio would then be assessed.
• Advantages:
  • Helps to identify potential risks and vulnerabilities.
  • Provides a framework for contingency planning.
  • Can be used to communicate risks to stakeholders.
• Disadvantages:
  • Subjective and dependent on the scenarios chosen.
  • Can be difficult to estimate the impact of each scenario.
  • May not capture all possible risks.
• Practical Application: Stress tests mandated to banks and insurance companies often require scenario analysis to check capital levels.

Standard Error of the Estimate (SEE)

• Definition: SEE is a measure of the accuracy of a statistical model’s predictions. It quantifies the average difference between the predicted values and the actual values.
• Conceptual Explanation: SEE provides a forward-looking measure of risk based on an econometric forecast. It leverages the relationships captured in the model to estimate the potential variability around the forecast.
• Distinction from Standard Deviation: Unlike historical standard deviation, which measures the variability of past data, SEE measures the variability of the forecast itself. The forecast attempts to capture cyclical dynamics, and the interaction of equations in the model prevents uniform expansion of standard deviation over time.

• Mathematical Representation: The SEE is the standard deviation of the regression residuals:

  SEE = sqrt[ Σ(yi - ŷi)² / (n - p) ]

  Where:

  • yi is the actual value of the dependent variable
  • ŷi is the predicted value of the dependent variable
  • n is the number of observations
  • p is the number of parameters in the model
  • Application with Vector Autoregressive (VAR) Models: To obtain SEE in a system of equations used to forecast, the system needs to be converted to a structured VAR model.
  • Advantages:
  • Forward-looking, capturing the cyclical dynamics of the market.
  • Considers the interdependencies between different variables.
  • Can incorporate information not reflected in historical data (e.g., square footage under construction).
  • Disadvantages:
  • More complex to implement than historical standard deviation.
  • Requires a well-specified econometric model.
  • Can be less transparent to investors when the forecasting process is outsourced.
  • Practical Application: Using econometric forecasts for rent and vacancy inputs in a pro-forma analysis, SEE provides a relevant risk measure for capturing the cyclical dynamics of real estate, unlike a purely stochastic forecast.

Monte Carlo Simulation

• Definition: A Monte Carlo simulation is a computational technique that uses random sampling to obtain numerical results. It is used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables.
• Process:
  1. Define the Model: Create a mathematical model that represents the system or process being analyzed. This model will include variables and relationships between them.
  2. Identify Input Variables and Distributions: Determine the key input variables that influence the outcome of the model. Assign probability distributions to each of these variables. These distributions reflect the uncertainty associated with each variable.
  3. Generate Random Samples: Generate a large number of random samples from the probability distributions assigned to the input variables.
  4. Run the Simulation: For each set of random samples, run the model and calculate the outcome.
  5. Analyze the Results: After running the simulation a large number of times (e.g., thousands or millions), analyze the distribution of outcomes. This provides insights into the range of possible outcomes and their probabilities.
• Example: A Monte Carlo Simulation could use the means and standard deviations of value change to create an array of forecasts. Period-by-period growth is chosen from a distribution hundreds or thousands of times to produce an array of results. The probability distribution can then be examined for any particular time period.
• Advantages:
  • Handles complex models with many uncertain inputs.
  • Provides a distribution of possible outcomes, rather than a single point estimate.
  • Allows for sensitivity analysis to identify the most important risk factors.
• Disadvantages:
  • Computationally intensive.
  • Results are only as good as the underlying model and assumptions.
  • Can be difficult to interpret the results.

Combining Approaches

Tying a Scenario to a VaR Stress Level

1. Produce SEE distributions: Generate distributions of potential outcomes using the Standard Error of the Estimate (SEE) approach.
2. Generate a Scenario: Create a specific scenario that differs from the base case forecast (e.g., a double-dip recession).
3. Calculate Standard Deviations: For a specific time period, calculate the number of standard deviations the scenario’s forecasted value is away from the base case forecast.
4. Compute Percentage of Distribution: Determine the percentage of the SEE distribution that falls below the scenario’s forecasted value. For a normal distribution, you can use the standard normal distribution function in Excel to calculate this percentage based on the number of standard deviations. This percentage represents the VaR level associated with the scenario.

Utility of Including Historic Standard Deviations with Forecasts

• Using historical standard deviation can be thought of as estimating the error around the forecast.
• Historical standard deviations are often larger than SEE estimates (as the forecast captures the cycle), meaning the historic standard deviation would be a conservative estimate.
• Historical standard error and SEE are often related (‘noise’ will drive up both measures), meaning historical standard deviation can be a shortcut to the more complicated estimate.

Converting Forecast Distributions to Probability of Equity Loss

• Translates the use of distributions in the Monte Carlo or SEE approaches into the equity investor’s mindset.
• Instead of listing an estimate of mean and standard deviation of expected future returns, calculate the probability of beating a hurdle rate.
• Analysis can also be used to calculate the probability of a negative result such as an investment being underwater at any time in the life of its loan or at loan maturity.

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

1. Produce SEE distributions.
2. Separately, generate a scenario that differs from the 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.