Risk Management: Overview and Calculation of VaR and CVaR

This article introduces VaR (Value at Risk) and CVaR (Conditional Value at Risk), two key metrics used in investment risk management. These play an important role in evaluating the risk of a financial portfolio and are useful when constructing one, so I hope this helps you build a solid understanding of them.

VaR (Value at Risk)

Overview

VaR (Value at Risk) is a statistical method for measuring risk. It indicates “the maximum loss a portfolio could suffer over a given period (e.g., 1 day or 10 days) at a given confidence level (e.g., 95% or 99%).”

For example, if VaR is $1,000,000 at a 95$1,000,000. Calculation methods include the historical (percentile) method, Monte Carlo simulation, and the parametric (analytical distribution) method.

VaR is widely used in risk management, but it has several limitations. The main issue is that it does not capture extreme market movements well. Because VaR ignores the magnitude of losses beyond the confidence threshold, it may underestimate the risk of very large losses. Additionally, since VaR is calculated from historical data, its accuracy decreases when future market conditions differ significantly from the past.

Calculation Methods

I personally prefer to avoid the Monte Carlo method, so I will introduce the other two.

Variance-Covariance Method

The variance-covariance method calculates VaR by assuming a normal distribution and using the standard deviation of portfolio returns. The steps are as follows:

1. Calculate the mean return and standard deviation from historical price data.
2. Assuming a normal distribution, obtain the z-score corresponding to the specified confidence level (e.g., z = 1.65 for 95% confidence).
3. $\text{VaR} = \text{Mean Return} - (\text{z-score} \times \text{Standard Deviation})$

Historical Method

This method simply sorts the data and picks out the relevant value.

• Collect historical price data (e.g., daily prices for the past year).
• Calculate the return for each day.
• Sort the returns in ascending order.
• Find the return corresponding to the specified confidence level (e.g., the bottom 5% for 95% confidence).

CVaR (Conditional Value at Risk)

Overview

CVaR (Conditional Value at Risk) represents the expected value of losses that exceed a given confidence level (typically 95% or 99%). While VaR is the maximum loss at a specific confidence level, CVaR calculates the average of losses beyond that threshold. The advantage of CVaR is that it gives a more accurate picture of the risk of extreme losses caused by severe market movements — losses that fall outside the confidence interval.

Calculation Method

The calculation follows these steps:

1. Calculate VaR.
2. Calculate the average of losses that exceed that VaR — this is CVaR.

Use Cases for VaR and CVaR

Risk Management

Since you know the maximum loss, you can evaluate whether it is acceptable. For individuals in particular, the range of tolerable losses tends to be limited, so judging whether a portfolio exceeds your risk tolerance is an important step.

Performance Evaluation

These metrics are sometimes used in performance calculations. Specifically, they are used when computing RVaR (Return over VaR):

$$\text{RVaR} = \frac{\text{Portfolio Return}}{\text{VaR}}$$

By dividing returns by VaR, you can evaluate the return per unit of risk, measuring risk-adjusted performance.

Stress Testing

VaR is also used as part of stress testing, where portfolio performance is evaluated under abnormal market conditions. For example, you can reproduce past financial crises or market shocks, calculate VaR, and assess a portfolio’s vulnerabilities and potential losses.

Conclusion

VaR and CVaR serve the same fundamental purpose but each has its own strengths and weaknesses.

Both can be used for risk evaluation, but it is best to choose according to your personal goals — specifically whether you want to account for rare, large losses or not.