Value-at-Risk

Value-at-Risk

Value-at-Risk (VaR) estimates the potential loss in the value of a portfolio or investment over a specific time horizon and under a given level of confidence. It provides a way for financial institutions and investors to assess and manage the potential downside risk associated with their investments. VaR is defined as a threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceeds this value (assuming normal markets and no trading in the portfolio) of the given probability level.

The breakdown of the key components of Value-at-Risk

• Time Horizon. VaR is measured over a specific time interval, indicating the potential loss within that timeframe. Common time horizons include one day, one week, or one month
• Confidence Level. The confidence level represents the probability that the actual loss will not exceed the calculated VaR. For example, a 95% confidence level implies that there is a 5% chance that the actual loss will be greater than the VaR.

VaR Interpretation

VaR estimates the loss with the specified confidence level over a given time horizon. For example VaR of -7% (1-month time horizon and 5% confidence level) means that theref is a 5% chance that the losses over 1-month period will be greater than -7%.

VaR Pros

• It does not rely on the assumption of normality in the distribution of returns.
• VaR can be used for scenario analysis and stress testing.
• Regulatory Compliance. Many regulatory frameworks require financial institutions to assess and report their risk exposures using VaR as a key metric.

VaR Cons

• Like all risk statistics derived from distribution models, Value-at-Risk (VaR) doesn't consider the sequence of asset returns. Consequently, assets with identical VaR values may experience different drawdowns within the designated time horizon.
• Relying solely on VaR can yield misleading results and potentially underestimate risks. For instance, an asset may exhibit a seemingly low risk according to its VaR. However, if this low-risk scenario repeats consistently across numerous consecutive periods, the cumulative negative returns may ultimately prove to be unacceptable.
• Calculation difficulties.
• VaR does not provide information about the actual losses that could occur beyond the calculated VaR.
• Subadditivity problems. The sum of individual VaRs for multiple portfolios or positions is not always equal to the VaR of the combined portfolio.
• Non-convex function. One of the implications of non-convexity is the problem of portfolio optimization using VaR as an objective function: an optimization model may require a multi-extreme optimization, i.e. the objective function has multiple local optima.
• Difficulties of calculating VaR ratios, similar to the Sortino or Sharp ratios, because VaR values can be poisitive or negative. Risk Shell proprietary statistics include a special VaR ratio incorporating a linear transformation of VaRs of underlying assets to ensure that all the VaRs are either positive or negative.

VaR Claculation Pitfalls

In some instances, the calculation of Value-at-Risk (VaR) is simplified by multiplying the standard deviation of asset returns by a multiplier associated with a specific confidence level (e.g., 1.65 for a 5% confidence level). This approach, often referred to as the 'delta-normal' or 'parametric' method, assumes the normality of the distribution of asset returns. However, this simplification raises several issues and may not provide meaningful benefits when compared to using the standard deviation as a risk measure.

• Assumption of Normality. The delta-normal method relies on the assumption that the distribution of asset returns is normal. In reality, financial markets frequently exhibit non-normal behaviors, with returns displaying fat tails and skewness. Relying on normality assumptions may lead to inaccurate estimations of extreme events and potential losses.
• Limited Accuracy in Extreme Events. VaR calculated using the delta-normal method may lack accuracy, particularly during extreme market conditions. Since the method assumes a normal distribution, it may underestimate the likelihood and magnitude of tail events, leading to a potential underestimation of the true risk.
• Risk Concentration Overlooked. The delta-normal method treats the risk of individual assets independently, potentially overlooking the concentration of risk within a portfolio. In reality, the correlations between assets and the overall portfolio structure play a crucial role in determining the actual risk exposure.
• Static Nature of Multiplier. Using a fixed multiplier for a given confidence level assumes a static relationship between standard deviation and VaR. In dynamic and changing market conditions, this fixed multiplier may not accurately capture the evolving nature of risk, especially during periods of high volatility.
• Limited Informational Content. The delta-normal VaR calculation does not provide additional informational content beyond what is already captured by the standard deviation. In essence, it may not offer significant benefits in terms of risk assessment compared to using more sophisticated risk measures that consider the entire distribution of returns.

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