Adjusted value at risk

What Is Adjusted Value at Risk?

Adjusted Value at Risk (Adjusted VaR) is a sophisticated measure within quantitative finance that refines the traditional Value at Risk (VaR) calculation by incorporating higher moments of the probability distribution of returns, specifically skewness and kurtosis. While standard VaR estimates the maximum potential loss of a portfolio over a specified period with a given confidence level, it often assumes that financial returns follow a normal distribution. However, real-world financial data frequently exhibits "fat tails" (leptokurtosis) and asymmetry (skewness), meaning extreme events are more probable than a normal distribution would suggest²⁴, ²⁵, ²⁶. Adjusted VaR addresses these shortcomings, providing a more accurate representation of potential losses, especially during periods of market turbulence. This advanced metric is a crucial tool in modern risk management practices, aiming to capture the true distribution of financial losses.

History and Origin

The concept of Value at Risk (VaR) gained prominence in the financial industry in the mid-1990s, though its origins can be traced back to capital requirements imposed by the New York Stock Exchange on member firms as early as 1922²¹, ²², ²³. Early forms of VaR-like metrics were also seen in the U.S. Securities and Exchange Commission's Uniform Net Capital Rule in 1975, which included "haircuts" based on statistical analysis of historical market data to safeguard against potential losses¹⁹, ²⁰. By the late 1980s, driven partly by the 1987 stock market crash and the need for firm-wide risk aggregation, VaR began to emerge as a distinct concept, notably at Bankers Trust and J.P. Morgan, the latter publicly releasing its "RiskMetrics" methodology in 1994¹⁸.

Despite its widespread adoption, especially for regulatory capital requirements, VaR faced significant criticism, particularly after the 2008 financial crisis¹⁶, ¹⁷. Critics argued that traditional VaR models, relying heavily on historical data and the assumption of normally distributed returns, failed to adequately capture "tail risk" – the risk of extreme, low-probability events. ¹⁴, ¹⁵This led to an underestimation of potential losses during severe market downturns. The development of Adjusted Value at Risk stemmed from the recognition that financial returns often exhibit significant skewness (asymmetry) and kurtosis (fat tails), deviations from normality that standard VaR overlooks. ¹¹, ¹², ¹³Researchers and practitioners began developing adjustments to VaR to incorporate these higher-order statistical moments, providing a more robust measure of risk that better reflects the true likelihood and magnitude of extreme losses. The Basel Committee on Banking Supervision, for instance, has since moved towards using Expected Shortfall instead of VaR for calculating market risk capital requirements, partly due to VaR's limitations in capturing tail risk.
⁹, ¹⁰

Key Takeaways

• Adjusted Value at Risk (Adjusted VaR) extends traditional VaR by accounting for the non-normal characteristics of financial returns, such as skewness and kurtosis.
• It provides a more realistic estimate of potential losses, especially during periods of extreme market movements, by acknowledging "fat tails" and asymmetry in data.
• Adjusted VaR is particularly relevant for portfolios with assets whose returns deviate significantly from a normal distribution.
• While more complex to calculate, it offers a more comprehensive view of downside risk compared to standard VaR.
• It helps financial institutions and investors make more informed decisions by providing a better measure of potential extreme losses.

Formula and Calculation

The calculation of Adjusted Value at Risk typically modifies the standard VaR formula to account for the third and fourth moments of the return distribution: skewness and kurtosis.

For a standard VaR under the assumption of normal distribution, the formula might be:

Where:

• (\mu) = Mean of the portfolio returns
• (z_{\alpha}) = Z-score corresponding to the chosen confidence level (\alpha) (e.g., 1.645 for 95% confidence)
• (\sigma) = Standard deviation of the portfolio returns (representing volatility)

Adjusted VaR incorporates a Cornish-Fisher expansion, which approximates the quantiles of a non-normal distribution based on its mean, standard deviation, skewness, and kurtosis. The adjusted Z-score ((z_{\alpha}^{adj})) is calculated as:

Where:

• (z_{\alpha}) = Standard normal variate (Z-score) at the (\alpha) confidence level.
• (S) = Skewness of the return distribution.
• (K) = Excess kurtosis of the return distribution (kurtosis - 3 for normal distribution).

Once the adjusted Z-score is determined, the Adjusted Value at Risk is calculated as:

This formula accounts for the asymmetry and fat tails present in many financial returns, providing a more accurate assessment of potential losses, especially in volatile markets.

Interpreting the Adjusted Value at Risk

Interpreting Adjusted Value at Risk requires an understanding of its underlying components. While traditional VaR provides a single number representing the maximum expected loss at a given confidence level, Adjusted VaR refines this by recognizing that real-world financial asset returns often deviate from a symmetric, bell-shaped normal distribution.

• Skewness: If the return distribution exhibits negative skewness (a long tail on the left side), it implies that large negative returns are more frequent than large positive returns. Standard VaR would likely underestimate the potential for such large losses. Adjusted VaR, by incorporating negative skewness, would yield a larger (more conservative) loss estimate, signaling higher downside risk than implied by a normal distribution.
• Kurtosis: Excess kurtosis (leptokurtosis or "fat tails") indicates that extreme events, both positive and negative, occur more frequently than predicted by a normal distribution. This means a higher probability of very large losses. Adjusted VaR accounts for this by adjusting the quantile to reflect the increased likelihood of tail events, leading to a higher loss estimate than conventional VaR.

Therefore, a higher Adjusted VaR compared to a standard VaR at the same confidence level suggests that the portfolio's returns have significant negative skewness and/or excess kurtosis, indicating a greater exposure to extreme downside movements. This refined measure helps investors and financial institutions gauge their vulnerability to severe market dislocations more accurately. It highlights the importance of going beyond simple mean-variance analysis when assessing risk.

Hypothetical Example

Consider a hypothetical investment fund, "Global Macro Fund X," which holds a portfolio of diverse assets. The fund manager wants to calculate the one-day Adjusted Value at Risk for the portfolio with a 99% confidence level.

Step 1: Gather Historical Data and Calculate Moments
Over the past year, the daily returns of Global Macro Fund X have the following characteristics:

• Mean daily return ((\mu)): 0.05% (0.0005)
• Standard deviation ((\sigma)): 1.5% (0.015)
• Skewness ((S)): -0.8 (indicating a fatter left tail)
• Excess Kurtosis ((K)): 2.5 (indicating fatter tails than a normal distribution)

For a 99% confidence level, the standard normal variate ((z_{\alpha})) is approximately 2.326.

Step 2: Calculate the Adjusted Z-score using the Cornish-Fisher Expansion

Using the formula:

Substitute the values:

Calculating the terms:

• ((2.326)^2 - 1 \approx 5.410 - 1 = 4.410)
• ((2.326)^3 - 3(2.326) \approx 12.585 - 6.978 = 5.607)
• ((2(2.326)^3 - 5(2.326)) \approx (2 \times 12.585) - 11.63 = 25.17 - 11.63 = 13.54)

Now, substitute back:

Step 3: Calculate the Adjusted VaR

Assuming the portfolio value is $100,000,000:

So, the one-day Adjusted Value at Risk is approximately 3.0715% of the portfolio value.
For a $100,000,000 portfolio, this translates to a potential loss of $3,071,500.

In comparison, a standard VaR calculation (assuming normal distribution) would be:

Or a loss of $3,439,000.

In this specific hypothetical example, the Adjusted VaR appears smaller than the standard VaR. This can occur if the combination of skewness and kurtosis pushes the adjusted quantile closer to the mean than the simple normal quantile, or if the initial mean return is high enough to offset the negative adjustments. However, typically, with negative skewness and positive excess kurtosis, Adjusted VaR tends to be larger (more conservative) than standard VaR because the adjusted Z-score often becomes larger in absolute magnitude in the left tail, reflecting the increased probability of extreme losses due to non-normality. The example highlights the numerical calculation; the interpretation relies on how (z_{\alpha}^{{adj}) shifts from (z_{\alpha}). In this case, the negative skewness and positive kurtosis terms had a net negative effect on the (z}{adj}) in this hypothetical, moving it closer to the mean. It's crucial for users to understand that the impact of skewness and kurtosis can vary, and the Cornish-Fisher expansion is an approximation that can sometimes yield unexpected results at very high confidence levels or with extreme moments.

Practical Applications

Adjusted Value at Risk is a valuable tool with several practical applications across the financial industry, particularly in areas where understanding the true nature of downside risk is paramount.

• Investment Portfolio Management: Fund managers utilize Adjusted VaR to assess and manage the market risk of their investment portfolios. By considering skewness and kurtosis, they gain a more realistic view of potential extreme losses, allowing for more precise asset allocation decisions and risk budgeting. It helps in constructing portfolios that are more resilient to adverse market movements, especially those characterized by fat tails, which traditional VaR might underestimate.
• Risk Reporting and Regulatory Compliance: While the Basel Committee has shifted its focus to Expected Shortfall for internal models in capital requirements, understanding Adjusted VaR remains important for internal risk reporting and stress testing. It can complement other risk measures, providing a more comprehensive view of risk exposure to regulators and internal stakeholders who require a nuanced understanding of potential losses under non-normal market conditions.
  ⁸* Derivatives and Structured Products: The returns of derivatives, such as options and complex structured products, often exhibit significant skewness and kurtosis. Adjusted VaR is particularly useful for assessing the risk of these instruments, where the assumption of normal returns can lead to substantial misestimations of risk.
• Hedge Fund and Alternative Investments: Many alternative investment strategies, including those employed by hedge funds, can generate returns with non-normal distributions. Adjusted VaR offers a superior measure for risk assessment in these complex strategies, allowing investors to better understand the true downside potential and the impact of low-probability, high-impact events.
• Enterprise Risk Management (ERM): Large financial institutions use Adjusted VaR as part¹², ³⁴, ⁵⁶