Parametric Value-at-Risk: A Quantitative Method to Risk Assessment

Value-at-Risk (VaR) is a statistical measure used to estimate the potential loss in the value of a portfolio at a certain confidence level over a specific time horizon. It provides a single risk number that helps traders, risk managers, and financial institutions assess how much they can potentially lose under normal market conditions.

Parametric Value-at-Risk (VaR)

The Parametric VaR method, also known as the Variance-Covariance Approach or Delta-Normal VaR, is a statistical model that estimates risk based on the assumed normal distribution of returns. It calculates the potential loss of a portfolio using the mean and standard deviation (volatility) of asset returns, assuming that returns are normally distributed and independent.

For VaR calculation using the parametric method, the z-score is determined (a statistical measurement that reflects the number of standard deviations from the mean a particular data point is). And to calculate the z-score, we use the standard normal distribution, whose statistical parameters such as mean is 0 and the standard deviation is 1.

Using the inverse of the standard normal distribution function, one can easily identify the z-score given the confidence level as probability as an argument to this function.

= NORM.S.INV(Probability)

= NORM.S.INV(99%) is 2.3263. It means that 2.3263 times the standard deviation from the mean of the distribution would correspond to a confidence level of 99%

The formula for parametric VaR calculation is:

Value at Risk (VaR) = Mean - (z-Score * Standard Deviation)

Parametric VaR Method With An Illustration

As per the past return outcomes, the daily mean return on a portfolio of stocks having exposure of $5m is 0.1056% with a standard deviation of 0.9815%. Calculate 1-Day 99% VaR and 10-Day 99% VaR assuming 252 days a year.

1-Day Value at Risk (VaR) = Daily Mean - (z-Score * Daily Standard Deviation)

Where,

• Daily Mean = 0.1024%

• Daily Standard Deviation = 1.0457%

• At 99% probability, the z-score is approximately 2.3263.

1-Day 99% Relative Value-at-Risk (VaR)

= Mean - ( z-Score * Standard Deviation ) = 0.1024% - ( 2.3263 * 1.0457% ) = -2.3302%.

Therefore, a 1-Day 99% Absolute VaR is -$0.1165m (-2.3302% of $5m) is the maximum potential loss that can occur in a single day with a 1% probability of being exceeded (in the 1% of the abnormal circumstances, the losses can exceed $0.1165m).

10-Day 99% VaR can be calculated as,

10-Day Value at Risk (VaR) = 10-Day Mean - (z-Score * 10-Day Standard Deviation)

Where,

• A 10-Day Mean = Daily Mean * 10 = 0.1024% * 252 = 1.0243%

• A 10-Day Standard Deviation = Daily Standard Deviation * √10 = 1.0457% * √250 = 3.3067%

• At 99% probability, the z-score is approximately 2.3263.
  

10-Day 99% Relative Value-at-Risk (VaR)

= Mean - ( z-Score * Standard Deviation ) = 1.0243% - ( 2.3263 * 3.3067% ) = -6.6683%.

Therefore, a 10-Day 99% Absolute VaR is -$0.3334m (-6.6683% of $5m) is the maximum potential loss that can occur in 10 days with a 1% probability of being exceeded (in the 1% of the abnormal circumstances, the losses can exceed $0.3334m).

Efficient Market Hypothesis (EMH) Theory

The EMH is a foundational theory in finance that asserts that markets are informationally efficient. In the weak form of EMH, prices reflect all publicly available information, including historical prices and returns.

• It is impossible to generate consistent excess returns by relying on historical data.

• Technical analysis becomes ineffective as all patterns and trends from past price movements are already incorporated into the current price.

• The expected return on a daily basis, derived from past returns, is statistically zero, as price changes follow a random walk.

Since prices are assumed to follow a random walk with no autocorrelation, future returns are unpredictable, and the expected mean return is taken as 0%. This directly affects the Value-at-Risk (VaR) calculations, as VaR uses the mean and standard deviation of returns.

If the EMH theory holds in real financial markets, the expected mean return will be 0, and thus,

1-Day 99% Relative Value-at-Risk (VaR) Calculation, assuming that:

• Mean return = 0% (based on EMH – weak form),

• Standard deviation (σ) = 1.0457%,

• z-score for 99% confidence level = 2.3263,

The 1-day 99% Relative VaR is:

= Mean - ( z-Score * Standard Deviation ) = 0% - ( 2.3263 * 1.0457% ) = -2.4326%

Using the square-root-of-time rule, which assumes returns are independent and identically distributed (i.i.d) and volatility scales with the square root of time:

• Mean return = 0% (based on EMH – weak form),

• Standard deviation (σ) = 1.0457% x SQRT(10) = 3.3067%,

The 10-Day 99% Relative VaR is:

= Mean - ( z-Score * Standard Deviation ) = 0% - ( 2.3263 * 3.3067% ) = -7.6926%.

Or, 1-Day 99% Relative VaR * SQRT(10) = -2.4326% * SQRT(10) = -7.6926%.

The 10-Day 99% Absolute VaR is -7.6926% of $5,000,000 = -$384,630 (or -$0.3846m)

Under the EMH assumption with zero mean return,

There is 99% confidence that the maximum potential loss over 10 days will not exceed $0.3846 million. However, there is a 1% probability that losses could exceed this amount under abnormal market conditions—which are often ignored in traditional VaR frameworks but critical in stress testing.

Practical Implications

• EMH's assumption of a zero mean return simplifies VaR modeling but ignores potential drifts in asset prices, especially in trending or volatile environments. In reality, small but persistent positive or negative drifts (returns) can impact long-term VaR.

  
  

• The square root of time (√T) rule is valid only under i.i.d. returns. It breaks down during volatility clustering or structural breaks, such as during crises.

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Limitations of the Parametric Value-at-Risk Method

The parametric value-at-risk method has some advantages over other VaR methods, such as its simplicity and ease of calculation. Still, it also has limitations that must be taken into account.

• Underestimates Extreme Risk (Tail Risk): Parametric Value-at-Risk (VaR) assumes that asset returns follow a normal distribution, which significantly underestimates the probability of extreme losses. In reality, financial return distributions exhibit fat tails—extreme events are more frequent than predicted by the normal distribution. for example, during the 2008 financial crisis, equity markets dropped more than 10% in a single day, a move that, under a normal distribution, would be considered a near impossibility (a 6+ standard deviation event). Parametric VaR did not anticipate such tail events, leaving institutions exposed.

  
  

  Solution: Use Expected Shortfall (ES), which captures the average loss beyond the VaR threshold, providing a fuller picture of tail risk, or Extreme Value Theory (EVT), which specifically models the tails of the distribution using the Generalized Pareto Distribution (GPD), offering a more accurate assessment of rare, catastrophic losses..

• Ignores Volatility Clustering (Ineffective in Crisis Periods): Parametric VaR models typically assume constant volatility. However, empirical evidence shows that financial markets exhibit volatility clustering—periods of high volatility are often followed by more high volatility. for example, In the COVID-19 market crash (March 2020), market volatility exploded over a short span (markets became highly volatile). Parametric VaR, using a fixed standard deviation or an exponentially weighted average, lagged behind the rapid changes and underestimated the true risk during the escalation phase.

  
  

  Solution: GARCH models can improve risk estimation by modeling time-varying volatility. They dynamically model time-varying volatility by capturing autocorrelation in variance, making VaR estimates more responsive to market stress and regime changes.

• Fails for Non-Linear Portfolios (Options): Parametric VaR assumes linear relationships between changes in risk factors and portfolio value. This is a poor assumption for instruments with non-linear payoffs such as options, structured products, and convex instruments, where small changes in underlying assets can have amplified effects on portfolio value due to Greeks like Gamma and Vega. for example, a portfolio containing long-dated call options (non-linear payoffs due to Gamma risk) may show minimal risk in a linear VaR framework, but as the underlying asset approaches the strike price, Gamma risk increases significantly, which linear Parametric VaR fails to account for.

  
  

  Solution: Use Delta-Gamma VaR that extends the linear model by incorporating second-order (Gamma) effects, offering better accuracy for convex instruments or Monte Carlo simulation for options portfolios that allows for full revaluation of the portfolio under randomly generated scenarios, capturing all non-linear dynamics and tail risks.

• Assumes Correlations Are Constant (Breaks in Stress Events): Parametric VaR models often rely on historical correlation matrices, assuming that asset relationships remain stable. However, during market stress, correlations tend to spike, especially among asset classes that were previously uncorrelated and become highly correlated. for example, equities and government bonds typically have a negative correlation. But during global crises (2008 or March 2020), both markets declined simultaneously as investors sold off all assets to raise cash—rendering historical correlations invalid.

  
  

  Solution: Use stress testing, which evaluates portfolio sensitivity under extreme but plausible market conditions where correlations are assumed to break, and dynamic correlation models (DCC-GARCH) to adjust correlations over time to reflect evolving market conditions and co-movement risk more realistically and model's improve accuracy.

• Not Suitable for Fat-Tailed Markets: In some markets, especially commodities, emerging markets, and cryptocurrencies, returns often exhibit heavy skewness and kurtosis. The normal distribution assumption in Parametric VaR fails to account for these frequent and extreme jumps. for example, Oil prices are susceptible to geopolitical events and supply-demand shocks. In 2020, WTI crude oil futures even turned negative—an event that would be impossible under a Gaussian assumption.

  
  

  Solution: Use the Extreme Value Theory (EVT), which models the tails explicitly and is well-suited for markets prone to jumps and discontinuities, or historical simulation for fat-tailed markets that rely on actual historical return distributions, capturing extreme movements without relying on parametric assumptions.