Marginal VAR Calculator
This tool calculates the **Marginal Value-at-Risk (MVaR)** for a specific asset position you hold or are considering adding to an existing portfolio. MVaR estimates how the portfolio's overall Value-at-Risk (VaR) is expected to change if you make a small change to the value of that specific asset.
It helps assess an asset's contribution to portfolio risk, considering its own volatility and its correlation with the rest of the portfolio.
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Understanding Marginal VAR (MVaR)
What is Marginal VAR?
Marginal VaR (MVaR) is a risk metric used to estimate how the overall Value-at-Risk (VaR) of a portfolio would change if the value of a specific asset position within that portfolio were to change by a small amount. It's essentially the sensitivity of the portfolio's VaR to a change in a particular asset's position.
While standard VaR tells you the maximum expected loss for the *entire portfolio* at a certain confidence level over a specific period, MVaR tells you the *impact* of adding or removing a specific asset on that total portfolio VaR. It's crucial for understanding which assets are contributing the most (or least) to your portfolio's total risk, especially considering diversification effects (captured by correlation).
MVaR vs. Component VAR vs. Incremental VAR
- Marginal VAR (MVaR): The sensitivity of portfolio VaR to the *value* of an asset (∂VaR_p / ∂V_a). Approximates the change in VaR from a small change in asset value.
- Component VAR (CVaR): A theoretical value representing the part of the total portfolio VaR that is "attributable" to a specific asset. It's often defined such that the sum of all Component VaRs equals the total Portfolio VaR. The relationship is often CVaR_i ≈ MVaR_i * V_i.
- Incremental VAR (IVaR): The actual change in portfolio VaR when a position is *added or removed* (not a small change). Calculating IVaR requires calculating the portfolio VaR before and after the change and taking the difference. It's generally more complex than MVaR.
This calculator uses the definition of MVaR based on the asset's correlation to the portfolio, which is a common simplification focusing on systematic risk contribution.
Marginal VAR Formula Used
The formula used in this calculator for Marginal VAR is:
MVaR = Z * Specific Asset Value * Specific Asset Volatility * Correlation
Where:
Zis the Z-score corresponding to the chosen Confidence Level (e.g., 1.645 for 95%, 2.326 for 99%).Specific Asset Valueis the current market value of the asset position.Specific Asset Volatilityis the standard deviation of the asset's returns for the chosen time period (input as a percentage, used as a decimal in calculation).Correlationis the correlation coefficient between the asset's returns and the portfolio's returns (-1 to +1).
This formula essentially calculates the asset's standalone risk (Z * Value * Volatility) and scales it by its correlation to the portfolio, reflecting how much of that risk is systematic (undiversifiable) from the portfolio's perspective.
Marginal VAR Examples
Explore how MVaR changes with different inputs:
Example 1: Highly Correlated Stock
Scenario: Calculating the MVaR of a large-cap stock (highly correlated with the market portfolio).
Inputs:
- Specific Asset Value: $50,000
- Specific Asset Volatility: 1.2% (daily)
- Correlation with Portfolio: 0.85
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $50,000 * (1.2 / 100) * 0.85
MVaR = 1.645 * $50,000 * 0.012 * 0.85
MVaR ≈ $839.03
Conclusion: At a 95% confidence level, this stock position contributes approximately $839.03 to the portfolio's daily VaR.
Example 2: Low Correlated Asset (Gold)
Scenario: Calculating the MVaR of a gold position, often having low correlation with equity portfolios.
Inputs:
- Specific Asset Value: $25,000
- Specific Asset Volatility: 0.9% (daily)
- Correlation with Portfolio: 0.20
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $25,000 * (0.9 / 100) * 0.20
MVaR = 1.645 * $25,000 * 0.009 * 0.20
MVaR ≈ $74.03
Conclusion: Due to low correlation, the gold position contributes relatively little ($74.03) to the portfolio's daily VaR compared to the stock in Example 1, despite having similar value and volatility.
Example 3: Uncorrelated Asset (Hypothetical)
Scenario: Calculating MVaR for a perfectly uncorrelated asset (correlation = 0).
Inputs:
- Specific Asset Value: $10,000
- Specific Asset Volatility: 0.5% (daily)
- Correlation with Portfolio: 0.00
- Confidence Level: 99% (Z = 2.326)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 2.326 * $10,000 * (0.5 / 100) * 0.00
MVaR = 2.326 * $10,000 * 0.005 * 0
MVaR = $0.00
Conclusion: A perfectly uncorrelated asset (according to this simplified MVaR model focusing only on systematic risk contribution) theoretically contributes $0 to the portfolio's VaR. Its standalone risk is diversified away from the portfolio's perspective.
Example 4: Negatively Correlated Asset
Scenario: Calculating MVaR for an asset that is negatively correlated with the portfolio (e.g., some hedge fund strategies, options). Note: Negative MVaR implies adding the asset might *decrease* portfolio VaR.
Inputs:
- Specific Asset Value: $5,000
- Specific Asset Volatility: 0.8% (daily)
- Correlation with Portfolio: -0.30
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $5,000 * (0.8 / 100) * -0.30
MVaR = 1.645 * $5,000 * 0.008 * -0.30
MVaR ≈ -$19.74
Conclusion: A negative MVaR suggests that this asset's position, due to its negative correlation, is expected to *reduce* the overall portfolio's VaR at the 95% confidence level. This highlights its diversification benefit.
Example 5: Higher Confidence Level
Scenario: Recalculating Example 1 with a higher confidence level (99%).
Inputs:
- Specific Asset Value: $50,000
- Specific Asset Volatility: 1.2% (daily)
- Correlation with Portfolio: 0.85
- Confidence Level: 99% (Z = 2.326)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 2.326 * $50,000 * (1.2 / 100) * 0.85
MVaR = 2.326 * $50,000 * 0.012 * 0.85
MVaR ≈ $1185.78
Conclusion: As expected, increasing the confidence level increases the calculated MVaR, reflecting a concern for more extreme potential losses (or gains/diversification effects for negative correlation).
Example 6: Lower Asset Value
Scenario: Calculating MVaR for a smaller position in the asset from Example 1.
Inputs:
- Specific Asset Value: $10,000
- Specific Asset Volatility: 1.2% (daily)
- Correlation with Portfolio: 0.85
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $10,000 * (1.2 / 100) * 0.85
MVaR = 1.645 * $10,000 * 0.012 * 0.85
MVaR ≈ $167.81
Conclusion: The MVaR is directly proportional to the asset value. A smaller position contributes less to the portfolio's VaR.
Example 7: Lower Asset Volatility
Scenario: Calculating MVaR for an asset with lower volatility than Example 1.
Inputs:
- Specific Asset Value: $50,000
- Specific Asset Volatility: 0.6% (daily)
- Correlation with Portfolio: 0.85
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $50,000 * (0.6 / 100) * 0.85
MVaR = 1.645 * $50,000 * 0.006 * 0.85
MVaR ≈ $419.52
Conclusion: Lower asset volatility directly reduces its marginal contribution to portfolio VaR.
Example 8: Asset is the Whole Portfolio (Correlation = 1)
Scenario: Calculating the MVaR when the "specific asset" is actually the entire portfolio itself. Correlation is 1.
Inputs:
- Specific Asset Value: $100,000 (This is the portfolio value)
- Specific Asset Volatility: 1.0% (daily - this is the portfolio volatility)
- Correlation with Portfolio: 1.00 (An asset is perfectly correlated with itself)
- Confidence Level: 99% (Z = 2.326)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 2.326 * $100,000 * (1.0 / 100) * 1.00
MVaR = 2.326 * $100,000 * 0.01 * 1
MVaR ≈ $2326.00
Conclusion: When the asset is the whole portfolio, MVaR calculated this way equals the total Portfolio VaR (Z * Portfolio Value * Portfolio Volatility). This simplified MVaR approach highlights the asset's contribution to portfolio risk.
Example 9: Zero Asset Value
Scenario: Calculating MVaR for an asset where the position value is zero.
Inputs:
- Specific Asset Value: $0
- Specific Asset Volatility: 2.0% (daily)
- Correlation with Portfolio: 0.50
- Confidence Level: 95% (Z = 1.645)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 1.645 * $0 * (2.0 / 100) * 0.50
MVaR = 1.645 * 0 * 0.02 * 0.50
MVaR = $0.00
Conclusion: A position with zero value contributes zero MVaR. This makes sense, as there's no capital at risk in that specific asset yet.
Example 10: Very High Volatility Asset
Scenario: Calculating MVaR for a highly volatile asset (like a small-cap stock or cryptocurrency) with moderate correlation, using a high confidence level.
Inputs:
- Specific Asset Value: $10,000
- Specific Asset Volatility: 5.0% (daily)
- Correlation with Portfolio: 0.60
- Confidence Level: 99.9% (Z = 3.090)
Formula: MVaR = Z * Value * Volatility * Correlation
Calculation: MVaR = 3.090 * $10,000 * (5.0 / 100) * 0.60
MVaR = 3.090 * $10,000 * 0.05 * 0.60
MVaR ≈ $927.00
Conclusion: High volatility combined with moderate correlation and a high confidence level can result in a significant MVaR, indicating this asset is a substantial contributor to the portfolio's tail risk.
Frequently Asked Questions about Marginal VAR
1. What is Marginal VAR (MVaR)?
MVaR is a risk metric that estimates how a portfolio's overall Value-at-Risk (VaR) would change if the value of a specific asset position within it were slightly increased or decreased. It measures the sensitivity of portfolio VaR to the size of an individual asset holding.
2. How is MVaR different from total Portfolio VaR?
Total Portfolio VaR gives you a single number representing the maximum expected loss for the entire portfolio at a given confidence level. MVaR breaks down the contribution of individual assets to that total VaR, considering their correlation with the rest of the portfolio. While sum of MVaRs * Asset_Values often approximates total VaR, it's not exactly equal like Component VaR.
3. Why is MVaR useful?
MVaR helps portfolio managers understand which assets are the biggest contributors to their portfolio's risk, accounting for diversification effects. It's valuable for portfolio construction, risk budgeting, and identifying assets that might disproportionately increase portfolio risk.
4. Does MVaR account for diversification?
Yes, the formula used (MVaR = Z * Value * Volatility * Correlation) explicitly includes the correlation with the portfolio. A lower or negative correlation reduces the asset's MVaR, reflecting its diversification benefit within the portfolio context.
5. What inputs do I need for this calculator?
You need the value of the specific asset position, the volatility of that asset's returns, the correlation between the asset's returns and the portfolio's returns, and the desired confidence level for the VaR calculation.
6. What units should I use for Asset Value and Volatility?
Asset Value should be in your reporting currency (e.g., USD, EUR). Asset Volatility should be a percentage (%) representing the standard deviation of returns over a specific period (e.g., daily, weekly, annually). The output MVaR will be in the same currency as the Asset Value, and the time horizon implicitly matches the period of your volatility input.
7. What does a negative MVaR mean?
A negative MVaR indicates that the asset is negatively correlated with the portfolio. Adding more of this asset (or increasing its value) is expected to *decrease* the portfolio's overall VaR, highlighting its potential role as a hedge or diversifier.
8. What does MVaR = 0 mean?
A MVaR of zero can occur if the asset value is zero, the volatility is zero, or the correlation with the portfolio is zero. A zero correlation implies the asset's movements are uncorrelated with the portfolio, and its standalone risk might be considered fully diversified away from the portfolio's perspective using this specific MVaR formula.
9. How is the Confidence Level used?
The confidence level determines the Z-score multiplier. A higher confidence level (e.g., 99%) uses a larger Z-score than a lower one (e.g., 95%). This results in a higher MVaR value because you are measuring the potential loss at a more extreme point in the distribution of returns.
10. Can this calculator determine if I should add an asset?
MVaR provides valuable insight into the risk contribution of an asset. A low or negative MVaR might suggest the asset is a good diversifier. However, investment decisions should also consider expected returns, transaction costs, liquidity, and other factors, not just MVaR in isolation. MVaR is a risk measure, not an investment recommendation.
