Marginal VaR: How One Position Moves Risk

Key Takeaways

• Marginal value at risk is the change in portfolio VaR from a tiny increase in one position, a sensitivity measure.
• It depends on how the position co-moves with the portfolio, not on the position's standalone volatility.
• The common mistake is treating the riskiest standalone asset as the biggest risk contributor, which ignores diversification.
• Marginal VaR drives risk budgeting and trade selection, since it shows where adding exposure costs the most risk.

What It Is

Marginal value at risk, often shortened to marginal VaR or MVaR, is the partial derivative of portfolio VaR with respect to a single position. In plain terms, it answers: if I add one more dollar to this holding, how much does the portfolio's total VaR rise or fall?

This is described in standard risk texts such as Jorion's work on value at risk. The key insight is that marginal VaR is not about how risky an asset is on its own. It is about how the asset interacts with everything else you hold.

A position that is volatile but moves opposite to the rest of the portfolio can have a small or even negative marginal VaR, because adding it reduces total risk. That is the diversification effect made measurable.

The Intuition

Total portfolio risk is not the sum of the parts. Because assets partly offset each other, the whole is usually less risky than its pieces added up. So when you ask which holding contributes most to risk, the standalone volatility is the wrong answer.

Marginal VaR fixes this. It isolates the marginal effect: the extra risk one more unit of a position brings, after accounting for how that position correlates with the portfolio. A small allocation that is highly correlated with the rest can contribute more marginal risk than a large allocation that diversifies.

This makes marginal VaR the natural tool for decisions at the edge. Before adding to a position, you want to know its risk cost, not its risk in isolation. Marginal VaR is that cost.

How It Works

Marginal VaR for asset i is the derivative of portfolio VaR with respect to the weight or dollar amount in i. With a normal-based VaR, it has a clean form driven by the covariance between the asset and the portfolio:

Marginal VaR_i = z * Cov(R_i, R_p) / sigma_p

Where:

z              = z-score for the confidence level (about 1.65 at 95 percent)
Cov(R_i, R_p)  = covariance of asset i with the portfolio
sigma_p        = standard deviation of the portfolio
R_p            = portfolio return

Note the term Cov(R_i, R_p) / sigma_p. That ratio is the asset's beta to the portfolio times the portfolio volatility. Marginal VaR therefore scales with how much the asset amplifies portfolio moves. An asset uncorrelated with the portfolio has near-zero marginal VaR; a hedge can have negative marginal VaR.

Worked Example

Suppose a portfolio has a 95 percent VaR of 1,000,000 dollars and a standard deviation of 2 percent. Asset A has a covariance with the portfolio that gives it a beta to the portfolio of 1.5.

Using the relationship between marginal VaR and portfolio beta, asset A's marginal VaR per dollar is 1.5 times the portfolio's VaR per dollar of exposure. So adding 100,000 dollars of asset A raises portfolio VaR by roughly 1.5 times what an average-beta dollar would add.

Now compare asset B, a hedge with a portfolio beta of negative 0.3. Its marginal VaR is negative. Adding 100,000 dollars of B lowers portfolio VaR. A trader choosing between A and B for the same expected return would prefer B, because it reduces total risk while A concentrates it.

Common Mistakes

1. Confusing standalone risk with marginal risk. The most volatile asset is not always the largest risk contributor. Marginal VaR depends on co-movement, so a calm but highly correlated asset can matter more.

2. Ignoring the sign. Marginal VaR can be negative for a genuine hedge. Treating all positions as risk-additive misses the offsetting positions that lower total VaR.

3. Using it for large trades. Marginal VaR is a derivative, valid for small changes. For a big position change, the linear approximation breaks down and you should compute incremental VaR instead.

4. Forgetting it shifts as the portfolio changes. Marginal VaR is measured at the current portfolio. Once weights change, every asset's marginal VaR changes too, so it must be recomputed.

5. Assuming normality blindly. The clean covariance formula assumes normal returns. With fat tails, marginal contributions to a tail measure can differ, so historical or simulation methods may be more reliable.

Frequently Asked Questions

What is marginal value at risk in simple terms? Marginal value at risk is how much your portfolio's total risk goes up if you add a little more to one holding. It measures a position's risk cost given everything else you own.

How does marginal value at risk affect investment decisions? It guides which position to add or trim. If two assets offer similar expected returns, you add the one with lower marginal VaR, because it raises portfolio risk the least. It is the core input to risk budgeting.

What is a real-world example of marginal value at risk? A fund manager deciding whether to add an energy stock or a bond computes each one's marginal VaR. The bond, moving opposite to the equity-heavy book, may show negative marginal VaR, signaling it would actually cut total portfolio risk.

How can investors use marginal VaR effectively? Use it for small allocation tweaks and to rank positions by risk contribution. For large trades, switch to incremental VaR, and recompute marginal VaR after each rebalance since it depends on current weights.

How is marginal VaR different from component VaR? Marginal VaR is a per-unit sensitivity, the risk added by one more dollar. Component VaR multiplies that sensitivity by the position size to give each holding's actual share of total portfolio VaR, and the components sum to the whole.

Sources

1. AnalystPrep. "Describe Extensions of VaR." https://analystprep.com/study-notes/cfa-level-2/describe-extensions-of-var/
2. CFA Institute. "Measuring and Managing Market Risk." https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/measuring-managing-market-risk
3. Ryan O'Connell, CFA. "Portfolio VaR & Risk Decomposition: Component and Marginal VaR." https://ryanoconnellfinance.com/portfolio-var-risk-decomposition/
4. Garman, M. "Decomposing Portfolio Value-at-Risk: A General Analysis." Erasmus University repository. https://repub.eur.nl/pub/7723/1999-0342.pdf

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.