Portfolio Risk Management with Marginal and Incremental VaR and Expected Shortfall

A practitioner’s guide to risk decomposition and trade sizing, from theory to Python implementation

Photo by Sean Pollock on Unsplash

1. Introduction

Value at Risk (“VaR”) is one of the most widely used measures of market risk, summarizing in a single number the worst loss a portfolio is expected to suffer over a given horizon at a given confidence level. While the portfolio-level number is useful for reporting, it tells a risk manager very little about where the risk is coming from or how it would change if the portfolio is re-weighted.

Two complementary risk decomposition tools answer those questions. Marginal VaR measures the sensitivity of portfolio VaR to a small change in a position — effectively a partial derivative of VaR with respect to a holding. Incremental VaR measures the discrete change in portfolio VaR caused by adding, removing, or resizing a position in full. Together they let a risk manager attribute total VaR to its sources, identify hedging candidates, and evaluate trades before they are put on.

This article develops both measures, shows how they relate to one another, and walks through a Python implementation on a small equity portfolio using historical simulation, parametric method and Monte Carlo simulation. The same logic extends to Expected Shortfall (“ES”) with minor changes, which we touch on at the end.

2. Understanding Incremental VaR

Incremental VaR (“IVaR”) is the change in portfolio VaR that results from adding a new position, removing an existing one, or changing the size of a holding. Formally, if VaR(P) is the VaR of the current portfolio P and VaR(P + Δ) is the VaR after the proposed change Δ, then:

Because VaR is computed on the full repriced portfolio in both states, IVaR captures the position’s standalone risk and the diversification (or concentration) effect with the rest of the book.

A trade that looks risky in isolation can have a small or even negative IVaR if it hedges existing exposures; on the other hand, a position that looks small can have a large IVaR if it amplifies dominant risk factors.

IVaR is computed by full revaluation i.e. building the modified portfolio, running the same VaR engine, and then differencing the two numbers. This makes it accurate but slightly more computationally expensive than marginal measures, and the result depends on the size of Δ — IVaR is not linear in trade size.

Applications of Incremental VaR

1. Pre-trade risk check: Before executing a trade, traders and risk managers compute IVaR to confirm that the post-trade portfolio remains within VaR limits. A small notional that pushes the portfolio over its limit will be flagged here.

2. Hedge sizing: Candidate hedges are evaluated by IVaR rather than by their standalone volatility. The hedge that produces the most negative IVaR per unit of cost is preferred.

3. Position unwinds: To free up risk budget, we can compute IVaR for unwinding each existing position and ranks them — the largest positive IVaR positions are the most attractive to cut.

4. Capital allocation reviews: When deciding whether to grow or shrink a strategy, IVaR on a representative scaling of that book gives a direct read on the marginal risk of the change.

3. Understanding Marginal VaR

Marginal VaR (“MVaR”) measures the sensitivity of portfolio VaR to an infinitesimal change in a position’s weight. If wᵢ is the dollar position in asset i and VaR(P) is the portfolio VaR, then:

Under the standard assumption that returns are jointly elliptical (e.g. multivariate normal) and using the parametric method, MVaR has a clean closed form solution.

Let σ(P) be portfolio standard deviation, Σ the covariance matrix, w the vector of dollar positions, and z(α) the relevant normal quantile (for example 1.645 at 95%). Then:

The derivation of this closed form solution is set out in the appendix.

An asset’s marginal VaR is proportional to its covariance with the portfolio, scaled by the confidence-level multiplier and inverse portfolio volatility.

Assets that are uncorrelated with the rest of the book have a marginal VaR near zero; assets that are heavily affected by the portfolio’s dominant risk factors have a large marginal VaR.

Marginal VaR has an important property: weighting it by position size and summing gives the total VaR exactly. The product wᵢ · MVaRᵢ is called the component VaR of asset i, and the component VaRs sum to portfolio VaR — providing an additive risk decomposition that marginal VaR alone does not. This is known as the Euler’s theorem on homogeneous functions, since VaR is positively homogeneous of degree one in the weights.

For non-parametric methods such as historical simulation and Monte Carlo simulation, marginal VaR is typically estimated by finite differences — bumping each position by a small amount and re-running VaR.

Applications of Marginal VaR

1. Risk attribution: By computing component VaR (wᵢ · MVaRᵢ) for every position, a risk manager produces an additive decomposition of portfolio VaR that answers “where is our risk?” cleanly.

2. Risk-adjusted performance: Returns divided by component VaR give a per-position Sharpe-like ratio that accounts for diversification, unlike standalone volatility-based measures.

3. Optimal portfolio construction: At the global minimum-VaR portfolio the marginal VaRs are equal across all positions (or if a return target is imposed, the optimality conditions instead require each position’s marginal VaR to line up linearly with its expected return stem from the first-order condition from the optimisation) — a useful diagnostic when reviewing whether a book is well-balanced.

4. Limit setting: Position-level limits derived from marginal VaR scale naturally with the rest of the portfolio, automatically tightening when correlations rise.

4. Comparing Incremental VaR and Marginal VaR

Marginal and Incremental VaR answer related but distinct questions. Marginal VaR is a derivative — it tells you the rate of change of portfolio VaR with respect to a small change in a position. Incremental VaR is a finite difference — it tells you the actual change in VaR for a specific, sized trade.

For very small trades the two converge: IVaR(Δ) ≈ MVaR · Δ. For meaningful trade sizes they diverge because portfolio VaR is non-linear in position size — covariances, tail dependence, and the identity of the worst-case scenario — the particular historical day or simulated draw that sits at the VaR quantile, which can jump to a different scenario as positions are resized — all shift as the book changes.

A second difference is decomposability. Component VaR (wᵢ · MVaRᵢ) sums to total VaR by Euler’s theorem, so marginal VaR supports a clean additive attribution. This additivity is a property of VaR’s positive homogeneity in the position vector and therefore holds under parametric, historical-simulation, and Monte Carlo VaR alike; what differs across methods is the noise in the MVaR estimates themselves, not whether the components sum to the total.

IVaR is path-dependent: the IVaR of removing two positions one after the other depends on the order, and the sum of single-position IVaRs does not in general equal the IVaR of removing them together. IVaR is the right tool for evaluating an actual trade; MVaR (and component VaR) is the right tool for ongoing portfolio attribution.

A third difference is computational cost. Marginal VaR has a closed form under elliptical assumptions and can be computed in one pass per position even under historical simulation. IVaR requires a full re-pricing and re-aggregation of the portfolio for each scenario considered, which scales poorly when many candidate trades are being screened.

5. Worked Example in Python

The example below builds a small four-asset equity portfolio — US, European, and emerging-market equity books (EQ_US, EQ_EU, EQ_EM) together with a technology sleeve (TECH) — held in fixed dollar positions that total $10 million. Three years of daily returns (750 observations) are simulated from a Cholesky factor and a drift vector, chosen to give the portfolio a more realistic correlation structure.

On this portfolio the script computes 95% one-day VaR by all three standard methods — historical simulation, the parametric closed form, and Monte Carlo simulation — and then derives both marginal and incremental VaR for each position.

Marginal VaR is computed three ways — by finite difference under historical simulation, by the parametric closed form on the same covariance, and by finite difference under Monte Carlo on common random draws.

Incremental VaR is evaluated for two candidate trades — adding $1 million of TECH and cutting the emerging-market book by $750,000 as a hedge — and each trade is priced four ways: by full revaluation under historical simulation, by full revaluation under Monte Carlo (re-simulating the base and post-trade books on common random numbers), and by the parametric method in both its exact form and its first-order linear approximation MVaR · Δw. Comparing the exact and linear parametric numbers shows how closely the marginal-VaR approximation tracks the true incremental VaR at this trade size.

The Monte Carlo engine draws correlated returns from the same Cholesky factor and mean used to generate the historical returns, and uses common random numbers across the base and bumped books so that marginal and incremental differences reflect the trade rather than simulation noise.

The table below sets out the resulting figures — portfolio VaR, the marginal VaR of each position, and the incremental VaR of the two candidate trades — side by side across the historical-simulation, parametric, and Monte Carlo methods. Presenting the three methods alongside one another makes the key consistency checks easy to read off: the three portfolio VaR numbers agree closely on this near-Gaussian sample, the marginal VaR estimates line up across all three approaches, and the component VaRs sum to the parametric portfolio VaR by construction.

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6. Marginal and Incremental Expected Shortfall

Expected Shortfall (“ES”), also known as Conditional VaR or “CVaR”, is the average loss in the worst (1 − α) tail of the return distribution. Where VaR answers “how bad is the threshold loss?”, ES answers “how bad is the average loss once that threshold is breached?”. ES is sub-additive (the ES of a combined portfolio never exceeds the sum of the parts), which makes it a coherent risk measure.

Marginal and incremental decompositions extend to ES with essentially the same machinery developed for VaR. The interpretations carry over directly: marginal ES is the sensitivity of portfolio ES to a small change in a position, and incremental ES is the actual change in portfolio ES from a finite trade.

Marginal ES. Under joint ellipticity, marginal ES has a closed form analogous to marginal VaR. If φ is the standard normal density and Φ its CDF, define the ES multiplier:

Then portfolio ES is:

And marginal ES is:

The only change from the marginal VaR formula is the multiplier — z(α) (the normal quantile, ≈ 1.645 at 95%) is replaced by k(α) (the conditional tail mean, ≈ 2.063 at 95%). Component ES (wᵢ · MESᵢ) sums to portfolio ES by Euler’s theorem just as component VaR sums to portfolio VaR, since ES is also positively homogeneous of degree one in the weights.

For historical simulation, marginal ES is naturally estimated by averaging each asset’s P&L over the same set of tail scenarios that define portfolio ES, rather than just the single threshold scenario that defines VaR. This averaging is the reason marginal ES is typically less noisy than marginal VaR — it draws information from the whole tail rather than from one observation.

Incremental ES. Defined exactly as for Incremental VaR:

Same workflow: build the modified portfolio, re-run the ES engine, and difference. Because ES is sub-additive, incremental ES has a property incremental VaR lacks — the sum of the standalone IES values for two trades is an upper bound on the IES of doing both together, so component-by-component IES screening cannot understate combined risk.

The table below sets out the resulting figures — portfolio ES, the marginal ES of each position, and the incremental ES of the two candidate trades — side by side across the historical-simulation, parametric, and Monte Carlo methods. The ES figures sit above their VaR counterparts throughout, reflecting the larger tail multiplier k(α) ≈ 2.063 against z(α) ≈ 1.645 at 95%, and the marginal ES estimates display the lower sampling noise expected from averaging over the whole tail rather than a single threshold scenario.

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7. Conclusion

Marginal VaR and Incremental VaR are the two natural ways of asking “what does this position contribute to my risk?”. Marginal VaR, with its companion measure component VaR, is the right tool for ongoing attribution and for diagnosing whether a portfolio is well-balanced — it is fast to compute, additive across positions by Euler’s theorem, and has a clean closed form under elliptical assumptions.

Incremental VaR is the right tool for sizing actual trades. It captures the non-linear effect of a finite change to the book, including diversification benefits that pure standalone measures miss, at the cost of a full revaluation per scenario considered.

Used together — marginal VaR for attribution, incremental VaR for trade decisions — they give a risk manager a coherent picture of where risk currently sits and how it would shift under any proposed change. The same machinery extends without surprises to Expected Shortfall, which is increasingly the preferred tail measure given its sub-additivity and better behavior under fat-tailed return distributions.

8. Appendix: Derivation of Marginal VaR under Parametric Method

Under joint ellipticity with zero mean (the mean term drops out for short horizons and is carried through unchanged otherwise), portfolio loss is:

and portfolio VaR at confidence α is:

Marginal VaR is the partial derivative of this with respect to the dollar position wᵢ:

Differentiating σ(P) by the chain rule, and for symmetric Σ using the product rule:

The partial derivative can be expressed as:

Substituting back gives the first equality:

The second equality follows because (Σw)ᵢ is, by definition the covariance of asset i’s P&L with the portfolio P&L:

It is more intuitive to see with a two-assets (n=2) example:

Substituting back gives:

As a consistency check, multiplying through by wᵢ and summing gives:

Euler’s theorem applied to the degree-one homogeneous function σ(P), and the reason component VaR is additive.

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