VaR Component Contributions and Differentiability

Why VaR Contributions Are Local

A portfolio VaR is a single capital measure for all included risks. Allocation asks how that total changes when a business unit’s exposure changes. For VaR, the answer depends on the portfolio’s loss distribution at its chosen quantile—not only on the unit’s own loss distribution. This makes VaR allocation a local sensitivity calculation.

Let $X_i$ be the loss per unit of exposure for business unit $i$, and let $w_i$ be its exposure weight. The aggregate loss is $L(\boldsymbol{w})=\sum_{i=1}^n w_iX_i$. For confidence level $\alpha$, define portfolio VaR as the $\alpha$-quantile of $L$. The definitions below use losses, so a larger VaR means a larger loss threshold.

Definition: For a loss $L$ with distribution function $F_L$, define $\operatorname{VaR}_\alpha(L)=\inf\{x:F_L(x)\geq\alpha\}$. When the function $V(\boldsymbol{w})=\operatorname{VaR}_\alpha(L(\boldsymbol{w}))$ is differentiable with respect to the weights, the marginal VaR for unit $i$ is $\partial V/\partial w_i$. Its component VaR contribution is $C_i=w_i\,\partial V/\partial w_i$.

Marginal VaR measures the change in portfolio VaR per additional unit of exposure, holding the other weights fixed. Component VaR scales that sensitivity by the unit’s current weight. It is an allocation at the current portfolio position, not a separate calculation of the unit’s own capital.

Conditions for Differentiating VaR

VaR is a quantile, and quantiles do not always vary smoothly with model inputs. A useful sufficient setting is a portfolio loss with a density that is positive at the selected quantile, together with enough regularity of the joint loss distribution to differentiate the quantile equation. In particular, assume that the relevant derivatives exist, that differentiation can be passed through the distribution calculation, and that the conditional expectation at the quantile is well-defined.

Write $q=V(\boldsymbol{w})$, and let $f_L(q)>0$ be the density of portfolio loss at $q$. Under these conditions, the marginal VaR has a useful conditional-expectation representation:

Result: For $L=\sum_j w_jX_j$, when the loss distribution is sufficiently smooth at $q=\operatorname{VaR}_\alpha(L)$ and $f_L(q)>0$, $\displaystyle \frac{\partial V}{\partial w_i}=E[X_i\mid L=q]$. The conditional expectation is understood through the conditional distribution at the loss level $q$; it is not an ordinary average over the event $L=q$, which typically has probability zero for a continuous loss.

The formula says that a unit’s marginal contribution is its expected loss in scenarios lying at the portfolio VaR boundary. It is not generally the unit’s unconditional expected loss, nor its own VaR. Dependence matters because the other units determine which scenarios put the whole portfolio at that boundary.

Why the Conditional-Expectation Formula Holds

The quantile $q$ satisfies $F_L(q;\boldsymbol{w})=\alpha$ when the distribution is continuous and strictly increasing at $q$. Differentiate this equation with respect to $w_i$. The change in $q$ contributes $f_L(q)\,\partial q/\partial w_i$. Increasing $w_i$ changes the aggregate loss by $X_i$; differentiating the probability that aggregate loss remains below the fixed threshold $q$ gives $-f_L(q)E[X_i\mid L=q]$. Thus:

0=f_L(q)\frac{\partial q}{\partial w_i} -f_L(q)E[X_i\mid L=q].

Because $f_L(q)>0$, divide both sides by $f_L(q)$ to obtain $\partial q/\partial w_i=E[X_i\mid L=q]$. The boundary-probability derivative used here requires the stated smoothness and integrability conditions; it should not be applied automatically to distributions with atoms or other points of non-differentiability.

Normal-Model Calculation

A particularly convenient case is a jointly normal vector of unit losses. Suppose $X_i$ has mean $\mu_i$, and the portfolio loss $L=\sum_j w_jX_j$ has mean $\mu_L$ and standard deviation $\sigma_L>0$. At confidence level $\alpha$, let $z_\alpha=\Phi^{-1}(\alpha)$, where $\Phi$ is the standard normal distribution function. Then $q=\mu_L+z_\alpha\sigma_L$.

For jointly normal variables, the conditional mean is linear: $E[X_i\mid L=q]=\mu_i+\operatorname{Cov}(X_i,L)(q-\mu_L)/\sigma_L^2$. Substituting $q-\mu_L=z_\alpha\sigma_L$ gives:

\frac{\partial V}{\partial w_i} =\mu_i+z_\alpha\frac{\operatorname{Cov}(X_i,L)}{\sigma_L}, \qquad C_i=w_i\left(\mu_i+z_\alpha\frac{\operatorname{Cov}(X_i,L)}{\sigma_L}\right).

This formula assumes joint normality and positive portfolio standard deviation. It is not a general formula for arbitrary loss distributions. Outside the normal model, use a valid distribution-specific derivative or the conditional-expectation result when its regularity conditions hold.

Worked Example: Two Correlated Normal Business Units

Method and purpose. Use the jointly normal conditional-mean formula to allocate portfolio VaR. Two hypothetical business units have unit losses $X_1\sim N(10,16)$ and $X_2\sim N(20,9)$, where the second argument is the variance. Their correlation is $0.5$, and both exposure weights are 1. Calculate VaR at $\alpha=\Phi(2)$, approximately $0.9772$, and each component contribution.

Portfolio distribution. The covariance is $0.5(4)(3)=6$. The portfolio mean and variance are:

\mu_L=10+20=30, \qquad \sigma_L^2=16+9+2(6)=37, \qquad \sigma_L=\sqrt{37}=6.0828.

Therefore $V=30+2\sqrt{37}=42.1655$, or $42.17 when rounded to the nearest cent.

Marginal and component contributions. The covariances with aggregate loss are $\operatorname{Cov}(X_1,L)=16+6=22$ and $\operatorname{Cov}(X_2,L)=6+9=15$. Hence:

\frac{\partial V}{\partial w_1}=10+\frac{2(22)}{\sqrt{37}}=17.2336, \qquad C_1=1(17.2336)=17.2336,

\frac{\partial V}{\partial w_2}=20+\frac{2(15)}{\sqrt{37}}=24.9320, \qquad C_2=1(24.9320)=24.9320.

The unrounded contributions sum to $17.2336+24.9320=42.1656$, agreeing with VaR up to the displayed intermediate rounding. The component amounts are $17.23 and $24.93. Their separately rounded values sum to $42.16; the one-cent difference from rounded portfolio VaR is only a rounding effect.

Common mistake avoided. Using each unit’s standalone VaR would ignore the portfolio dependence and would not produce these local sensitivities. The calculation uses each unit’s covariance with the aggregate loss.

Euler Allocation and Full Allocation

For positive exposure weights, portfolio VaR is positively homogeneous: multiplying every exposure by $c>0$ multiplies every portfolio loss and its VaR by $c$. If $V$ is differentiable at $\boldsymbol{w}$, Euler’s theorem for differentiable homogeneous functions gives:

V(c\boldsymbol{w})=cV(\boldsymbol{w}) \quad\Longrightarrow\quad \sum_{i=1}^n w_i\frac{\partial V}{\partial w_i}=V(\boldsymbol{w}).

The left-hand side is the sum of component VaR contributions. Thus, under differentiability and positive homogeneity, Euler component contributions fully allocate portfolio VaR. This is a reconciliation property; it does not imply that each contribution must be positive or equal to the unit’s standalone VaR.

Calculate aggregate VaR.
Determine the portfolio quantile at the required confidence level under the specified joint loss model.

Find each marginal sensitivity.
Use the normal-model expression when joint normality applies, or the conditional loss at the VaR boundary under suitable smoothness conditions.

Scale by current exposure.
Multiply each marginal VaR by its weight to obtain the component contribution.

Reconcile before rounding.
Check that the unrounded components sum to portfolio VaR when the differentiability and homogeneity conditions hold.

Worked Example: Unequal Exposures and Standalone VaR

Method and purpose. Apply the normal formula with unequal weights, then compare the local components with standalone VaRs. Two hypothetical independent unit losses are $X_1\sim N(4,9)$ and $X_2\sim N(6,16)$. Their weights are $w_1=2$ and $w_2=1$. Calculate VaR and contributions at $\alpha=\Phi(1)$, approximately $0.8413$.

The portfolio mean is $2(4)+1(6)=14$. Independence gives portfolio variance $2^2(9)+1^2(16)=52$, so $\sigma_L=\sqrt{52}=7.2111$. Thus $V=14+\sqrt{52}=21.2111$, or $21.21 to the nearest cent.

Independence gives $\operatorname{Cov}(X_1,L)=2(9)=18$ and $\operatorname{Cov}(X_2,L)=1(16)=16$. The marginal sensitivities and weighted contributions are:

\frac{\partial V}{\partial w_1}=4+\frac{18}{\sqrt{52}}=6.4962, \qquad C_1=2(6.4962)=12.9923, $$ $$ \frac{\partial V}{\partial w_2}=6+\frac{16}{\sqrt{52}}=8.2188, \qquad C_2=1(8.2188)=8.2188.

Before rounding, $C_1+C_2=21.2111=V$. The contributions are $12.99 and $8.22. By contrast, the standalone VaRs per unit are $4+3=7$ and $6+4=10$; multiplying these by the current weights gives $14 and $10. Those weighted standalone amounts are not the portfolio’s Euler components.

Common mistake avoided. Do not substitute a unit’s standalone VaR for its marginal VaR and then multiply by its weight. Marginal VaR measures sensitivity of the total portfolio quantile, not the unit’s own quantile.

Worked Example: A Smooth Non-Normal Portfolio

Method and purpose. Use the conditional-expectation result rather than a normal approximation. Let $X_1$ and $X_2$ be independent hypothetical losses, each uniformly distributed on $[0,10]$, with weights both equal to 1. Find VaR and component contributions at $\alpha=0.75$.

For $10\leq s\leq20$, the portfolio sum has distribution function $F_L(s)=1-(20-s)^2/200$. Set this equal to $0.75$:

1-\frac{(20-s)^2}{200}=0.75 \quad\Longrightarrow\quad (20-s)^2=50 \quad\Longrightarrow\quad q=20-\sqrt{50}=12.9289.

The relevant density is $f_L(q)=(20-q)/100=\sqrt{50}/100>0$, so it is positive at the quantile. Given $X_1+X_2=q$, the possible values of $X_1$ range from $q-10$ to $10$, and the conditional density is uniform over that interval. Its mean is $((q-10)+10)/2=q/2$. By symmetry, $E[X_2\mid L=q]=q/2$ as well.

Therefore each marginal VaR is $q/2=6.4645$, and, because both weights are 1, each component is also $6.4645$. The unrounded components sum to $q=12.9289$. Rounded to the nearest cent, VaR is $12.93 and each contribution is $6.46; separately rounded components need not sum to the rounded total.

Common mistake avoided. The units’ marginal contributions are not their unconditional means, which are both 5. The relevant conditional means are evaluated at the portfolio’s VaR boundary.

Exam Traps and Interpretation

A frequent error is to confuse “marginal” with “standalone.” For example, in the unequal-exposure example, a mistaken calculation might assign $2(7)=14$ to unit 1 because its weighted standalone VaR is $14. The correction is to calculate the derivative of portfolio VaR first: its marginal is approximately $6.50 per unit of weight, so its current component is $2(6.4962)$, or about $12.99.

Check the loss convention. These formulas treat larger values as larger losses. A convention based on gains may reverse signs or require a different quantile.
Keep the exposure weight in the component. The marginal derivative alone is not the component unless the weight is 1.
Use a model-specific formula. The normal expression requires joint normality; it is not justified merely because means and variances are known.
Check regularity before differentiating. A positive density and suitable smoothness at the quantile support the local derivative formula. If differentiability fails, the conditional-expectation and Euler arguments above do not automatically apply.
Reconcile unrounded amounts. Small discrepancies after rounding individual contributions are not failures of full allocation.

These methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. In an allocation calculation, state the loss model and confidence level, identify the derivative being used, scale it by the current exposure, and check the total before rounding.

Check Your Understanding

For each problem, identify the applicable model and check whether the component contributions reconcile to portfolio VaR.

Independent losses are $A\sim N(5,4)$ and $B\sim N(8,9)$, with both weights equal to 1. At $\alpha=\Phi(1)$, find portfolio VaR and both component contributions. (Answer: VaR $=13+\sqrt{13}\approx \$16.61$; contributions $5+4/\sqrt{13}\approx \$6.11$ and $8+9/\sqrt{13}\approx \$10.50$.)
Jointly normal losses $A$ and $B$ have means 2 and 5, variances 1 and 4, and covariance 1. Weights are 1 and 2. At $\alpha=\Phi(1)$, calculate VaR and the two components. (Answer: VaR $=12+\sqrt{21}\approx \$16.58$; contributions $2+3/\sqrt{21}\approx \$2.65$ and $2(5+9/\sqrt{21})\approx \$13.93$.)
Independent losses $X_1,X_2$ are each uniform on $[0,6]$, with weights 1 and 1. At $\alpha=0.25$, find VaR and component contributions. (Answer: VaR $=3\sqrt{2}\approx \$4.24$; each component $=3/\sqrt{2}\approx \$2.12$.)
Independent losses $U\sim N(3,16)$ and $W\sim N(7,9)$ have weights 1 and 1. At $\alpha=\Phi(1)$, calculate the components and compare them with the standalone VaRs. (Answer: portfolio VaR $=\$15.00$; components are $\$6.20$ and $\$8.80$; standalone VaRs are $\$7.00$ and $\$10.00$.)

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