Computing Portfolio VaR from Dependent Risks with a Copula
Summary
For a portfolio formed by summing two dependent random variables, the note explains that VaR is still a one-dimensional quantile: first find the distribution of their sum, then take its target quantile. It gives an integral for the sum’s cumulative distribution, using the copula’s partial derivative and the marginal distribution functions. This accounts for dependence through the copula rather than treating the component distributions as independent.
The derivation integrates the joint density over outcomes whose sum is below a threshold, then reduces that calculation to an integral with respect to the first variable’s marginal distribution. VaR can be computed once this cumulative distribution is available or estimated. The displayed derivation assumes differentiable distributions and copula; the note does not provide a numerical example or discuss estimation choices, discrete marginals, or methods for handling cases where the derivative or density does not exist.
Key ideas
- VaR for a portfolio sum is the quantile of the sum’s univariate distribution.
- The sum’s cumulative distribution can be computed by integrating the copula-based conditional contribution across one marginal.
- Dependence enters through the copula’s partial derivative, alongside both marginal distribution functions.
- The derivation uses density and differentiability assumptions that may not hold for every model.
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# Portfolio VaR with Copula?
# Portfolio VaR with Copula?
Let the portfolio be given by: $$X=X_1+X_2$$ $(X_1,X_2)$ are dependent through a Copula function $C(u_1,u_2)$, such that the joint distribution is given by: $$F(x_1,x_2)=C(F(x_1),F(x_2))$$
What is the VaR of this portfolio?
Usually VaR is the inverse quantile: $VaR_\alpha=F^{-1}_X(\alpha)$.
I am not sure how to determine it in this multivariate case?
## Answer by Ulysses (score 5, accepted)
https://quant.stackexchange.com/a/15603
You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that $$ \operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\} $$ and since in your case $X = X_1+X_2$ you just need to compute $F_X$ in terms of $X_1$ and $X_2$. For the notation of partial derivatives, I denote the generic variables of the copula function by $u_1$ and $u_2$. $$ F_X(v) = \mathbb P(X_1+X_2\leq v) = \int\limits_{-\infty}^\infty \frac{\partial C}{\partial u_1}\left(F_{X_1}(x_1),F_{X_2}(v-x_1)\right)\mathrm dF_{X_1}(x_1). \tag{1} $$ As long as you can compute/estimate this function, you can get a value/estimate for VaR. The formula $(1)$ can be obtained as follows: $$ \begin{align} F_X(v) &= \mathbb P(X_1+X_2\leq v) = \int\limits_{-\infty}^\infty \mathrm dx_1 \int\limits_{-\infty}^{v-x_1}\frac{\partial^2 C}{\partial u_1\partial u_2}\left(F_{X_1}(x_1),F_{X_2}(x_2)\right)F'_{X_1}(x_1)F'_{X_2}(x_2)\mathrm dx_2 \\ &= \int\limits_{-\infty}^\infty \mathrm dx_1 \int\limits_{-\infty}^{v-x_1}\frac{\partial}{\partial x_2}\left(\frac{\partial C}{\partial u_1}\left(F_{X_1}(x_1),F_{X_2}(x_2)\right)F'_{X_1}(x_1)\right)\mathrm dx_2 \\ &= \int\limits_{-\infty}^\infty \left.\frac{\partial C}{\partial u_1}\left(F_{X_1}(x_1),F_{X_2}(x_2)\right)F'_{X_1}(x_1)\right|_{x_2=-\infty}^{x_2=v-x_1}\mathrm dx_1 \\ &= \int\limits_{-\infty}^\infty \frac{\partial C}{\partial u_1}\left(F_{X_1}(x_1),F_{X_2}(v-x_1)\right)F'_{X_1}(x_1)\mathrm dx_1 \\ &= \int\limits_{-\infty}^\infty \frac{\partial C}{\partial u_1}\left(F_{X_1}(x_1),F_{X_2}(v-x_1)\right)\mathrm dF_{X_1}(x_1). \end{align} $$ For the partial derivatives notation, consider the following example. If $g(u_1,u_2) = u_1 + u_2$ then $$ \frac{\partial }{\partial u_1}g(x_1^2,x_2^2) = 1,$$ $$ \frac{\partial}{\partial x_1}g(x_1^2,x_2^2) = 2x_1. $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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