Why Normal VaR Is Subadditive Under Joint Normality
Summary
The document explains why Value at Risk can satisfy subadditivity when portfolio profits are jointly normally distributed. It expresses VaR at a confidence level below one-half using the position’s mean and standard deviation, then compares the VaR of two positions with the VaR of their combined return. The combined standard deviation includes their correlation, so it cannot exceed the sum of the individual standard deviations; this yields the diversification inequality under the stated assumptions.
The argument applies to two normally distributed positions, with the same conclusion stated for elliptically distributed risk factors in the question. It assumes the specified VaR convention based on portfolio profits and a confidence parameter below one-half. The document cautions that subadditivity need not hold for other distributions, and notes that position weights can be incorporated. It gives an analytic derivation rather than empirical evidence, and does not establish the claim for arbitrary portfolios or every VaR convention.
Key ideas
- For jointly normal positions, portfolio profit remains normally distributed, with variance determined by both individual variances and their correlation.
- Under the document’s profit-based VaR convention and stated confidence assumption, the VaR of the combined positions does not exceed the sum of their individual VaRs.
- The diversification inequality follows because combined volatility is bounded by the sum of individual volatilities.
- The result may fail when the positions do not follow the assumed distribution.
- Portfolio weights can be included when defining the combined position.
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# Parametric VaR, Normality and Subadditivity
# Parametric VaR, Normality and Subadditivity
Good evening; I just have a simple question about Value at Risk and the subadditivity property, and I know that it may sound silly
I got that, in general, VaR is not subadditive. However, if a portfolio contain elliptically distributed risk factors, then VaR will be subadditive.
What I do not understand is why an ellipticall distribution, such as the Normal distribution, implies subadditivity of VaR.
Thanks in advance.
## Answer by Kevin (score 8, accepted)
https://quant.stackexchange.com/a/61669
Suppose $X\sim N(\mu_X,\sigma_X^2)$ and $Y\sim N(\mu_Y,\sigma_Y^2)$ are correlated jointly normal random variables. Then, $$X+Y\sim N(\mu_X+\mu_Y,\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y).$$
Suppose $X$ and $Y$ denote the profit of your portfolio returns (so negative values for $X,Y$ mean losses). Then, the 5% value at risk is the 0.05 quantile of the profit distribution (the minimum amount you lose in the worst five percent of cases $\Leftrightarrow$ in 95% of the cases you're sure to lose less than this value-at-risk). Thus, using the inverse function of the normal distribution (see here), \begin{align*} \text{VaR}(X,\alpha)&=-\mu_X+\sigma_X\Phi^{-1}(1-\alpha),\\ \text{VaR}(Y,\alpha)&=-\mu_Y+\sigma_Y\Phi^{-1}(1-\alpha), \\ \text{VaR}(X+Y,\alpha)&=-\mu_X-\mu_Y+\sqrt{\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y}\Phi^{-1}(1-\alpha). \end{align*} Thus, \begin{align*} \text{VaR}(X,\alpha) + \text{VaR}(Y,\alpha) &= -\mu_X-\mu_Y+(\sigma_X+\sigma_Y)\Phi^{-1}(1-\alpha) \\ &\geq -\mu_X-\mu_Y+\sqrt{\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y}\Phi^{-1}(1-\alpha) \\ &= \text{VaR}(X+Y,\alpha), \end{align*} because for $\rho\in(-1,1)$, \begin{align*} \sigma_X+\sigma_Y = \sqrt{\sigma^2_X+\sigma_Y^2+2\sigma_X\sigma_Y} \geq \sqrt{\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y}. \end{align*}
Thus, value-at-risk (just like the standard deviation) is always sub-additive for two normally distributed loss distributions, if we assume $\alpha<0.5$, (``benefit of diversification''): a portfolio containing $X$ and $Y$ is less risky than the sum of the individual risks. If we assume a different distribution for $X$ and $Y$, then this result may no longer hold.
You can, of course, fiddle in some weights and consider $wX+(1-w)Y$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.