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I have been reading up on VaR and get very confused by the subadditivity concept.

On wikipedia, it says "VaR is not subadditive: VaR of a combined portfolio can be larger than the sum of the VaRs of its components."

However, as I am reading through John Hull's Options, Futures and other Derivatives, it has the following example talking about the benefit of diversification with VaR:

The 10-day 99% VaR for the portfolio of Microsoft shares is $1,473,621.

The 10-day 99% VaR for the portfolio of AT&T shares is $368,405.

The 10-day 99% VaR for the portfolio of both Microsoft and AT&T shares is $1,622,657.

The amount (1,473,621+368,405) - 1,622,657 = $219,369

So my confusion is that, in this case, isn't the portfolio VaR less than the sum of individual VaR? It looks like it is subadditive here. So where does the conflict come from? Thanks in advance.

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    $\begingroup$ it is sometimes sub-additive. For example if the distributions are Gaussian then it is. When we say VAR is not sub-additive, we mean that it is possible to find cases where it fails. That doesn't mean it always fails. $\endgroup$ – Mark Joshi May 9 '17 at 5:40
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VaR is not sub-additive in general.

Relying on Mark Joshi comment, there are particular cases where it can be. Such cases occur for portfolios containing elliptically distributed risk factors. Of course the normal distribution is among the elliptical distributions family.

The latter can be helpful for analytical VaR modelling as an elliptical model is usually a reasonable approximation for instruments such as equity or FX returns. Then the sub-additive property can be applied.

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Simple example where sub-additivity fails

Let there be four possible outcomes $i=1,2,3,4$ that occur with equal probability $\frac{1}{4}$. Payoffs for $X$, $Y$, and $X + Y$ are given by:

$$ X = \begin{bmatrix}-1\\0\\1\\2 \end{bmatrix} \quad Y = \begin{bmatrix}0\\-1\\1\\2 \end{bmatrix} \quad X + Y = \begin{bmatrix}-1\\-1\\2\\4 \end{bmatrix}$$

What's the 75% value at risk (VAR) for each?

  • $\operatorname{VAR}(X, .75) = 0$
  • $\operatorname{VAR}(Y, .75) = 0$
  • $\operatorname{VAR}(X+Y, .75) = 1$

Making it explicit:

Value at risk (VAR) can be mathematically defined as:

$$\operatorname{VAR}\left(X, \alpha \right) = -\sup_x \left\{ x \in \mathbb{R} : P(X < x ) \leq 1 -\alpha \right\}$$

  • The set of $x \in \mathbb{R}$ where $P(X<x) \leq .25$ is the open set $(-\infty, 0)$.
  • The least upper bound of this set is 0, hence the supremum is 0, and the 75% VAR of $X$ is 0.
  • The set of $x \in \mathbb{R}$ where $P(X + Y < x) \leq .25$ is the open set $(-\infty, -1)$.
  • The least upper bound of this set is -1, hence the supremum is -1, and the 75% VAR of $X+Y$ is 1.
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