Friday, February 19, 2010

Value at Risk is Neither Subadditive Nor Distributive

Let's assume that a trio of financial institutions with portfolio holdings X, Y, and Z represent their respective Value at Risk (VaR) measures as

VaR (X) + VaR (Y) + VaR (Z),

and that the three portfolios are subsequently merged into one by the method

VaR (X + Y + Z).

Of course, coherence would require that

VaR (X + Y + Z) = VaR (X) + VaR (Y) + VaR (Z).

However, VaR is neither subadditive nor distributive, and depending on the composition of portfolios X, Y, and Z, the post-merger VaR can exceed the summation of VaR for the segregated portfolios, resulting in

VaR (X + Y + Z) ≥ VaR (X) + VaR (Y) + VaR (Z).

The point of this exercise is to caution that merging portfolios can increase VaR. Yet, our nation’s financial regulators and leadership continue to defend and justify such capital formations (e.g., bank mergers) as necessary and expedient within the rubric of "globalization," "too big to fail," or some other convolution of fear mongering. Given the risk calculus above, I maintain that financial institutions that become "too big to fail" are likewise too risky to keep around. Bank mergers and acquisitions that expand VaR make no sense in today's business climate.

1 comment:

Jon 'JB' Beckett said...

That's a good question William; am I right to understand then that composite VaR as described canoot hold any diversification value whatsover (i.e. that the sum of VaR could be < that the sum of the parts)? Assuming VaR is normally distributed that the moment of loss could occur at different horizons?

In the same way that composite STDEV is said to be lower than the sum of the parts - that the interactions and non-correlated variances smooth each other. (if you agree with bell curve of course)..

On the premis that the conbination of VaR creates a new distribution and probably changes the tails (Sk/K) - I can appreciate the notion of concentration of VaR, what I'm struggling with is that it could only increase risk.

Best rgds

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