The discussion of VaR in the SA3 notes is insufficient / incorrect in two respects:
(1.) In the example on page 14, chapter 7, we have two (I assume independent) identical portfolios each with a probability of 99.1% of a profit of 10 (i.e. a loss of -10) and a probability of 0.9% of a loss of 100. Following the definition of VaR on page 14 (but see (2.) ), this gives:
VaR @ 1%, single portfolio = -10
VaR @ 1%, two portfolios = 90, which is greater than -20.
The example thus has its signs incorrect: it quotes 10 and 20. In addition, it neglects to mention the assumption of independence between the two portfolios - the results above are certainly not true if we simply double the size of a single portfolio! It would be more pedagogically instructive to explicitly state the probabilities of the three outcomes in the two-portfolio case, assuming independence, and then apply the VaR formula to these.
(2.) The definition of VaR given in the formula on page 13 is not correct. Note that, applying it to the example on page 14 we would be looking to find m such that:
P(X >= m) = 1%
...which has no solution for this discrete distribution (the formula given is really only applicable to continuous distributions which, since much VaR work is done in the context of Monte-Carlo simulation, is not particularly useful).
A better definition of VaR in the context given would be:
"VaR @ y% = the smallest m such that P(X > m) < y%"
...which will hold equally well in the discrete case.
Apologies for being somewhat pedantic here ( 
), but this is very important. You will quickly find that a practical implementation of the method proposed in the notes in a stochastic model will go awry. Since the proposed Solvency II internal model SCR is based upon a VaR measure, for example, it is important to phrase it exactly right.
It would also be useful to have some greater discussion of the controversy surrounding VaR as a risk measure, as well as a mention of its lack of smoothness in a Monte Carlo setting (this latter can cause problems with numerical optimisation methods that rely on such smoothness - for instance, when maximising expected net profit subject to a 99.5% VaR constraint on risk capital).
Last edited by a moderator: Apr 12, 2010
