Hi David,
Can you please clarify the bold and underlined portion in your explanations above?
what is Jorion's logic of choosing 90% VaR as -10%?
how do we calculate 90% VaR as - 7.5% as per Linda Allen? Why do we use cumulative weight here?
90% VaR: According to Dowd, it's arguably -5% (loss of 5%); according to Jorion, it is -10%. Dowd acknowledges the mathematically ambiguity of selecting a VaR when, essentially, the quantile falls exactly between two densities (see quote below). The analogy is HS with, say, n = 1,000; for a 95% VaR, Dowd selects the "less conservative" 51st worst loss and Jorion selects the more conservative (ie, higher VaR) 50th worst loss. Further, in this case, the hybrid approach (to me) fits very well (this is Hull's or Linda Allen's): here, the worst loss observation (-10% loss) is a random variable with its own distributional center (ie, loss of -10%) at the cumulative weight of 10%/2 = 5.0%. In this way: loss at -10% --> weight of 10%/2 = 5.0%; loss at -7.5% = average (-10%, -5%) --> weight of 10%; and loss at -5% --> weight of 20% = 10% + 20%/2. With implied: 95% VaR = -10.0%, 90% VaR = -7.5%, 80% VaR = -5%. In summary, 90% VaR here equals -10% (Jorion), -5% (Dowd) or even -7.5% (L Allen).
Regards,
Trilo
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