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I am struggling on I think a quite simple issue.

Let's take a portfolio of 100 loans.

If we assume they are independent, each loan’s default is a Bernoulli with parameter $p=0.01$ over a certain time horizon (eg. 1 year) and, of course, the overall portfolio number of defaults is a Binomial with the further parameter $n=1000$.

$P(Y=k) = \binom{100}{k}(0.01)^k(0.99)^{100-k}.$

To find $\alpha-$VaR, we solve for the smallest integer j∗ such that

$\sum_{k=j^*}^{100}\binom{100}{k}(0.01)^k(0.99)^{100-k} \leq 1-\alpha$

What happens if I change the time horizon from 1-year to 10-years? The uncertainty of the loss increases or decreases?

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  • $\begingroup$ Assuming that, for a longer time horizon (T=10), your PD will increase as well (say from 1% to 10%), then your VaR will increase as well. Say at alpha=1%, your original VaR amounts to 3 or more losses; with a higher PD, of course, your VaR amounts to 17. I feel that this is not your desired answer - Maybe you could add a bit more to your question? $\endgroup$ Oct 6, 2020 at 12:39
  • $\begingroup$ I mean, everything equal, including the 1yr-PD (i.e. the PD from 0 to 1yr is equal to the PD from 9 to 10 yr), just changing the time horizon: on one hand, I believe that on a greater time horizon, the uncertainty of the loss should increase; on the other hand, over the time there will be more incurred losses so as the incurred losses increase the uncertainty of future losses should decrease. What between these two reasoning is correct and why from a mathematical perspective? $\endgroup$
    – Nasser Bin
    Oct 6, 2020 at 14:13

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