How Loan Default Loss VaR Changes with the Time Horizon

Summary

The document sets up a credit-loss example in which each loan defaults independently with a stated one-year Bernoulli probability. It models the number of defaults across a portfolio with a binomial distribution and expresses an upper-tail quantile condition for finding value at risk at a chosen confidence level. The question is how the uncertainty in portfolio losses changes when the horizon is extended from one year to ten.

No solution or evidence is provided, so the effect of the longer horizon remains unresolved. The setup also contains an inconsistency: it describes a portfolio of 100 loans, uses 100 in the displayed binomial probability, but then states a binomial trial count of 1000. It does not specify how default probabilities evolve over multiple years, whether loans that default remain at risk, or how loss severity and dependence are handled. Those assumptions would be needed to compare horizon-specific loss distributions.

Key ideas

• The example models independent loan defaults as Bernoulli events over a fixed horizon.
• The portfolio default count is represented with a binomial distribution.
• Value at risk is framed as an upper-tail quantile of the number of defaults.
• The document asks how extending the time horizon affects loss uncertainty but gives no answer.
• The portfolio size is inconsistent between the prose and the stated binomial trial count.

Tags

# Longer / Shorter period loss


# Longer / Shorter period loss












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?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.