Does anyone have the derivation for the Unexpected Loss formula? It's formula 5.5 from Schroeck but the derivation is only available in Ong, Michael, Internal Credit Risk Models (Risk Books, 1999).
HI @Kosuke Yamaji It is based on the variance of the product of two independent random variables, see https://en.wikipedia.org/wiki/Produ...f_the_product_of_independent_random_variables
You can see that σ^2(XY) = σ^2(X)*σ^2(Y) + σ^2(X)*μ^2(Y) + σ^2(Y)*μ^2(X). So the assumption here is independence between PD and LGD such that σ^2(PD*LGD) = σ^2(PD)*σ^2(LGD) + σ^2(PD)*μ^2(LGD) + σ^2(LGD)*μ^2(PD). But as a Bernoulli μ^2(PD) = PD^2 and μ^2(LGD) = LGD^2 so we have really this version of the variance of a product:
σ^2(PD*LGD) = σ^2(PD)*σ^2(LGD) + σ^2(PD)*LGD^2 + σ^2(LGD)*PD^2
= ( σ^2(PD)*σ^2(LGD) + σ^2(LGD)*PD^2 ) + σ^2(PD)*LGD^2
= ( σ^2(LGD)*[σ^2(PD) + PD^2] ) + σ^2(PD)*LGD^2, because σ^2(PD) = PD*(1-PD):
= ( σ^2(LGD)*[PD*(1-PD) + PD^2] ) + σ^2(PD)*LGD^2
= ( σ^2(LGD)*[PD - PD^2 + PD^2] ) + σ^2(PD)*LGD^2
= σ^2(LGD)*PD + σ^2(PD)*LGD^2; i.e., since this is the variance, we can see that UL is the standard deviation of the product PD*LGD. I hope that's helpful.
Hat tip to @MarekH who actually showed me (because I previously assumed it was more difficult) at https://www.bionicturtle.com/forum/threads/the-origin-of-ongs-unexpected-loss-ul.1792/post-78444