What's new

# Ong (1999) - Unexpected Loss derivation

#### zer0

##### New Member
Subscriber
Hello,

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).

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Hello,

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).
Hello @zer0

I've tagged your post at the top with "unexpected-loss" so if you click on that, it will bring up many posts in the forum that discuss unexpected loss. This may prevent you from having to wait for someone to answer your question since it is most likely already provided in the forum. The search and tag functions are located here: https://www.bionicturtle.com/forum/search/?type=post

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks @Nicole Seaman @zer0 Such a search will produce this recent thread where we showed the derivation of UL https://www.bionicturtle.com/forum/...nd-risk-contribution-schroeck.8534/post-83181 (hat tip to @MarekH ) i.e.,
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

Subscriber