# BT2012.P2.16. Risk contribution of credit exposure

Discussion in 'BT Part 2' started by David Harper CFA FRM, May 6, 2012.

1. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

Question:

16. Michael Ong writes that "risk contribution is the single most important risk measure for assessing credit risk." The risk contribution (RC) of an asset is defined as the incremental risk that the exposure of a single asset contributes to the portfolio's total credit risk. The practical formula for the risk contribution of an exposure, RC(i), is given by:

Consider a credit portfolio with three exposures. We are given their respective unexpected losses (UL), the portfolio's unexpected loss (portfolio UL), and a default correlation matrix:

What is the risk contribution of Exposure #1?

a. $172.0 million b.$205.3 million
c. $288.4 million d.$300.0 million

16. A. $172.0 million RC (exposure #1) =$300 MM * ($300 +$500*0.30 + $900*0.40)/$1,412.80 = $172.0 million RC(exposure #2) =$399.92, and
RC(exposure #3) = $840.88 such that sum of risk contributions is equal to portfolio UL of$1,412.80

2. ### W0LFNew Member

Is the above formula missing a + sign as it implies to me we should be multiplying 300*(150+360) not 300+(150+360)??

3. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

Hi @W0LF The formula looks okay to me. Inside the summation is UL(j)*rho(ij), such that we have the summation of three terms: 300*1.0 + 500*0.30 + 900*0.40, where 1.0 is the correlation of Exposure #1 with itself (the three correlations used are the first column of the correlation matrix). I hope that explains and that I didn't miss your point. Thanks,

• Like x 1
4. ### W0LFNew Member

Yes that does explain it Thanks. I just didn't to look at it again and work out what is doing!

• Like x 1
5. ### Xin ZhanMember

I used sqrt(500^2+900^2 + 2*.6*500*900) = 1264.91, and 1412.80-1264.91 = 147.89. Could you please point out what did I do wrong? thank you!

6. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

Hi @cindyzhan As unexpected loss (UL) is multiple of standard deviation, your sqrt(500^2+900^2 + 2*.6*500*900) is correct but for a different question: what is the unexpected loss of the portfolio consisting of exposure #2 plus exposure #3? That is, we can use UL(A + B) = sqrt[UL(A)^2 + UL(B)^2 + 2*UL(A)*UL(B)*default_ρ(A,B)]. Okay, but the subset portfolio(#2 + #3) does not play a direct role here. The question is for the risk contribution of exposure #1 to the entire portfolio of all three exposures, which is solved by the formula given. I hope that is helpful!

Last edited: Nov 4, 2016
7. ### Xin ZhanMember

thank you, David, I got confused by "The risk contribution (RC) of an asset is defined as the incremental risk", thinking incremental risk for #1 is the difference between total risk and the risk without #1.

• Like x 1