Online appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory

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1 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8A: Formulas for EE, PFE ad EPE for a ormal distributio Cosider a ormal distributio with mea (expected future value) ad stadard deviatio (of the future value). Let us calculate aalytically the two differet exposure metrics discussed. Uder the ormal distributio assumptio, the future value of the portfolio i questio (for a arbitrary time horizo) is give by: where Z is a stadard ormal variable. i) Potetial future exposure (PFE) V Z, This measure is exactly the same as that used for value-at-risk calculatios. The PFE at a give cofidece level, ( PFE ) tells us a exposure that will be exceeded with a probability of o more tha 1. For a ormal distributio, it is defied by a poit a certai umber of stadard deviatios away from the mea: - PFE 1 ( ), 1 where (.) represets the iverse of a cumulative ormal distributio fuctio (this is the fuctio NORMSINV(.) i Microsoft Excel TM ). For example, with a cofidece level of 99%, we have 1 (99%). 33 ad the worse case exposure is.33 stadard deviatios above the expected future value. ii) Expected exposure (EE) Exposure is give by: E max( V,0) max( Z,0) The EE defies the expected value over the positive future values ad is therefore: / EE ( x) ( x) dx ( / ) ( / ), where (.) represets a ormal distributio fuctio ad (.) represets the cumulative ormal distributio fuctio. We see that EE depeds o both the mea ad the stadard deviatio; as the stadard deviatio icreases so will the EE. I the special case of 0 we have EE (0) / iii) Expected positive exposure The above aalysis is valid oly for a sigle poit i time. Suppose we are lookig at the whole profile of exposure defied by V ( t) tz where ow represets a 1

2 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a aual stadard deviatio (volatility). The EPE, itegratig over time ad dividig by the time horizo, would be: EPE T 1 1/ 1/ 0 tdt / T T 3 0.7T.

3 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8B: Example exposure calculatio for a forward cotract. Suppose we wat to calculate the exposure o a forward cotract ad assume the followig model for the evolutio of the future value of the cotract V ) : - ( t dv dt, t dw t where represets a drift ad is a volatility of the exposure with dw t represetig a stadard Browia motio. Uder such assumptios the future value at a give time s i the future will follow a ormal distributio with kow mea ad stadard deviatio: - V s ~ N s, We therefore have aalytical expressio for the PFE ad EE followig from the formulas i Appedix 8A. s PFE 1 s s s ( ). EE s s s s s. 3

4 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8C: Example exposure calculatio for a swap. Followig the example i Appedix 8B, a approximatio to a swap cotract is to assume that the future value at a give time s is ormally distributed accordig to: V s ~ N 0, s( T s), where the ( T s) factor correspods to the approximate duratio of the swap of maturity T at time s. This assumes that the expected future value is zero at all future dates which i practice is the case for a flat yield curve. We ca show that the maximum exposure is at s = T / 3 by differetiatig the volatility term: d ds 1 s( T s) ( T s) s 0 s 4

5 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8D: Example exposure calculatio for a cross-currecy swap Combied the results i the two previous Appedices, we cosider a cross currecy swap to be a combiatio of the approximate FX forward ad iterest rate swap positios. The FX forward future value follows Vs N0, FX s follows V ~ N0, s( T s) ~ ad the iterest rate swap s IR. Assumig a correlatio of betwee future value of each, the approximate cross-currecy swap future value will be give by: V s ~ N 0, s s( T s) s( T s), FX which is used to compute the PFE show i Spreadsheet 8.4. IR FX IR 5

6 Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8E: Simple ettig calculatio We have already show i Appedix 8A that the EE of a ormally distributed radom variable is: EEi / ) ( / ). i ( i i i i i Cosider a series of idepedet ormal variables represetig trasactios withi a ettig set (). They will have a mea ad stadard deviatio give by: i i ij i j i1 where ij is the correlatio betwee the future values. Assumig ormal variables with zero mea ad equal stadard deviatios,, we have that the overall mea ad stadard deviatio are give by: i1 i1 ji 0 ( 1), where is a average correlatio value. Hece, sice ( 0) 1/, the overall EE will be: EE ( 1) / The sum of the idividual EEs gives the result i the case of o ettig (NN): Hece the ettig beefit will be: EE NN / EE / EE NN ( 1) I the case of perfect positive correlatio, 100%, we have: EE ( 1) / EENN 100% The maximum egative correlatio is bouded by 1/( 1) ad we therefore obtai: EE ( 1) /( 1) / EENN 0% 6

Online appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy.

Online appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy. APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1