Portfolio Loss Distribution

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1 Portfolo Loss Dstrbuton

2 Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment A commtment s an amount the bank has commtted to lend. Should the borrower encounter fnancal dffcultes, t would draw on ths commtted lne of credt.

3 Adjusted exosure and exected loss Let α be the amount of drawn down or usage gven default. Outstandng + α commtment, Rsky Asset value at later tme H, V H Adjusted exosure s the rsky art of V H. (1 α) commtment, Rskless Exected loss adjusted exosure loss gven default robablty of default * Normally, racttoners treat the uncertan draw-down rate as a known functon of the oblgor s end-of-horzon credt class ratng.

4 Examle calculaton of exected loss Commtment Outstandng Internal rsk ratng Maturty Tye Unused drawn-down on default (for nternal ratng 3) Adjusted exosure on default EDF for nternal ratng 3 Loss gven default for non-secured asset Exected loss $10,000,000 $3,000, year Non-secured 65% $8,250, % 50% $6,188

5 Unexected loss Unexected loss s the estmated volatlty of the otental loss n value of the asset around ts exected loss. AE EDF LGD 2 2 σ LGD + σ 2 EDF where σ EDF 2 EDF (1- EDF). Assumtons * The random rsk factors contrbutng to an oblgor s default (resultng n EDF) are statstcally ndeendent of the severty of loss (as gven by LGD). * The default rocess s two-state event.

6 Examle on unexected loss calculaton Adjusted exosure $8,250,000 EDF 0.15% σ EDF 3.87% LGD 50% σ LGD 25% Unexected loss $178,511 * The calculated unexected loss s 2.16% of the adjusted exosure, whle the exected loss s only 0.075%

7 Comarson between exected loss and unexected loss * The hgher the recovery rate (lower LGD), the lower s the ercentage loss for both EL and. * EL ncreases lnearly wth decreasng credt qualty (wth ncreasng EDF) * ncreases much faster than EL wth ncreasng EDF. Percentage loss er unt of adjusted loss 10% 5% 10% EL EDF

8 Assets wth varyng terms of maturty * The longer the term to maturty, the greater the varaton n asset value due to changes n credt qualty. * The two-state default rocess aradgm nherently gnores the credt losses assocated wth defaults that occur beyond the analyss horzon. * To mtgate some of the maturty effect, banks commonly adjust a rsky asset s nternal credt class ratng n accordance wth ts terms to maturty.

9 Portfolo exected loss where EL EL AE LGD EDF EL s the exected loss for the ortfolo, AE s the rsky orton of the termnal value of the th asset to whch the bank s exosed n the event of default. We may wrte EL AE w EL AE where the weghts refer to AE w AE AE AE.

10 EL AE EL AE AE AE EL AE w EL AE AE w EL EL /AE 1 $10 M 0.5 $ $4 M 0.2 $ $6 M 0.3 $ AE $20M 1 w EL AE

11 Portfolo unexected loss ortfolo unexected loss j ρ j w w j j where AE EDF σ LGD + GD σ E L DF and ρ j s the correlaton of default between asset and asset j. Due to dversfcaton effect, we exect <<.

12 Rsk contrbuton The rsk contrbuton of a rsky asset to the ortfolo unexected loss s defned to be the ncremental rsk that the exosure of a sngle asset contrbutes to the ortfolo s total rsk. RC and t can be shown that RC j j ρ j.

13 Undversfable rsk The rsk contrbuton s a measure of the undversfable rsk of an asset n the ortfolo the amount of credt rsk whch cannot be dversfed away by lacng the asset n the ortfolo. RC To ncororate ndustry correlaton, usng ndustry α and j ndustry β RC (1 ) + α ραα β α k β k ρ αβ.

14 Calculaton of EL, and RC for a two-asset ortfolo ρ EL default correlaton between the two exosures ortfolo exected loss EL EL 1 + EL 2 ortfolo unexected loss RC 1 RC rsk contrbuton from Exosure 1 rsk contrbuton from Exosure 2 + ρ RC1 1(1 + ρ2) / RC2 2(2 + ρ1) / RC 1 + RC 2 << 1 + 2

15 Fttng of loss dstrbuton The two statstcal measures about the credt ortfolo are ortfolo exected loss; ortfolo unexected loss. At the smlest level, the beta dstrbuton may be chosen to ft the ortfolo loss dstrbuton. Reservaton A beta dstrbuton wth only two degrees of freedom s erhas nsuffcent to gve an adequate descrton of the tal events n the loss dstrbuton.

16 Beta dstrbuton The densty functon of a beta dstrbuton s > > < < Γ Γ + Γ otherwse 0 0 0, 1 0, ) (1 ),, ( 1 1 ) ( ) ( ) ( β α β α β α β α β α x x x x F Mean and varance β α α µ +. 1) ( ) ( β α β α αβ σ 1 x f(x, α, β)

17 Economc Catal If X T s the random varable for loss and z s the ercentage robablty (confdence level), what s the quantty v of mnmum economc catal EC needed to rotect the bank from nsolvency at the tme horzon T such that Pr[ X T v] z. Here, z s the desred debt ratng of the bank, say, 99.97% for an AA ratng.

18 frequency of loss X T EL EC

19 Catal multler Gven a desred level of z, what s EC such that Pr[ X EL EC] T z. Let CM (catal multler) be defned by EC CM then Pr X T EL CM z.

20 Monte Carol smulaton of loss dstrbuton of a ortfolo 1. Estmate default and losses 2. Estmate asset correlaton between oblgors Assgn rsk ratngs to loss facltes and determne ther default robablty + Assgn LGD and σ LGD Determne arwse asset correlaton whenever ossble OR Assgn oblgors to ndustry groungs, then determne ndustry ar correlaton

21 3. Generate random loss gven default 4. Generate correlated default events Determne stochastc loss gven default + Correlated default events + Decomoston of covarance matrx + Smulate default ont

22 5. Loss calculaton 6. Loss dstrbuton Calculate faclty loss for each scenaro and obtan ortfolo loss Construct smulated ortfolo loss dstrbuton

23 Generaton of correlated default events Generate a set of random numbers drawn from a standard normal dstrbuton. Perform a decomoston (Cholesky, SVD or egenvalue) on the asset correlaton matrx to transform the ndeendent set of random numbers (stored n the vector e ) nto a set of correlated asset values (stored n the vector e ). Here, the transformaton matrx s M, where e M e. The covarance matrx and M are related by M T M.

24 Calculaton of the default ont The default ont threshold, DP, of the th oblgor can be defned as DP N 1 (EDF, 0, 1). The crteron of default for the th oblgor s default f e < ' DP no default f e ' DP.

25 Generate loss gven default The LGD s a stochastc varable wth an unknown dstrbuton. A tycal examle may be Recovery rate (%) LGD (%) σ LGD (%) secured unsecured LGD LGD + σ s s f LGD where f s drawn from a unform dstrbuton whose range s selected so that the resultng LGD has a standard devaton that s consstent wth hstorcal observaton.

26 Calculaton of loss Summng all the smulated losses from one sngle scenaro Loss Adjusted exosure Oblgors n default LGD Smulated loss dstrbuton The smulated loss dstrbuton s obtaned by reeatng the above rocess suffcently number of tmes.

27 Features of ortfolo rsk The varablty of default rsk wthn a ortfolo s substantal. The correlaton between default rsks s generally low. The default rsk tself s dynamc and subject to large fluctuatons. Default rsks can be effectvely managed through dversfcaton. Wthn a well-dversfed ortfolo, the loss behavor s characterzed by lower than exected default credt losses for much of the tme, but very large losses whch are ncurred nfrequently.

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