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Martha Lyons
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1 CreditRis + Dowload documet from CSFB web site:

2 Features of CreditRis+ pplies a actuarial sciece framewor to the derivatio of the loss distributio of a bod/loa portfolio. Oly default ris is modelled, ot dowgrade ris. Default ris is ot related to the capital structure of the firm. No assumptio is made about the causes of default. obligor is either i default with probability P ad that ot i default with probability 1 P.

3 ssumptios o probability of default The probability distributio for the umber of defaults durig a give period, is approximated by Poisso distributio where e P defaults for 0,1,2,!! average umber of defaults over the period Statioary assumptio o the probability of default. Number of defaults that occur i ay give period is idepedet of the umber of defaults that occur i ay other period. P.

4 CreditRis+ ris measuremet framewor Iput default rates exposure default rates/volatilities recovery rates Stage Oe What is the frequecy of defaults? What is the severity of the losses? Stage Two Distributio of default losses

5 Frequecy of default evets Oe year default rate Credit Ratig verage % Stadard deviatio % aa a Baa Ba B source: Carty ad Lieberma 1996 Ratig B: stadard deviatio of default rate versus 5.1 CreditRis + assumes that the mea default rate is Gamma distributed.

6 Distributio of default evets

7 Historical default rates of corporate bod issuers Seiority ad security verage Stadard Derivatio Seior secured ba loas Seior secured public debt Seior usecured public debt Seior subordiated public debt Subordiated public debt Juior subordiated public debt

8 Severity of the losses I CreditRis +, the exposure for each obligor is adusted by the aticipated recovery rate i order to produce a loss give default exogeous to the model. Example of data set O b lig o r Exp o s u re Cre d it ra ti g 1 358,475 H 2 1,082, ,124,342 B 4 135,423 G 5 278,477 D

9 Credit ratig Mea default rate Stadard deviatio 1.50% 0.75% B 1.65% 0.80% C 3.00% 1.50% D 5.00% 2.50% E 7.50% 3.75% F 10.00% 5.00% G 15.00% 7.50%

10 Divisio ito exposure bads Losses exposures, et of recovery are divided ito bads, with the level of exposure i each bad beig approximated by a sigle umber. Notatio Obligor Exposure et of recovery Probability of default L P Expected loss λ L P

11 Obligor Exposure $ Exposure Roud-off Bad loss give i $100,000 exposure v default i $100, , , , , , , v *L $100,000

12 v umber of obligors v commo exposure i bad i uits of L expected loss i bad i uits of L expected umber of defaults i bad

13 Expected loss for obligor i uits of L e λ L. The expected loss over oe-year i bad : v v. The expected umber of default i bad v /. The portfolio is divided ito m exposure bads to simplify calculatios. The expected umber of defaults i the portfolio m 1.

14 Probability geeratig fuctios The probability geeratig fuctio of a discrete radom variable K is a fuctio of the auxiliary variable such that the probability that K is give by the coefficiet of i the polyomial expasio of the probability geeratig fuctio. The pgf of the sum K 1 + K 2 Example For a sigle obligor, F 1 P + P. is simply the product of the two pgf s. For the whole portfolio, F p defaults. 0

15 Probability geeratig fuctio of the distributio of loss amouts for bad exp.! defaults loss v v v e P L P G + Probability geeratig fuctio for the etire portfolio exp. 1 + m v G We the have. 1,2,,! 1 loss of 0! d G d L P

16 From, exp m v m G we write, 1 1 m m v v v p the. 1] [ p e G Here, gives the Poisso radomess of the icidece of default evets ad p gives the variability of exposure amouts withi the portfolio.

17 Correlatio i defaults Observed default probabilities are volatile over time, eve for obligors havig comparable credit quality. The variability of default probabilities ca be related to uderlyig variability i a umber of bacgroud factors, lie the state of the ecoomy. Two obligors are sesitive to the same set of bacgroud factors with differig weights, their default probabilities will move together. These co-movemets i probabilities give rise to correlatios i defaults. CreditRis + does ot attempt to model correlatios explicitly but captures the same cocetratio effects through the use of default rate volatilities ad sector aalysis.

18 Sector alysis Write S, 1,, for the sectors, each of which should be thought of as a subset of the set of obligors. Each sector is drive by a sigle uderlyig factor, which explais the variability over time i the average total default rate measured for that sector. The uderlyig factor iflueces the sector through the total average rate of defaults i that sector, which is modeled as a radom variable x with mea ad stadard deviatio σ.

19 Let x be Gamma distributed with mea ad stadard deviatio σ dx x e dx x f dx x x x p x Γ + < α α β α β Now, x p p dx x e e dx x f x p p F x x α α α α β β Γ 1 1 defaults defaults where. ad, p β β σ β σ α + The pgf for default evets from the whole portfolio p p F F α

20 ρ Geeral sector aalysis The default rate of a idividual obligor depeds o more tha oe factors. Sector decompositio ssigmet of θ K represets the udgemet of the extet to which the state of sector ifluece the fortues of obligor. Pairwise correlatio B ρ I, I B where I σ 1 0 if θ θ σ obligor otherwise. 2 defaults σ 2 1 ρ B B θ θ B. 1 If obligors ad B have o sector i commo, the ρ B 0. withi time horio

21 Ris cotributio The ris cotributio of obligor havig exposure E is the margial effect of the presece of E o the stadard deviatio of the distributio of the portfolio credit loss. 2 + p p E E E E RC θ σ σ σ

22 dvatages ad limitatios of CreditRis + Closed form expressios are derived for the probability of portfolio bad/loa losses. Margial ris cotributios by obligor ca be easily computed. Focuses oly o default, requirig relatively few iputs to estimate. ssumes o maret ris. Igores migratio ris so that the exposure for each obligor is fixed ad does ot deped o evetual chages i credit quality. Credit exposures are tae to be costat.

23 CreditMetrics CreditRis + Methodlology ad dataset Methodology Costly CreditMaager software Implemeted i spreadsheet Simulatio-based portfolio approach alytic-based portfolio approach B a sed o p ro b ab ilities o f ratigs Based o default rate associated with trasitio s a d co rre la tio s o f these probabilities Geerates sewed loss distributio same for calculatio of expected loss, uexpected loss ad ris capital ra tigs ad the vo la tility o f the se rate s

24 Extesio I To icorporate the effects of ratigs chages i CreditRis + without the eed of Mote Carlo simulatio as i CreditMetrics. Profits ad losses are used as et exposures. Default rate correspods to the migratio rate. Referece Good migratios, by B. Rolfes ad F. Broeer, Ris, Nov p

25 Extesio II Correlated credit evets such as defaults ca be studied ad aalyed i a closed form fashio without the eed of simulatios. Data set N: The umber of differet types of exposures. The type of the exposure could be based o the ratig of the exposure or BB, for example, the sector of the exposure baig or aerospace, for example, or the geographical regio or a combiatio of all three. i : The umber of exposures of the i th asset type. e i : The dollar amout of the th exposure of the i th asset type. p i : The probability of default for assets of i th type. c i : The default correlatio betwee exposures of i th type ad th type. q i : The oit default probability of a exposure of type i ad a exposure of type. Give the defiitios above, the followig idetity holds: q p p + c p 1 p p 1 p. i i i i i

26 If the portfolio is made up of N asset types, the the loss distributio uder correlated defaults ca be obtaied by combiig loss distributio of at most 2N scearios. Refereces 1. K. M. Nagpal ad R. Bahar, aalytical approach for credit ris aalysis uder correlated defaults, CreditMetrics Moitor pril 1999 p K. M. Nagpal ad R. Bahar, Modellig default correlatio, Ris pril 2001 p

27 Review articles S. Paul-Choudhury, Choosig the right box of credit trics, Ris Nov H. U. Koyluoglu ad. Hicma, Recocilable differeces, Ris Oct p M. Crouchy, D. Galai ad R. Mar, comparative aalysis of curret credit ris models, Joural of Baig ad Fiace, vol p

How the Default Probability is Defined by the CreditRisk+Model?

Iteratioal Joural of Global Eergy Marets ad Fiace, 28, Vol, No, 2-25 vailable olie at http://pubssciepubcom/igefm///4 Sciece ad Educatio Publishig DOI:269/igefm---4 How the Default Probability is Defied