Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies we have recenly inroduced in he sofware for modelling couner-pary risk. Collaeralised lending represens a relaively complex ype of exposure for risk analysis. Essenially, he lender provides funds wih he proviso ha in he even of defaul by he borrower, repaymen will be he minimum of he ou-sanding loan amoun and he value of some designaed asses. The arrangemens may be such ha a pool of loans is secured agains a pool of collaeral asses. Collaeralised Lending Payoffs In he simple case of a single loan, one may hink of he oal exposure held by he lender as a safe loan, L, plus a credi derivaive which, in he even of defaul, requires a paymen by he lender of he maximum of zero and he safe loan minus he value of he collaeral, A, i.e.,: Payoff in even of defaul a = L max { L A,0} Prior o defaul, he cash flows are simply he cash flows on he defaul-free loan and here is no cash flow on he noional derivaive posiion. If here are N collaeralised loans wih no collaeral pooling, he payoff in he even of defaul will be: Payoff in even of defaul a = N i= 1 { L A, } Li, max i, i, 0
When here are N collaeralized loans and pooling, in he even of defaul, he payoff o he lender will be: Payoff in even of defaul a Mapping o he Couner-pary Risk Approach N N = Li, max Li i= 1 i= 1 We can model his ype of exposure using he se-up we have creaed for counerpary risk modelling. We would inroduce ino he model a defaul free bond and a collaeral exposure. In a preliminary Mone Carlo, we would simulae he payoffs on he derivaive posiion up o he final dae of he loan and hen discoun his back o he VaR horizon and regress he discouned payoffs on sae variables o creae a condiional pricing funcion. In he main, VaR Mone Carlo, we would hen simulae up o he VaR horizon, and evaluae he pricing funcion a he sae variables o ge he price of he derivaive posiion. Fiing his ino he collaeral modelling approach we have requires some rickery as he se up righ now consiss of working wih a porfolio of asses which pay off sreams of income prior o defaul and which are hen valued, need and muliplied by a fracional loss rae in he even of defaul. In he collaeralised lending case jus described, he collaeral asse does no pay any cash flows prior o defaul. One can handle his by inroducing a pair of exacly offseing collaeral asse posiions and making jus one of hem par of he collaeral pool. The cash flows and values will always offse each oher so here is no impac on he direcly held porfolio. Bu, in he even of defaul by he couner-pary exposure, one of he pair of mached long and shor posiions would conribue o he payou on he derivaive posiion, i.e., he couner-pary exposure. To be specific wihin Risk Conroller, consider a porfolio consising of (i) a safe loan, L, (ii) a long and (iii) a shor posiion in a collaeral asse wih respecive prices A and -A, and finally (iv) a couner-pary exposure wih value V. The safe loan and he shor collaeral exposure are linked o he couner-pary exposure. The counerpary exposure has a raing ha evolves over ime. There are no cash flows on he couner-pary exposure unless i defauls. If i defauls, he maximum loss is he maximum of L A and zero. We may presume ha here is a fracion loss rae, (1-γ), applied o his maximum loss because he holder of he couner-pary exposure may be successful in exracing some value from he bankrupcy selemen. So he payoff is ( 1 γ ) max{ L A,0} This is exacly he payoff assumed in our sandard modelling of couner-pary risk. Example Calculaions, A i,,0
Here, we consider a simple example of en collaeralised deb exposures. All exposures are 4 year loans wih par value 100. The borrowers have raings varying beween A and AA. Daa for he couner-paries is shown in Table 1 while informaion on he underlying loans and collaeral are shown in Table 2. For simpliciy, we have assumed ha he collaeral consiss of FRNs wih bank indusry and UK risk facors. (More elaborae exercises could be implemened including parameers appropriae for UK RMBS. These laer securiies could be modelled eiher as quasi bonds or using our look-hough approach o modelling srucured producs.) Table 1: Couner-paries Indusry Counry Idiosyncra Recovery Transiion id Obligor id Counry Indusry Currency Raing facor facor ic weigh rae marix Mauriy 5000 Goldman Sachs US Banks Serling A Banks US 0.8 0 All Obligor 5 5001 Lehman Brohers US Banks Serling AA Banks US 0.8 0 All Obligor 5 5002 Deusche Bank UK Banks Serling AA Banks US 0.8 0 All Obligor 5 5003 ABN AMRO UK Banks Serling AA Banks US 0.8 0 All Obligor 5 5004 Morgan Sanley US Banks Serling AA Banks US 0.8 0 All Obligor 5 5005 J P Morgan Chase US Banks Serling A Banks US 0.8 0 All Obligor 5 5006 Dresdner Kleinwor Benson Germany Banks Serling AA Banks US 0.8 0 All Obligor 5 5007 Commerz Bank Germany Banks Serling AA Banks US 0.8 0 All Obligor 5 5008 ING UK Banks Serling AA Banks US 0.8 0 All Obligor 5 5009 Unicredi Ialy Banks Serling AA Banks US 0.8 0 All Obligor 5 Table 2: Loans and collaeral id Counry Mean recovery rae Indusry Counry Idiosyn Transiion Reference Counerpa ype Principal Currency Raing facor facor weigh marix rae Mauriy ry link 10022 UK FRN 100 Serling BB Banks UK 0.8 0.56 All Obligor Libor GBP 4 10023 UK FRN 100 Serling Banks UK Libor GBP 4 5000 10025 UK FRN 100 Serling BB Banks UK 0.8 0.56 All Obligor Libor GBP 4 10026 UK FRN 100 Serling Banks UK Libor GBP 4 5001 10028 UK FRN 100 Serling BBB Banks UK 0.8 0.3 All Obligor Libor GBP 4 10029 UK FRN 100 Serling Banks UK Libor GBP 4 5002 10031 UK FRN 100 Serling B Banks UK 0.8 0.4 All Obligor Libor GBP 4 10032 UK FRN 100 Serling Banks UK Libor GBP 4 5003 10034 UK FRN 100 Serling A Banks UK 0.8 0.6 All Obligor Libor GBP 4 10035 UK FRN 100 Serling Banks UK Libor GBP 4 5004 10037 UK FRN 100 Serling BB Banks UK 0.8 0.46 All Obligor Libor GBP 4 10038 UK FRN 100 Serling Banks UK Libor GBP 4 5005 10040 UK FRN 100 Serling BB Banks UK 0.8 0.56 All Obligor Libor GBP 4 10041 UK FRN 100 Serling Banks UK Libor GBP 4 5006 10043 UK FRN 100 Serling BBB Banks UK 0.8 0.35 All Obligor Libor GBP 4 10044 UK FRN 100 Serling Banks UK Libor GBP 4 5007 10046 UK FRN 100 Serling B Banks UK 0.8 0.2 All Obligor Libor GBP 4 10047 UK FRN 100 Serling Banks UK Libor GBP 4 5008 10049 UK FRN 100 Serling A Banks UK 0.8 0.3 All Obligor Libor GBP 4 10050 UK FRN 100 Serling Banks UK Libor GBP 4 5009 We performed a baseline run of he model over a one year horizon wih 1,000,000 Mone Carlo replicaions. The reference currency for values and risk measures is Serling. All he exposures being denominaed in Serling, FX risk plays no role. We also conduc he simulaion assuming no ineres rae risk. The simulaion horizon is one year bu he mauriy of he defaul-free bond and of he collaeral is 4 years. The model uses reference rae spread of LIBOR GBP for his paricular se of exposures. The saisics ha have been employed reflec he curren very subsanial spread of LIBOR over Treasuries currenly presen in he marke. Figure 1 below shows he hisogram of he porfolio value disribuion. As is apparen, he disribuion is highly lef-skewed. There is some small chance ha he raing of he couner-paries rise; bu, apar from his, mos variaion in he porfolio value is associaed wih possible defauls or falls in he value of he collaeral.
Figure 1: Two-Bond Porfolio Value a 1Year Horizon Table 3: Risk Saisics for Two-Loan Porfolio Descripion Value Mean value (discouned) 190.45 Volailiy of value (discouned) 1.89 Skewness coefficien -4.88 Kurosis coefficien 65.49 Value-a-risk 300bp, % of mean 2.11% Value-a-risk 200bp, % of mean 2.58% Value-a-risk 100bp, % of mean 3.50% Value-a-risk 50bp, % of mean 4.57% Value-a-risk 300bp (discouned) 4.01 Value-a-risk 200bp (discouned) 4.91 Value-a-risk 100bp (discouned) 6.66 Value-a-risk 50bp (discouned) 8.71 Expeced shorfall 300bp (discouned) 6.76 Expeced shorfall 200bp (discouned) 7.92 Expeced shorfall 100bp (discouned) 10.18 Expeced shorfall 50bp (discouned) 12.82 Risk saisics for he porfolio as a whole are shown in Table 3. The 100-basis-poin (bps) VaR for he porfolio is 3% of he mean payoff on he porfolio or 6.7 in moneary unis. (Absolue number unis are deermined by he unis of he exposure daa. Recall ha each of he wo loans in his case has a par value of 100.) The Skewness and kurosis of he porfolio are -4.9 and 65 respecively. This indicaes he exremely fa-ailed naure of he disribuions. We also exhibi he 100-bps Expeced Shorfall (ES), also commonly referred o as he Tail-VaR or Condiional VaR by differen auhors, a risk measure which ends o
reflec exreme risk beer han simple VaR. In his case, he ES is 10.2. This is disincly greaer han he VaR, a fac ha again underlines he fa-ailed naure of he risk. Figure 2: Ten- Porfolio Value a 1Year Horizon Table 4: Risk Saisics for Ten- Porfolio Descripion Value Mean value (discouned) 949.85 Volailiy of value (discouned) 5.96 Skewness coefficien -3.88 Kurosis coefficien 31.31 Value-a-risk 300bp, % of mean 1.50% Value-a-risk 200bp, % of mean 1.85% Value-a-risk 100bp, % of mean 2.48% Value-a-risk 50bp, % of mean 3.18% Value-a-risk 300bp (discouned) 14.25 Value-a-risk 200bp (discouned) 17.54 Value-a-risk 100bp (discouned) 23.59 Value-a-risk 50bp (discouned) 30.21 Expeced shorfall 300bp (discouned) 23.34 Expeced shorfall 200bp (discouned) 27.13 Expeced shorfall 100bp (discouned) 34.07 Expeced shorfall 50bp (discouned) 41.67 We hen perform calculaions for a porfolio comprising all en collaeralised deb exposures. The disribuion of he fuure value is shown in Figure 2. The disribuion is spread ou over a wider suppor range as one migh expec and here is sill a chance of caasrophic losses. Risk saisics for he porfolio are shown in Table 4. The
Skewness and kurosis are now -3.9 and 31. Aggregaing over 10 insead of 2 exposures has somewha reduced he skewness and kurosis. The 100-bps VaR is now 2.5% of he mean value, reflecing diversificaion across he differen exposures. The 100-bps ES is sill subsanially greaer han he 100-bps VaR, again reflecing he exreme naure of he ail risk. Table 5 shows individual exposure risk saisics for he 10-exposure porfolio. These risk saisics include volailiies of individual posiions, heir correlaion wih he porfolio as a whole and he Marginal VaRs and ES of he individual exposures. Recall ha an MVaR is he amoun by which he oal porfolio VaR decreases when he exposure in quesion is dropped from he porfolio. These laer saisics are an imporan guide as o he source of he main sources of risk in a complex porfolio wih a large number of exposures. Effecively, hey reflec a combinaion of he riskiness of he exposure on a sandalone basis and he degree o which he exposure is posiively correlaed wih he res of he porfolio. (Greaer correlaion resuls in larger MVaRs.) Risk Conroller supplies risk saisics for individual exposures and aggregaed across differen caegories such as indusry, currency, counry ec, every ime he model is run. Table 5: Individual Risk Saisics Marginal VaR 100bp Marginal Expeced Shorfall 100bp Name Mean value Volailiy of value Correlaion 5005 J P Morgan Chase -8.37 2.45 0.65 4.79 5.57 5008 ING -10.84 1.80 0.54 2.87 3.88 5000 Goldman Sachs -6.95 1.60 0.53 2.30 2.78 5003 ABN AMRO -7.92 1.40 0.47 2.30 3.29 5002 Deusche Bank -4.23 1.13 0.38 1.38 2.36 5007 Commerz Bank -3.91 0.92 0.37 1.26 1.84 5009 Unicredi -6.87 1.09 0.40 1.14 1.78 5001 Lehman Brohers -5.24 0.83 0.35 1.02 1.44 5006 Dresdner Kleinwor Benson -5.07 0.78 0.31 0.90 1.31 5004 Morgan Sanley -3.98 0.46 0.35 0.51 0.63 The marginal VaR of a paricular exposure depends on he raing, he qualiy of he collaeral and he degree o which he risk is correlaed wih he main risks driving he porfolio as a whole. The resuls in Table 5 are consisen wih his. The four larges sources of risk as measured by he 100-bps MVaRs include he wo A-raed US banks. The fac ha hey are boh US banks adds o heir risk as four of he en counerparies are from he US. (In inerpreing he resuls in he able, one should noe ha he exposures here are sylised in he sense ha he raings of A given o Goldman Sachs and JP Morgan Chase in our daa are no he acual ones.) The wo oher of he larges four exposures as recorded by he 100-bps MVaR are ING and ABN-Amro which have very low qualiy collaeral.