Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 Publshed onlne Aprl 0, 014 (http://www.scencepublshnggroup.com/j/sjams) do: 10.11648/j.sjams.01400.11 Measurng portfolo loss usng approxmaton methods Ose Antw Mathematcs & Statstcs Department, Accra Polytechnc, Accra, Ghana Emal address: oseantw@yahoo.co.uk To cte ths artcle: Ose Antw. Measurng Portfolo Loss Usng Approxmaton Methods. Scence Journal of Appled Mathematcs and Statstcs. Vol., No., 014, pp. 4-5. do: 10.11648/j.sjams.01400.11 Abstract: One of the approaches to determnng and quantfyng the credt rsk of a loan portfolo s by obtanng the dstrbuton of losses of the portfolo and determnng the rsk quanttes from such dstrbutons. In ths paper, we descrbe the challenges to usng ths approach and llustrate a practcal soluton where smulaton methods are used to obtan loss dstrbuton for a two oblgor portfolo. Ths s then extended to ten and hundred oblgor portfolos. Exstng probablty dstrbutons wth specfed parameters are then used to approxmate the loss dstrbutons obtaned. Usng such parameters of the exstng probablty dstrbutons, we obtan the rsk quanttes assocated wth the loan portfolo ncludng Expected and Unexpected losses. We realzed that dependng on the confdence nterval for whch we measure the Unexpected Loss, Stress Losses are needed to account for the total loss of the portfolo. Keywords: Economc Captal, Expected Loss, Unexpected Loss, Oblgor, Loss Gven Default, Exposure at Default, Stress Loss 1. Introducton In the last two decades, credt rsk management has become a topc of paramount mportance n fnance. Its mportance became greatly sgnfcant, when n 007 Lehman Brothers, one of the bggest nvestment banks n the US, collapsed after falng to honour ts oblgatons. The collapse trggered a domno effect that led to several corporate falures leadng to what became known as the credt crunch. If one consders the level of corporate debt, soveregn debt, retal loans, debt securtzaton vehcles and nstruments that suffered durng ths perod then the extent of the credt market and ts effect on corporate and most natonal economes s overwhelmng. Although banks n developng countres were not drectly ht by the huge losses that characterzed the European and Amercan banks durng the perod, t s essental that ther fnancal nsttutons develop robust credt rsk management models as t contnues to expose ts busnesses further n the nternatonal market. The dramatc growth n the credt market n developng countres especally n the last decade provdes the drve and need for development and study of a wealth of new credt models. Recent advances n credt rsk models have been reflected n several reforms proposed by the Basel commttee on bankng supervson. Under the Basel proposal, banks can determne ther own estmates of some of the components of rsk measure: Probablty of Default (PD), Loss Gven Default (), Exposure at Default (EAD) and Maturty (M). The goal of these regulatons s to defne rsk weghts by determnng cut-off ponts between and wthn areas of Expected Loss and Unexpected Loss where regulatory captal s to be held, n the event of a default. Such determnaton of rsk measures depends on the level of sophstcaton of the bank s credt models. These measures nclude the Standardzed Approach, the Internal Ratngs Based Approach (Foundaton) and the Advanced Internal Ratng Based Approach. Currently, most central banks n developng countres contnue to use the Standardzed Approach whch smply specfy the mnmum captal requrements as a rsk weght percentage (usually 8-10% of debt), to be held as captal [1]. Such models n bankng supervson are not robust and sophstcated enough to wthstand smlar catastrophc losses such as whch occurred n Europe and Amerca. It s therefore essental that more advanced credt models are used to determne regulatory captal especally n Afrca. The semnal paper on credt rsk modelng s Merton (1974). In Merton s model, the value of the total assets of a frm s modeled by a geometrc Brownan moton. The frm defaults f ts assets fall below the (fxed) level of ts labltes at a pre specfed tme horzon. An opton prcng approach can then be used to value the frm s equty as a call opton on ts assets wth strke prce equal to the level
Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 43 of ts labltes []. The dea of lnkng a frm s default behavour wth the value of ts assets s ntutvely appealng, and has spawned the broad class of models referred to as structural models. The common feature of all of these models s the assumpton that the frm s credt qualty s determned by ts captal structure and the values of ts assets and labltes. Jont default behavour can then be ntroduced by lnkng the values of the assets and labltes to macroeconomc factors. Examples of ths approach nclude the CredtMetrcs and Moody s-kmv models popular n ndustry [3]. In factor models, the prce of a credt senstve nstrument s drven by the values of a set of fundamental random varables (common to all securtes) as well as an dosyncratc factor. The key property s the condtonal ndependence of credt losses, gven the values of the fundamental factors. Often, one-factor models admt a decomposton of the portfolo loss varable nto a monotonc functon of the factor and a resdual. The former part of the decomposton s called systematc rsk whereas the latter part s called specfc or dosyncratc rsk. In ths paper we wll assume that the random varable X s a macroeconomc factor affectng all defaults, and there exst an dosyncratc factor ε ndependent of X, affectng the ndustry wthn whch the oblgor operates [3]. Our approach n ths study begns by outlnng the rsk components of a credt rsk nstrument of a loan portfolo whch ncludes Expected Loss, Unexpected Loss, Loss Gven Default, Exposure at Default and Maturty. As the number of oblgors n a portfolo ncreases, the portfolo loss s then drven by the jont default behavor of the oblgors. If all oblgors default ndependently and the Loss Gven Default nstruments are ndependent, then the Central Lmt Theorem mples that portfolo losses wll tend to be normally dstrbuted. However, n most cases ths s not so as the oblgor defaults do not occur ndependently. Default correlaton, the tendency of defaults to occur together, can come from the fact that oblgors may be drawn from same or smlar ndustres, or affected by the overall health of the economy. Defaults can also be related due to contagon effect from drect tes between frms. Ths mples that oblgor defaults are not ndependent. For these reasons, real portfolo losses tend to exhbt skewness and fat tals [3]. The jont probablty of default s acheved by determnng the asset value correlaton of oblgors. As true asset returns of ndvdual oblgors are not drectly observable, equty prces are used as a proxy for asset returns to determne asset value correlaton of oblgors. Monte Carlo smulaton, a probablty smulaton technque, s employed to understand the mpact or how lkely the resultng outcomes or default behavour s dependent on the macroeconomc factors of systematc and dosyncratc rsk. Monte Carlo smulaton s wdely employed n fnance. The most common use of Monte Carlo smulatons n fnance s when we need to calculate an expected value of a functon gven a specfed dstrbuton densty [4]. Phelm f (x) (1977) developed a Monte Carlo smulaton method for solvng opton valuaton problems. The method smulates the process generatng the returns on the underlyng asset and nvokes the rsk neutralty assumpton to derve the value of the opton [5]. Recently, Monte Carlo smulaton approach to generate loss dstrbutons has also become popular n operatonal rsk models. Enrque Navarrate developed operatonal rsk quantfcaton usng Monte Carlo smulaton [6]. A more detaled approach to credt rsk usng Monte Carlo smulaton was outlned as n [7]. For an extensve dscusson of credt rsk modelng see as outlned n [8]. In ths paper, we employ Monte Carlo smulaton by generatng scenaros for the portfolo loss. Ths s acheved by smulatng the common macroeconomc factor X and the dosyncratc factor ε. Ths procedure s repeated a large number of tmes to produce a sample from the true portfolo dstrbuton. The credt rsk measures of the dstrbuton are then determned by methods of approxmaton usng probablty dstrbutons. Although ths procedure s computatonally ntensve (for example, n a partcular scenaro for a portfolo of sze hundred there are 7,04,000 smulatons), t can be appled to portfolo of any sze. Thus, we begn wth a portfolo of sze two, extend t to ten and then to a portfolo of sze hundred. After obtanng frequency of loss usng Monte Carlo smulaton, we do not know the exact probablty dstrbuton followed by the loss dstrbuton. To obtan the correspondng probablty dstrbuton, we employ analytcal approxmaton method to determne the nature of dstrbuton followed by the loss dstrbuton. Methods of Approxmaton usng a probablty dstrbuton have been successfully used n fnance and engneerng to determne solutons to analytc problems that do not have closed form solutons. It has many applcatons n rsk analyss, qualty control and cost schedulng. In fnance, Govann Ades provded a smple, analytc approxmatons method for prcng exchange-traded Amercan call and put optons wrtten on commodtes and commodty futures contracts [9]. The reasons for fttng a dstrbuton to a set of data nclude the desre for objectvty, the need for automatng the data analyss, and nterest n the values of the dstrbuton parameters [10]. One problem of analyzng data through the process of smulaton and fttng a statstcal dstrbuton s that many other classcal dstrbuton functons could ft the sample and consequently, goodness-of-ft tests are needed to be performed on the constructed probablty dstrbuton functons [11]. However, recent developments n computer software offer programs that automatcally assess the goodness-of-ft and provde the best dstrbuton for a gven data.. Man Body We begn by outlnng the buldng blocks of credt rsk modelng and develop a smple framework for a portfolo
44 Ose Antw: Measurng Portfolo Loss Usng Approxmaton Methods of two assets. We wll frst use these components to generate loss dstrbuton from whch we wll calculate the rsk quanttes for a two oblgor portfolo and then extend the framework to a portfolo of 10 or more oblgors..1. Components of Credt Rsk Models We begn by lookng at the varous components of credt rsk models, defne these components and outlne ther mathematcal contrbutons to the framework..1.1. Default There exst varous defntons of default. Throughout the text we shall refer to default as falure to pay promptly Interest or prncpal on a loan agreement when due.e. payment default. Bascally, there are two methods of measurng losses due to credt rsk: mark-to-market and default methods. The mark-to-market paradgm recognzes losses when the credt qualty of the oblgor (also referred to as debtor), deterorates.e. mgrates to a lower credt ratng. Such losses are not pad out, but only recognzed when the portfolo of the bank s marked-to-market. Credt mgraton s the approach used by JP Morgan Chase Bank [1]. On the other hand, the default method recognzes losses only when they are realzed n the form of default. Gven that an oblgor defaults, the loan provder or the oblgee suffers a loss referred to as Loss Gven default. Mathematcally, the default method s a specal case of the mark-to-market method (consderng only two ratng classes-default or survve). However, from a rsk management vew the two approaches are qute dstnct. Throughout the text we shall focus on defaults only and gnore the credt rsk related to credt mgraton. That s, we shall gnore the probablty that the oblgor would move from one credt qualty to another ncludng default wthn a gven tme horzon..1.. Probablty of default (PD) Ths ndcates the degree of lkelhood or the probablty that promsed payments such as nterest and coupon payments and prncpal repayments wll not be pad by the oblgor. The assgnment of a default probablty to every customer n a credt portfolo s far from an easy task. There are essentally two approaches to ths assgnment: Calbraton of default probabltes from market data The most famous representaton of ths type s the concept of Expected Default Frequences (EDF) developed by KMV Corporaton. Ths method for calbratng default probabltes from market data s based on credt spreads of traded products bearng credt rsk, e.g., corporate bonds and credt dervatves (swaps, etc.) Calbraton of default probabltes from ratngs Bascally, ratngs descrbe the credtworthness of customers. Ratngs are assgned to customers ether by external ratng agences lke Moody Investors Servces, Standard & Poor (S&P) or FITCH, or by a bank nternal ratng methodologes. Quanttatve as well as qualtatve nformaton s used to evaluate the clent. In practce, the procedure s more often based on the judgment and experence of the ratng analyst than on pure mathematcal procedures wth strct defned outcome. In ratngs, many drvers of the consdered frm s economc future are consdered. These nclude future earnngs and cash flows, debt, short and long term labltes [5]..1.3. Exposure at Default (EAD) Exposure at Default (EAD) estmates the amount that the oblgor owes n case of default. EAD actually specfes the exposure a bank does have to ts borrowers. For example, f a bank loans a 1000 to an oblgor, then the bank s Exposure at Default s 1000. In general, the exposure conssts of two major parts; the outstandngs and commtments. The outstandngs refer to the porton of the exposure already drawn by the oblgor. In case of borrower s default, the bank s exposed to the total amount of the outstandngs. The commtments can be dvded nto two portons, drawn and un-drawn, n the tme of default. EAD can be defned as: EAD = Outstandngs + γ Commtments 1.1 Where γ s the expected porton of the commtment lkely to be drawn pror to default..1.4. Loss Gven Default () The Loss Gven Default () represent an estmate of the porton of the exposure-at-default that wll not be recovered as a result of a default event,.e., the quantfes the porton of loss the lendng entty wll really suffer n case of default. The estmaton of loss quotes s somehow complcated because recovery rates depend on many drvng factors, for example on the qualty of the collateral (securtes, mortgages, guarantees, etc.) and on the senorty of the lenders clam on the borrower s assets. Loss Gven Default s thus consdered as a random varable descrbng the severty of the loss of a faclty type n case of default. thus refers to the expectaton of the severty of loss = Expected [Severty] 1. Most lendng nsttutons depend on ratng agences as a source for data of defaulted bonds..1.5. Expected Loss (EL) Expected Loss (EL) gves an ndcaton of the portfolo loss that we can expect to occur over the comng year. The Expected Loss of a transacton s an nsurance or loss reserve to cover losses that a lendng nsttuton expects from hstorcal default experence. In a captal charge requrement of a loan portfolo, the EL s provded for by prcng and provsons. As n probablty theory, the attrbute expected always refers to an expectaton or mean value, and ths s the case n credt rsk. In assessng expected loss of a loan, a customer s assgned a Probablty of Default (PD), a Loss Gven Default () and an Exposure at Default (EAD). A loss of any oblgor s then defned by a loss varable
Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 45 Expected Loss = PD EAD 1.3 Snce the expected loss s not predctable to a certan degree we should menton that underlyng our model s some probablty space ( Ω, F, Ρ) where Ω s a sample space of a set of outcomes. F s a collecton of subset of Ω that form a - feld and Ρ s a probablty measure. The elements of F are measurable events of the model and by ntuton t makes sense to clam that the event of default should be measurable. For obtanng the representaton above we need some assumptons that EAD and are constant values. Ths s not necessarly the case under all crcumstances. There are varous stuatons n whch for example EAD has to be modeled as a random varable. In such cases the EL s stll gven by Equaton 1.3 f one can assume that the exposure, the and the PD are ndependent and EAD and are the expectatons of some underlyng random varable. But whether the consttuents of equaton 1.3 are ndependent or not the basc concept of EL s stll the same and for reasons of smplcty our conventon wll be that EAD s always determnstc (non-random) quantty whereas wll be consdered as a random varable. Equaton 1.3 s used to calculate the EL of a sngle loan. The portfolo EL s smply the sum of the ELs of the loans n the portfolo [7] Example 1. We shall look at the Expected Loss of two oblgors; oblgor 1 and oblgor that owe us 1000 and 000 pounds respectvely. Ths means our EAD s 1000 and 000 for oblgor 1 and oblgor respectvely. Oblgor 1 has a PD on ts loan of 1%, whle oblgor has a PD of %. The for both loans s 50%. The Expected Loss of the loan for oblgor 1 = 1% * 50% * 1000 = 0.01*0.5*1000 = 5 The Expected Loss of the loan for oblgor = % * 50% * 1000 = 0.0*0.5*000 = 0 The Expected loss of the portfolo s therefore = 5 + 0 = 5.1.6. Unexpected Loss We now look at the key components of credt rsk modelng, startng wth the standard devaton of loss. At the begnnng, we ntroduced Expected Loss (EL) and ponted out that the EL of a transacton s an nsurance or loss reserve to cover losses that a lendng nsttuton expects from hstorcal default experence. But holdng captal aganst expected losses s not enough. In fact, the lendng nsttuton should n addton to the expected loss reserve, also save money to cover unexpected losses exceedng the average experenced losses from past hstory. Ths arses because most of the tme actual losses are not equal to the EL. We therefore need a measure of the devaton of actual losses around ther expected levels. The standard devaton of loss measures ths credt rsk of transacton, typcally called the Unexpected Loss (UL) [7]..1.7. Unexpected Loss of a Sngle Credt Rsk Transacton The unexpected loss of a credt rsk transacton s a functon of the PD,, and EAD and ther varances. PD, and EAD are ndependent random varables. The unexpected loss s gven n Equaton 1.4 UL= + EAD PD EAD + EAD PD + PD PD EAD PD + PD + EAD EAD + PD EAD 1.4 All the functons are ndependent and we wll assume that the probablty of default has a Bernoull dstrbuton, PD by ( PD PD ) so that we can substtute. The Loss Gven Default has a βeta dstrbuton, whch allows us to replace by *(1-)/4, and EAD s assumed to be determnstc, so that EAD = 0. Ths leads to Equaton 1.5. Unexpected Loss = EAD [ * PD * (1 PD) + PD * * (1 ) / 4] 1.5.1.8. Unexpected Loss of a Portfolo Due to dversfcaton (we can spread our nvestment over varous postons n dfferent ndustry sectors and regons), the unexpected loss of the portfolo s gven by Equaton 1.6. Unexpected Loss of a Portfolo =,..., n = = 1 1,..., n * ULj *, j, UL ρ 1.6 Where ρ, j represents the default correlaton between oblgor and oblgor j n a portfolo of n oblgors. For a portfolo of two oblgors, n =, the unexpected loss of the two credt rsk transacton s a functon of the ULs of the transactons and ther correlaton and s gven n Equaton 1.7 Portfolo Unexpected Loss = UL 1 + UL + UL 1UL ρ1, 1.7
46 Ose Antw: Measurng Portfolo Loss Usng Approxmaton Methods ρ 1, s the default correlaton between oblgor 1 and oblgor. The default correlaton gves an ndcaton of the tendency of the two loans to default at the same pont n tme. The default correlaton between oblgor and oblgor j s determned by Equaton 1.8. JPD PD * PD j ρ, j = 1.8 PD * (1 PD )(1 PD ) Where JDP s the jont probablty of default of oblgor and j ad PD and PD j are the probablty of default of oblgor and j respectvely. The Jont Probablty of Default s a functon of the PD s of the oblgors and ther asset correlaton. JDP = JDP PD, PD, asset correlaton ) 1.9 ( j, j The Jont Probablty of Default s establshed by determnng the volume under the asset value dstrbuton up to the default threshold of the two oblgors. Mathematcally, the JDP s a double ntegral whch s approxmated by a Vsual Basc program n Excel [7]. The Unexpected Loss determned n ths procedure represents credt rsk at one standard devaton of loss. Ths provdes confdence level of around 90% whch s not suffcent to capture all the losses assocated wth the loan portfolo. For hghly rated fnancal nsttutons wth confdence level of 99.9%, ths addtonal loss, referred to as Stress Loss (SL) s hghly sgnfcant. We would determne the Stress Loss assocated wth each portfolo after the smulaton procedure outlned n 3. 3. Smulaton Procedure We wll now use a Monte Carlo smulaton procedure to generate frequency of loss of the portfolo by followng the followng steps: 1. Specfy PD, and EAD and R (systematc factor) for the portfolo.. Smulate changes n the state of the economy by generatng random varables from 0 to 1; Macroeconomc factor for scenaro = NORMSINV (RAND ( )) 3. Smulate changes n the oblgor specfc rsk ε by generatng random varables from 0 to 1; Specfc Factor for oblgor = NORMSINV (RAND ( )) 4. Obtan asset return of each oblgor gven by: Asset return oblgor = r = ( R )* Y + ( 1 R )* ε where Y s the state of economy and ε s the oblgor specfc rsk and Y and ε are obtaned from steps and 3 respectvely and R s as specfed n step 1. 5. Set Default Pont d, where d = NORMSINV (PD). j If r < d then t follows that oblgor s asset value has dropped below the default threshold and hence oblgor defaults, otherwse the oblgor survves [7]. By repeatng the smulaton of defaults many tmes, we wll obtan frequency of losses from whch we obtan hstograms of the loss frequences. We shall ntally perform the Monte Carlo smulaton for a portfolo of two oblgors and then extend t to a portfolo of 10 and 100 oblgors. The results of the smulaton for a two oblgor portfolo are shown n Fgure 1. Smlar smulaton procedures are obtaned for 10 and 100 oblgor portfolos. 3.1. Analytcal Approxmaton Methods We have so far used Monte Carlo smulaton technque to obtan an emprcal loss dstrbuton of an underlyng portfolo consstng of two and ten and hundred oblgors. However, we do not know the exact probablty dstrbuton followed by the frequency loss dstrbutons. To obtan the correspondng probablty dstrbuton we shall use an analytcal approxmaton method to ft a probablty dstrbuton functon to the loss frequency. Typcally, the analytcal approxmaton method maps an actual portfolo wth unknown loss dstrbuton to an equvalent portfolo wth known loss dstrbuton. The loss dstrbuton of the equvalent portfolo s then taken as a substtute for the true loss dstrbuton of the orgnal portfolo. In practce ths s often acheved by choosng a famly of dstrbutons characterzed by ts frst and second moments showng the typcal shape of loss dstrbuton (.e., rght-skewed wth fat tals). We can then choose from the parameterzed famly of loss dstrbutons, the dstrbuton best matchng the orgnal portfolo wth respect to frst and second moments [1]. Thus, the analytcal approxmaton method works by approxmatng the loss dstrbuton of the orgnal portfolo by a known dstrbuton and matchng the frst and second moments of the orgnal portfolo to the parameters of the known portfolo. Thus, suppose we match our loss dstrbuton (unknown) to say beta dstrbuton (known), then bascally we are lookng for a random varable X ~ β ( α, β ), representng the percentage portfolo loss such that the parameters α and β solve the frst and second moments of the beta dstrbuton. The probablty densty functon of the beta dstrbuton s gven as: wth frst moment and second moment a + b) a 1 b 1 β a, b ( X ) = X (1 X ) 1.11 a) b) α E [X ] = 1.1 α + β αβ V [ X ] = 1.13 ( α + β ) ( α + β + 1)
Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 47 The analytcal approxmaton thus takes the random varable X as a proxy for the unknown loss dstrbuton of the portfolo we started wth. Followng ths assumpton, the rsk quanttes of the orgnal portfolo can be approxmated by the respectve quantles of the random varable X.. For example, the quantles of the loss dstrbuton of our portfolo can be calculated as quantles of the beta dstrbuton [13]. Thus, the true loss dstrbuton s substtuted by a closed-form, analytcal and a wellknown dstrbuton. Ths s very useful for computatonal purposes. We shall now perform Monte Carlo smulaton for Two (), Ten (10) and Hundred (100) oblgor portfolos. 4. Results and Dscussons 4.1. -Oblgor Portfolo Smulaton For the two-oblgor oblgor portfolo we generate a scenaro by settng the followng rsk measures: PD = 5% (Set by bank usng hstorcal default rates) EAD = 3,000 ( 1000 to oblgor 1, 000 to oblgor ) = 60% (obtaned from hstorcal average of s) M = Maturty = 1 year ρ = Jont Default Probablty of portfolo = 0.5 The smulaton result s summarzed n Table 1. Table 1. Smulaton Results for -Oblgor Portfolo. Smulaton Macro factor Specfc factor Oblgor 1 1-1.6413 0.3537 0.6115-1.949 3-1.0385-0.5481 4 1.769 0.314 5-0.1714-0.0585 6 0.654 1.640 7-1.7894 0.7774 8-1.119 0.4465 9-0.13-0.061 10-0.4145 1.366 11-0.9587.5191 1 0.7650 1.9007 13 0.8657 1.4769 Specfc factor Oblgor Return Oblgor 1 Return Oblgor Default Pont Oblgor 1 -.363 -.363 Default Oblgor 1-1.6019-1.3099 -.1336 0 0.613-0.3139 0.8013 0 0.4594-1.1740-0.7795 0-0.5896 1.6843 1.3638 0 0.145-0.1795-0.1018 0-0.968 0.9709 0.197 0 0.5050-1.58-1.454 0-1.5877-0.8013-1.6467 0-0.1971-0.81-0.71 0-0.600.5-0.6145 0.991 0.691 0.0066 0 0.161 1.534 0.768 0.7657 1.4348 1.0947 0 Default Pont Oblgor 1 Default Oblgor Portfolo Loss The loss frequency obtaned s summarzed n the Table and the resultng hstogram s shown n Fgure 1. Table. Frequency Dstrbuton of -Oblgor Portfolo. Losses Frequency Percent of Frequency 0.00 1955 97.98% 100.00 41.0137% 00.0 0 300.0 0 400.0 0 500.0 0 600.0 0 700.0 0 800.0 0 900.0 0 1,000.0 0 Cumulatve Frequency 97.98% Fgure 1. Frequency Dstrbuton of -Oblgor Portfolo. The probablty dstrbuton functon that best ft the - oblgor portfolo s the bounded gamma dstrbuton wth parameters α=.77 and θ=0. Ths gamma dstrbuton s completely characterzed by two parameters, α and θ. These quanttes are lnked as follows: µ = αθ = αθ The probablty densty functon of a gamma dstrbuton s:
48 Ose Antw: Measurng Portfolo Loss Usng Approxmaton Methods 1 α 1 P( x) = x exp( x θ) dx α α ) θ Where x s the dstrbuton varable, and Γ s the standard gamma functon evaluated at the relevant parameters and defned by the functon x) = 0 u x 1 exp( x) du. By usng gamma functon to approxmate the dstrbuton of losses t follows that there exst a random varable x ~ g( α, θ ), representng the percentage portfolo loss such that the parameters α and θ solve the frst and second moments of the gamma dstrbuton. The total portfolo loss s smply the value of x when P (x) = Confdence nterval (chosen to comply wth bank s rsk appette). Thus, suppose we set the confdence nterval at 95%, then the total loss of the portfolo s the value of x such that: 1 α 1 0.95 = x exp( x θ )dx α α ) θ Several other dstrbutons can be ftted to the mean and standard devaton of loss rate data but the gamma dstrbuton provdes the best ft for all the three portfolos. The gamma dstrbuton of the -Oblgor portfolo s shown n Fgure. Fgure. Graph of Gamma Dstrbuton for -Oblgor Portfolo. The data statstcs and percentle dstrbuton of losses obtaned from the gamma dstrbuton are gven n Table 3 and Table 4 respectvely. Statstc Range Mean Varance Table 3. Data Statstcs of -Oblgor Portfolo. Standard Devaton Coeffcent Of Varaton Standard Error Skewness Value( ) 1000 500 1.1000E+5 331.66 0.6633 10-1. Excess Kurtoss Table 4. Percentle Dstrbuton of -Oblgor Portfolo. Percentle Mn 5% 10% 5% (QI) 50% (Medan) 75% (Q3) 90% 95% Max Value( ) 0 500 800 980 1000 1000 From Table 3: Mean = 500 Standard devaton = 331.66 Hence Expected Loss of portfolo = 500 Unexpected Loss of portfolo = 331.66 The total loss of the portfolo s 1000. Thus by our conventon; Stress Loss =Total Loss Expected Loss Unexpected Loss Stress Loss = 1000 831.66 = 168.34 For ths portfolo there s no loss beyond the 95 th percentle, that s, 95 th percentle concdes wth the maxmum loss of the portfolo. Thus, n ths portfolo settng a rsk tolerance of 95% wll be suffcent to total loss of the portfolo.
Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 49 4.. 10-Oblgor Portfolo Smulaton For the 10-Oblgor portfolo we generate a scenaro by settng the followng rsk measures: PD = 5% (Set by bank usng hstorcal default rates) EAD = 0,150 = 60% (Obtaned from hstorcal average of s) M = Maturty = 1 year ρ = Jont Default Probablty of portfolo = 0.5 The loss frequences are shown n Table 5 and the correspondng hstogram generated s shown n Fgure 3. The probablty dstrbuton that ft the data n the 10- Oblgor portfolo s the bounded Gamma dstrbuton wth parameters α =.77 and θ = 0. Ths s shown n Fgure 4. Fgure 3. Frequency Dstrbuton of 10-Oblgor Portfolo. Fgure 4. Graph of Gamma dstrbuton for 10-Oblgor Portfolo. The data statstcs and percentle dstrbuton of loss obtaned from the gamma dstrbuton are gven n Table 5 and Table 6 respectvely. Table 5. Data Statstcs of 10-Oblgor Portfolo. Statstc Range Mean Varance Standard Devaton Coeffcent of Varaton Standard Error Value( ) 4500 50.917E+6 1513.8 0.6781 478 Skewness Excess Kurtoss 0-1. Table 6. Percentle Dstrbuton of 10-Oblgor Portfolo. Percentle Mn 5% 10% 5% (QI) 50% (Medan) 75% (Q3) 90% 95% Max Value( ) 50 875 50 365 4450 4500 4500 From Table 5: Mean =,50 Standard devaton = 1,513.8 Hence
50 Ose Antw: Measurng Portfolo Loss Usng Approxmaton Methods Expected Loss of the portfolo =,50 Unexpected Loss of the portfolo = 1,513.80 Stress Loss = 4500 3763.80= 736.0 In ths portfolo, there s no loss beyond the 95 th percentle as the 95 th percentle concdes wth the maxmum value of the losses whch s 4500. In ths portfolo settng a rsk tolerance of 95% wll be suffcent to cover the total loss of the portfolo. 4.3. Hundred (100) - Oblgor Portfolo Smulaton The smulaton procedure s extended to a portfolo contanng hundred oblgors as obtaned n real lfe bankng operaton. For the 100-oblgor portfolo we generate a scenaro by settng the followng rsk measures Asset value correlaton s set at 0.5. PD = 5% (Set by bank usng hstorcal default rates) EAD = 10,000,000,000 = sum of loans to the 100 oblgors = 60% (obtaned from hstorcal average of s ) M = 1 year ρ = Jont Default Probablty of portfolo = 0.5 The loss frequency generated from the smulatons and the correspondng hstogram s shown n Table 7 and Fgure 5 respectvely. Fgure 5. Frequency Dstrbuton of 100-Oblgor Portfolo. The probablty dstrbuton functon that best descrbe the data n the 100-Oblgor portfolo s the bounded gamma dstrbuton wth parameters α = 1.63 and θ =.9367E+7. Ths s shown n Fgure 6. Fgure 6. Graph of Gamma Dstrbuton for 100-Oblgor Portfolo. The data statstcs and percentle dstrbuton of loss obtaned from the gamma dstrbuton are gven n Table
Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 51 7and Table 8 respectvely. Statstc Range Mean Varance Table 7. Data Statstcs of 100-Oblgor Portfolo. Standard Devaton Coeffcent of Varaton Standard Error Skewness Excess Kurtoss Value( ).5800E+8.886E+7 1.3693E+15 3.7003E+7 1.837 1.1547E+6.3151 6.896 Table 8. Percentle Dstrbuton of 100-Oblgor Portfolo. Percentle Mn 5% 10% 5% (QI) 50% (Medan) 75% (Q3) 90% 95% Max Value( ) 0 6.0000E+6 1.8000E+7 3.6000E+7 7.8000E+7 1.000E+8 1.5800E+8 From Table 8: Mean =.8800E +7 Standard devaton = 3.7003E+7 Hence Expected Loss of the portfolo =.8800E +7 Unexpected Loss of the portfolo = 3.7003E+7 In ths portfolo, there are losses beyond the 95 th percentle as the 95 th percentle does not concde wth the maxmum loss of the portfolo. The cumulatve loss up to the 95 th percentle s 1.0E+8. The cumulatve loss up to 99.9 th percentle s 1.58E+8. Ths means that a bank or an nsttuton keepng such a portfolo whose rsk tolerance s 95% CI wll keep a sum of 10m as Unexpected Captal. However, a bank wth a hgh rsk tolerance such as AAA bank, wth rsk tolerance of 99.9% CI wll keep a sum of 58m as Unexpected Captal. For ths portfolo, the Stress Loss s computed as: Stress Loss=Total Loss Expected Loss Unexpected Loss Stress Loss = 1.5800E+8.8800E +7-3.7003E+7 = 9,197,000 5. Conclusons We have so far outlned and developed a model that can determne potental portfolo loss n excess of the UL. We have shown that based on your confdence level a bank can hold captal far n excess of the UL. It now les wth the bank s credt rsk management team to determne whether to hold captal aganst losses n excess of the unexpected loss, beyond say 90 th percentle of the dstrbuton. Usually, hghly rated banks such as AAA banks would lke to hold reserves to cover stress losses. However, banks wth lower ratngs wll deem t too expensve to hold captal aganst such huge potental losses whch has very low probablty of occurrence. It must be emphaszed here that although losses beyond the 90 th percentle are huge, they have very low probablty of occurrng. For example, from Fgure 5, t can be observed that beyond the 90 th percentle there s a loss of around 40,000,000, but the probablty of such loss occurrng s about 0.01. However, should such loss (catastrophc) arse, t could lead to the collapse of the nsttuton. In a clmate of hgh level of defaults, uncertanty and corporate bankruptcy, t wll be prudent and rsk sensble for a bank to keep captal to cover such losses. As we have seen, smulaton methods are smple to mplement and nterpret. It s able to determne all the losses n a loan portfolo and percentles can be determned at dfferent confdence levels. In summary, the methods we have provded here wll smplfy the processes nvolved n the determnaton of portfolo rsk. References [1] Martn, Hansen, Dr. Gary, van Vuuren and Mararosa, Verde Basel II Correlaton Values., An Emprcal Analyss of EL, UL and the IRB Model, Credt Market Research Fnancal Insttutons Specal Report, Ftch Ratng, 008, pp3 [] Merton, R. (1974): On the Prcng of Corporate Debt: The Rsk Structure of Interest Rates," Journal of Fnance, 9, 449-470. [3] Davd Saunders, Costas Xouros, Stavros A. Zenos, Credt rsk optmzaton usng factor models. Annals of Operatons Research, (007), pp.49-77 [4] Peter J ackel, Monte Caro methods n fnance Wley Fnance & Sons Ltd. Chchester, England (00), PP 1-6 [5] Phelm P. Boyle, Journal of Fnancal Economcs, Volume 4, Issue 3, May 1977, Pages 33 338 [6] Enrque Navarrete, Practcal Calculaton of Expected and Unexpected Losses on Operatonal Rsk by Smulaton Methods, Banca & Fnanzas: Documentos de Trabajo, 006 Vol. I, pp. 7-9 [7] Hoogbrun, Peter P, Journal of Global Assocaton of Rsk Professonals, (September/October 006), pp. 34-39. [8] Loffler, Gunter and Posch, Peter N. Credt Rsk Modelng usng Excel and VBA, Wley Fnance Seres, John Wley & Sons, Ltd., New York, NY, USA, 007. [9] GIOVANNI BARONE-ADESI, Effcent Analytc Approxmaton of Amercan Opton Values, The Journal of Fnance, Volume 4, Issue, 1987, pp 301-30. [10] John S. Ramberg, Pandu R. Tadkamalla, Edward J. Dudewcz, Edward F. Mykytka, A Probablty Dstrbuton and Its Uses n Fttng Data. Amercan Statstcal Assocaton and Amercan Socety for Qualty Technometrcs, v ol. 1, no., may, 1979 [11] AbouRzk, S., Halpn, D., and Wlson, J. Fttng Beta Dstrbutons Based on Sample Data Journal of Constructon Engneerng and Management Volume 10, Issue (June 1994). Pp 88-89 [1] Bluhm, Ludger Overbeck and Wagner, C. An Introducton to Credt Rsk Modelng, Chrstan Chapman & Hall/CRC, London, UK, 003.
5 Ose Antw: Measurng Portfolo Loss Usng Approxmaton Methods [13] Barreto, Humberto and Howland, Frank M. Introductory Econometrcs, Cambrdge Unversty Press, Cambrdge, UK, 006; pp 15-35. [14] Berenson, Mark L., Levne, Davd M. Basc Busness Statstcs, Prentce-Hall Internatonal nc New Jersey, NJ, USA, 1999; pp 45-75. [15] Davd Vose, Rsk Analyss a Quanttatve Gude, John Wley & Sons Ltd., New York, NY, USA, 003; pp 59 [16] Hagh, J. Probablty Models, Sprnger Undergraduate Mathematcal Seres, Sprnger, New York, NY, USA, 005, pp. 1-86