RISK ANALYSIS FOR LARGE POOLS OF LOANS

Transcription

1 RISK AALYSIS FOR LARGE POOLS OF LOAS JUSTI A. SIRIGAO AD KAY GIESECKE Absrac. Financial insiuions, governmen-sponsored enerprises, and asse-backed securiy invesors are ofen exposed o delinquency and prepaymen risk from large numbers of loans. Examples include morgages, credi cards, auo, suden, and business loans. Due o he size of he pools, he measuremen and managemen of hese exposures is compuaionally expensive. This paper develops and ess efficien numerical mehods for he analysis of large pools of loans as well as asse-backed securiies backed by such pools. For a broad class of dynamic, daa-driven models of loan-level risk, we develop a law of large numbers and a cenral limi heorem ha describe he behavior of pool-level risk. The asympoics are hen used o consruc efficien Mone Carlo approximaions for a large pool. The approximaions aggregae he full loan-level dynamics, making i possible o ake advanage of he deailed loan-level daa ofen available in pracice. To demonsrae he effeciveness of our approach, we implemen i on a daa se of over 25 million acual subprime and agency morgages. The resuls show he accuracy and speed of he approximaion in comparison o brue-force simulaion of a pool, for a variey of pools wih differen risk profiles. 1. Inroducion Financial insiuions, governmen sponsored enerprises such as Fannie Mae and Freddie Mac, and invesors have credi exposures o large pools of loans, including morgages, credi card receivables, auo loans, suden loans, and business loans. A major US bank migh direcly own a million morgages. The GSEs are exposed o ens of millions of morgages, eiher hrough direcly owning he morgages or providing credi guaranees agains defaul for he morgages. Major loan servicers can service up o en million morgages and, even hough hey do no direcly own he loans, morgage servicing cashflows are sill srongly affeced by defaul and prepaymen risk. Morgage-backed securiies (MBS) ypically have hundreds o ens of housands of loans. Large pools of loans are also common for many oher ypes of loans. For insance, a major US bank migh easily have on he order of 2, wholesale loans and 1, mid-marke and commercial loans. A major credi card company can have ens of millions of credi card accouns. Over half of all consumer credi is evenually securiized, and each deal can consis of ens of housands of credi cards. Due o heir size, hese loan pools and he securiies hey back are compuaionally challenging o analyze. A simulaion of a ypical pool migh ake many hours. This paper develops and ess efficien numerical mehods for he analysis of large, heerogeneous pools of loans. We focus on a broad class of dynamic, discree-ime models of loan-level risk. A loan migh be in one of several saes, such as 3 days pas due, prepaid, or foreclosure. The condiional ransiion probabiliies could come from a range of saisical or machine learning formulaions such as generalized linear models (e.g., logisic regression), neural neworks, decision rees, and suppor vecor machines. The ransiion probabiliy is allowed o depend on a vecor of loan-level feaures such as credi score and loan-o-value raio, a vecor of common risk facors influencing muliple loans, such as he unemploymen rae, as well as he pas behavior of he loans in he pool (hrough a mean-field erm). These daa-driven models are widely used in pracice (see [33], [9], [19], [17], and [16]) and are fied from hisorical loan performance daa ha are colleced inernally or acquired from daa vendors. For his imporan class of models, we prove a law of large numbers and a cenral limi heorem ha describe pool-level risk. These limi heorems are hen used o consruc efficien Mone Carlo approximaions Dae: December 22, 215. The auhors graefully acknowledge suppor from he aional Science Foundaion hrough Mehodology, Measuremen, and Saisics Gran o We hank seminar paricipans a he Universiy of Illinois a Urbana-Champaign, Imperial College London, London-Paris Bachelier Workshop on Mahemaical Finance, and he Inernaional Moneary Fund for commens. Special hanks are due o Kosas Spiliopoulos and Sefan Weber for insighful commens. We would also like o hank paricipans a he 214 Annual IFORMS Meeing and he 214 SIAM Conference on Financial Mahemaics and Engineering, where his paper won he 214 SIAM Financial Mahemaics and Engineering Conference Paper Prize. 1

2 of he loss, prepaymen, and cash flow disribuions for a large loan pool. We also prove a uniform inegrabiliy resul, which guaranees he convergence of our approximaion for a large class of coninuous funcions of pool-level quaniies such as loss, prepaymen, and cashflow. The approximaions accoun for he full loanlevel dynamics, making i possible o ake advanage of he deailed loan-level daa ofen available for loans. Very imporanly, he cos of our approximaion remains consan no maer he dimension of he loan-level feaures. This is essenial since loan-level daa is ofen high-dimensional (he dimension could be in he hundreds). Furhermore, since he law of large numbers and cenral limi heorem are dynamic, he approximaion provides he loss, prepaymen, and cash flow disribuions across all ime horizons a no exra compuaional cos. Given hese disribuions, risk analysis, pricing, and hedging of asse-backed securiies such as MBSs backed by he pool is immediae. We es he performance of our approximaions on a deailed loan-level morgage daa se which includes over 25 million prime and subprime morgages. We compare he approximae disribuion wih he rue disribuion (obained by brue-force simulaion) for various pools drawn from his daa se. The comparison is performed using model parameers fied o he daa se. The approximaion s compuaional cos is ofen several orders of magniude less han he cos of brue-force simulaion of he acual pool. I has a similar level of accuracy, in he cener as well as he ail of he disribuion. Alhough he approximaion s accuracy increases wih he size of he pool, i is highly accurae even for a pool having as lile as 5 loans Lieraure. The compuaional expense associaed wih loan-by-loan models for large pools is widely recognized. Prior research has analyzed several approaches o ackle his issue. [46], [52], and [54] propose parallel compuing approaches o MBS valuaion. [2], [53], [43], [31] and [36] develop op-down models of morgage pool behavior, and [2], [25], [15], [18] and ohers develop op-down models of corporae credi pools. The models are formulaed wihou reference o he consiuen loans and only model he pool in aggregae. The approximaion ha we develop for loan-level models is as racable as a op-down model while aking full advanage of he loan-level feaure daa. Limi heorems for credi pools have previously been proven in oher model frameworks; see [7], [11], [12], [14], [23], [24], [42], [45], and ohers. In conras o his lieraure, we consider a discree-ime formulaion ha is naural given he daa srucure common in pracice, where evens are repored on a monhly or quarerly basis. Unlike earlier resuls, our limi heorems cover a wide range of saisical and machine learning models ha are in widespread indusry use. Our formulaion is well-adaped o seings wih high-dimensional loanlevel feaure daa, where he feaures can be coninuously valued. Earlier model frameworks for which limiing laws have been proven do no include loan-level daa as a model inpu. Unlike he aforemenioned papers, we allow for muliple ypes of evens which may be muually exclusive. In pracice, many loans are subjec o several ypes of evens, such as delinquency, prepaymen, and foreclosure. Finally, limiing laws in he above papers do no cover imporan applicaions where he funcion of ineres is unbounded. This includes he expecaion, variance, higher-order momens, and uiliy funcions of he pool-level loss, prepaymen, or cashflow. To address hese applicaions, we prove uniform inegrabiliy, which allows our convergence resuls o cover a large class of funcions of pracical ineres. Discree-ime ineracing paricle models have been previously sudied in he conex of filering; see [39], [4], and [1]. The seing hey sudy is differen han he one in his paper. They prove limiing laws for a sysem of paricles where, given a sequence of observaions, he paricles ransiion according o paricle filering updaes. In effec, hey use he finie sysem o approximae he limiing law, while his paper uses he limiing law o racably approximae he finie sysem Srucure of he paper. The class of models we consider is described in Secion 2. The limi heorems as well as uniform inegrabiliy for his class of models are presened in Secion 3. These resuls are used o develop an efficien Mone Carlo approximaion for large pools. umerical mehods o compue he approximaion are described in Secion 4. In Secion 5, he approximaion is esed on acual morgage daa. There are several echnical appendices, one conaining he proofs of our resuls. 2. Class of Loan-Level Models We consider a broad family of dynamic models of loan-level risk in a pool a imes I = {, 1,..., T }. We fix a probabiliy space (Ω, F, P) and an informaion filraion (F ) I. P is he acual probabiliy measure. The oal number of loans iniially in he pool is. The process U n = (U n ) I prescribes he sae of he 2

3 n-h loan. The variable U n akes values in a finie discree space U. Common saes are curren, 3, 6, and 9+ days delinquen, prepaid, charge-off, and foreclosure. Some saes migh be absorbing. Sae ransiions are influenced by several ypes of risk facors. Each loan has an F -measurable loan-level covariae vecor Y n Y R d Y, 1 which can conain saic variables such as loan-o-value (LTV) raio, credi score, geographic locaion, ype of loan, and hisorical loan performance up unil he iniial ime of ineres = (for insance, how many days behind paymen he loan is or wheher i is in foreclosure). These loan-level facors are specific o each loan and are sources of idiosyncraic risk. We also consider sysemaic risk facors V which will have a common influence across many loans in he pool. The vecor process V = (V ) I, where V R d V, migh represen he behavior of local and naional economic condiions such as he unemploymen rae, housing prices, and ineres raes. 2 The risk facor V is exogeneous in he sense ha is dynamics are no affeced by he saes U n nor he Y n. Finally, we allow borrowers pas behavior o influence sae ransiions. Define he mean-field process H = (H ) I as H = 1 f H (U n, Y n ), where f H = ( ) f1 H,..., fk H and f H k : U R d Y R. We le H:s = (H,..., Hs ) denoe he pah of H beween imes and s. A specific applicaion for H would be o model a conagion effec for morgages where pas defauls of morgages in areas geographically close o he n-h morgage increase he likelihood of he n-h morgage defauling. The process H would hen keep rack of he number of defauls a each geographic locaion. In ligh of he morgage meldown, such a feedback mechanism has been suppored by several recen empirical papers; see [1], [26], [28], [34], and [49]. The dynamics of U n are prescribed by he ransiion funcion: (1) P[U n = u F 1 ] = h θ (u, U n 1, Y n, V 1, H τ: 1), = 1,..., T, where τ 1 is some fixed ineger and H is se o some predeermined consan (independen of ) for <. Therefore, he dynamics of he saes U 1,..., U can poenially depend upon he hisory of he mean field erm H. Exension of he model (and convergence resuls) when h θ depends upon he hisory of he common facor V is sraighforward. oe ha even condiional on he pah of he common facor V, he dynamics of he loans U 1,..., U are no independen due o he mean-field erm H. The funcion h θ is specified by a parameer θ, which akes values in he compac Euclidean space Θ, and which mus be esimaed from daa on loan performance. Equaion (1) gives he marginal probabiliy for he ransiions of he loans from heir sae a ime 1 o ime. Furhermore, we sipulae ha condiional on F 1, he saes U 1,..., U are independen. This fully specifies he dynamics of he loans n = 1,...,. In addiion, we model he financial losses suffered by he loan originaor, servicer, or invesor upon cerain sae ransiions. Given a sae ransiion U 1 n d U n = d, where d U represens an absorbing defaul sae such as 9+ days delinquen, charge-off, or foreclosure, he n-h loan suffers a loss l n (Y n, V ) [, 1]. Here, l n (Y n, V ) is iself a random variable condiional on Y n and V, herefore allowing for idiosyncraic losses. 3 Condiional on V, he loan-level feaure Y n, and he n-h loan defauling a ime, he loss given defaul is l n (Y n, V ) where P[l n (y, v) A] = ν,y,v (A). Condiional on V, Y n, Y n, and boh loans n, n defauling, l n (Y n, V ) and l n (Y n, V ) are independen. Since he defaul sae is absorbing, losses for he same loan canno occur wice. More generally, one could consider coss or losses associaed wih each of he sae ransiions. For example, here migh be servicer coss associaed wih loans which are behind paymen bu have no defauled ye. The resuls in his paper could be exended o his more general case. Typical loss given defaul models include logi models or neural neworks; an overview can be found in [35]. Many models commonly used o describe loan delinquency and prepaymen fall under he model class (1): see [32], [3], [4], [5], [8], [47], [51], and many ohers. The model framework (1) is popular in pracice (see [33], [9], [19], [17], and [16]) because i allows for deailed modeling of loan-level dynamics, inclusion of high-dimensional loan-level daa, flexible choice of ransiion funcions from saisical and machine learning, 1 Y n includes boh coninuous and caegorical variables. Caegorical variables are encoded as a vecor whose elemens are each in {, 1}. Of course, {, 1} is a subse of he real line, so Y R d Y. 2 The ime can also be included in he sysemaic facor vecor V. 3 Here we have implicily assumed a uni noional for each loan. This paper s resuls can be exended o he case of loans wih differen noional sizes as long as he loans noional sizes are bounded (which hey are in pracice). The noional size of he loan would be included in he Y n variable. 3

4 and he abiliy o incorporae imporan characerisics of loan pools such as correlaion beween delinquency and prepaymen raes for differen ypes of loans, geographic diversiy, and burnou. A ypical choice for h θ migh be a generalized linear model (GLM), alhough many oher choices are available. An example of a GLM is logisic regression. The assumpions required for he resuls in he paper require only mild assumpions on he form of he funcion h θ ; hese assumpions are saisfied by many sandard models. Some examples are presened below. Example 2.1 (Logisic regression model). Le he elemens of U be u 1,..., u K. If a loan is in sae u k, denoe he saes o which i can ransiion as u 1 k,..., u k k. Then, h θ (u n exp(θ u n k, u k, y, v, H) = k,u k (y, v, H)) 1 + k 1 n =1 exp(θ u n k,u (y, v, H)), n < k, k where θ = {θ u n k,u k } k=1:k,:k 1. h θ (u k k, u k 1 k, y, v, H) = 1 h θ (u n k, u k, y, v, H), n =1 Example 2.2 (eural nework model). Le he elemens of U be u 1,..., u K. If a loan is in sae u k, denoe he saes o which i can ransiion as u 1 k,..., u k k. Then, h θ (u n exp(σ θ (u n k k, u k, y, v, H) =, u k, y, v, H)) 1 + k 1 n =1 exp(σ θ(u n k, u k, y, v, H)), n < k, h θ (u k k, u k, y, v, H) = 1 The acivaion funcion σ θ is given by k 1 n =1 h θ (u n k, u k, y, v, H). σ θ (u n k, u k, y, v, H) = W 2 θ q(w 1 θ Q n ), where Q n = (u n k, u k, y, v, H) R dq 1, and Wθ 1 Rd h d Q and Wθ 2 R1 d h are weighs on he inpu Q n and he oupu of he hidden layer q, respecively. The hidden layer q : R dh 1 R dh 1 is he funcion q(x) = (q 1 (x 1 ), q 2 (x 2 ),..., q dh (x dh )). Typical choices for q 1,..., q dh are sigmoid funcions and d h is he number of neurons in he hidden layer. See [29] for more deails on neural neworks. Example 2.3 (Decision ree model). Divide he space U Y R d V R K ino he regions {Rθ m}m m=1, wih he condiion ha hese regions are disjoin and heir union covers he enire space. The ransiion probabiliies in region m are p m θ (u, u, y, v, H). The ransiion probabiliy h θ hen is M h θ (u, u, y, v, H) = p m θ (u, u, y, v, H)1 (u,y,v,h) Rθ m. m=1 The regions {R m θ }M m=1 are chosen by sequenially spliing he space via recursive binary spliing. For each region m, he ransiion probabiliies p m θ (u, u, y, v, H) could be a logisic regression or, more simply, a consan ransiion marix beween he differen saes. See [29] for more deails on decision rees. For risk managemen and oher applicaions, one is ineresed in he behavior of a pool of loans under he model class (1). The behavior of he pool a ime can be described by he empirical measure of he variables (U 1, Y 1 ),..., (U, Y ), which is denoed by µ B = P(U Y), where P is he space of probabiliy measures. Formally, µ = 1 (2) δ (U n,y n ), where δ is he Dirac measure. The empirical measure µ (u, A) = A µ (u, dy) gives he fracion of he loans in pool a ime which are in sae u and which have loan-level feaures in he se A R d Y. I compleely encodes, up o permuaions, he pool dynamics and feaures. For insance, he pool-level defaul rae can be expressed as Y µ (d, dy). Represening prepaymen by an absorbing sae p U, he pool-level prepaymen 4

5 rae akes he form Y µ (p, dy). The process H R K specifying he mean-field erm in (1) can also be expressed in erms of he empirical measure: H = f H, µ. Here and below, we use he noaion U Y f, ν E = E f(x)ν(dx).4 For noaional convenience, le f(u, y), ν(u, dy) = f(u, ), ν(u, ) Y. The defaul loss rae L a ime is: L = 1 (3) l n (Y n, V )(1 U n =d 1 U n 1 =d). The cumulaive loss from defaul up unil ime is simply s=1 L s. Le he loss process over all imes be L = (L ) I. Given he empirical measure and he loss from defaul of a pool, analysis of securiies backed by he pool is immediae. For insance, he disribuion of he cashflow from a pool of loans can be analyzed, an example of which is given below. Example 2.4 (Cashflow from a Loan Pool). Le U = {c, p, d}, where c represens a loan which is curren on is paymens, p represens prepaymen, and d represens defaul. The oal cashflow rae from a pool of loans a ime is: C = r,v, µ U Y + b, µ (p, ) µ 1(p, ) + (1 L ). The cashflow for a loan wih feaure y in sae u a ime, when he common facor V = v, is r,v (u, y). I is comprised of boh ineres and principal paymens. The dependence on V allows one o capure a variable loan ineres rae ied o a benchmark rae. The ousanding balance for a loan wih feaure y a ime is b (y). Upon prepaymen, he borrower will pay off he enire ousanding balance. The funcions r,v (u, y) and b (y) are deerminisic. The recovery from defaul a ime is 1 L. The oal cashflow rae is jus he summaion of C over all imes I. 3. Limiing Laws and an Efficien Mone Carlo Approximaion For large pools of loans (i.e., large ), he compuaion of he disribuion of he defaul rae, prepaymen rae, and loss from defaul is compuaionally burdensome. We develop an efficien Mone Carlo approximaion of hese disribuions for large pools. The approximaion is based on a law of large numbers (LL) and a cenral limi heorem (CLT) for he empirical measure µ and he loss process L. The proofs for he heorems below are given in Appendix A. Assumpion 3.1. Suppose ha µ converges in disribuion o µ in B = P(U Y), where µ is deerminisic, and ha he feaure space Y is compac. Also, h θ is coninuous in is hird and fifh argumens and he funcion f H is coninuous and bounded in is second argumen. Theorem 3.2. Provided Assumpion 3.1, he empirical measure µ converges in disribuion o µ in B T +1 as, where µ saisfies: (4) µ (u, dy) = h θ (u, u, y, V 1, H τ: 1 ) µ 1 (u, dy), where H = f H, µ U Y and H :s = ( H,..., H s ). The assumpions required for Theorem 3.2 are relaively mild and are saisfied by sandard models. For insance, logisic regression and neural neworks boh saisfy Assumpion 3.1. The compacness assumpion for he feaure space Y is also saisfied by commonly used daa feaures (such as credi score, LTV raio, ineres rae, and original balance). The assumpion ha he funcion f H is bounded and coninuous sill allows he mean-field erm H o keep rack of he fracion of he pool in any combinaion of saes in U and caegories in Y. For example, H could rack he fracion of he pool which has defauled a each geographic locaion as well as he fracion of he pool which has defauled for each loan produc ype (such erm loan, credi line, ec.). I is imporan o noe ha he LL in Theorem 3.2 is dynamic and is also a random equaion. Randomness eners hrough he pah of he common facor V : he quaniies H and µ are deerminisic funcions of 4 For insance, f, νu Y = u U Y f(u, y)ν(u, dy) and f(u, ), ν(u, ) Y = Y f(u, y)ν(u, dy). 5

6 V : 1 = (V,..., V 1 ). If he ransiion funcion h θ depends on he pas pool dynamics hrough H τ: 1, he LL is nonlinear. We supplemen he LL wih a CLT. Define he empirical flucuaion process Ξ = (µ µ ), which akes values in W = U u=1 S (R d Y ), where S is he space of empered disribuions. An elemen in he space of empered disribuions is he finie-order (disribuional) derivaive of some coninuous funcion wih polynomial growh. The space of measures is a subse of he space of empered disribuions. Convergence of Ξ mus be considered in a larger space han he space of measures since Ξ can ake negaive values and A Ξ (u, dy) is no bounded. Alhough Ξ is no an elemen of P(U R d Y ), i does exis in he larger space W. Moreover, is limi Ξ also exiss in W. The CLT saisfies an equaion linearized around he nonlinear dynamics of he LL. Randomness eners boh hrough V and a maringale erm. Like he LL, he CLT is also dynamic. Assumpion 3.3. (µ µ ) converges in disribuion o Ξ in W. In addiion, he funcion f H is smooh in is second argumen and h θ is smooh in is hird and fifh argumens. Theorem 3.4. Provided Assumpions 3.1 and 3.3, Ξ converges in disribuion o Ξ in W T +1 as, where Ξ saisfies he equaion: Ξ (u, dy) = u U h θ (u, u, y, V 1, H τ: 1 ) Ξ 1 (u, dy) + ( ) (5) H h θ(u, u, y, V 1, H τ: 1 ) Ē τ: 1 µ 1 (u, dy) + M (u, dy), where E = f H, Ξ U Y, Ē = f H, Ξ, and U Y Ē:s = (Ē,..., Ēs). Given V, M(u, dy) is a condiionally Gaussian process wih zero mean and covariance: Cov[ M (u 1, dy), M (u 2, dy) V : 1 ] = h θ (u 1, u, y, V 1, H τ: 1 )h θ (u 2, u, y, V 1, H τ: 1 ) µ 1 (u, dy), Var[ M (u, dy) V : 1 ] = h θ (u, u, y, V 1, H τ: 1 )(1 h θ (u, u, y, V 1, H τ: 1 ) µ 1 (u, dy), where u 1 u 2 and V : = (V,..., V ). Given V, M2 is independen of M1 for 2 1 and M (u, dy 1 ) is independen of M (u, dy 2 ) for y 1 y 2. The addiional assumpions necessary for Theorem 3.4 are mild. The main addiional assumpion is ha h θ and f H are smooh in he loan feaure y. Typical models for h θ such as logisic regression, neural neworks, probi, generalized linear models wih a smooh link funcion, kernel mehods wih smooh kernels (e.g., Gaussian kernel), and Gaussian process regression saisfy Assumpions 3.1 and 3.3. Due o heir disconinuiies, decision rees do no saisfy Assumpions 3.1 and 3.3. However, if he Y n are i.i.d., Theorems 3.2 and 3.4 will hold if h θ is piecewise smooh in is hird argumen and smooh in is fifh argumen, which would cover decision rees. Alhough we prove convergence in S, whose dual is he space of Schwarz funcions, smoohness of h θ and f H on Y suffices for Theorem 3.4 o hold since Y is compac. Finally, he resricion on f H is no sringen; he mean-field erm H can sill keep rack of he fracion of he pool in any combinaion of saes in U and caegories in Y. Theorems 3.2 and 3.4 can be exended o he loss from defaul (3). To his end, define he empirical loss flucuaions Λ = (L L ) R. Assumpion 3.5. The firs and second momens of he loss given defaul, g 1 (, v, y) = 1 zν,y,v(dz) and g 2 (, v, y) = 1 z2 ν,y,v (dz), are smooh in y for each, v. Proposiion 3.6. Le Assumpions 3.1, 3.3, and 3.5 hold. The loss process L converges in disribuion o L in [, 1] T +1 as, where L saisfies: 1 (6) L = zν,y,v (dz), µ (d, dy) µ 1 (d, dy). 6

7 The flucuaions for he loss Λ converge in disribuion o Λ in R T +1 as where Λ saisfies: 1 (7) Λ = zν,y,v (dz), Ξ (d, dy) Ξ 1 (d, dy) + Z, where, condiional on V, Z is a mean-zero Gaussian wih variance: 1 1 Var[ Z V : ] = [ z 2 ν,y,v (dz) ( zν,y,v (dz)) 2 ], µ (d, dy) µ 1 (d, dy). Given V, Z1 is independen of Z2 for 1 2 as well as Ξ. Assumpion 3.5 requires smoohness of he firs and second momens of he loss given defaul. Sandard models of he loss given defaul such as neural neworks (see [35] for an overview) saisfy his assumpion. Corollary 3.7. Under Assumpions 3.1, 3.3, and 3.5, (V, µ, L, Ξ, Λ ) converges in disribuion o (V, µ, L, Ξ, Λ) in (R d V ) T +1 B T +1 R W T +1 R as. Using Corollary 3.7, he law of large numbers and cenral limi heorem can be combined o form an approximaion for a finie pool of loans: µ = µ + 1 Ξ d µ = µ + 1 Ξ, (8) L = L + 1 Λ d L = L + 1 Λ. The LL µ is a firs-order approximaion of µ while he CLT Ξ provides a second-order correcion. The large-pool approximaions (8) are condiionally Gaussian given V and can be uilized o efficienly esimae pool-level quaniies such as prepaymen, defaul, loss, and cashflow raes for large pools of loans. Secion 4 below explains he compuaion of he approximaion. The heorems 3.2 and 3.4 prove ha he disribuion of he finie pool and is flucuaions converge o he disribuion of he LL and CLT. However, his does no guaranee ha he saisics, such as he momens, of he disribuion of he finie pool and is flucuaions converge o he saisics of he LL and CLT. 5 In paricular, X d X only implies E[f(X )] E[f( X)] for f coninuous and bounded. In order o prove E[f(X )] E[f( X)] for coninuous f, one needs o show f(x ) is uniformly inegrable. Since Y is assumed o be compac, P(Y U) is also compac. Consequenly, he empirical measure µ lies in a compac space (he space of measures on a compac space is compac). The loss L also lies in he compac se [, 1]. f(µ, L ) is herefore uniformly inegrable for any coninuous funcion f and E[f(µ, L )] E[f( µ, L)] for any coninuous funcion f. The main challenge resides in showing ha f(ξ, Λ ) is also uniformly inegrable. Unlike µ, Ξ and Λ are no resriced o compac ses. For insance, convergence in disribuion is no sufficien for he approximaion (8) o hold for he second momen of he loss a ime : (9) E[(L ) 2 ] = E[ L 2 ] + 2 E[ L Λ ] + 1 E[(Λ ) 2 ]. In order o jusify he approximaion E[(L ) 2 ] E[( L ) 2 ], one needs he convergence resuls E[ L Λ ] E[ L Λ ] and E[(Λ ) 2 ] E[( Λ ) 2 ] as. Due o L being bounded, i is sufficien o show ha Λ and (Λ ) 2 are uniformly inegrable. Assumpion 3.8. For funcions of he form f(ξ, Λ ) = g( T =1 u U φ,u (y), Ξ (u, dy) + T =1 c Λ ) where c R, g is coninuous and polynomially bounded, 6 and he funcions φ,u are coninuous. In addiion, V :T = (V, V 1,..., V T ) akes values in a compac se V, and for any coninuous φ, here are posiive consans K 1,,u and K 2,,u such ha (1) [ P (sup φ(y) ) 1 φ(y), Ξ (u, dy) ] > α V :T K 1,,u exp ( K 2,,u α 2). y Y 5 For insance, consider he random variable X where X = 2 wih probabiliy 1 and is oherwise. X p bu E[X ] as. 6 I.e., here is some posiive consan C such ha g(x) x k for x C and k. 7

8 Theorem 3.9. Under Assumpions 3.1, 3.3, 3.5, and 3.8, f(ξ, Λ ) is uniformly inegrable. The class of funcions covered by Assumpion 3.8 can represen any funcion g of he defaul, loss, prepaymen, or cashflow raes where g does no have faser han polynomial growh in is ails. We are no aware of any applicaions wih exponenial growh in he ails, so his class of funcions is quie general. A sufficien condiion for he bound (1) is ha he loan feaures Y n are independen; hen, by he Azuma- Hoeffding inequaliy, he bound is saisfied. The ideas and resuls in his paper, which are developed for discree ime, can be exended o a coninuousime framework where evens are governed by couning processes wih inensiies. The law of large numbers and cenral limi heorem for he coninuous-ime framework saisfy a random ODE and an SDE, respecively. Their forms are very similar o he forms given above for he discree-ime model considered in his paper. In fac, he discree-ime limiing laws presened here are he Euler discreizaion for he coninuous ime limiing laws. Some examples of coninuous-ime loan models which would be covered under such an exension are he morgage models sudied by [13] and [36]. 4. Compuing he Approximaion The approximaion (8) can be compued by Mone Carlo simulaion. Firs, many pahs are simulaed from he sysemaic process V. Condiional on each pah of V, µ can be calculaed deerminisically and he condiional covariance of Ξ can be evaluaed in closed-form. Since Ξ is condiionally Gaussian, his direcly yields he condiional disribuion of Ξ. Then, he uncondiional disribuion for he approximaion µ can be found by averaging he condiional disribuions across he pahs of V. We highligh ha he approximae loss L is simply a linear funcion of ( µ, Ξ) plus some independen Gaussian noise, and herefore is also very sraighforward o simulae. In effec, he approximae loss L can be simulaed (a no exra cos) on op of he simulaed pahs of ( µ, Ξ) Algorihm. The numerical evaluaion of µ and Ξ condiional on each pah of V requires discreizing R d Y ino compuaional cells and hen calculaing µ (u, dy) and Ξ (u, dy) a he cener of each cell. Le he se of cells be C = {c 1,..., c K } where K k=1 c k = Y wih associaed ceners (i.e., grid poins) R = {y 1,..., y K } where y k c k. The iniial measures µ would be calibraed o he daa of he pool and he second-order correcion Ξ would be se o zero. For each V, Ξ s disribuion on he se of grid poins can be compued in closed-form (since i is condiionally Gaussian). The approach is oulined below: Define µ Ψ and Ξ Ψ o saisfy he LL and CLT evoluion equaions, bu wih iniial condiion µ Ψ = = Ψ(u, dy) where Ψ(u, dy) = K k=1 µ =(u, c k )δ yk. 7 Simulae pahs V 1,..., V L from he random variable V. Condiional on a pah V l, le µ Ψ and Ξ Ψ be µ Ψ,l and Ξ Ψ,l, respecively. For each V l : Calculae µ Ψ,l (u, y k ) for all y k R a imes = 1,..., T. Ψ,l Calculae he (closed-form) covariance of Ξ across he grid poins y 1,..., y K a = 1,..., T. 8 For each V l and any funcion f, a numerical approximaion can be made: f, µ,l d f, µ l U Y + 1 f, Ξl U Y U Y K k=1 u U f(u, y k )[ µ Ψ,l (u, y k ) + 1 ΞΨ,l (u, y k )] (m l, Σ l ), where µ,l is he empirical measure µ for he pah V l, m l = K k=1 u U f(u, y k) µ Ψ,l (u, y k ), and Σ l = 1 Var[ K k=1 u U f(u, y k) Ξ Ψ,l (u, y k )] (which can be calculaed in closed-form). 7 This is he equivalen of solving he LL and CLT if Y was finie-dimensional, i.e., a a finie se of grid poins y1,..., y K. A generalizaion allowing differen ses of grid poins for differen saes u is sraighforward. 8 Provided Cov[ Ξ 1 (u, y i ), Ξ 1 (u, y j ) V l ] and Cov[ Ξ 1 (u, y i ), Ē: 1 V l ] for all y i, y j R and u, u U, one can calculae he quaniies Cov[ Ξ (u, y i ), Ξ (u, y j ) V l ] and Cov[ Ξ (u, y i ), Ē: 1 V l ] in closed-form. Thus, one can march forward Ψ,l hrough ime, calculaing he (closed-form) covariance of Ξ across he grid poins y 1,..., y K a each ime = 1,..., T. 8

9 Collecing he condiional disribuions for each V l, he densiy of f, µ can be direcly U Y compued as 1 L L l=1 φ(z, ml, Σ l ) where φ(, m, Σ) is he densiy of Gaussian random variable wih mean m and covariance Σ. Alernaively, Ξ could be direcly simulaed insead of finding is closed-form covariance condiional on each V l. In his laer approach, one would simply simulae Ξ Ψ for each V l. Such simulaion is sraighforward since Ξ is condiionally Gaussian given V. The simulaion scheme above has he advanage of lower variance per sample wih he same bias. The convergence for he numerical error of he simulaion scheme is analyzed in Appendix B. We show ha as he sizes of he compuaional cells end o zero (i.e., he number of grid poins ends o infiniy), he error also converges o zero. Moreover, a specific convergence rae is provided which can guide he choice of he locaion of grid poins, number of grid poins, and number of Mone Carlo rials. Given a fixed compuaional budge, here is a radeoff beween more rials (reducion of variance) and more grid poins (reducion of bias). In Appendix B, we derive a closed-form formula for he opimal allocaion beween he number of grid poins and he number of rials provided a fixed budge. The compuaional performance of he LL and CLT can be furher improved by he use of a non-uniform grid for discreizing R d Y. Using a non-uniform grid, more poins would be placed where µ (u, dy) is large and less poins would be placed where µ (u, dy) is small. Appendix C provides an example of a paricular non-uniform grid well-adaped o our problem. The proposed grid is highly accurae even wih only a small number of grid poins. A sparse grid can also be used in order o furher decrease compuaional ime. Appendix D describes his approach. Using a sparse grid, simulaion is performed a only a few poins and hen he soluion is evaluaed on a finer grid via inerpolaion. A final advanage of he approximaion is is reusabiliy; one se of simulaions on a pre-chosen grid can be used o calculae he disribuion for many differen pools by re-weighing he simulaed CLT and LL. Anoher choice, implemened in Secion 5.7, is a non-uniform sparse grid using k-means clusering Efficien Low-dimensional Approximaion. There is a poenial drawback o he law of large numbers (4) and cenral limi heorem (28). If he loan-level feaure space Y is very high-dimensional, he law of large numbers will be high-dimensional and compuaions using radiional uniform grids can become expensive due o he curse of dimensionaliy. The number of mesh poins in a uniform grid will grow exponenially wih he dimension d Y of he loan-level feaure space. Forunaely, for a reasonably large subclass of models, one can perform a ransformaion of he LL in equaion (4) which convers he LL ino a low-dimensional problem. Suppose ha here is a funcion g θ such ha h θ (u, u, y, v, H) = g θ (u, u, f(y), v, H) where f : Y R d W. Then, h θ is invarian under he coordinae ransformaion w = f(y) and one can reduce he high-dimensional equaion for µ (u, dy) o he low-dimensional equaion: (11) µ (u, dw) = g θ (u, u, w, V 1, H τ: 1 ) µ 1 (u, dw), w R d W. The same coordinae ransformaion can be used o arrive a a low-dimensional CLT in erms of w. o maer how large he original dimension d Y is, he ransformaion reduces he dimension d Y of he LL and CLT o a consan dimension d W, paving he way for racable compuaions. An example of a funcion h θ which saisfies he necessary requiremen is any generalized linear model (GLM), such as logisic regression (see Example 2.1). GLMs are widely used in pracice for defaul and prepaymen modeling. Furhermore, i is o be emphasized ha he ransformaion allows arbirary complexiy wih respec o he inpu facors y and v; he sole resricion is ha v can only inerac wih y hrough he funcion f(y). For insance, one can always make a GLM arbirarily nonlinear in he inpu space y by adding feaures which are nonlinear funcions of he iniial se of feaures. A common choice is o use basis funcions; a very simple example migh be polynomials while a more complicaed choice migh be waveles. This can grealy increase he dimension d Y. However, such an expansion of he feaure space does no increase he dimension d W of he low-dimensional LL (11) and is compuaional cos remains he same. The dimension d W does no depend on how large he original dimension d Y is. The iniial disribuion µ (u, dw) can be calibraed direcly from he daa available for he pool of loans. Simply ransform he feaures Y = (Y 1,..., Y ) o W = (W 1,..., W ) via he map f and hen calculae 9

10 he empirical disribuion of W. Due o heir broad applicabiliy and compuaional racabiliy, he lowdimensional LL and CLT are very useful in pracice. The combinaion of he low-dimensional ransformaion and he non-uniform grid schemes menioned in he previous secion make simulaing he pool loss and prepaymen levels via he LL and CLT very compuaionally racable. For models of h θ which do no fall under he subclass of models for which he lowdimensional ransformaion is applicable, one can sill compue he approximaion (8) for high-dimensions (wihou any ransformaion) via sparse non-uniform grids. We describe and numerically implemen his laer approach in Secion 5.7, which also srongly ouperforms brue-force simulaion of he acual pool. 5. Compuaional Performance of he Efficien Mone Carlo Approximaion We compare he accuracy and compuaional cos of he efficien Mone Carlo approximaion wih brueforce Mone Carlo simulaion using acual morgage daa. The approximaion has very high accuracy even for small pools in he hundreds of morgages. Is compuaional cos is ypically orders of magniude lower han brue-force Mone Carlo simulaion of he acual pool Models and Daa. We es he performance of he approximaion using several logisic regression and neural nework models of he ransiion funcion h θ in (1); see Example 2.1 and 2.2, respecively. We consider hree saes: curren, prepaymen, and defaul. The loss given defaul is normalized o one. We fi he model parameers θ o he daa via maximum likelihood esimaion; see Appendix E for deails. 9 Our daa is comprised of wo loan-level daa ses. The firs is a subprime daa se consising of over 1 million morgages during he ime period , obained from he Trus Company of he Wes. The morgages are spread across he enire US, covering over 36, differen zip codes. The feaure daa includes zip code, FICO score, loan-o-value (LTV) raio, iniial ineres rae, iniial balance, ype of morgage, deal ID, and ime of originaion. Defaul, foreclosure, modificaion, real esae owned (REO) and prepaymen evens, if hey occur, are also recorded in he daa se on a monhly basis. The second daa se, obained from Freddie Mac, conains agency morgages over he ime period and consiss of 16 million morgages. The feaure daa includes FICO score, firs ime homebuyer indicaor, ype of morgage, mauriy dae, number of unis, occupancy saus, combined loan-o-value, original deb-o-income raio, LTV raio, iniial ineres rae, prepaymen penaly morgage indicaor, loan purpose (purchase, cash-ou refinance, or no cash-ou refinance), original loan erm, and number of borrowers. The zip codes for each morgage have been parially anonymized in his daa se, bu he meropolian saisical area (MSA) is repored for each morgage. There are roughly 43 differen MSAs repored in he daa se. Finally, evens such as prepaymen, foreclosure, 18+ days delinquen are also recorded on a monhly basis Performance of LL and CLT. Firs, we demonsrae he accuracy of he approximae disribuion by comparing i wih he acual disribuion for he pool. The acual (or rue ) disribuion is found via brue-force Mone Carlo simulaion. The pool is drawn a random from he subprime daa se. Here and in Secions 5.3 hrough 5.6, we use a logisic regression model for h θ. The parameers θ are fied using he enire subprime daa se prior o January 1, 212. The loan-level feaures Y n used for boh defaul and prepaymen are FICO score, LTV, iniial balance of he morgage, and iniial ineres rae for he morgage. The naional unemploymen rae is used as he sysemaic facor for defaul. Boh he naional unemploymen rae and naional morgage rae are used as sysemaic facors for prepaymens. Unless oherwise saed, we do no include a mean field erm H. A mean-field erm is included in Secion 5.4 where we sudy he accuracy of he approximaion for he momens of he loss. We perform 25, Mone Carlo rials for boh he brue-force simulaion of he acual pool as well as he simulaion of he approximaion. We simulae he pool for a one-year ime horizon, wih monhly discreizaion (T = 12). The morgage rae and unemploymen rae are simulaed as (independen) discreeime random walks wih sandard deviaions fied o heir hisorical values prior o January 1, We model boh defaul and prepaymen, and hence d W = 2 for he efficien Mone Carlo approximaion. Our implemenaion of he approximaion uses non-uniform grids; see Appendix C. 9 Parameer esimaes are available upon reques. 1 More complicaed models for V may be worhwhile o implemen in pracice, which incorporae correlaion beween sysemaic facors as well as seasonal effecs or rends. 1

11 Figure 1 compares he acual disribuion wih he LL disribuion for he loss from defaul for pools of sizes = 5,, 1,, 25, and 1,, respecively. The LL is very accurae. The LL can be combined wih he CLT o creae a second-order accurae approximaion. The approximaion is accurae even for relaively small pools in he hundreds of morgages. Figure 2 compares he approximae disribuion (using boh he LL and CLT) wih he acual disribuion for pools wih sizes = 5, 1,, 2, 5, and 5,. Using he LL alone can underesimae he ails of he disribuion for small. By including he CLT in addiion o he LL, one is able o accuraely capure he ail of he disribuion. In he nex secion we assess he accuracy in he ails more horoughly Acual loss disribuion Large pool approximaion Loss from Defaul for = 5, Loss from Defaul for = 1, Loss from Defaul for = 25, Loss from Defaul for = 5, Figure 1. Comparison of acual loss disribuion wih LL loss disribuion (does no include CLT in approximaion). Loss repored as fracion of pool which defauled. The horizon is 12 monhs. Compuaional coss are repored in Table 1 for differen sized pools. In general, a rough approximaion of he raio of he compuaional coss is (12) Cos of LL Cos of Simulaion of Acual Pool = g, where g is he number of grid poins needed for he numerical soluion of he LL and CLT equaions across he space R d W and is he number of morgages in he acual pool. For insance, if one only needs 25 grid poins, he raio of compuaional coss is 25. For a million morgages, ha leads o a reducion in compuaional ime of well over 4 orders in magniude. Compuaional imes lised in Table 1 for brue-force simulaion of he pool, simulaion of he LL, and simulaion of he full approximaion (LL combined wih he CLT) are for d W = 2 (model includes boh prepaymen and defaul). These compuaional imes are for a welve-monh ime horizon umerical Performance across Acual Deals. The subprime daa se covers over 6, deals. 11 As menioned earlier, for each morgage, a deal ID is available. One can herefore reconsruc he acual 11 Deal refers o he securiizaion of a pool of morgages ino an MBS. An MBS srucure may vary widely: i could be anyhing from a pass-hrough o a collaeralized morgage obligaion (CMO). A CMO has a ranched srucure, wih differen paymen rules for he differen ranches. 11

12 Acual loss disribuion Efficien MC Approximaion Loss from defaul for = Loss from defaul for = 1, Loss from defaul for = 2, Loss from defaul for = 5, Figure 2. Comparison of acual disribuion wih approximae disribuion (using boh LL and CLT). Loss repored as fracion of pool which defauled. The horizon is 12 monhs. Time for Brue-force Simulaion Time for LL Time for LL and CLT 1, , , , , 2, ,, 28, Table 1. Comparison of compuaional imes (seconds) for efficien Mone Carlo approximaion and brue-force Mone Carlo simulaion of he pool. pools for deals. We will perform some numerical ess o sudy he efficien Mone Carlo approximaion s accuracy across he wide diversiy of deals in he daa universe. Figure 3 compares he approximae disribuion from he efficien Mone Carlo approximaion wih he rue disribuion (found via brue-force Mone Carlo simulaion) a a ime horizon of 12 monhs for en acual deals in he daa se. The deals are chosen a random and each conains beween 5, and 1, morgages. I is ineresing ha he defaul rae can vary considerably beween deals as a consequence of he qualiy of he underlying morgages, demonsraing how imporan i is for a model o consider he loan-level characerisics of he morgages in he pool. To furher assess he accuracy of he approximaion, a se of deals is seleced a random from he daa se and he approximaion s 99% value a risk (VaR) is compared wih he rue 99% value a risk. In oal, we look a 185 deals and repor he average error of jus he LL by iself as well as he average error for he full efficien Mone Carlo approximaion (LL and CLT combined). In addiion, Figure 4 shows he disribuion of he efficien Mone Carlo approximaion s error across he se of deals. 5, Mone Carlo simulaions are performed and he ime horizon is again welve monhs. For deals wih 5, 1, morgages, he average error for he approximaion (LL and CLT) is.22%. The average error jus using he LL (no 12

13 8 7 Acual loss disribuion Efficien MC Approximaion Loss from defaul Figure 3. Comparison of acual disribuion wih approximae disribuion for en acual deals wih 5, < < 1,. Loss repored as fracion of pool which defauled. The horizon is 12 monhs. CLT) is 2.25%. For deals wih more han 1, morgages, he average error for he approximaion (LL and CLT) is is.18%. The average error jus using he LL is 1.25% MBS pools wih 5, < < 1, Disribuion of error Mean error Percen Error for 99% VaR 12 MBS pools wih > 1, Percen Error for 99% VaR Figure 4. Disribuion across deals of error for 99% VaR from he efficien Mone Carlo approximaion. The ime horizon is 12 monhs Accuracy of Approximaion for Momens. We now sudy he accuracy of he Mone Carlo approximaion for he momens for pools of subprime morgages. In addiion o he loan-level and common facors used in he previous secions, a mean-field erm is also included. The mean-field erm is he pool-level 13

14 defaul rae from he previous monh. Table 2 repors he percen error of he Mone Carlo approximaion for he momens of he pool loss for differen sizes a a T = 12 monh ime horizon. The acual values for he momens are calculaed via brue-force Mone Carlo simulaion wih 5, rials. 5, Mone Carlo rials are also used for he approximaion of he momens. There is lower comparaive accuracy for higherorder momens which represen he non-gaussian characerisics of he finie pool. These non-gaussian characerisics fade as becomes large. Firs Momen Second Momen Third Momen Fourh Momen Fifh Momen , , , Table 2. Percen error for Mone Carlo approximaion for momens of he loss One-dimensional Efficien Mone Carlo Approximaion. So far, we have focused on he case where d W = 2. In his case, one models boh defaul and prepaymen. However, some ypes of loans only have defaul risk and do no have prepaymen risk. In addiion, for very high qualiy morgage pools, defaul risk is small and i may be reasonable o consider only prepaymen risk. Finally, for agency morgage pools, he GSEs insure agains any defaul losses, so prepaymen and defaul can be reaed as he same even. To demonsrae he one-dimensional approach, we fi a logisic regression model only including prepaymen o he agency morgage daa. For prepaymen, we consider a number of loan-level feaures: FICO score, wheher a firs ime homebuyer, he number of unis, occupancy saus (owner occupied, invesmen propery, or second home), combined loan-o-value, loan-o-value, iniial ineres rae for he morgage, deb-o-income raio, wheher here is a prepaymen penaly in he morgage conrac, propery ype (condo, leasehold, PUD, manufacured housing, 1 4 fee simple, Co-op), loan purpose (purchase, cash-ou refinance, or no cash-ou refinance), and number of borrowers. These feaures amoun o 22 dimensions in he feaure space. We also include he meropolian saisical area (MSA); here are 43 meropolian saisical areas in he daa se. Therefore, in oal, d Y = We emphasize ha, even hough d Y = 452, d W only equals 1. The sysemaic facors are he naional unemploymen rae and he naional morgage rae, as above. Figure 5 compares he approximaion wih he acual prepaymen disribuion for a randomly drawn pool of agency morgages for a ime horizon of 12 monhs. 5, Mone Carlo simulaions are used Precompuaion for Financial Insiuions. In Appendix D, we propose o pre-simulae µ on a pre-chosen grid and hen use his one se of simulaions on he single grid in order o find he disribuion for many differen morgage-backed securiies. Using his approach, he efficien Mone Carlo approximaion can provide grea compuaional cos savings even for very small morgage-backed securiies, as long as a financial insiuion is dealing wih many of hese small morgage-backed securiies in aggregae. We now illusrae his approach using he parameer fis from Secion pools, each of size = 2, 5, are drawn a random from he agency morgage daa se. Each of hese pools is simulaed 5, imes using brue-force Mone Carlo simulaion. Using he efficien Mone Carlo approximaion, we also presimulae on a pre-chosen grid (as described in Secion 5.5). 5, Mone Carlo pahs are also used for he efficien Mone Carlo simulaion on his grid. The pre-chosen grid only has 2 grid poins, placed a uniform inervals. The disribuion of µ is smooh in w, so one can numerically inerpolae from he sparse grid poins o ge a finer soluion. We use piecewise cubic spline inerpolaion. 13 This saves compuaional ime since i allows one o simulae on a very sparse grid bu sill ulimaely achieve a very accurae soluion on a fine grid. Figure 6 is a hisogram of he percen error over he 4 pools for he 99% VaR from he efficien 12 The parameer esimaes are available upon reques. 13 An ineresing quesion is wheher one is guaraneed ha he inerpolaed soluion a any poin y converges o he correc value as he number of Mone Carlo samples increases and he disance beween grid poins decreases. If h is coninuously differeniable and X akes values in a compac se, such convergence holds (by he applicaion of a Taylor expansion, boundedness of coninuous funcion on a compac meric space, and he dominaed convergence heorem). More general cases may require addiional echnical condiions. 14

15 .8 8 Acual prepaymen disribuion Efficien MC Approximaion w ("Prepaymen risk space") Toal prepaymen for = Toal prepaymen for = 1, Toal prepaymen for = 2,5 Figure 5. Top righ plo and boom plos compare he acual prepaymen disribuion wih he efficien Mone Carlo approximaion for = 5, = 1,, and = 2, 5. The oal prepaymen is repored as he fracion of he pool which prepays. The ime horizon is 12 monhs. The op lef plo shows he disribuion of he pool in he prepaymen risk space. Mone Carlo approximaion using pre-simulaion. The ime horizon is 12 monhs. The efficien Mone Carlo approximaion is very accurae; he average percen error across he 4 pools for he 99% VaR from he approximaion is only.25%. Brue-force simulaion of he 4 pools akes 36, seconds while he efficien Mone Carlo approximaion of he 4 pools akes 4.72 seconds. The approximaion provides cos savings of nearly 4 orders of magniude versus brue-force simulaion Efficien Approximaion wihou Low-dimensional Transformaion. The previous numerical soluions of he law of large numbers and cenral limi heorem rely upon an exac low-dimensional ransformaion, which requires a resricion on he ineracion beween he facors Y n and V. Alhough ha low-dimensional ransformaion sill covers a wide range of models, for compleeness, we now presen some mehods for he compuaion of he LL and CLT for he full class of models where a high-dimensional feaure space Y may make a radiional Caresian grid infeasible. We le h θ be a funcion of wo neural neworks, one for defauls and one for prepaymens (see Example 2.2). Boh neural neworks have a single hidden layer of five neurons. The loan-level feaures are FICO score, LTV raio, original balance, and iniial ineres rae. The neural nework for defauls also akes as an inpu he naional unemploymen rae while he neural nework for prepaymens akes boh he naional unemploymen rae and he naional morgage rae as inpus. The model is rained on he subprime morgage daa se. We implemen wo mehods for he evaluaion of he LL and CLT for his case where he low-dimensional ransformaion is no applicable. In he firs mehod, we cluser he pool ino K clusers using k-means clusering on Y 1,..., Y. The K cenroids are chosen as he grid poins. This is a sparse non-uniform grid in a high-dimensional space. The second mehod is an approximae low-dimensional ransformaion via an addiional muli-layer neural nework h θ (u, u, y, V ). The muli-layer neural nework has hree layers: he firs (muli-neuron) layer akes as an inpu y, he second layer has d W neurons (where d W d Y ), and he 15

16 Percen error for 99% VaR Figure 6. Disribuion across deals of error for 99% VaR from he efficien Mone Carlo approximaion using pre-simulaion. The ime horizon is 12 monhs. hird layer akes as inpus he oupu of he d W neurons from he second layer as well as V. This mulilayer neural nework is rained o mach he oupu of h θ using he Levenberg-Marquard algorihm. oe ha h θ (u, u, y, V ) saisfies he exac low-dimensional ransformaion and y R d Y can be ransformed ino w R d W. In he case we numerically implemen, we use d W = 1 and he w-space is discreized using k-means clusering. Figure 7 compares he LL and CLT wih he acual disribuion. 1, Mone Carlo rials were performed and K = 5 clusers were used for he sparse grids. The ime horizon is 12 monhs Acual loss disribuion 5 MC Approx. (K means clusering) MC Approx. (Approx. low dimensional ransf.) Loss from Defaul for = Loss from Defaul for = 1, Loss from Defaul for = 2, Loss from Defaul for = 5, Figure 7. Comparison of acual loss disribuion wih Mone Carlo approximaion using sparse grids when h θ is a neural nework. 16

17 6. Conclusion This paper develops an efficien Mone Carlo approximaion for a general class of loan-by-loan defaul and prepaymen models for pools of loans. The approximaion is based upon a law of large numbers and a cenral limi heorem for he pool. We exensively es our approach on acual morgage daa. The approximaion is highly accurae even for relaively small pools (as lile as 5 loans). In pracice, pools commonly range from a few housand o hundreds of housands of loans. Brue-force simulaion for large pools can be compuaionally expensive; our approximaion can save several orders of magniude in compuaional ime. The efficien Mone Carlo approximaion accouns for he full loan-level dynamics, aking advanage of he deailed loan-level informaion ypically available (such as credi score, loan-o-value raio, iniial ineres rae, and ype of loan). A key feaure of our approximaion is ha is compuaional cos is consan no maer he dimension of he loan-level daa. This feaure is desirable since loan-level daa can be high-dimensional. Appendix A. Proofs for LL, CLT, and Uniform Inegrabiliy This appendix proves he LL, CLT, and uniform inegrabiliy. The LL for he empirical measure is proven in A.1. The CLT for he empirical measure is proven in A.2. The LL and CLT for he loss from defaul are proven in A.3. Uniform inegrabiliy is proven in A.4. Exisence and uniqueness for he LL and CLT equaions for he empirical measure are proven in A.5. The proofs of he limi heorems will make use of an auxiliary sysem of paricles (U 1,v,..., U,v ) which are indexed by v = (v,..., v T ) R d V T +1 : (13) where H,v = 1 P[U = u F 1] v = h θ (u, U 1, Y n, v 1, H,v τ: 1 ), = 1,..., T, U = U n, f H (U, Y n ), and F v is he sigma-field generaed by (U 1,v,..., U,v, Y 1,..., Y ). (Y n, v ). The sysem (13) is simply he original,..., U,V has he same disribuion as U 1,..., Similarly, he loss given defaul for he n-h loan is l sysem (1) given a realizaion v of V. In paricular, U 1,V. Define: U (14) µ,v = 1 δ U,Y n. The empirical measures µ,v and µ have he same disribuion. In addiion, µ,v = µ for any v. We prove limiing laws for µ,v = (µ,v, µ,v 1,..., µ,v T ) for any v = (v,..., v T ) R d V T +1, which in urn will yield he desired limiing laws for µ = (µ, µ 1,..., µ T ). A.1. Law of Large umbers (Theorem 3.2). Le P(E) be he space of measures on a complee separable space E. The opology for P(E) is he opology of weak convergence, which is merized by he Prokhorov meric. A random variable X P(E) iself has a probabiliy measure P P(P(E)). The measure P for he random variable X weakly converges o he measure P of a random variable X if: (15) E[Φ(X )] E[Φ( X)], for every Φ S where S is he class of funcions which separaes P(E). In our case, E = U Y. We le S : P(U Y) R be he collecion of funcions of he form Φ(µ) = φ 1 (f 1, µ E,..., f M, µ E ) for φ 1 C(R M ), f m C(E). (Recall ha any funcion on a discree meric space is coninuous.) Then, S separaes P(U Y) and is a convergence deermining class for P(P(E)) (see [48] or [23]). Recall ha we have defined f, ν E = E f(x)ν(dx). For insance, if E = U Y, hen we have f, ν U Y = u U Y f(u, y)ν(u, dy). Also, for noaional convenience, define f, ν(u, dy) = f(u, ), ν(u, ) Y which is equal o f(u, y)ν(u, dy). Y We prove several facs, culminaing in he law of large numbers µ d µ. The proof uses an inducion argumen and he convergence deermining class S. 17

18 Lemma A.1. Define M 1,,v (u) o be he maringale (16) M 1,,v (u) = 1 φ(y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )), where φ C(R d Y ). Then, for each u U and any v R d V T +1, we have M 1,,v (u) p. Proof. The variance of M 1,,v (u) converges o zero: (17) Var[M 1,,v (u)] = E[( 1 = φ(y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 ))])2 ] E[(φ(Y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )))2 ] C. The firs and second equaliies use he ower propery and he independence of he processes U condiional on F 1. v The fourh equaliy uses he fac ha φ(y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )) is bounded. This follows from he facs ha φ is a coninuous funcion, Y n Y where Y is compac, and h θ is a probabiliy ransiion funcion (and so is bounded). Recall ha a coninuous funcion on a compac space is bounded. By Chebyshev s inequaliy, M 1,,v (u) p. Of course, if M 1,,v (u) p for each, u, M 1,,v = {M 1,,v p (u)} u=1: U,=1:T {}u=1: U,=1:T. Proof of Theorem 3.2. For each u U and any φ C b (R d Y ), we have ha φ, µ,v (u, dy) = (18) φ(y)h θ (u, u, y, V 1, H,v τ: 1 ), µ,v 1 (u, dy) + M 1,,v (u). Define µ v = E[ µ V = v] and H v = E[ H V = v]. oe ha µ V = µ and H V = H since µ v and H v are simply µ and H condiional on V = v: (19) µ v (u, dy) = h θ (u, u, y, v 1, H v τ: 1) µ v 1(u, dy), H v = f H, µ v U Y µ v = µ. We will now use an inducion argumen o prove he law of large numbers for µ,v = (µ,v,..., µ,v T ). Assuming µ,v p : 1 µ v : 1, we have ha for any v: lim φ, µ,v (u, dy) = (2) φ(y)hθ (u, u, y, v 1, H τ: 1), v µ v 1(u, dy) = φ, µ v (u, dy). The resul (2) is a consequence of he following facs. Firs, M 1,,v p by Lemma A.1. Secondly, since µ,v p : 1 µ v : 1, H,v p τ: 1 H τ: 1 v (f H is coninuous by assumpion). Thirdly, φ(y)h θ (u, u, y, v 1, H,v p (21) τ: 1 ), µ,v 1 (u, dy) φ(y)hθ (u, u, y, v 1, H τ: 1), v µ v 1(u, dy). This las fac follows from: φ(y)h θ (u, u, y, v, H,v (22) + τ: 1 ), µ,v 1 (u, dy) = φ(y)[h θ (u, u, y, v 1, H,v τ: 1 ) h θ(u, u, y, v 1, H τ: 1)], v µ,v 1 (u, dy). } {{ } ( ) 18 φ(y)h θ (u, u, y, v 1, H v τ: 1), µ,v 1 (u, dy)

19 The funcion h θ is a coninuous funcion, by assumpion. Since Y is compac and H,v τ: 1 is bounded (due o he assumpion ha f H is coninuous and U Y is compac), h θ is in fac uniformly coninuous on he space ha is argumens live on (a coninuous funcion a compac space is uniformly coninuous). Therefore: ( ) sup y Y p φ(y) sup y Y,u U, h θ (u, u, y, V 1, H,v τ: 1 ) h θ(u, u, y, v 1, H τ: 1) v p, since H,v τ: 1 H τ: 1 v and he desired resul (21) holds. By Lemma A.8, µ v from (4) is he unique measure which saisfies (2). Le Φ S where S is he convergence deermining class specified earlier. Then, for any Φ S, lim E[Φ(µ,v )] = lim E[φ 1( = lim E[φ 1( u U = E[Φ( µ v )]. f 1, µ,v B,..., f 1 (u, y), µ,v f M, µ,v (u, dy) B )],..., u U f M (u, y), µ,v (u, dy) )] The above resul follows from equaion (2), he coninuous mapping heorem, P(U Y) being compac (and hus any coninuous funcion of µ P(U Y) is bounded), and he dominaed convergence heorem. Therefore, for any v, µ,v d µ v and, since µ v is deerminisic, µ,v also converges in probabiliy o µ v. This also means ha µ,v p : µ v : for any v. oe ha in Assumpion 3.1, µ,v converges in disribuion o µ, where µ is deerminisic. By Assumpion 3.1 and inducion, i follows ha µ,v converges in probabiliy o µ v for any v. Since he convergence in probabiliy holds for any v, we cerainly have ha µ converges in disribuion o µ, since: (23) lim E[f(µ )] = lim E[f(µ,V )] = lim E[E[f(µ,V ) V ]] = E[ lim E[f(µ,V ) V ]] = E[E[f( µ V ) V ]] = E[f( µ V )] = E[f( µ)], for any coninuous bounded f : P(U R d Y ) R. We have used above he ower propery, dominaed convergence, and previous convergence resuls for µ,v. A.2. Cenral Limi Theorem (Theorem 3.4). Le S be he space of Schwarz funcions, and le is opological dual S = S (R d Y ) be he space of empered disribuions on R d Y (equipped wih he weak opology). S is a nuclear Fréche space and herefore reflexive (see [44]). A random variable X S iself has a probabiliy measure P P(S ). The measure P for he random variable X weakly converges o he measure P of a random variable X if: (24) for every φ S. 14 Of course, (24) is implied by: (25) E[exp ( iαφ(x ) ) ] E[exp ( iαφ( X) ) ], φ(x ) d φ( X), for every φ S. S, he space of Schwarz funcions, is defined as: S(R d Y ) = {φ C (R d Y ) : φ α,β < α, β }, φ α,β = sup y α D β φ(y). y R d Y The requiremen ha φ be rapidly decreasing for large y does no have an impac on our seing since Y is assumed o be compac. 14 This resul was originally proven for boh he space of empered disribuions S and he space of disribuions D in [21]. The resul was also independenly proven by [5] (Proposiion 3.3 and corresponding Supplemenary Commens on pg. 247), [41], and [27]. Finally, such resuls have been exended o coninuous-ime processes aking values in S and D, see [38] and [22]. See [3] for a similar resul for separable Banach spaces. 19

20 We wish o show ha Ξ (u, dy) S (R d Y ) for each u. To prove convergence in disribuion of Ξ (u, dy) for each u, one needs o show (24) or (25) for each φ S: φ(y), Ξ (u, dy) d (26) φ(y), Ξ(u, dy), where, as previously defined, f, ν(u, dy) = Y f(y)ν(u, dy). oe ha since he Y n Y R d Y, f, Ξ (u, dy) = Y f(y)ξ (u, dy) = R d Y f(y)ξ (u, dy) (and similarly for µ, µ, and Ξ). Since he dual of he caresian produc of finiely many spaces is he caresian produc of he duals, he crierion (24) or (25) also of course covers he case of Ξ T U i=1 S, where i is sufficien o show (24) for φ T U i=1 S, or more simply convergence in disribuion for he random vecor ( φ,u, Ξ (u, dy) ) I,u U for every ( ) φ,u I,u U T U i=1 S. Define: (27) Ξ,v = (µ,v µ v ), and noe ha Ξ = Ξ,V. We also define Ξ v : Ξ v (u, dy) = h θ (u, u, y, v 1, H τ: 1) Ξ v v 1(u, dy) u U + ( H h θ(u, u, y, v 1, H τ: 1) v Ēv v τ: 1 ) µ 1 (u, dy) + M v (u, dy), (28) Ξ v = Ξ. where E,v = f H, Ξ,v, Ēv = f H, Ξ v, and U Y U Y Ēv τ: 1 = (Ēv τ,..., Ēv 1). Mv (u, dy) is a Gaussian process wih zero mean and covariance: [ Cov Mv (u 1, dy), Mv (u 2, dy)] = h θ (u 1, u, y, v 1, H τ: 1)h v θ (u 2, u, y, v 1, H τ: 1) µ v v 1(u, dy), [ Var Mv (u, dy)] = h θ (u, u, y, v 1, H τ: 1)(1 v h θ (u, u, y, v 1, H τ: 1) µ v v 1(u, dy), where u 1 u 2. Mv 2 is independen of Mv 1 for 2 1 and M v (u, dy 1 ) is independen of Mv (u, dy 2 ) for y 1 y 2. Ξ v is Ξ condiional on V = v; Ξ V = Ξ. Lemma A.2. For every u and, Ξ,v (u, dy) S (R d Y ) for any v. Proof. X S if for any φ S here exiss a k, C k such ha (29) where he norm k is defined as: (3) φ k = Then, we cerainly have ha: φ, Ξ (31) = ( φ, µ,v This complees he proof. φ, X C k φ k, max sup α + β k y R d Y y α D β φ(y). φ, µ v ) 2 sup φ(y) 2 φ. y Y Lemma A.3. For any φ S, α, and v, he maringale difference array 15 Z, (α) = 1 α u [φ(y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )] u U saisfies he following properies for every α R U : (i) sup E[ ( max n Z, (α) ) 2 ] is uniformly bounded, 15 See [37] for deails on maringale difference arrays. 2

21 (ii) max n Z, (iii) For each v, (α) p, ( Z, (α) ) 2 p Var[ α u φ, Mv (u, dy) ]. Proof. Firs, recognize ha: Z, (α) 2 u U u U α u φ u (Y n ) 2 u U α u sup φ u (y) y Y C. This cerainly implies properies (i) and (ii). Finally, for each v, (Z, (α)) 2 = 1 α u φ u (Y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )]2 (32) Here, M 3,,v M 3,,v = 1 = 1 [ u U u,u U α u α u φ u (Y n )φ u (Y n )[1 U =u1 U =u + h θ (u, U 1, Y n, v 1, H,v τ: 1 )h θ(u, U 1, Y n, v 1, H,v τ: 1 ) = 1 = p 1 U =u h θ(u, U 1, Y n, v 1, H,v u,u U τ: 1 ) 1 U =uh θ (u, U 1, Y n, v 1, H,v τ: 1 )] α u α u φ u (Y n )φ u (Y n )[1 u=u h θ (u, U 1, Y n, v 1, H,v τ: 1 ) h θ (u, U 1, Y n, v 1, H,v τ: 1 )h θ(u, U 1, Y n, v 1, H,v τ: 1 )] + M3,,v α u α u [1 u=u φ u (y)φ u (y)h θ (u, u, y, v 1, H,v τ: 1 ), µ,v u,u U u U φ u (y)φ u (y)h θ (u, u, y, v 1, H,v τ: 1 )h θ(u, u, y, v 1, H,v τ: 1 α u α u [1 u=u φu (y)φ u (y)h θ (u, u, y, v 1, H τ: 1), v µ v 1 u,u U u U 1 ), µ,v 1 φ u (y)φ u (y)h θ (u, u, y, v 1, H v τ: 1)h θ (u, u, y, v 1, H v τ: 1), µ v 1 ]. is he remainder erm: u,u U α u α u φ u (Y n )φ u (Y n )[1 U 1 U =u h θ(u, U 1, Y n, v 1, H,v τ: 1 ) 1 U ] + M 3,,v =u1 U =u 1 u=u h θ(u, U 1, Y n, v 1, H,v τ: 1 ) =uh θ (u, U n 1, Y n, v 1, H,v τ: 1 ) + 2h θ (u, U n 1, Y n, v 1, H,v τ: 1 )h θ(u, U 1, Y n, v 1, H,v τ: 1 )]. This is of he same form as M 1,,v, and he exac same procedure as used in Lemma A.1 can be applied here o show ha M 3,,v p Using he exac same argumen as was employed in he proof of Theorem 3.2, he final line in equaion (32) follows from µ,v p µ v for any v and he uniform coninuiy of h θ on he compac se Y (a coninuous funcion on a compac se is uniformly coninuous). Finally, noe ha he las line is exacly he covariance of Mv, whose disribuion is given in Lemma A.4. Lemma A.4. Le M 2,,v (33) be: M 2,,v (u, dy) = 1 d [δ (U,Y n )(u, dy) h θ (u, U 1, Y n, v 1, H,v τ: 1 )δ Y n(dy)]. Then, M 2,,v M v where M v is a mean-zero Gaussian wih covariance: φ1 Cov[, Mv (u 1, dy), φ 2, Mv (u 2, dy) ] [ φ1 = Cov, Mv (u 1, dy), φ 2, Mv (u 2, dy) ] 21

22 = φ1 (y)φ 2 (y)h θ (u 1, u, y, v 1, H τ: 1)h v θ (u 2, u, y, v 1, H τ: 1), v µ v 1, φ, V ar[ Mv (u, dy) ] [ φ, = V ar Mv (u, dy) ] = φ(y) 2 h θ (u, u, y, v 1, H 1)(1 v h θ (u, u, y, v 1, H τ: 1), v µ v 1, where u 1 u 2. Furhermore, φ, Mv 1 is independen of φ, Mv 2 for 1 2. Proof. We firs show ha: ( φ 1, M 2,,v (u 1, dy),..., φ U, M 2,,v for any φ 1,..., φ U S. oe ha: α u φ u, M 2,,v (u, dy) = 1 u U u U (u U, dy) ) d ( φ 1, Mv (u 1, dy),..., φ U, Mv (u U, dy) ), α u [φ u (Y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 )] = Z,(α). Lemma A.3 shows ha he condiions of he maringale cenral limi heorem of [37] hold. The maringale cenral limi heorem of [37], in conjuncion wih he Cramer-Wold heorem, gives: ( φ 1, M 2,,v (u 1, dy),..., φ U, M 2,,v (u U, dy) ) d ( φ 1, Mv (u 1, dy),..., φ U, Mv (u U, dy) ), for any φ 1,..., φ U S. Furhermore, φ 1, Mv 1 is independen of φ2, Mv 2 for 1 2. To see his, assume 1 < 2, and firs apply he ower propery o wrie: E[exp ( iα 1 (34) φ, M 2,,v 1 (u, dy) + iα 2 φ, M 2,,v 2 = E [ exp ( )E[exp iα 1 φ 1, M 2,,v ( (u, dy) iα2 1 (u, dy) ] φ 2, M 2,,v 2 ex, we can express he inner expecaion as a produc of exponenials: E[exp ( ) µ iα 2 φ 2, M 2,,v (u,v, dy) = = = iα 2 E[e 2 φ 2(Y n )(1 U 2 (1 α2 2 2 E[φ 2(Y n ) 2 (1 U 2 (1 α2 2 2 : 2 1 ] ) µ (u,v, dy) : ]] 2 1. =u h θ(u,u 2 1,Y n,v 2 1,H,v 2 τ: 2 1 )) H,v 2 τ: 2 1, U 2 1, Y n ] γ(v, H,v 2 τ: 2 1, U 2 1, Y n ) + =u h θ(u, U 2 1, Y n, v 2 1, H,v 2 τ: 2 1 ))2 H,v 2 τ: 2 1, U 2 1, Y n ] + C 3/2, ) C ), 3/2 where we have used a Taylor expansion for he exponenial, sup y Y 2φ 2 (y) 3 < C <, and he quaniy γ(v, H,v, U 2 τ: 2 1, Y n 2 1 ) = E[φ 2 (Y n ) 2 (1 U =u h 2 θ(u, U, Y n 2 1, v 2 1, H,v 2 τ: 2 1 ))2 H,v, U 2 τ: 2 1, Y n 2 1 ]. Using he sandard inequaliy w n z n w n z n for w n, z n 1, we have ha: (1 α2 2 γ(v, H,v 2, U 2 τ: 2 1, Y n 2 1 ) + C ) (1 1 3/2 2 α2 2γ(v, H,v, U 2 τ: 2 1, Y n 2 1 )) (35) < C 1/2 a.s.. Then, all he remains o be shown is ha (1 1 2 α2 2γ(v, H,v, U 2 τ: 2 1, Y n 2 1 )) converges in probabiliy o exp( ) as. We have α 2 2 2Var[φ 2, M v (u,dy)] 2 (1 1 2 α2 2γ(v, H,v, U 2 τ: 2 1, Y n 2 1 )) = exp( log(1 1 2 α2 2γ(v, H,v, U 2 τ: 2 1, Y n 2 1 ))) 22

23 = exp( [ 1 2 α2 2γ(V, H 2 τ: 2 1, U n 2 1, Y n ) + C 2 ]) = exp( 1 2 α2 2 γ(v, H,v, 2 τ: 2 1 u, y), µ,v 2 1 (u, dy) + C ) p exp( 1 2 α2 2Var[ φ 2, Mv 2 ]) The las line follows from he previous convergence resul µ,v p µ v (see Proof of Theorem 3.2), he fac ha γ(v, H,v, U 2 τ: 2 1, Y n 2 1 ) = φ 2 (Y n ) 2 h θ (u, U, Y n 2 1, v 2 1, H v 2 τ: 2 1)(1 h θ (u, U, Y n 2 1, v 2 1, H v 2 τ: 2 1)), and he coninuous mapping heorem. Wih his resul in hand, we reurn o equaion (34). Since E[exp ( ) µ iα 2 φ 2, M 2,,v 2 (u,v ) : ] converges o a consan and exp ( ) 2 1 iα 1 φ 1, M 2,,v d ( ) 1 (u) exp iα1 φ1, Mv 1, Slusky s heorem yields ha exp ( )E[exp iα 1 φ 1, M 2,,v ( (u, dy) iα2 1 φ 2, M 2,,v 2 ) µ (u,v, dy) : 2 1 ] converges in disribuion o he quaniy exp ( ) iα 1 φ1, Mv 1 exp( 1 2 α2 2Var[ φ 2, Mv 2 ]). By he bounded convergence heorem, we hen have ha: E[exp ( )] iα 1 φ, M 2,,v (u, dy) + iα 2 φ, M 2,,v 2 (u, dy) exp( 1 2 α2 1Var[ φ 1, Mv 1 ]) exp( 1 2 α2 2Var[ φ 2, Mv 2 ]) Consequenly, φ 1, Mv 1 is independen of φ2, Mv 2 for Since hese resuls hold for any choice of φ S, we also have he sronger resul ha M 2,,v d M v, using he crierion (24). Lemma A.5. Suppose ha Ξ,v d : 1 Ξ v : 1. Then, (Ξ,v : 1, M2,,v ) d ( Ξ v : 1, Mv (u, dy)). Proof. The proof for his lemma is he same as in Lemma A.4. amely, we show ha Ξ v : 1 and are independen. For <, E[exp ( )] i φ 1,u,, Ξ,v (u, dy) + i φ 2, M 2,,v (u, dy) (36) = E [ exp ( i <,u U <,u U φ 1,u,, Ξ,v (u, dy) )E[exp ( i φ 2, M 2,,v ow, using he exac same seps as in Lemma A.4, he desired resul can be obained. ) µ (u,v, dy) : 1 ]]. Proof of Theorem 3.4. For any u U, φ S, and v, we have ha φ(y), Ξ,v (u, dy) = [ φ(y), h θ (u, u, y, v 1, H,v τ: 1 )µ,v 1 (u, dy) M v (u, dy) (37) φ(y), h θ (u, u, y, v 1, H τ: 1) µ v v 1(u, dy) ] + φ, M 2,,v (u), where M 2,,v (38) saisfies: φ, M 2,,v (u, dy) = 1 φ(y n )(1 U =u h θ (u, U 1, Y n, v 1, H,v τ: 1 ). Using a Taylor expansion, one can obain φ(y), Ξ,v (u, dy) = φ(y)hθ (u, u, y, v 1, H τ: 1), v Ξ 1(u, dy) + φ(y)h θ,h (u, u, y, v 1, H τ: 1) v E,v τ: 1, µ,v 1 (u, dy) 1 (39) + R,v + φ, M 2,,v (u, dy), 23

24 where R,v is he Taylor remainder erm: R,v = 1 φ(y)(e,v τ: 1 2 ) h HH (u, u, y, v 1, H τ: 1)E v,v τ: 1, µ,v 1 (u, dy) 1 sup φ(y)(e,v τ: 1 2 ) h HH (u, u, y, v 1, H τ: 1)E v,v τ: 1, µ,v 1 (u, dy) y Y K E,v τ: 1 sup 2 φ(y)h HH (u, u, y, v 1, H τ: 1), v µ,v 1 (u, dy) y Y (4) E,v τ: 1 2 C, µ,v 1 (u, dy) = C E,v τ: 1 2. Here, h HH denoes he second parial derivaive of h θ wih respec o is fifh argumen. We have bounded h HH (u, u, y, v 1, H τ: 1) v due o he coninuous differeniabiliy of h θ and he compacness of Y. Similarly, φ(y) is bounded due o he coninuiy of φ and he compacness of Y. Since 2 is a coninuous funcion, we have by he coninuous mapping heorem ha E,v τ: 1 2 d Ēv τ: Consequenly, R,v d. We will now use an inducion argumen o prove he desired resul. Assume ha Ξ,v : 1 converges in disribuion o Ξ v : 1. Since convergence in disribuion o a consan implies convergence in probabiliy 1 o a consan, R,v converges in probabiliy o zero. The second erm in equaion (39) also converges in disribuion since µ,v 1 converges in probabiliy o µv 1 for any v (from he law of large numbers) and E,v d τ: Ēv τ: 1 (since we assumed Ξ,v d : 1 Ξ v : 1 and f H is coninuous). The hird erm converges in disribuion due o Lemma A.4. Then, assuming Ξ,v : 1 converges in disribuion o Ξ v : 1, φ, Ξ,v d φ, Ξv for any φ S where φ, Ξv saisfies he evoluion equaion: φ(y), Ξv (u, dy) = φ(y)hθ (u, u, y, v 1, H τ: 1 ), Ξ v 1(u, dy) (41) + φ(y) H h θ(u, u, y, v 1, H τ: 1) v Ēv τ: 1, µ v 1 + φ, Mv (u, dy), φ S. This is sufficien o show ha he Ξ,v d d Ξ v, by he crierion (24). However, o make he sronger sae- Ξ v :, we need an addiional fac, namely ha (Ξ,v men of he join convergence Ξ,v : : 1, M2,,v ) d ( Ξ v :, Mv (u, dy)). This has been proven in Lemma A.5, and herefore we have he desired resul ha Ξ,v : d Ξ v :. By inducion, we have ha Ξ,v d Ξ v for any v. Therefore, Ξ d Ξ. Proof of Corollary 3.7. Since (µ,v, L,v p ) ( µ v, L v ) where ( µ v, L v ) are consans and (Ξ,v, Λ,v d ) ( Ξ v, Λ v ), we have ha (v, µ,v, L,v, Ξ,v, Λ,v ) d (v, µ v, L v, Ξ v, Λ v ). Since his holds for any v, he desired convergence follows. A.3. Convergence for Loss From Defaul (Proposiion 3.6). Lemma A.6. Define Z,v o be he maringale Z,v = 1 1 l n (Y n, v )(1 U =d 1 U 1 =d ) where d is he defaul sae. Then, for any v, Z,v Proof. Firs, rewrie Z,v Z,v = 1 as: p. l n (Y n, v )(1 U =d 1 U 1 =d ) 1 24 zν,y,v (dz), µ,v (1 U =d 1 U 1 =d ) (d, dy) µ,v (d, dy), 1 zν,y n,v (dz).

25 Similar o Lemma A.1, we will show ha he variance of Z,v ends o zero as. Var[Z,v ] = E[E[( 1 (1 U =d 1 U 1 =d )(l n (Y n, v ) 1 E[ 1 2 zν,y n,v (dz))) 2 U 1,v,..., U,v, U 1,v 1,..., U,v 1, Y 1,..., Y ]] E[(l n (Y n, V ) 1 zν,y n,v (dz)) 2 U 1,v,..., U,v, Y 1,..., Y ]] C. We have again used he ower propery, he independence of he l n (Y n, v ) given (U 1,v,..., U,v, Y 1,..., Y ), and he fac ha he loss given defaul is bounded. By Chebyshev s inequaliy, Z,v p for any v. Using Lemma A.6, we can show ha L d L. Define L v = E[ L V = v] and noe ha L V = L. We also define: 1 (42) L,v = zν,y,v (dz)µ,v (d, dy) µ,v 1 (d, dy) + Z,v. For any v, Z,v and bounded for each y, v: 1 (43) p by Lemma A.6. oe ha L,V = L. Since g 1 (, v, y) = 1 zν,y,v(dz) is coninuous zν,y,v (dz), µ,v (d, dy) µ,v 1 (d, dy) 1 p zν,y,v (dz), µ v (d, dy) µ v 1(d, dy), for any v. This proves L,v p L v for each and any v, which (like before) implies ha (L,v,..., L,v T ) p ( L v,..., L v T ) for any v. By similar reasoning as in (23), his implies ha (µ, L ) d ( µ, L). We now urn o proving he CLT for he loss. Define Λ v : 1 (44) Λ v = zν,y,v (dz), Ξ v (d, dy) Ξ v 1(d, dy) + Z v, where Z v is a mean-zero Gaussian wih variance: Var[ Z v ] = 1 [ 1 z 2 ν,y,v (dz) ( zν,y,v (dz)) 2 ], µ v (d, dy) µ v 1(d, dy). Z v 1 is independen of Zv 2 for 1 2 as well as Ξ v. Lemma A.7. Suppose ha (Ξ,v :, Z,v : 1 ) d ( Ξ v :, Z : 1). v Then, ({Ξ,v : }, Z,v : ) d ( Ξ v :, Z :). v Z v is a mean zero Gaussian wih variance: 1 1 (45) Var[ Z v ] = Var[ Z v ] = z 2 ν,y,v (dz) ( zν,y,v (dz)) 2, µ v (d, dy) µ v 1(d, dy). Z v is independen of Zv : 1 and Ξ v :. Proof. We have (46) E[exp ( i = E [ exp ( i,u U,u U φ 1,u,, Ξ,v (u, dy) + i α Z,v + iα Z,v )] < φ 1,u,, Ξ,v (u, dy) + i α Z,v) ( E[exp iα Z,v ) ) µ :] ]. < The resul follows via he exac same procedure as used in Lemma A.4. (47) Define: Λ,v = (L,v L v ). 25

26 Using Lemma A.7, we have ha (Ξ,v, Λ,v ) d ( Ξ v, Λ v ) since: 1 Λ,v = zν,y,v (dz), Ξ,v (d, dy) Ξ,v d 1 (d, dy) 1 zν,y,v (dz), Ξ v (d, dy) Ξ v 1(d, dy) + Z,v + Z v = Λ v. This resul follows since Ξ,v d Ξ v, he assumpion of coninuiy in y for 1 zν,y,v (dz) for any, v, and Lemma A.7. Since he resuls holds for any v, (Ξ, Λ ) d ( Ξ, Λ). A.4. Uniform Inegrabiliy of Flucuaions (Theorem 3.9). Proof of Theorem 3.9. oe ha he bound (1) implies [ P (sup φ(y) ) 1 φ(y), Ξ (u, dy) ] > α K 1,,u exp ( K 2,,u α 2). y Y Consider he following bound for α > : [ T (48) P φ,u (y), Ξ (u, dy) + =1 u U For noaional convenience, define X = T holds and K > C, hen: [ g(x E ) ] 1 g(x ) = >K K K K =1 T c Λ ) ] > α C 1 exp ( C 2 α 2). =1 u U [ g(x P ) ] α ] P[ X k α φ,u (y), Ξ (u, dy) + T =1 c Λ. If he bound (48) dα dα = C 1 exp ( C 2 α 2/k) dα = C 1 exp ( C 2 x ) x k 1 K K k dx C 1 [ X P α 1/k] dα = C 1 k Ck 2 Γ(k), where Γ is he gamma funcion. Then, for any ɛ >, here is a K > such ha: exp ( C 2 x ) x k 1 k dx g(x E[ ) ] 1 X < ɛ. >K [ X Consequenly, he bound (48), P ] > α C 1 exp ( C 2 α 2) for α >, implies uniform inegrabiliy for g(x ). ex, recognize ha he following bound implies he bound (48): [ P φ,u(y), Ξ (u, dy) ] (49) + c Λ > α C 1,,u exp ( C 2,,u α 2). The bound (49) implies he bound (48) since: [ T P φ,u (y), Ξ (u, dy) + =1 u U T c Λ =1 [ P φ,u (y), Ξ (u, dy) + c Λ u [ P { φ,u (y), Ξ (u, dy) + c Λ,u 26 > ] > α ] > α α ] U T }

27 =,u,u,u [ P { φ,u (y), Ξ (u, dy) + c Λ [ P φ,u (y), Ξ (u, dy) + c Λ C 1,,u exp ( C 2,,u α 2) > > ] α U T } α U T T U max C 1,,u exp ( min C 2,,uα 2) = C 1 exp ( C 2 α 2).,u,u I is herefore sufficien o prove an exponenially decaying bound: [ P φ,u(y), Ξ (u, dy) ] (5) + c Λ > α C 1,,u exp ( C 2,,u α 2), for each and u. We prove his resul using inducion. From equaion (39): φ(y), Ξ (u, dy) = φ(y)hθ (u, u, y, V 1, H τ: 1 ), Ξ 1(u, dy) + φ(y)hθ,h (u, u, y, V 1, H τ: 1 ) E τ: 1, µ 1(u, dy) ] (51) + 1 R,V + φ, M 2,,V (u, dy), where R,V C supy Y φ(y) E τ: 1 2 (see equaion (4)) and h θ,h is he gradien of h θ wih respec o H. M 2,,V (u, dy) is defined in Lemma A.4. Firs, suppose ha: [ P (sup φ(y) ) 1 φ(y), Ξ (u, dy) ] (52) > α V K 1,,u exp ( K 2,,uα 2), y Y for any coninuous φ and all 1. Here and below, we use he noaion V = V :T. This is equivalen o P[ φ(y), Ξ (u, dy) ] > α V C 1,,u exp ( C 2,,uα 2) where C 2,,u = K 2,,u(sup y Y φ(y) ) 2 and K 1,,u = C 1,,u. Since φh θ is coninuous on a compac space, we also have ha he firs erm of he RHS of (51) saisfies: [ P φ(y)hθ (u, u, y, V 1, H τ: 1 ), Ξ 1(u, dy) ] > α V [ P φ(y)hθ (u, u, y, V 1, H τ: 1), Ξ 1(u, dy) > α ] U V (53) K 1, 1,u exp ( K 2, 1,u (sup φ(y)h(u, u, y, V 1, H 2 α2 ) τ: 1 ) ), u y Y U 2 U U max K u 1, 1,u U exp ( (sup φ(y) ) 2 min K α 2 ) u 2, 1,u U U 2 y Y C1,,u 1 exp ( (sup φ(y) ) 2 C2,,uα 1 2) y Y where he second-o-las inequaliy uses he fac ha h θ 1. oe ha he final coefficiens do no depend upon V nor φ. Recall ha E = f H, Ξ where f H is coninuous. Then, E has an exponenial bound, which allows one o exponenially bound he second erm of he RHS of (51). [ P φ(y)hθ,h (u, u, y, V 1, H τ: 1 ) E τ: 1, µ 1(u, dy) ] > α V 27

28 (54) [ P [ P [ P sup (y,v) Y V sup y Y sup y Y φ(y) φ(y) K φ(y)h θ,h (u, u, y, V 1, H τ: 1 ) sup (y,v) Y V 1 = τ 1 = τ h θ,h (u, u, y, V 1, H τ: 1 ) ] E > α V ] E > α V 1 = τ ] E > α V C1,,u 2 exp ( (sup φ(y) ) 2 C2,,uα 2 2), y Y where sup (y,v) Y V h θ,h(u, u, y, V 1, H τ: 1 ) K, a consan ha does no depend upon V nor φ, since Y V is compac and h is smooh. Similarly, he hird erm on he RHS of (51) can also be exponenially bounded by C1,,u 3 exp ( (sup y Y φ(y) ) 2 C2,,uα 3 2). The fourh erm on he RHS of (51) can be exponenially bounded via he Azuma-Hoeffding inequaliy. Define S,n = 1 n n =1 φ(y n )[1 U n h θ (u, U 1, n Y n, V 1, H τ: 1)] for n = 1,...,. Condiional on F 1, he sequence S,1, S,2,... is a maringale and has bounded differences: S,n S,n 1 = 1 φ(y n )[1 U n u h θ (u, U 1, n Y n, V 1, H τ: 1)] 2 (55) sup φ(y). Therefore, condiional on F 1, S,n saisfies he Azuma-Hoeffding inequaliy. oe ha S, = Then, by he Azuma-Hoeffding inequaliy: [ P φ(y), M 2,,V (u, dy) > α ] [ [ V = E P φ(y), M 2,,V (u, dy) > α ] ] F 1 V [ 2E exp( 1 ] 4 (sup φ(y) 2 ) 2 α 2 ) V y Y = 2 exp( 1 8 (sup φ(y) ) 2 α 2 ). y Y y Y u φ(y), M 2,,V Combining he exponenial bounds for he four erms on he RHS of (51) and applying he sandard union bound, we hen have a bound for he LHS of (51): [ P φ(y), Ξ (u, dy) ] (56) > α V C 1,,u exp ( C 2,,u sup φ(y) ) 2 α 2), y Y where C 1,,u and C 2,,u do no depend upon V nor φ. This of course implies P[ φ(y), Ξ (u, dy) > α ] C 1,,u exp ( C 2,,u sup y Y φ(y) ) 2 α 2). I also proves he inducive sep: [ P (sup φ(y) ) 1 φ(y), Ξ (u, dy) ] [ > α V = P φ(y), Ξ (u, dy) ] > (sup φ(y) )α V y Y y Y (57) = C 1,,u exp ( C 2,,u α 2) K 1,,u exp ( K 2,,u α 2). [ Since P (sup y Y φ(y) ) 1 φ(y), Ξ (u, dy) ] > α V K 1,,u exp ( K 2,,u α 2) by assumpion, his complees he inducion proof o show ha P[ φ,u (y), Ξ (u, dy) > α ] C 1,,u exp ( C 2,,u α 2) for each and u. The final sep requires o proving a similar bound for c Λ. Recall ha: Λ = 1 (58) (L L ) = zν,y,v (dz), Ξ (d, dy) Ξ 1(d, dy) + Z,V. Le s firs consider he firs erm on he RHS of (58). The funcion 1 zν,y,v (dz) is coninuous on y (by assumpion) and sup y Y 1 zν,y,v (dz) 1 since ν is a measure on [, 1]. Using he (now proven) bound 28 (u, dy).

29 P[ φ(y), Ξ (u, dy) ] > α V C 1,,u exp ( C 2,,u α 2) and he sandard union bound, we have ha: (59) [ 1 ] P zν,y,v (dz), Ξ (d, dy) Ξ 1(d, dy) α V A 1,,u exp ( A 2,,u α 2). ex, le s consider he second erm on he RHS of (58), Z,V [. Define Q,n = 1 n n =1 ln (Y n, V )(1 U n 1 U n =d) n 1 n =1 (1 U n =d 1 U 1 n =d) 1 zν,y n,v (dz) ]. Condiional on F, he sequence Q,1, Q,2,... is a maringale. The sequence also has bounded differences since: (6) Q,n Q,n 1 1 = [l n (Y n, V )(1 U n =d 1 U 1 n =d ) (1 U n =d 1 U n 1 =d) 2. 1 zν,y n,v (dz)] Therefore, condiional on F, Q,n saisfies he Azuma-Hoeffding inequaliy. oe ha Q, = Z,V. Then, by he Azuma-Hoeffding inequaliy: [ ] [ [ ] P Z,V > α V = E P Z,V > α ] F V [ E 2 exp( 1 ] 8 α2 ) V =d (61) = 2 exp( 1 8 α2 ). Combining he bounds (59) and (61), we have by he sandard union bound ha: [ ] (62) P c Λ α V B 1,,u exp ( B 2,,u α 2). Finally, again using he union bound, bounds (62) and (56) can be combined o prove he bound (5). A.5. Exisence and Uniqueness for LL and CLT. Lemma A.8. The soluion µ B T +1 o he law of large numbers equaion in Theorem 3.2 is unique, where B = P(U R d Y ). Proof. Recall ha µ v is he soluion o equaion (19), which is he LL (4) condiional on V = v. Suppose ha µ v : 1 is unique; hen, i can be proved by conradicion ha µ v mus be unique as well. Suppose here are differen soluions µ 1,v and µ 2,v, and le ν v = µ 1,v µ 2,v. From equaion (2), his implies ha φ, ν v B = for every bounded, coninuous φ. Since φ C b (U R d Y ) is separaing (see [6] or [48]) for P(U R d Y ), µ 1,v = µ 2,v. This is a conradicion and herefore µ v is unique. Since µ is unique (by Assumpion 3.1), µ v is unique by inducion. Of course, exisence of a soluion in B T +1 for (19) follows from he fac ha h θ is a probabiliy ransiion funcion and inducion. Assume µ v B. Since h θ is a probabiliy ransiion funcion, µ v +1 B as well. By assumpion, µ B and herefore we have exisence of a soluion µ v B T +1 by inducion. Finally, since µ = µ V, i immediaely follows ha µ exiss and is unique. Lemma A.9. There exiss a unique soluion Ξ W T +1 o equaion (28) in Theorem 3.4. Proof. We firs show exisence of Ξ v in W T +1 for any v. Suppose Ξ v < W. We now seek o show ha Ξ v W or, equivalenly, Ξ v (u, dy) S for each u. We recall (28): Ξ v (u, dy) = u U h θ (u, u, y, v 1, H v τ: 1) Ξ v 1(u, dy) + ( H h θ(u, u, y, v 1, H τ: 1) v Ēv v τ: 1 ) µ 1 (u, dy) + M v (u, dy). Using he assumpion ha Ξ v < W, h θ is smooh, and Y is compac, i immediaely follows via definion (29) ha he firs wo erms of are in S. 29

30 I remains o verify ha Mv (u, y) S for each u. For his, we use he Bochner-Minlos heorem (see [5]) for S which saes ha a necessary and sufficien condiion for he exisence of a random variable X in S (R d Y ) wih a characerisic funcional g(φ) = E[exp(i φ, X)] is: (i) g() = 1 (ii) g is posiive definie in he sense ha n j,l=1 z j z l g(φ j φ l ) for any z j C, φ j S. (iii) g is pseudo-coninuous. 16 Since M v (u, y) is Gaussian where is covariance is known in closed-form (see Lemma A.4), he characerisic funcional for Mv (u, y) is: [ g Mv(φ) = E exp (i φ(y), Mv (u, y) )] ( = exp 1 φ(y) 2, h θ (u, u, y, v 1, 2 H τ: 1)(1 v h θ (u, u, y, v 1, H τ: 1) µ v v 1(u, dy) ) (63). Propery (i) is rivially saisfied. To show Propery (iii), firs noe ha h θ (u, u, y, v 1, H v τ: 1)(1 h θ (u, u, y, v 1, H v τ: 1) µ v 1(u, dy) η(u, dy) S. Since η(u, dy) S,, η(u, dy) : S R is a coninuous linear operaor. Since S is merizable, i suffices o check sequenial coninuiy in order o prove Propery (iii). Recall ha η(u, dy) has compac suppor on he compac se Y. Therefore, φ 2 m φ 2 on Y since he produc operaion Cc (R d Y ) Cc (R d Y ) Cc (R d Y ) is coninuous where Cc is he space of smooh funcions wih compac suppor. Since, η(u, dy) is a coninuous linear operaor, φ m φ implies ha φ m (y) 2, η(u, dy) φ(y) 2, η(u, dy) for any v. ( This in urn implies ha exp 1 2 φm (y) 2, η(u, dy) ) ( exp 1 2 φ(y) 2, η(u, dy) ) for every V. The disribuion η(u, dy) is nonnegaive and φ m (y) 2 >, so 1 2 φm (y) 2, η(u, dy) ( and exp φm (y) 2, η(u, dy) ) 1. Therefore, g Mv(φ m ) g Mv(φ) by he dominaed convergence 1 2 heorem, which proves Propery (iii). I remains o show Propery (ii). We need o show ha he following quaniy is nonnegaive for any z j C and φ j S, j = 1,..., n: n ( z j z l exp 1 (φj φ l ) 2, η(u, dy) ) (64) 2 j,l Le he marix Σ have elemens Σ l,j = φ jφ l, η(u, dy). The marix Σ is posiive semi-definie: n n z j z l Σ l,j = z j z l φ j φ l, η(u, dy) = n z j z l φ j φ l, η(u, dy) l,j l,j l,j = n z l φ l 2 (65), η(u, dy). l The las inequaliy comes from η(u, dy). Consider he mean-zero Gaussian random variables Z = (Z 1,..., Z n ) wih covariance marix Σ l,j. Z s characerisic funcional is posiive-definie, meaning ha: n ( ) (66) z j z l E[exp i(e l e j ) Z ], l,j for e j R n. Le e j = (1 j=1, 1 j=2,..., 1 j=n ). Then, (66) is exacly he quaniy (64), which proves Propery (ii). Therefore, assuming Ξ v < W, Mv (u, y) S. By inducion and Ξ v W, Ξ v (u, dy) S for each u. 1,v 2,v The uniqueness of he soluion can proven by supposing here are wo soluions Ξ and Ξ. Le heir difference be ν = Ξ 1 Ξ 2. Due o he lineariy of (28), subsiuing ν ino equaion (28) yields ha φ, ν U R d Y = for every φ, which implies uniqueness. 16 A funcion f : E R is pseudo-coninuous if is resricion o any finie-dimensional subspace of E is coninuous. 3

31 Finally, since Ξ = Ξ V, i immediaely follows ha Ξ exiss and is unique. Appendix B. Convergence Rae of a Quadraure Scheme for he Evaluaion of he LL In Secion 4, a quadraure scheme was proposed in order o simulae he Mone Carlo approximaion µ (which is a linear combinaion of a law of large numbers µ and a cenral limi heorem Ξ). Under some echnical condiions, we show a convergence rae for ha quadraure scheme for he law of large numbers. The convergence rae for he simulaion scheme for he cenral limi heorem can be proven in a similar fashion. The resuls in his secion can be used as a pracical guideline o deermine he number of grid poins for he simulaion scheme in Secion 4. Le he maximum radius of he compuaional cells be: r K,i = 1 2 max max k=1,...,k y i,y i c k y i y i, where y i is he i-h elemen of he vecor y Y R d Y. As in Secion 4, for a sample V l of he process X: f, µ l U Y K k=1 f(u, y k ) µ Ψ,l (u, y k ) m l, where µ Ψ,l is he law of large numbers under he quadraure scheme, µ l is he law of large numbers condiional on he pah V l, and we define H Ψ = u U Y f θ h(u, y) µψ (u, dy). For noaional convenience, we have suppressed he Mone Carlo sample noaion l for H Ψ and H. Under cerain condiions, we will prove a convergence rae for he mean-squared error (MSE) of he simulaion scheme given in Secion 4 for he law of large numbers µ: MSE = E[(E[g(f, µ U Y )] 1 L (67) g(m l )) 2 ] L The convergence rae will give insigh ino how o opimally design a grid for he simulaion scheme described in Secion 4. Assumpion B.1. µ (u, dy) = if u c (i.e., all of he pool is iniially in he {curren} sae), V V R d V where V is compac, Y is compac, g, f, f H are coninuously differeniable, and h θ is wice coninuously differeniable. (68) Le ζ l,h (u, y) saisfy: l=1 ζ l,h (u, y) = h θ (u, u, y, V 1, l H 1 )ζ l,h 1 (u, y), ζ l,h (c, y) = 1, ζ l,h (u, y) =, u c. Then, µ l l, H (u, dy) = ζ (u, y) µ (c, dy) and µ l,ψ (u, y k ) = ζ l,hψ (u, y k ) µ (c, c k ). Since h θ is coninuously differeniable, ζ l,h (u, y) is coninuously differeniable in y for each u. ζ l, H (u, y k ) ζ l,hψ (u, y k ) ζ l, H (u, y k ) h θ (u, u, y k, V l 1, H 1 )ζ l,hψ 1 (u, y k ) + h θ (u, u, y k, V 1, l H 1 )ζ l,hψ 1 (u, y k ) ζ l,hψ (u, y k ) l, H h θ (u, u, y k, V 1, l H 1 ) ζ 1 (u, y k ) ζ l,hψ 1 (u, y k ) + h θ H (u, u, y k, V l 1, h ) ζ l,hψ 1 (u, y k ) H Ψ 1 H 1 l, H h θ (u, u, y k, V 1, l H 1 ) ζ 1 (u, y k ) ζ l,hψ 1 (u, y k ) 31

32 + h θ H (u, u, y k, V l 1, h ) H 1 H Ψ 1, (69) max u,y k l, H h θ (u, u, y k, V 1, l H 1 ) ζ 1 (u, y k ) ζ l,hψ 1 (u, y k ) + K 1 H 1 H 1 Ψ l, H ζ 1 (u, y k) ζ l,hψ (u, y k) + K 1 H 1 H 1 Ψ 1 where we have bounded h θ H using he compacness of he space is argumens live on and is coninuiy. In addiion, we have used he fac ha h(, u, ) is a probabiliy kernel and herefore sums o one. Taking he maximum over he u U and he grid poins y k : (7) max u,y k ζ l, H (u, y k ) ζ l,hψ (u, y k ) max u,y k l, H ζ 1 (u, y k) ζ l,hψ 1 (u, y k) + K 1 H 1 H 1. Ψ ex, we find a bound for H H Ψ in erms of ζ l, H (u, y k ) ζ l,hψ (u, y k ). H Ψ = u U Using a Taylor expansion: (71) K k=1 f h θ (u, y k ) µ Ψ,l (u, c k ) = H H Ψ = f H, µ l K U Y u U k=1 K fθ h (u, y)ζ l, H c k + u U 2 2 u U k=1 K u U k=1 d Y i=1 d Y i=1 K k=1 fθ h (u, y)ζ l, H c k fθ h (u, y k )ζ l, H c k r K,i sup [fθ h (u, y)ζ y i y Y u U C 1,,i (X l )r K,i + C 3 max u,y k Reurning o equaion (7), we now have: max u,y k ζ l, H (u, y k ) ζ l,hψ (u, y k ) max (72) u,y k l, H ζ + 2K 1 d Y K k=1 u U f h θ (u, y k ) µ Ψ,l (u, c k ) f(u, y k )ζ l,hψ (u, y k ) µ (c, c k ). (u, y) fθ h (u, y k )ζ l,hψ (u, y k ) µ (c, dy) (u, y) f h θ (u, y k )ζ l, H (u, y k ) µ (c, dy) (u, y k ) fθ h (u, y k )ζ l,hψ (u, y k ) µ (c, dy). ζ l, H l, H (u, y)] + C 3 max u,y k ζ l, H (u, y k ) ζ l,hψ (u, y k ) 1 (u, y k) ζ l,hψ 1 (u, y k) i=1 2K 1 d Y i=1 C 1, 1,i (V l )r K,i + C 3 K 1 max u,y k l, H ζ 1 C 1, 1,i (V l )r K,i + (1 + C 3 K 1 ) (u, y k ) ζ l,hψ (u, y k ) 1 (u, y k) ζ l,hψ 1 (u, y k) = max u,y k d Y 1 2K 1 C 1, 1,i (V l )r K,i + 2K 1 (1 + C 3 K 1 ) d Y i=1 i=1 C 2,,i (V l )r K,i 32 = ζ l, H (u, y k) ζ l,hψ (u, y k) d Y e ( 1)(1+C 3K 1) i=1 C 1,,i(V l )r K,i,

33 where we have used Gronwall s lemma for he las inequaliy. We noe ha numerical scheme converges as he size of he cells r K,1,..., r K,dY. The error depends upon he magniude of he derivaive of he law of large numbers is wih respec o each dimension i and how fine he grid is along he dimension i. The sensiiviy of he error o he magniude of he derivaive of he law of large numbers wih respec o he dimension i is capured in he erm C 2,,i (V l ). By using a Taylor expansion in he exac same manner as shown previously, his of course implies ha: (73) g( f, µ l f, U Y ) g( µ l,ψ U Y ) d Y i=1 C 3,,i (V l )r K,i. We also noe ha C 3,,i (v) < C 4 < since i is a coninuous funcion on a compac se (due o assumpions ha V is compac and h is wice differeniable). We now find he desired convergence rae: (74) E [(E[g(f, µ U Y )] 1 L + E[( 1 L L l=1 L g(m l )) 2 ] E[(E[g(f, µ U Y )] 1 L l=1 g( f, µ l U Y ) 1 L L l=1 g( f, µ l U Y ))2 ] L g(m l )) 2 ] 1 L Var[g(f, µ d Y U Y )] + C 5 d Y E[C 3,,i (V ) 2 ]rk,i 2 } {{ } i=1 variance } {{ } bias l=1 The assumpion ha V is compac was made in order ha C 3,,i (V ) 2 < C3 2 and hus C 3,,i (V ) 2 would be inegrable. This assumpion can be relaxed o simply requiring ha C 3,,i (V ) 2 be inegrable. As a consequence of equaion (74), he mean-squared error of he numerical approximaion converges o zero as L and max i r K,i. Equaion (74) also provides insigh ino he facors driving he numerical error of he simulaion scheme. The mean-squared error is composed of a variance and a bias erm. The variance erm is he variance of a sample wihou numerical error; i.e., he variance erm would remain even if one could produce samples wih no numerical error. The bias erm is a consequence of he error produced by he quadraure scheme. I is he sum of he maximum lengh of he cells along a paricular dimension muliplied by he average of he squared parial derivaive of a funcion of he law of large numbers µ wih respec o ha dimension (i.e., E[C 3,,i (V ) 2 ]). Therefore, i is desirable o have a finer grid wih respec o he dimensions along which he parial derivaives of he law of large nubmers µ are mos rapidly changing. Overall, one can reduce he MSE by eiher choosing a finer grid wih respec o a paricular dimension (hus reducing he bias) or by generaing more Mone Carlo samples (hus reducing he variance). The convergence rae (74) suggess an opimal allocaion of a compuaional budge. Given a fixed compuaional cos (i.e., maximum allowed compuaional ime), one mus choose he opimal number of Mone Carlo rials L and he cell radii r K,1,..., r K,K in order o minimize he mean-squared error. For insance, assuming a recangular grid (which is no he bes approach in higher-dimensions; see Secion 5.7 for a beer alernaive) wih a oal compuaional budge B, he budge equaion is: L d Y i=1 1 r K,i = B C 6, where he consan C 6 involves he cos of each simulaion as well as he size of he space Y which one is discreizing over. For noaional convenience, define C variance = Var[g(f, µ )] and C bias,i = C 5 d Y E[C 3,,i (V ) 2 ]. The consan C bias,i is larger for dimensions i along which he soluion varies more rapidly. The opimal choices for L and r K,1,..., r K,K saisfy a sysem of hyperbolic equaions which can be solved explicily. For insance, in wo dimensions (d Y = 2), he opimal choices are: r K,1 = (C variance C 6 ) 2/1 1 ( ) 3/1 1 ( ) 1/1, 2BC bias,1 2BC bias,2 r K,2 = (C variance C 6 ) 2/1 1 ( ) 3/1 1 ( ) 1/1, 2BC bias,2 2BC bias,1 L = B 3/5 C 4/5 6 C 4/1 variance ( 1 ) 1/5 1 ( ) 1/5. 2C bias,1 2BC bias,2 33

34 As expeced, he opimal number of Mone Carlo rials L increases wih he variance. Similarly, he fineness of he grid in dimension i decreases he larger he soluion s derivaive is wih respec o ha dimension. Appendix C. on-uniform Grids To increase compuaional efficiency for he low-dimensional LL and CLT, we recommend non-uniform grids. In a non-uniform grid, more poins would be placed where µ (u, dw) is large and less poins would be placed where µ (u, dw) is small. In he case where h θ (y) = g θ (w) is a logisic funcion, we propose he following non-uniform grid for R d W : Divide R d W ino K boxes, each wih equal mass ν(dw) = 1/K where we ake ν(dw) = box k µ (c, dw). In one dimension, his can be done by finding he quaniles of he disribuion ν. I is assumed ha R µ d W (c, dw) = 1. In he k-h box, choose he grid poin w k = log K box k ew ν(dw). Evaluae he soluion µ (u, dw) a he grid poins w 1,..., w K. If g θ is locally linear (a leas wihin he k-h box) in e w, he grid poins y k can make his scheme highly accurae. We demonsrae for one ime-sep o explain he choice of he poins y k. Define he funcion q θ such ha q θ (u, e w ) = g θ (u, w). The exac mass wihin he k-h box a = 1 is box k µ 1(u, dw) = box k g θ(u, w) ν(dw) = box k q θ(u, e w ) ν(dw) where we have suppressed he oher argumens of g for noaional convenience. If g θ is approximaely locally linear in e w (i.e., q is approximaely linear) in he k-h box, one has ha µ 1 (u, dw) = q θ (u, e w ) ν(dw) 1 box k box k K q θ(u, K e w ν(dw)) box k = 1 K g θ(u, w k ) = g θ (u, w k ) µ (c, dw). Then, if g is close o locally linear in e w, he choice of he grid poin w k will lead o a very accurae soluion for he oal mass in he k-h box. In he end, he quaniy of ineres is he oal mass in each sae u (i.e., wha fracion of loans are sill alive, wha fracion have defauled, and wha fracion have prepaid), so his is highly useful. One can simply sum up he mass in each box o find he oal mass in sae u. Alhough his scheme has been specifically ailored o he case where h θ is a logisic funcion, generalizaions can be made o oher funcion choices. box k Appendix D. Pre-compuaion for Financial Insiuions Even for he risk analysis of smaller, individual pools, he efficien Mone Carlo approximaion can provide considerably faser compuaions. For insance, for a single pool of 1, loans, alhough he approximaion is accurae, i does no offer as large compuaional cos savings as for very large pools. However, a ypical financial insiuion will deal wih housands of such pools. As menioned earlier, a morgage rading desk a a major bank will on a daily basis analyze housands of MBSs and hundreds of CMOs. Assuming here is no mean field dependence in equaion (1), one can pre-simulae he LL and CLT a a se of grid poins R R d W. This pre-simulaion occurs only once. Then, one can find he disribuion for he k-h pool by aking a weighed combinaion of he pre-simulaed approximaion µ across he grid poins R, where he weighs are chosen o mach he k-h pool s loan-level feaure disribuion. If he series of pools have sizes 1,..., K wih = K, hen he compuaional cos of he efficien Mone Carlo approximaion compared wih brue-force Mone Carlo simulaion of he acual pool is g / where g = R is he number of grid poins. Furhermore, he mehod immediaely yields he correlaion beween he differen pools, which is essenial for risk managemen purposes. amely, for each pah of he sysemaic facor V, we simulaneously have he defaul and prepaymen behavior for all of he 1,..., K. The approach is summarized below: Pre-simulae he (finie-dimensional) LL µ Ψ and CLT Ξ Ψ on he grid R = {w 1,..., w I } wih iniial condiion Ψ(c, dw) = I i=1 δ w i. For each pool 1,..., K: Find he k-h pool s disribuion in he w-space and approximae i a he grid poins R; le z i be he fracion a he i-h grid poin. Then, he k-h pool s disribuion is µ k (u, w i ) = z i µ Ψ zi (75) (u, w i ) + ΞΨ (u, w i ),, i = 1,..., I, k 34

35 and zero oherwise. The mehod can be furher improved by aking a sparse grid R in order o reduce he number of calculaions and hen, afer he pre-simulaion, inerpolaing on a finer grid. Due o he smoohness of µ for ypical funcions h θ, only a few grid poins are usually needed in order o ge an accurae inerpolaed soluion. Using his approach, he efficien Mone Carlo approximaion can be highly useful even for small pools of loans as long as he financial insiuion is dealing wih many such pools in aggregae. The approach is implemened using acual morgage daa in Secion 5.6. Appendix E. Parameer Esimaion The parameer θ specifying he model (1) can be esimaed by he mehod of maximum likelihood. We are given observaions of Y = (Y 1,..., Y ) and (U 1,..., U, V ) =1,...,T. Collecively, he observaions of he saes up o ime T are D T, = (Z1,..., ZT ) where Z = (U 1,..., U ). The log-likelihood funcion for D T, given V and Y is L(θ) = log P θ (D T, V, Y ) = log P θ [Z 1,..., Z T V, Y ] = log (76) =1 T P θ [Z Z,..., Z 1, V, Y ] =1 T T = log P θ [Z Z 1, H τ: 1, V, Y ] = log h θ (U n, U 1, n Y n, V 1, H τ: 1) = T =1 =1 log h θ (U n, U 1, n Y n, V 1, H τ: 1). oe ha we have used he condiional independence of U 1,..., U wih respec o F 1 on he second line in equaion (76). The maximum likelihood esimaor ˆθ = arg max θ Θ L(θ). Sochasic gradien descen can be used o numerically opimize L(θ). Typically, one will also choose a separae model for he sysemaic facors V wih is own parameers. These parameers can be esimaed separaely from θ using sandard mehods; noe ha he likelihood for θ depends only on he observed values of V and is independen of V s exac form or parameerizaion since V is an exogenous process. References [1] B. Ambrose and C. Capone. Modeling he condiional probabiliy of foreclosure in he conex of single-family morgage defaul resoluions. Real Esae Economics, 26(3): , [2] M. Arnsdorf and I. Halperin. BSLP: markovian bivariae spread-loss model for porfolio credi derivaives. Journal of Compuaional Finance, 12:77 1, 28. [3] B. Baesens. eural nework survival analysis for personal loan daa. Journal of he Operaional Research Sociey, 56(9): , 25. [4] J. Banasik, J. Crook, and L. Thomas. o if bu when will borrowers defaul. Journal of Operaional Research Sociey, pages , [5] J. Basos. Forecasing bank loans loss-given-defaul. Journal of Banking and Finance, 34(1): , 21. [6] P. Billingsley. Convergence of Probabiliy Measures. John Wiley and Sons, 28. [7]. Bush, B. M. Hambly, H. Haworh, L. Jin, and C. Reisinger. Sochasic evoluion equaions in porfolio credi modelling. SIAM Journal of Financial Mahemaics, 2(1): , 211. [8] D. Capozza, D. Kazarian, and T. Thomson. Morgage defaul in local markes. Real Esae Economics, 25(4): , [9] CoreLogic. Riskmodel. Technical repor, 214. [1] D. Crisan, P. Del Moral, and T. Lyons. Discree filering using branching and ineracing paricle sysems. Technical repor, Laboraoire de Saisique e Probabiliies, Universie de Toulouse, [11] J. Cvianic, J. Ma, and J. Zhang. The law of large numbers for self-exciing correlaed defauls. Sochasic Processes and heir Applicaions, 122(8): , 212. [12] P. Dai Pra, W.J. Runggaldier, E. Sarori, and M. Toloi. Large porfolio losses: A dynamic conagion model. The Annals of Applied Probabiliy, 19(1): , 29. [13] Y. Deng, J. Quigley, and R. Van Order. Morgage erminaions, heerogeneiy, and he exercise of morgage opions. Economerica, 68(2):275 37, 2. [14]. Diener, R. Jarrow, and P. Proer. Relaing op-down wih boom-up approaches in he evaluaion of abs wih large collaeral pools. Inernaional Journal of Theoreical and Applied Finance, 15(2), 212. [15] Xiaowei Ding, Kay Giesecke, and Pascal Tomecek. Time-changed birh processes and muli-name credi derivaives. Operaions Research, 57(4):99 15,

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