A DAPTIVE INTEGRATION FOR MULTI - FACTOR PORTFOLIO CREDIT LOSS MODELS Xinzheng Huang, TU Delft and Rabobank, the Netherlands Mid-Term Conference on Advanced Mathematical Methods for Finance Vienna, Austria, September, 17th-22nd, 2007 1 Joint work with Cornelis W. Oosterlee at TU Delft and CWI. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
O UTLINE 1 Portfolio Credit Loss Modeling • Laten Factor Models 2 Globally Adaptive Algorithms for Numerical Integration • Adaptive Genz-Malik Rule • Adaptive Monte Carlo Integration 3 Numerical Results 4 Conclusions Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
P ORTFOLIO CREDIT LOSS • A credit portfolio consisting of n obligors with exposure w 1 , w 2 ,..., w n . • Default indicator D i = 1 { X i < γ i } , X i standardized log asset value and γ i default threshold. • Default probability p i = P ( X i < γ i ) . • Portfolio loss L = ∑ n i = 1 w i D i . • Tail probability P ( L > x ) for some extreme loss level x . • Value at Risk (VaR): the α -quantile of the loss distribution of L for some α close to 1. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
L ATENT FACTOR MODELS X i = α i 1 Y 1 + ··· + α id Y d + β i Z i = α i Y d + β i Z i , • Y 1 ... Y d : systematic factors that affect more than one obligor, e.g., state of economy, effects of different industries and geographical regions. • Z i : idiosyncratic factor that only affects an obligor itself. • Y d and Z i are independent for all i . � Y d � � Y d � • D i and D j are independent. � Y d � � Y d � • L = ∑ w i D i becomes a weighted sum of independent Bernoulli random variables. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
T AIL PROBABILITY : A NUMERICAL INTEGRATION PROBLEM � � L > x | Y d � dP ( Y d ) P ( L > x ) = P L > x | Y d � � The integrand P can be calculated accurately by • the recursive method - Andersen et al (2003) • the normal approximation - Martin (2004) • the saddlepoint approximation - Martin et al (2001a, b), Huang et al (2007a) • review of various methods - Glasserman and Ruiz-Mata (2006), Huang et al (2007b) Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
P ROPERTIES OF THE CONDITIONAL TAIL PROBABILITY Assuming the factor loadings, α ik , i = 1 , ··· , n , k = 1 , ··· , d are all nonnegative, • The mapping y k �− → P ( L > x | Y 1 = y 1 , Y 2 = y 2 ,..., Y d = y d ) , k = 1 , ··· , d is non-increasing in y k . ⇓ ∀ Y d ∈ [ a 1 , b 1 ] × [ a 2 , b 2 ] ... × [ a d , b d ] � L > x | Y d � P ( L > x | b 1 ,... b d ) ≤ P ≤ P ( L > x | a 1 ,... a d ) • P ( L > x | Y 1 , Y 2 ,..., Y d ) is continuous and differentiable with respect to Y k , k = 1 , ··· , d . Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A G AUSSIAN ONE - FACTOR EXAMPLE 1 0.9 0.8 P(L>100|Y) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 Y F IGURE : The integrand P ( L > 100 | Y ) as a function of the common factor Y for portfolio A, which consists of 1000 obligors with w i = 1, p i = 0 . 0033 and √ α i = 0 . 2, i = 1 ,..., 1000. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
T HE G AUSSIAN MULTI - FACTOR MODEL � � I ( f ) = P ( L > x ) = ··· f ( Y 1 ,... Y d ) φ ( Y 1 ,... Y d ) dY 1 ... dY d , where f ( Y 1 ,... Y d ) = P ( L > x | Y 1 ,... Y d ) . • curse of dimensionality: The product quadrature rule becomes impractical because the number of function evaluations grows exponentially with d . • (quasi-) Monte Carlo methods: sample uniformly in the cube [ 0 , 1 ] d . • focus on the subregions where the integrand is most irregular ⇒ adaptive integration. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
G LOBALLY ADAPTIVE ALGORITHMS integration rule ✲ error estimate ❄ ✛ Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
G LOBALLY ADAPTIVE ALGORITHMS FOR NUMERICAL INTEGRATION 1 Choose a subregion from a collection of subregions and subdivide the chosen subregion. 2 Apply an integration rule to the resulting new subregions; update the collection of subregions. 3 Update the global integral and error estimate; check whether a predefined termination criterion is met; if not, go back to step 1. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
T HE G ENZ -M ALIK RULE • A polynomial interpolatory rule of degree 7, which integrates exactly all monomials x k 1 1 x k 2 2 ... x k d n with ∑ k i ≤ 7 and fails to integrate exactly at least one monomial of degree 8. • All integration nodes are inside integration domain. • Requires 2 d + 2 d 2 + 2 d + 1 integrand evaluations for a function of d variables, most advantageous for problems with d ≤ 8. Gauss-Legendre quadrature of degree 7 would need 4 d integrand evaluations. • A degree 5 rule embedded in the degree 7 rule is used for error estimation, no additional integrand evaluations are necessary. ε = | I 7 − I 5 | . Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
T HE G ENZ -M ALIK RULE • Bounded integral in each subregion. ∀ Y d ∈ [ a 1 , b 1 ] × [ a 2 , b 2 ] ... × [ a d , b d ] � L > x | Y d � L ≤ P ≤ U ⇒ d d ∏ ∏ (Φ( b i ) − Φ( a i )) ≤ I ( f ) ≤ U (Φ( b i ) − Φ( a i )) . L i = 1 i = 1 • Local bounds aggregate to a global upper bound and a global lower bound for the whole integration region. • Asymptotic convergence: I 7 → I ( f ) if we continue with the subdivision until the global upper bound and lower bound coincide. • Error estimate not so reliable, cf. Lyness & Kaganove (1976), Berntsen (1989). Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A DAPTIVE M ONTE C ARLO INTEGRATION • Globally adaptive algorithm using Monte Carlo simulation as a basic integration rule. • Asymptotically convergent. • Unbiased estimate for the tail probability. • Practical variance estimate, probabilistic error bounds available. √ � � • Error convergence rate at worst O 1 / N . • Number of sampling points in each subregion independent of number of dimensions d . Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A 2D EXAMPLE 5 4 1 3 0.8 2 P(L>x|Y 1 ,Y 2 ) 0.6 1 Y 2 0.4 0 −1 0.2 −2 0 −5 −3 −2.5 5 −4 0 2.5 Y 1 0 −5 2.5 Y 2 −2.5 −5 −4 −3 −2 −1 0 1 2 3 4 5 Y 1 5 −5 F IGURE : Adaptive Genz-Malik rule for a 2 factor model. (left) integrand P ( L > x | Y 1 , Y 2 ) ; (right) centers of the subregions generated by adaptive integration. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A FIVE - FACTOR MODEL • 1000 obligors with w i = 1, p i = 0 . 0033, i = 1 ,..., 1000, grouped into 5 buckets of 200 obligors. • Factor loadings � � 1 6 , 1 6 , 1 6 , 1 6 , 1 , i = 1 ,..., 200 , √ √ √ √ √ 6 � � 1 5 , 1 5 , 1 5 , 1 √ √ √ √ 5 , 0 , i = 201 ,..., 400 , � � 1 4 , 1 4 , 1 α i = 4 , 0 , 0 , i = 401 ,..., 600 , √ √ √ � � 1 3 , 1 3 , 0 , 0 , 0 , i = 601 ,..., 800 , √ √ � � 1 √ 2 , 0 , 0 , 0 , 0 , i = 800 ,..., 1000 . • Benchmark: plain MC with hundreds of millions of scenarios. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A FIVE - FACTOR MODEL : A DAPTIVE GM 10% 5% Relative Error 2% 1% MC 0.1% QMC AI MC 95% CI 0.01% 100 150 200 250 300 350 400 450 500 550 loss level F IGURE : Estimation relative errors of adaptive GM, plain MC and quasi-MC methods with around N = 10 6 evaluations for various loss levels. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
A FIVE - FACTOR MODEL : A DAPTIVE MC 5 x 10 −4 Benchmark 4.8 ADMC 4.6 P(L>400) 4.4 4.2 2 4 6 8 10 # of integrand evaluations x 10 5 F IGURE : Tail probability P ( L > 400 ) computed by adaptive MC integration with number of integrand evaluations ranging from 50 , 000 to 10 6 and their corresponding 95 % confidence intervals (dotted lines). The dashed line is our Benchmark. Xinzheng Huang, TU Delft and Rabobank, the Netherlands Adaptive integration for multi-factor portfolio credit loss models
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