A Generalization of Panjers Recursion and Numerically Stable Risk - PDF document

A Generalization of Panjer’s Recursion and Numerically Stable Risk Aggregation Workshop on Credit Risk Universit´ e d’Evry Val d’Essonne, June 25–27, 2008 Uwe Schmock (Joint work with S. Gerhold and R. Warnung) Christian Doppler Laboratory for Portfolio Risk Management (PRisMa Lab) Financial and Actuarial Mathematics Vienna University of Technology, Austria www.fam.tuwien.ac.at Workshop on Portfolio Risk Management Mo., Sept. 29, 2008, Vienna, Austria Organized by Research Group for Financial and Actuarial Mathematics and CD-Laboratory for Portfolio Risk Management Institute for Mathematical Methods in Economics Vienna University of Technology A-1040 Vienna, Austria www.fam.tuwien.ac.at/prisma2008 � June 25, 2008, U. Schmock, FAM, TU Vienna c 2

Motivation: The Collective Model Task: Calculate the distribution of the random sum S = X 1 + · · · + X N of N losses, where the loss sizes { X i } i ∈ N are i. i. d. and independent of N . Applications: • Claims in a homogeneous insurance portfolio • Losses in a credit portfolio ( → extended CreditRisk + ) • Operational losses (Basel II), aggregation for every line of business and loss type. Standard tool: Panjer’s recursion for specific distribu- tions of N , when the X i are N 0 -valued. 3 Loss Number Distributions in the Panjer Class Definition: A probability distribution { q n } n ∈ N 0 is said to belong to the Panjer( a, b, k ) class with a, b ∈ R and k ∈ N 0 if q 0 = q 1 = · · · = q k − 1 = 0 and a + b � � q n = q n − 1 for all n ∈ N with n ≥ k + 1. n Important examples: (all distributions are known) • Poisson( λ ) ∈ Panjer(0 , λ, 0) with intensity λ > 0 • NegBin( α, p ) ∈ Panjer( q, ( α − 1) q, 0) with α > 0, probability p ∈ (0 , 1) and q := 1 − p • Log( q ) ∈ Panjer( q, − q, 1) with q ∈ (0 , 1) and q n q n = − n log(1 − q ) for all n ∈ N � June 25, 2008, U. Schmock, FAM, TU Vienna c 4

Extended Logarithmic Distribution For k ∈ N \{ 1 } and q ∈ (0 , 1] define q 0 = · · · = q k − 1 = 0 , � − 1 q n � n k q n = for n ≥ k . � − 1 q l � ∞ � l l = k k ExtLog( k, q ) is in the Panjer( q, − kq, k ) class. Extended Negative Binomial Distribution For k ∈ N , α ∈ ( − k, − k + 1) and p ∈ [0 , 1) define q = 1 − p , q 0 = · · · = q k − 1 = 0 and � α + n − 1 q n � n q n = for n ≥ k . p − α − � k − 1 � α + j − 1 � q j j =0 j ExtNegBin( α, k, p ) is in the Panjer( q, ( α − 1) q, k ) class. 5 Extended Panjer Recursion If L ( N ) ∈ Panjer( a, b, k ), independent of the i. i. d. N 0 -valued sequence { X n } n ∈ N , and a P ( X 1 = 0) � = 1, then S := X 1 + · · · + X N satisfies � � P ( S = 0) = ϕ N P ( X 1 = 0) with ϕ N the probability generating function of N , and � 1 P ( S = n ) = P ( S k = n ) P ( N = k ) 1 − a P ( X 1 = 0) n � a + bj � � � + P ( X 1 = j ) P ( S = n − j ) n j =1 for all n ∈ N , where S k = X 1 + · · · + X k . � June 25, 2008, U. Schmock, FAM, TU Vienna c 6

Example for Numerical Instability Take N ∼ ExtNegBin( α, k, p ) with k ∈ N , ε, p ∈ (0 , 1) and α := − k + ε . Consider the loss distribution P ( X 1 = 1) = P ( X 1 = l ) = 1 / 2 with l ≥ 3. Then � q k p k + l = P ( S = k + l ) = q k ( l − 1) + εk 2 k +1 + q k + l − 1 � k 2 k + l k + l − q k ( l − 1) − εl q k 2 k +1 . k + l With ε = 1 / 10 000, k = 1, l = 5, p = 1 / 10: p 6 = 0 . 1499 926 − 0 . 1499 701 = 0 . 0000 225 . Panjer recursion with five significant digits gives p 6 = 0 . 0000 400 . . . ( ≈ 78% relative error) . 7 Panjer Recursion Replaced by Weighted Convolution Fix l ∈ N , consider N ∼ { q n } n ∈ N 0 and ˜ N i ∼ { ˜ q i,n } n ∈ N 0 , define S = X 1 + · · · + X N ∼ { p n } n ∈ N 0 and ˜ S ( i ) = X 1 + · · · + X ˜ N i ∼ { ˜ p i,n } n ∈ N 0 for i ∈ { 1 , . . . , l } . Assume there exist k ∈ N 0 and a 1 , . . . , a l , b 1 , . . . , b l ∈ R such that l a i + b i � � � q n = q i,n − i ˜ for n ≥ k + l n i =1 and ˜ q i, 0 = · · · = ˜ q i,k + l − i − 1 = 0 for i ∈ { 1 , . . . , l } . � � Then p 0 = ϕ N P ( X 1 = 0) and, for n ∈ N , k + l − 1 l n a i + b i j � � � � � p n = P ( S j = n ) q j + P ( S i = j )˜ p i,n − j . in j =1 i =1 j =0 � June 25, 2008, U. Schmock, FAM, TU Vienna c 8

Combination of Truncated Distributions Fix k ∈ N 0 , l ∈ N . For all i ∈ { 1 , . . . , l } assume that α i ≥ 0, β i ≥ − iα i (at least one � =) and that the N 0 -valued ˜ N i satisfies P ( ˜ N i < k + l − i ) = 0. Consider q 0 , . . . , q k + l − 1 ≥ 0 with q 0 + · · · + q k + l − 1 ≤ 1. Define l α i + β i � � � P ( ˜ q n = c N i = n − i ) for n ≥ k + l, n i =1 k + l − 1 l � �� � � �� 1 � � c = 1 − q n α i + β i E . i + ˜ N i n =0 i =1 Then { q n } n ∈ N 0 is a probability distribution satisfying the recursion condition with a i = cα i and b i = cβ i and the calculation of { p n } n ∈ N 0 is numerically stable. 9 Weighted Convolution for ExtLog Let k ∈ N and q ∈ (0 , 1). Let ˜ N ∼ ExtLog( k, q ) and N ∼ ExtLog( k + 1 , q ), where ExtLog(1 , q ) means Log( q ). Define ˜ S = X 1 + · · · + X ˜ N and S = X 1 + · · · + X N . Then, with an explicit b 1 > 0, the weighted convolution n P ( S = n ) = b 1 � j P ( X 1 = j ) P ( ˜ S = n − j ) , n ∈ N , n j =1 is numerically stable. Numerically stable algorithm: • Panjer recursion for Log( q ) • k − 1 weighted convolutions: Log( q ) → ExtLog(2 , q ) → · · · → ExtLog( k − 1 , q ) → ExtLog( k, q ) � June 25, 2008, U. Schmock, FAM, TU Vienna c 10

Numerically Stable Algorithm for ExtLog(2,1) Let N ∼ ExtLog(2 , 1). For S = X 1 + · · · + X N we have P ( S = 0) = P ( X 1 = 0) + P ( X 1 ≥ 1) log P ( X 1 ≥ 1) with 0 log 0 := 0 and, in the case P ( X 1 ≥ 1) > 0, n P ( S = n ) = 1 � j P ( X 1 = j ) r n − j , n ∈ N , n j =1 where r 0 = − log P ( X 1 ≥ 1) and, recursively for n ∈ N , n � � 1 P ( X 1 = n ) + 1 � r n = j P ( X 1 = n − j ) r j P ( X 1 ≥ 1) n j =1 11 Weighted Convolution for ExtNegBin Let k ∈ N 0 , α ∈ ( − k, − k + 1) and p ∈ (0 , 1). Let ˜ N ∼ ExtNegBin( α, k, p ) and N ∼ ExtNegBin( α − 1 , k +1 , p ), where ExtNegBin( α, 0 , p ) means NegBin( α, p ). Define ˜ S = X 1 + · · · + X ˜ N and S = X 1 + · · · + X N . Then n P ( S = n ) = b 1 � j P ( X 1 = j ) P ( ˜ S = n − j ) , n ∈ N , n j =1 with an explicit b 1 > 0 is numerically stable. Algorithm: • Panjer recursion for NegBin( α + k, p ) • k weighted convolutions: NegBin( α + k, p ) → ExtNegBin( α + k − 1 , 1 , p ) → · · · → ExtNegBin( α + 1 , k − 1 , p ) → ExtNegBin( α, k, p ) � June 25, 2008, U. Schmock, FAM, TU Vienna c 12

Stable Algorithm for ExtNegBin( α − 1, 1, 0) Let N ∼ ExtNegBin( α − 1 , 1 , 0) with α ∈ (0 , 1). For S = X 1 + · · · + X N we have � 1 − α � P ( S = 0) = 1 − P ( X 1 ≥ 1) and in the case P ( X 1 ≥ 1) > 0 n P ( S = n ) = 1 − α � j P ( X 1 = j ) r n − j , n ∈ N , n j =1 � − α and, recursively for n ∈ N , � where r 0 = P ( X 1 ≥ 1) n 1 n − j + αj � r n = P ( X 1 = j ) r n − j . P ( X 1 ≥ 1) n j =1 13 Tempered α -Stable Distributions on [0, ∞ ) Let Y be α -stable on [0 , ∞ ) with Laplace transform L Y ( s ) = E [exp( − sY )] = exp( − γ α,σ s α ) , s ≥ 0 , � απ where α ∈ (0 , 1), σ > 0 and γ α,σ := σ α / cos � . 2 For τ ≥ 0 define τ -tempered α -stable distribution F α,σ,τ ( y ) := E [ e − τY 1 { Y ≤ y } ] / E [ e − τY ] , y ∈ R . Let Λ ∼ F α,σ,τ . Then for τ > 0 ( s + τ ) α − τ α �� � � L Λ ( s ) = exp − γ α,σ , s ≥ − τ, E [Λ] = −L ′ Λ (0) = αγ α,σ τ α − 1 , Λ (0)) 2 = α (1 − α ) γ α,σ τ α − 2 . Var(Λ) = −L ′′ Λ (0) − ( L ′ � June 25, 2008, U. Schmock, FAM, TU Vienna c 14

1.2 Α � 0.4 1 Α � 0.5 Α � 0.6 Gamma 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 3 Density of Λ ∼ F α,σ,τ in comparison with gamma distribution, where α ∈ (0 , 1) and σ, τ > 0 satisfy E [Λ] = 1 and Var(Λ) = 0 . 3. 15 2 1.75 Τ � 0.5 Τ � 0.75 1.5 Τ � 1 Gamma 1.25 1 0.75 0.5 0.25 0.5 1 1.5 2 2.5 3 Density of Λ ∼ F α,σ,τ in comparison with gamma distribution, where α ∈ (0 , 1) and σ, τ > 0 satisfy E [Λ] = 1 and Var(Λ) = 0 . 3. � June 25, 2008, U. Schmock, FAM, TU Vienna c 16

Application: Poisson–Tempered α -Stable Mixtures For α ∈ (0 , 1), λ, σ > 0 and τ ≥ 0 consider Λ ∼ F α,σ,τ a.s. and the Poisson mixture L ( N | Λ) = Poisson( λ Λ). Representation as compound Poisson distribution: If M ∼ Poisson( γ α,σ (( λ + τ ) α − τ α )) and the i. i. d. sequence { N m } m ∈ N with τ � � N m ∼ ExtNegBin − α, 1 , λ + τ d are independent, then N = N 1 + · · · + N M . 17 Numerically Stable Algorithm for Poisson–Tempered α -Stable Mixtures • Apply Panjer’s recursion for ˜ S = X 1 + · · · + X ˜ N τ ˜ N ∼ NegBin(1 − α, p ) with p = λ + τ . • Use weighted convolution to pass from ˜ N ∼ NegBin(1 − α, p ) → N ∼ ExtNegBin( − α, 1,p). • Take the previous calculated distribution of S = X 1 + · · · + X N as new claim size distribution and apply Panjer’s recursion for M ∼ Poisson( γ α,σ (( λ + τ ) α − τ α )) . � June 25, 2008, U. Schmock, FAM, TU Vienna c 18