Problem of random summation and its role in risk aggregation models - PowerPoint PPT Presentation

Problem of random summation and its role in risk aggregation models Gregory Temnov School of Mathematical Sciences University College Cork Oct 14, 2011 PRisMa Day, TU Wien

Problem of random summation and its role in risk aggregation models N ( t ) � Y j j =1 • Y j are assumed to be iid random variables • N ( t ) is a counting process 2 of 25

Initial approach PhD topic : Insurance models with stochastic premium N ( t ) � S ( t ) = u + ct − Y j j =1 Ψ( u ) = P (inf { S ( t ) } < 0 | S (0) = u ) Proposed model: � N ( t ) N ( t ) � � S ( t ) = u + X i − Y j i =1 j =1 • G. Temnov. (2004). Risk process with random income. Journal of Mathematical Sciences. 121 (2), 236 - 244. 3 of 25

Solution to ruin problems Solution to classical ruin problem : Pollaczek − Khinchine Formula � λ 1 a X � k ∞ � Ψ cl ( z ) = (1 − λ 1 a X (1 − � F ∗ k ) X ( z )) , (1) c c k =0 � x where F X ( x ) = 1 � (1 − F X ( y )) dy , x ≥ 0 , (2) a X 0 Case with stochastic income : ∞ � (1 − q ) k (1 − F ∗ k Ψ( z ) = q h ( z )); k =0 � ∞ ∞ � 1 1 e isx dW n ∗ ( x ); ln = 1 − (1 − q ) � n f h ( s ) n =1 0+ � � k ∞ � 1 λ 2 /λ 1 ∗ k W ( x ) = F X ( x ) ∗ G Y ( x ) , G Y ( x ) = 1 − G Y ( − x − 0) . 1 + λ 2 /λ 1 1 + λ 2 /λ 1 k =0 4 of 25

Work with PRiSMa-Lab Initial task : Operational risk measurement The common scheme: for each BL L i ( i = 1 , . . . , m ) : � � • Modelling loss severity (single–loss df) F ( x ) = P X L 1 < x � � N L = n • Modelling loss frequency L = P N L � ⇒ F S L =? • Loss aggregation S L = X L j j =1 • Basic measure in the capital allocation problem VaR α ( S L ) = inf { s ∈ R : P ( S L > s ) ≤ 1 − α } , OpRisk: α = 0 . 999 5 of 25

Operational risk measurement - from basic to advanced • Methodology; numerical techniques : analyzing accuracy, speed . . . • Finding an optimal scheme Taking into account factors that affect regularity of data: • Peculiarity of severity distributions (e.g., presence of outlying data points) • Inflation, trends and other scaling factors • Dependence between different types of risks 6 of 25

Loss aggregation and ch.f. N ( t ) � S N ( t ) = X k , P ( X k < x ) =: F X ( x ) , P ( N ( t ) = k ) =: α k (3) h ( x ) = � k α k x k , k =1 Characteristic function (ch.f.) � ∞ � � = � � � a k � e iux dF X ( x ); f k f X ( u ) = g S ( u ) = h � f X ( u ) X ( u ) k −∞ � a k f ∗ k g S ( x ) = X ( x ) . (4) k � � � ( λ t � f X ( u )) k e λ t λ t ( � Poisson: g S ( u ) = = exp f X ( u ) − 1) (5) k ! k 7 of 25

Aggregate loss distribution 8 of 25

Aggregate loss – error bounds 9 of 25

Quantile as a function of the model parameters If Y = H ( X ), where contin. rv X and Y , cdf F Y ; pdf f X � F Y ( y ) = P ( Y ≤ y ) = P [ H ( X ) ≤ y ] = f X ( x ) dx . x : H ( x ) ≤ y � � F Y ( y ) = P ( Y ≤ y ) = P [ H ( X 1 , . . . , X n ) ≤ y ] = . . . f X ( x 1 , . . . x n ) dx 1 , . . . x n x : H ( x ) ≤ y ( α, σ ); Q θ : Ω ∈ R 2 − Q θ ≡ Q ( θ ) : → R . � � T Q ( y ) = P [ Q θ ( α, σ ) ≤ y ] = T θ ( α, σ ) d α d σ . . . θ : q ( θ ) ≤ y � � On the other hand, 0 . 95 = . . . T θ ( α, σ ) d α d σ . θ ∈ Ω 0 . 95 � � � � T θ ( α, σ ) d α d σ T θ ( α, σ ) d α d σ , . . . > . . . θ ∈ Ω 0 . 95 θ : q 1 ≤ q ( θ ) ≤ q 2 for such q 1 = inf { q ∈ R : Q ( θ ) = q , θ ∈ Θ (Θ ⊂ Ω) } and q 2 = sup { q ∈ R : Q ( θ ) = q , θ ∈ Θ (Θ ⊂ Ω) } . 10 of 25

Quantile as a function of the model parameters 11 of 25

Confidence intervals for the quantile Parameters / Error bounds lower upper Line N VaR (FFT) bound bound Line 1 ξ = 1 . 12 (0 . 95 , 1 . 29) 656 . 12 115 3738 β = 7460 (6326 , 8594) Line 4 ξ = 0 . 52 (0 . 58 , 0 . 46) 27 . 3 18 44 β = 1 . 38 · 10 6 (1 . 21 , 1 . 55) · 10 6 Line 7 ξ = 1 . 2 (1 . 1 , 1 . 3) 209 . 47 94 468 β = 15600 (14352 , 16848) Table: VaR bounds from confidence intervals 12 of 25

Bayesian inference and and MCMC for modelling VaR (accounting for uncertainty) f X | θ ( x | θ ) π ( θ ) � π θ | X ( θ | x ) = f X | θ ( x | θ ) π ( θ ) d θ, (6) π θ | X ( θ | x ) — posterior, π ( θ ) — prior Even in the case of Pareto ( F ( x ) = 1 − (1 + x β ) − 1 /ξ ) and Poisson joint model, π θ | X ( θ | x ) = explicit( π ( θ )) is not always possible 13 of 25

Quantiles of the full predictive distribution A sample from the predictive distribution is considered is simulated by MCMC. As the size N of the observed sample X = { X i } i =1 ,..., N increases, asymptotically, h ( z | X ) − N →∞ g ( z | θ ) − − − → 14 of 25 Figure: The histogram for the � Q P 0 . 999 obtained given a sample of the size 50 and 300

Varying threshold π Θ | X ( θ | x ) ∝ f X | Θ ( x | θ ) π ( θ ) . (7) � N f ( T ) ( X i | L t i , α ) g λ i ( τ i | σ ), f ( T ) ( · ) = f ( X i | α ) f X | Θ ( x | θ ) = 1 − F ( L i | α ) . i =1 15 of 25

Influence of the inflation Investigations of mis-specified models Peter Grandits & GT, 2010 � � − α − 1 � � − α n � l α,σ ( Y ) = α n 1 + y 1 + Y i � G α,σ ( y ) = 1 − , σ σ n σ i =1 The system of ML equations  n � α n − = 0 , � �  n  � � Yi  ln 1+ � σ n i =1 (8) � � n n � �  1 Y i 1 Y i 1 + Y i − σ n = 0 .  � �  n � � σ 2 σ n � n σ 2 σ n � � n + � n + � Yi � Y i � ln 1+ i =1 i =1 � σ n i =1 Y i ≡ Y i q i = Y i e rTi Inflation incoming Y i → � n r yearly inflation rate (if you do not take into account inflation at all) or an error in the estimation of inflation 16 of 25

Results: Influence of inflation and trends Y i = X i q ( r , T i ) + d ( r , T i ) Inflation rTi Y i = X i e n (trend in scaling and location parameter) A special case: Y i = X i e rT i + ArT i ( α ∗ + 1)( α ∗ ) 2 A rT max ∆ α r = � ∆ α r σ ∗ 2 = 0 , � α ∗ + (1 + α ∗ ) 2 � σ ∗ rT max A rT max r σ ∗ T ∆ σ r = ∆ σ r = + . . 2 2 2 P. Grandits, GT, 2010 P. Grandits, R. Kainhofer & GT, 2010 17 of 25

Influence of inflation : case with positive threshold ( α ∗ ) 2 · T · A 2 C 1 − A 1 C 2 α r (0) = ( α ∗ ) 2 A 2 2 1 − A 2 σ ∗ · T · α ∗ A 1 C 1 − C 2 σ r (0) = ( α ∗ ) 2 A 2 2 1 − A 2 � � σ ∗ 1 := · A 1 σ ∗ + L α ∗ + 1 � � 2 σ ∗ α ∗ A 2 := · σ ∗ + L α ∗ + 2 L := A 1 + C 1 L + σ ∗ � � 2 A 2 + α ∗ L σ ∗ := C 2 σ ∗ L + σ ∗ 18 of 25

Inflation and threshold - Illustration of the impact Infl. rate 0 . 03, period 5 years, true param. (1 , 3) True quantile 2999 , True aggregate quantile 11374 ( λ = 7) Misconsideration Resulting effects Trunc. Loss Threshold Quantile Aggregate before scaling scaling quantile α σ scaling X X X 0 . 94 4 . 16 6075 . 4 23242 X X 0 . 94 3 . 78 5520 19730 X X 1 3 . 75 3748 15200 X X 1 3 . 4 3399 13720 19 of 25

Actuarial and Financial data : difference and similarities Financial data : • Evidence of stability (”scale-invariance”, ”self-similarity”) • Relevance of the whole distribution Actuarial data : • ”Heavy-tailedness”, relevance of the right tail Similarities : • Aggregation of Actuarial data and Increments of Financial data 20 of 25

Extensions of Stable distribution Regular stability Characteristic function e λ ( − it ) γ =: f St ( t ) d X 1 + · · · + X n = S n = b n X + a n ; Stability under random summation = X ( n ) d + · · · + X ( n ) ν n ; Characteristic function L γ ( − ln f St ( t )) X 1 Discrete stability � X ”Binomial operation” B j , where B j ∼ Bernoulli( α ) α ◦ X = j =1 and X is some discrete r.v. ( X ∈ Z + ) Then the r.v. X is discrete stable if 1 Ch. f. e λ ( e it − 1) γ d X = N 1 /γ ◦ ( X 1 + · · · + X N ) ; 21 of 25

Discrete models for financial data (random summation revisited) Recall: characteristic function of a Poisson process g ( t ) = exp { λ ( e it − 1) } , . ”Discrete Brownian Motion” Its characteristic function is g ( t ) = exp { λ 1 ( e ia 1 t − 1) + λ 2 ( e ia 2 t − 1) } . Advantages : • Explicit link both with random summation and with regular Brownian Motion • Simple analytic results for first hitting time etc. 22 of 25

Real data example 23 of 25

Simulated ”Discrete Br. Motion” 24 of 25

Thanks Thank you very much for your attention! 25 of 25