A New Approach to Assessing Model Risk in High Dimensions Carole Bernard (University of Waterloo) and Steven Vanduffel (Vrije Universiteit Brussels) ARIA meeting 2014, Seattle Carole Bernard A new approach to assessing model risk in high dimensions 1
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Objectives and Findings • Model uncertainty on the risk assessment of the sum of d dependent risks (portfolio). ◮ Given all information available in the market, what can we say about the maximum and minimum possible values of a given risk measure of a portfolio? • A non-parametric method based on the data at hand. • Analytical expressions for the maximum and minimum Carole Bernard A new approach to assessing model risk in high dimensions 2
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Objectives and Findings • Model uncertainty on the risk assessment of the sum of d dependent risks (portfolio). ◮ Given all information available in the market, what can we say about the maximum and minimum possible values of a given risk measure of a portfolio? • A non-parametric method based on the data at hand. • Analytical expressions for the maximum and minimum • Implications: ◮ Current VaR based regulation is subject to high model risk, even if one knows the multivariate distribution almost completely. ◮ We can identify for which risk measures it is meaningful to develop accurate multivariate models. Carole Bernard A new approach to assessing model risk in high dimensions 2
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Model Risk 1 Goal: Assess the risk of a portfolio sum S = � d i =1 X i . 2 Choose a risk measure ρ ( · ): variance, Value-at-Risk... 3 “Fit” a multivariate distribution for ( X 1 , X 2 , ..., X d ) and compute ρ ( S ) 4 How about model risk? How wrong can we be? � d � d � �� � �� ρ + � ρ − � F := sup ρ , F := inf ρ X i X i i =1 i =1 where the bounds are taken over all other (joint distributions of) random vectors ( X 1 , X 2 , ..., X d ) that “agree” with the available information F Carole Bernard A new approach to assessing model risk in high dimensions 3
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Assessing Model Risk on Dependence with d Risks ◮ Marginals known and dependence fully unknown ◮ A challenging problem in d � 3 dimensions • Puccetti and R¨ uschendorf (2012): algorithm (RA) useful to approximate the minimum variance. • Embrechts, Puccetti, R¨ uschendorf (2013): algorithm (RA) to find bounds on VaR ◮ Issues • bounds are generally very wide • ignore all information on dependence. ◮ Our answer: • We incorporate in a natural way dependence information. Carole Bernard A new approach to assessing model risk in high dimensions 4
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Rearrangement Algorithm N = 4 observations of d = 3 variables: X 1 , X 2 , X 3 1 1 2 0 6 3 M = 4 0 0 6 3 4 Each column: marginal distribution Interaction among columns: dependence among the risks Carole Bernard A new approach to assessing model risk in high dimensions 5
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Same marginals, different dependence ⇒ Effect on the sum! X 1 + X 2 + X 3 1 1 2 4 0 6 3 9 S N = 4 0 0 4 6 3 4 13 X 1 + X 2 + X 3 16 6 6 4 10 4 3 3 S N = 3 1 1 2 0 0 0 0 Aggregate Risk with Maximum Variance comonotonic scenario Carole Bernard A new approach to assessing model risk in high dimensions 6
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Rearrangement Algorithm: Sum with Minimum Variance minimum variance with d = 2 risks X 1 and X 2 Antimonotonicity: var ( X a 1 + X 2 ) � var ( X 1 + X 2 ) How about in d dimensions? Carole Bernard A new approach to assessing model risk in high dimensions 7
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Rearrangement Algorithm: Sum with Minimum Variance minimum variance with d = 2 risks X 1 and X 2 Antimonotonicity: var ( X a 1 + X 2 ) � var ( X 1 + X 2 ) How about in d dimensions? Use of the rearrangement algorithm on the original matrix M . Aggregate Risk with Minimum Variance ◮ Columns of M are rearranged such that they become anti-monotonic with the sum of all other columns. ∀ k ∈ { 1 , 2 , ..., d } , X a � k antimonotonic with X j j � = k � � � � X a k + � X k + � ◮ After each step, var j � = k X j � var j � = k X j where X a k is antimonotonic with � j � = k X j Carole Bernard A new approach to assessing model risk in high dimensions 7
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Aggregate risk with minimum variance Step 1: First column ↓ X 2 + X 3 6 6 4 10 0 6 4 4 3 2 5 becomes 1 3 2 1 1 1 2 4 1 1 0 0 0 0 6 0 0 Carole Bernard A new approach to assessing model risk in high dimensions 8
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Aggregate risk with minimum variance ↓ X 2 + X 3 6 6 4 10 0 6 4 4 3 2 5 becomes 1 3 2 1 1 2 1 1 1 4 0 0 0 0 0 0 6 ↓ X 1 + X 3 0 6 4 4 0 3 4 1 2 3 becomes 1 2 3 6 4 1 5 4 1 1 1 6 0 0 6 6 0 0 ↓ X 1 + X 2 0 3 3 0 3 4 4 1 6 7 1 6 2 becomes 0 4 1 1 5 4 1 2 6 0 0 6 6 0 1 Carole Bernard A new approach to assessing model risk in high dimensions 9
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Aggregate risk with minimum variance Each column is antimonotonic with the sum of the others: ↓ X 2 + X 3 ↓ X 1 + X 3 ↓ X 1 + X 2 7 4 3 0 3 4 0 3 4 0 3 4 1 6 0 6 , 1 6 0 1 , 1 6 0 7 4 1 2 3 4 1 2 6 4 1 2 5 1 7 6 6 0 1 6 0 1 6 0 1 X 1 + X 2 + X 3 0 3 4 7 7 1 6 0 S N = 4 1 2 7 7 6 0 1 The minimum variance of the sum is equal to 0! (ideal case of a constant sum ( complete mixability , see Wang and Wang (2011)) Carole Bernard A new approach to assessing model risk in high dimensions 10
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Bounds on variance Analytical Bounds on Standard Deviation Consider d risks X i with standard deviation σ i 0 � std ( X 1 + X 2 + ... + X d ) � σ 1 + σ 2 + ... + σ d Example with 20 normal N(0,1) 0 � std ( X 1 + X 2 + ... + X 20 ) � 20 and in this case, both bounds are sharp and too wide for practical use! Our idea: Incorporate information on dependence. Carole Bernard A new approach to assessing model risk in high dimensions 11
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Illustration with 2 risks with marginals N(0,1) 3 2 1 X 2 0 −1 −2 −3 −3 −2 −1 0 1 2 3 X 1 Carole Bernard A new approach to assessing model risk in high dimensions 12
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Illustration with 2 risks with marginals N(0,1) 3 2 1 X 2 0 −1 −2 −3 −3 −2 −1 0 1 2 3 X 1 2 � Assumption: Independence on F = { q β � X k � q 1 − β } k =1 Carole Bernard A new approach to assessing model risk in high dimensions 13
Introduction Model Risk Bounds on variance Dependence Information Value-at-Risk bounds Conclusions Our assumptions on the cdf of ( X 1 , X 2 , ..., X d ) F ⊂ R d (“trusted” or “fixed” area) U = R d \F (“untrusted”). We assume that we know : (i) the marginal distribution F i of X i on R for i = 1 , 2 , ..., d , (ii) the distribution of ( X 1 , X 2 , ..., X d ) | { ( X 1 , X 2 , ..., X d ) ∈ F} . (iii) P (( X 1 , X 2 , ..., X d ) ∈ F ) ◮ When only marginals are known: U = R d and F = ∅ . ◮ Our Goal: Find bounds on ρ ( S ) := ρ ( X 1 + ... + X d ) when ( X 1 , ..., X d ) satisfy (i), (ii) and (iii). Carole Bernard A new approach to assessing model risk in high dimensions 14
Recommend
More recommend
