Model-free computation of risk contributions in credit portfolios ´ Alvaro Leitao and Luis Ortiz-Gracia Seminar Riskcenter IREA UB May 27, 2019 ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 1 / 35
Motivation In a financial institution, portfolio credit risk represents one of the most important sources of risk. The well-known VaR and ES risk measures are usually employed. Besides, the decomposition of the total risk into the individual risk contribution of each obligor is a problem of practical importance. Identification of risk concentrations, portfolio optimization or capital allocation are, among others, relevant examples of application. The problem of obtaining the risk contributions represents a great challenge from the computational standpoint. Commonly in practice: Monte Carlo methods. Easy to implement and understand, and attractive for practitioners, but rather expensive. ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 2 / 35
What we propose An alternative approach for computing risk contributions based on non-parametric density estimation based on wavelets. Once the density function of the loss variable is recovered, we derive closed-form solutions for VaR and ES . According to the Euler’s capital allocation principle, the risk contributions can be calculated by taking partial derivatives of the risk measures ( VaR or ES ) w.r.t. the individual exposures. Thanks to the wavelet properties, these partial derivatives can be efficiently computed, obtaining high precision. The presented methodology is model-free, in the sense that it applies in the same manner regardless of the model driving the losses. ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 3 / 35
Outline Problem formulation 1 Risk measures and risk contributions 2 Wavelet-based estimation of the loss distribution 3 Numerical experiments 4 Conclusions 5 ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 4 / 35
Problem formulation Let us consider a portfolio consisting in N obligors. Each obligor j is characterized by the exposure at default , E j , the probability of default , P j , and the loss given default , 100%. We follow the framework of Merton’s firm-value model. Let V j ( t ) denote the firm value of obligor j at time t < T , where T is the time horizon (typically one year). The obligor j defaults when its value at the end of the observation period, V j ( T ), falls below a certain threshold, τ j , i.e, V j ( T ) < τ j . We can therefore define the default indicator as D j = 1 { V j ( T ) <τ j } . Given D j , the individual loss of obligor j is defined as, L j = D j · E j , while the total loss in the portfolio reads, N � L = L j . j =1 ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 5 / 35
Factor models The firm (obligor) value V j is split into two terms: one common component called systematic factor, and an idiosyncratic component for each obligor. Depending on the number of factors of the systematic part, the model can be classified into the one- or multi-factor class. One-factor models : Gaussian copula and t -copula � ν V j = √ ρ j Y + � √ ρ j Y + � � � 1 − ρ j ε j , V j = 1 − ρ j ε j , W where ε 1 , · · · , ε N , Y ∼ N (0 , 1), W follows a chi-square distribution χ 2 ( ν ) with ν degrees of freedom and ε 1 , · · · , ε N , Y and W are mutually independent. The parameters ρ 1 , · · · , ρ j ∈ (0 , 1) are the correlation coefficients. ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 6 / 35
Factor models When we need to capture complicated correlation structures, extend the previous models to multiple dimensions. Multi-factor models : V j = ❛ T ❥ ❨ + b j ε j , j = 1 , · · · , N . where ❨ = [ Y 1 , Y 2 , . . . , Y d ] T denotes the systematic risk factors. Here, ❛ ❥ = [ a j 1 , a j 2 , . . . , a jd ] T represents the factor loadings satisfying ❛ T ❥ ❛ ❥ < 1, and b j , being the factor loading of the idiosyncratic risk � � � a 2 j 1 + a 2 j 2 + · · · + a 2 factor, b j = 1 − , ensuring V j ∼ N (0 , 1). jd Similarly, the multi-factor t -copula model definition reads, � ν � � ❛ T V j = ❥ ❨ + b j ε j , j = 1 , · · · , N , W where ❨ , ε j , ❛ ❥ and b j are defined as before, with W ∼ χ 2 ( ν ). ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 7 / 35
Risk measures We will use two well-known measures of risk, the value-at-risk , VaR , and the expected shortfall , ES . Definition Given a confidence level α ∈ (0 , 1) and the vector of exposures ❊ = [ E 1 , E 2 , . . . , E N ] T , we define the portfolio VaR , VaR α ( ❊ ) = inf { l ∈ R : P ( L ≤ l ) ≥ α } = inf { l ∈ R : F L ( l ; ❊ ) ≥ α } , where F L is the distribution function of the total loss random variable L (we emphasize the dependence of VaR with respect to the risk exposures). ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 8 / 35
Risk measures Definition Given the loss variable L with E [ | L | ] < ∞ and distribution function F L , the ES at confidence level α ∈ (0 , 1) is defined as, � 1 1 ES α ( ❊ ) = VaR u ( ❊ ) d u . 1 − α α When the loss variable is integrable with continuous distribution function, then the ES satisfies the equation, ES α ( ❊ ) = E [ L | L ≥ VaR α ( ❊ )] , or, in integral form, � + ∞ 1 ES α ( ❊ ) = xf L ( x ; ❊ ) d x , 1 − α VaR α ( ❊ ) where f L is the probability density function of the total loss random variable L . ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 9 / 35
Risk contributions The goal is allocating the risk to the elements of the portfolio, based on their individual contribution to the risk measure. This problem is also known as capital allocation and a solution to this problem is the Euler’s capital allocation principle , which states N N ∂ VaR α ∂ ES α � � ( ❊ ) = VaR α ( ❊ ) , and, ( ❊ ) = ES α ( ❊ ) . E j E j ∂ E j ∂ E j j =1 j =1 The contribution of obligor j to the VaR ( ES ) at confidence level α , ∂ VaR α ∂ ES α VaRC α, j := E j ( ❊ ) , and, ESC α, j := E j ( ❊ ) . ∂ E j ∂ E j It can be shown that, VaRC α, j = E [ L j | L = VaR α ( ❊ )] , j = 1 , . . . , N , and, ESC α, j = E [ L j | L ≥ VaR α ( ❊ )] , j = 1 , . . . , N . ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 10 / 35
Non-parametric density estimation by wavelets Given i.i.d samples from an unknown statistical distribution X . Apply the wavelet theory to approximate the density function f X . We consider the so-called linear wavelet estimator (or simply linear estimator ), � f X ( x ) ≈ c m , k φ m , k ( x ) , k where k varies within a finite range and φ m , k ( x ) = 2 m / 2 φ (2 m x − k ). The function φ is usually referred to as the scaling function or father wavelet . The coefficients c m , k are, by definition, given by, � f X ( x )¯ � ¯ � c m , k := � f X , φ m , k � = φ m , k ( x ) d x = E φ m , k ( X ) . R The last equality comes from the fact that f X is a density function. ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 11 / 35
Application to the loss distribution Estimate the density function of the loss variable L by means of the wavelet estimator. Generate n samples of the default indicator variable, D i j , of obligor j and sample i , where i = 1 , . . . , n , j = 1 . . . , N . Then, L i j = D i j · E j . Denote by L i the corresponding samples L i = � N j =1 L i j . For convenience, we consider the transformation Z = L − a b − a , and we define Z i = L i − a b − a , i = 1 , . . . , n , where, L i � L i � � � a = min , b = max . 1 ≤ i ≤ n 1 ≤ i ≤ n From the definition, we can obtain the following unbiased estimator for the wavelet series coefficients, n ≈ 1 � ¯ � φ m , k ( Z i ) =: ˆ � c m , k = E φ m , k ( Z ) c m , k . n i =1 ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 12 / 35
Application to the loss distribution Using the wavelet density estimation, the unknown density f L of L can be approximated as follows, K 1 � x − a � f L ( x ; ❊ ) ≈ ˆ � f L ( x ; ❊ ) := c m , k φ m , k ˆ , b − a b − a k =0 where, by construction, the lower bound for index k is equal to zero and the upper limit is K = 2 m − 1. Applying the definition, the distribution function of L can be estimated by � x F L ( x ; ❊ ) := f L ( y ; ❊ ) d y −∞ � x K 1 � y − a � � d y =: ˆ ≈ ˆ φ m , k F L ( x ; ❊ ) . c m , k b − a b − a a k =0 ´ A. Leitao & L. Ortiz-Gracia Model-free risk contributions May 27, 2019 13 / 35
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