Application to Climate Extremes and Contagion in Finance

Tools and models for the study of some spatial and network risks: application to climate extremes and contagion in ﬁnance

Erwan KOCH

ETH Zürich (Department of Mathematics, RiskLab)

November 5, 2015 Journée 100% Actuaires

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 1 / 34 Outline

1 Introduction

2 A frailty-contagion model for multi-site hourly precipitation

3 Estimation of max-stable processes by simulated maximum likelihood

4 Spatial risk measures and applications to max-stable processes

5 Conclusion

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 2 / 34 Introduction Weather-related catastrophes

Number of weather-related catastrophes, 1970-2013, according to the sigma 2013 report.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 3 / 34 Introduction Economic losses from extreme weather events

Economic losses from extreme weather events, 1970-2013, according to the sigma 2013 report.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 4 / 34 Introduction Insured catastrophe losses

Insured catastrophe losses, 1970-2013, according to the sigma 2013 report.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 5 / 34 Introduction Motivation, modeling and links between chapters

Motivation, modeling and links between chapters Crucial for authorities and insurance/reinsurance companies to precisely assess and predict the risk of extreme environmental events. =⇒ Adapted tools in Chapters 2, 3 and 4: Chapters 2 and 3 deal with models for environmental hazards; Chapter 4 deals with adapted risk measures, implying the conversion of environmental variables into economic costs. We also have to take systemic risk into account, i.e. risk at the scale of the whole ﬁnancial system. =⇒ See Chapter 5 (not presented here).

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 6 / 34 Introduction Motivation, modeling and links between chapters (continued)

Spatial statistics Meteorological and more generally environmental events have a natural spatial extent. Three main kinds of spatial statistics: Geostatistics (Matheron, 1965) =⇒ Used in Chapters 3 and 4; Markov random ﬁelds =⇒ Used in Chapters 2 and 5; Theory of point processes.

Another similarity between Chapters 2 and 5: an exogenous cause triggers consequences by endogenous contagion.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 7 / 34 A frailty-contagion model for multi-site hourly precipitation The data

Three weather stations in northern Brittany (France) with hourly winter precipitation records from 1995 to 2011.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 8 / 34 A frailty-contagion model for multi-site hourly precipitation Weather generators

Deﬁnition A model that can produce realistic (in a statistical sense) series of weather-related variables. =⇒ Very important in risk management (e.g. as inputs of ﬂooding models).

Literature To the best of our knowledge, no hourly precipitation generator so far; Generally, occurrence and intensity considered as independent (Katz, 1977; Kleiber et al., 2012, ...); Difﬁculty to reproduce the large precipitation amounts and the persistence of dry periods.

Weather stations Intensity range (mm) Brest-Guipavas Landivisiau Pleyber-Christ 0.2-0.8 0.63 0.65 0.68 0.8-1.4 0.74 0.75 0.79 1.4-2 0.81 0.82 0.82 2-4 0.90 0.91 0.91 > 4 0.94 0.94 0.94 Chi-Square test p-value 8.40 × 10−64 4.57 × 10−59 4.39 × 10−43

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 9 / 34 A frailty-contagion model for multi-site hourly precipitation The model

Let Pm,t (m = 1,..., M; t = 1,..., T ) be the precipitation amount at station m recorded during the tth hour.

The model

( 0 0 BmPt−1 + m,t if BmPt−1 + m,t ≥ u Pm,t = 0 0 if BmPt−1 + m,t < u where:

Bm is a vector of autoregressive coefﬁcients;

The m,t are independent and satisfy, conditionally to an observable common factor Ft , 0 m,t ∼ N 0, exp(θ Ft ) ;

Ft is a vector of d atmospheric explanatory variables at time t; The threshold u is positive.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 10 / 34 A frailty-contagion model for multi-site hourly precipitation Estimates

Matrix B Brest-Guipavas Landivisiau Pleyber-Christ Brest-Guipavas 0.65 -0.08 0.11 [0.59 ; 0.74] (87) [-0.15 ; 0.01] (86) [0.02 ; 0.19] (89) Landivisiau 0.47 0.25 0.02 [0.41 ; 0.53] (81) [0.17 ; 0.32] (92) [-0.06 ; 0.09] (91) Pleyber-Christ 0.22 0.10 0.36 [0.16 ; 0.27 ] (77) [0.02 ; 0.17] (82) [0.30 ; 0.43] (80)

Atmospheric variables Estimates Conﬁdence interval Coverage probability ˆ Regression intercept θ0 = 30.626 [28.02 ; 32.32] 80 ˆ Temperatures θ1 = 0.070 [0.064; 0.076] 38 ˆ Seal level pressures θ2 = −0.034 [-0.037 ; -0.031] 94 ˆ Humidity θ3 = 0.028 [0.022; 0.036] 94

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 11 / 34 A frailty-contagion model for multi-site hourly precipitation Forecast scheme

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 12 / 34 A frailty-contagion model for multi-site hourly precipitation Comparison of the dynamics: out-sample time series

Time (weeks) 60 61 62

Brest-Guipavas 6 5 4 3 2 Rainfall (mm) Rainfall 1 0

6 Landivisiau 5 4 3 2 1 0

Pleyber-Christ 6 5 4 3 2 Rainfall (mm) Rainfall 1 0 10001 10101 10201 Time 10301 10401 10501 Time (h)

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 13 / 34 A frailty-contagion model for multi-site hourly precipitation Out-sample intensities

Brest-Guipavas Landivisiau Pleyber-Christ 11 10 9 8 7 6 Median_OS Median_OS Median_OS 5 4 Simulated intensity (mm) intensity Simulated 3 2 1 0

0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 Quantiles_OS ObservedQuantiles_OS intensity (mm) Quantiles_OS

Quantile-quantile plot between observed rainfall amount (x-axis) and simulated one (y-axis) from the model. The gray color corresponds to the 98% conﬁdence interval and the solid line to the median.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 14 / 34 A frailty-contagion model for multi-site hourly precipitation Out-sample dry periods

Brest-Guipavas Landivisiau Pleyber-Christ 350 300 250 200 Median_OS Median_OS Median_OS 150 100 Simulated lengths of dry periods (h) of dry periods lengths Simulated 50 0

0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Quantiles_OS Observed lengthsQuantiles_OS of dry periods (h) Quantiles_OS

Quantile-quantile plot between observed dry periods length (x-axis) and simulated one (y-axis) from the model. The gray color corresponds to the 98% conﬁdence interval and the solid line to the median.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 15 / 34 Estimation of max-stable processes by simulated maximum likelihood Why max-stable processes?

∗ For d ∈ N , let us consider i.i.d. replications Ti (x), i = 1,..., n of a stochastic process {T (x)}x∈Rd having continuous sample paths. In the case where d ∈ {1, 2, 3}, T (x) can for instance be some environmental variable. Let {an(x)}x∈Rd > 0 and {bn(x)}x∈Rd be sequences of continuous functions.

Theorem

If there exists a non degenerate process {G(x)}x∈Rd such that n maxi=1 Ti (x) − bn(x) d → {G(x)}x∈Rd , for n → ∞, an(x) x∈Rd then G(x) is necessarily max-stable.

Deﬁnition (Max-stable process)

A stochastic process {G(x)}x∈Rd with continuous sample paths is said to be max-stable if there exist sequences of continuous functions{cT (x)}x∈Rd > 0 and {dT (x)}x∈Rd such that if Gt (x), t = 1,..., T are i.i.d. replications of G(x),

( T ) maxt=1 Gt (x) − dT (x) d = {G(x)}x∈Rd . cT (x) x∈Rd

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 16 / 34 Estimation of max-stable processes by simulated maximum likelihood Parametric max-stable models and estimation

Parametric max-stable models Using the spectral representations by de Haan (1984) and Schlather (2002), different parametric models have been introduced: The Smith model (Smith, 1990); The Schlather model (Schlather, 2002); The Brown-Resnick model (Kabluchko et al., 2009).

Estimation of these processes Multivariate density not available =⇒ Composite (pairwise or tripletwise) approaches are used (Padoan et al., 2010; Genton et al., 2011); =⇒ Corresponding estimators are not efﬁcient: a large number of temporal observations is required.

However, these processes can be simulated!

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 17 / 34 Estimation of max-stable processes by simulated maximum likelihood Multivariate density approximation

For t ∈ {1,..., T }, Zt denotes a max-stable vector of dimension M (the number of sites).

Multivariate kernel density estimator (Silverman, 1986; Scott, 1992)

S 1 X lS(Z , Σ) = K Z − ZS(Σ) t S H t s s=1 with: S the number of simulations; S Zs (Σ) the s-th simulated max-stable vector;

KH a multivariate kernel function associated to the multivariate bandwidth matrix H; We consider the bandwidth matrix H = h I, where h must be understood as a function of S tending to 0 as S tends to +∞.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 18 / 34 Estimation of max-stable processes by simulated maximum likelihood Simulated log-likelihood

Simulated log-likelihood: Lerman and Manski(1981), Fermanian and Salanié (2004)

T S X h S i h S i LT (Σ) = τS l (Zt , Σ) ln l (Zt , Σ) , t=1 where τS is a trimming function satisfying, for δ ≥ 0,

 0 if |u| < hδ  δ δ τS(u) = sufﬁciently regular if |u| ∈ [h , 2h ]  1 if |u| > 2hδ

The non parametric simulated maximum likelihood estimator

ˆ S S ΣT = arg max LT (Σ) Σ

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 19 / 34 Estimation of max-stable processes by simulated maximum likelihood Theoretical properties

Theorem 1: Consistency In the case of the Smith process, under assumptions on the bandwidth matrix ˆ S and the kernel and regularity assumptions on the real likelihood function, ΣT is strongly consistent. Almost surely,

ˆ S lim ΣT = Σ0. S,T →∞

Theorem 2: Asymptotic normality and efﬁciency In the case of the Smith process, under assumptions on the bandwidth matrix ˆ S and the kernel and regularity assumptions on the real likelihood function, ΣT is asymptotically normal and asymptotically efﬁcient: √ ˆ S d T (ΣT − Σ0) −→ N(0, Ω), S,T →∞ where Ω is the variance matrix of the true maximum likelihood estimator.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 20 / 34 Estimation of max-stable processes by simulated maximum likelihood Simulation results

Max-stable process to estimate We simulate a Smith process whose variance/covariance matrix is:

200 150 Σ = 0 150 300

In all the following experiments, S = 106. Statistics are computed on 20 replications.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 21 / 34 Estimation of max-stable processes by simulated maximum likelihood Simulation results (continued)

Experiment T = 5, M = 5.

True Simulated likelihood Pairwise likelihood

Mean Bias Rel Sd Mean Bias Rel Sd Cov11=200 244.0 44.0 0.22 95.8 197931.8 3480579.2 17402.9 15305579.7 Cov22=300 313.8 13.8 0.05 51.6 815606.1 197631.8 658.8 807980.5 Cov12=150 111.6 -38.4 -0.26 79.4 15305579.7 815456.1 5436.4 3515785.2

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 22 / 34 Estimation of max-stable processes by simulated maximum likelihood Simulation results (continued)

Experiment T = 30, M = 5.

True Simulated likelihood Pairwise likelihood

Mean Bias Rel Sd Mean Bias Rel Sd Cov11=200 211.7 11.7 0.06 22.5 211.8 11.8 0.06 61.8 Cov22=300 313.7 13.7 0.05 20.0 344.3 44.3 0.15 182.9 Cov12=150 147.2 -2.8 -0.02 12.7 163.1 13.1 0.09 109.3

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 23 / 34 Estimation of max-stable processes by simulated maximum likelihood The case where M is high: clustering

The PAM algorithm (Kaufman and Rousseeuw, 1987) Clustering algorithm: representatives are medoids.

We choose the F-madogram distance (Cooley et al., 2006).

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 24 / 34 Estimation of max-stable processes by simulated maximum likelihood Simulation results (continued)

Experiment T = 1, M = 30 (using PAM).

True Simulated likelihood Pairwise likelihood

Mean Bias Rel Sd Mean Bias Rel Sd Cov11=200 227.4 27.4 0.14 20.8 731.4 531.4 2.66 667.5 Cov22=300 317.4 17.4 0.06 29.3 1301.2 1001.2 3.34 931.8 Cov12=150 129.8 -20.2 -0.13 25.7 568.1 418.1 2.79 698.9

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 25 / 34 Spatial risk measures and applications to max-stable processes Motivation

The aims Disentangle the role of space and the role of the economic cost due to a speciﬁc hazard; Study the sensitivity of the risk with respect to space. =⇒ Spatial risk measures.

Literature Multivariate quantiles and Value-at-Risk (VaR): see e.g. Serﬂing (2008), Embrechts and Puccetti (2006); Spatial risk measures in a network of institutions: Föllmer (2014), Föllmer and Klüppelberg (2014).

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 26 / 34 Spatial risk measures and applications to max-stable processes Notations

We introduce: A = {A measurable : A ⊂ R2, |A| > 0 and |A| < ∞}, where |.| denotes the Lebesgue measure; P a family of distributions of stochastic processes on R2 having continuous sample paths.

Deﬁnition 1 (Normalized spatially aggregated loss) For A ∈ A and P ∈ P, the normalized spatially aggregated loss is deﬁned by Z CP (x) dx L (A, P) = A , (1) N |A| where the stochastic process {CP (x)}x∈R2 has distribution P.

Each process represents the economic or insured cost caused by the events belonging to a speciﬁed class and occurring during a given time period.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 27 / 34 Spatial risk measures and applications to max-stable processes Spatial risk measures

Deﬁnition 2 (Spatial risk measure)

A spatial risk measure is a function RΠ that associates to any region A ∈ A and to any distribution P ∈ P a real number

RΠ : A × P → R (A, P) 7→ RΠ(A, P) = Π[LN (A, P)], where LN (A, P) is deﬁned in (1) and Π is a classical risk measure.

If the distribution P of the economic (or insured) cost process is given, then the function RΠ(., P) summarizes, for any region belonging to A, the risk caused by the hazard characterized by P. RΠ(., P) is the spatial risk measure induced by P.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 28 / 34 Spatial risk measures and applications to max-stable processes Set of axioms for spatial risk measures

Natural to analyze how RΠ(A, P) evolves with respect to A for a given P.

Definition 3 (Set of axioms for spatial risk measures) For a fixed P ∈ P, we define the following axioms for the spatial risk measure induced by P:

1 Spatial invariance under translation: 2 For all v ∈ R and A ∈ A, RΠ(A + v, P) = RΠ(A, P), where A + v denotes the region A translated by the vector v; 2 Spatial sub-additivity: For all A1, A2 ∈ A, RΠ(A1 ∪ A2, P) ≤ min(RΠ(A1, P), RΠ(A2, P)); 3 Asymptotic spatial homogeneity of order −α, α > 0: For all A ∈ A, K2 1 RΠ(λA, P) = K1 + + o , λ→∞ λα λα where λA is the area obtained by applying an homothety of rate λ to A with respect to its center and K1, K2 ∈ R. Note that K1 and K2 can depend on A; 4 Spatial anti-monotonicity: For all A1, A2 ∈ A, A1 ⊂ A2 ⇒ RΠ(A2, P) ≤ RΠ(A1, P).

Proposition 1 The properties of spatial sub-additivity and spatial anti-monotonicity are equivalent.

In the case of a stationary process CP , there is spatial invariance under translation.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 29 / 34 Spatial risk measures and applications to max-stable processes Model for the economic cost process

The economic cost stochastic process {CP (x)}x∈R2 is modeled as follows:

CP (x) = E(x) D (Z(x)) , where:

{Z (x)}x∈R2 is the stochastic process of the environmental variable generating the cost (e.g. the temperature or the wind speed); D(.) is a damage function giving the destruction percentage associated to a given value of the environmental variable generating cost;

{E(x)}x∈R2 is the (deterministic) exposure process, involving especially demographic, economic and topographic conditions.

Simpliﬁcation: ∀x ∈ R2, E(x) = 1.

Final model for the economic cost process

CP (x) = D (Z (x)) , where D is deﬁned as

1 D (Z (x)) = I{Z(x)>u}, where u > 0 (appropriate for temperatures); 2 D (Z (x)) = Z (x)β , where β > 0 (appropriate for wind speeds); and {Z (x)}x∈R2 is a max-stable process with standard Fréchet margins (simple max-stable).

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 30 / 34 Spatial risk measures and applications to max-stable processes Application to max-stable processes

Spatial risk measures considered

R1(A, P) = E [LN (A, P)];

R2(A, P) = Var [LN (A, P)];

R3(A, P) = VaR [LN (A, P)].

Some results from the paper

Detailed study of R2(λA) for the Smith, the Schlather, the Brown-Resnick process and a new process, the tube model, in the case of the threshold damage function. =⇒ Analysis of the spatial diversiﬁcation;

Detailed study of R2(λA) for the Smith process in the case of the threshold damage function;

Detailed study of the axioms of R2 in the previous cases.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 31 / 34 Conclusion Some perspectives

Extension of our precipitation model:

Introduction of spatial dependence in the noise m,t ; Introduction of a nonparametric dependence between σt and Ft . Elaboration of new space-time max-stable models; Further study of spatial risk measures; CAT bonds pricing via an appropriate loss modeling (using Chapter 4); Study of high-risk scenarios.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 32 / 34 Conclusion References

Chapter 2

Katz, R. (1977). PRECIPITATION AS A CHAIN-DEPENDANTPROCESS. Journal of Applied Meteorology, 16:67-676. Koch, E. and Naveau, P. (2015). A FRAILTY-CONTAGION MODEL FOR MULTI-SITE HOURLY PRECIPITATION DRIVEN BY ATMOSPHERIC COVARIATES. Advances in Water Resources, 78:145-154. Kleiber, W., Katz, R., and Rajagopalan, B. (2012). DAILY SPATIOTEMPORAL PRECIPITATION SIMULATION USINGLATENTANDTRANSFORMEDGAUSSIANPROCESSES. Water Ressources Research, 48(1). Wilks, D. (1998). MULTISITE GENERALIZATION OF A DAILY STOCHASTIC PRECIPITATION GENERATION MODEL. Journal of Hydrology, 210(1):178-191.

Chapter 3

de Haan, L. (1984). A SPECTRAL REPRESENTATION FOR MAX-STABLE PROCESSES. The Annals of Probability, 12(4):1194–1204. Fermanian, J.-D. and Salanié, B. (2004). A NONPARAMETRIC SIMULATED MAXIMUM LIKELIHOOD ESTIMATION METHOD. Econometric Theory, 20(04):701-734. Koch, E., Ribereau, P. and Robert, C. (2015) ESTIMATION OF MAX-STABLE PROCESSES BY SIMULATEDMAXIMUMLIKELIHOOD. To be submitted. Padoan, S. A., Ribatet, M., and Sisson, S. A. (2010). LIKELIHOOD-BASEDINFERENCEFORMAX-STABLE PROCESSES. Journal of the American Statistical Association, 105(489):263-277. Smith, R. L. (1990). MAX-STABLE PROCESSES AND SPATIAL EXTREMES. Unpublished manuscript.

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 33 / 34 Conclusion References

Chapter 4

Embrechts, P. and Puccetti, G. (2006). BOUNDS FOR FUNCTIONS OF MULTIVARIATE RISKS. Journal of Multivariate Analysis, 97(2):526-547. Föllmer, H. (2014). SPATIAL RISK MEASURES AND THEIR LOCAL SPECIFICATION:THE LOCALLY LAW-INVARIANT CASE. Statistics and Risk Modeling, 31(1):79-101. Föllmer, H. and Klüppelberg, C. (2014). SPATIAL RISK MEASURES:LOCAL SPECIFICATION AND BOUNDARYRISK. Stochastic Analysis and Applications 2014, 307-326. Koch, E. (2015). SPATIAL RISK MEASURES AND APPLICATIONS TO MAX-STABLE PROCESSES. In revision.

Thank you for your attention!

Erwan KOCH (ETH Zürich) PhD thesis November 5, 2015 34 / 34