On the Systemic Nature of Weather Risk

Transcription

1 Martin Odening 1 Ostap Okhrin 2 Wei Xu 1 Department of Agricultural Economics 1 Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics 2 Humboldt Universität zu Berlin

2 Motivation 1-1 Agricultural Ins. Systems Ins. Premium Catastrophe Partici- Reinsu- Country coverage subsidies aid pation rance Germany hail, suppl. ins. none only for uninsureable risks approx.35% hail pri. ins. <1% MPCI France multiple peril crop ins. 60% government aid for natural disasters (drought, 20% pri. ins. earthquake, flooding) Greece comprehensive ins. 50% n.a. n.a. n.a. Italy hail, frost, drought 60% for hail 80% for MPCI only for uninsureable risks n.a. pri. ins. Luxem- comprehensive bourg ins. up to 50% n.a. 10% n.a. Austria comprehensive ins. 50% for hailand frost ins. only for uninsureable risks 78% hail 56% MPCI priv. ins. exclusively Spain comprehensive ins. 55% only for extreme disasters approx. 42% pri.and pub. ins. Canada multiple peril 50% for extreme and uninsurable 50% pri. and crop ins. disasters pub. ins. USA multiple peril crop ins. 60% only for uninsurable disasters 80% pri. and pub. ins. Table 1: Agricultural Insurances Systems

3 Motivation 1-2 Pearson Correlation Coefficients vs. Distance: normal yield years Figure 1: Goodwin, B.K.(2001)

4 Motivation 1-3 Pearson Correlation Coefficients vs. Distance: extreme yield years Figure 2: Goodwin, B.K.(2001)

5 Motivation 1-4 Objectives & Research Questions Quantification of the dependence structure of weather events at different locations Does the dependence of weather events fade out with increasing distance? Is spatial diversification of systemic weather risk possible? How to measure systemic weather risk correctly?

6 Motivation 1-5 Outline 1. Motivation 2. Copula: Modeling and Estimation 3. Application 4. Conclusion

7 Copula: Modeling and Estimation 2-1 Target: Buffer Fund n BF = Var α (NTL), NTL = w i (L i Π i ), i=1 L i = f (I i, K i ) V, Π i = E(L i ), BF buffer fund, NTL net total loss, L loss, Π fair premium, w weight, I weather index, K trigger level, V tick size, α confidence level, i region.

8 Copula: Modeling and Estimation 2-2 Copulae A copula maps a n-dimensional unit hypercube into the unit interval: C(u) = C(u 1,..., u n ), C : [0, 1] n [0, 1]. (1) A copula can be understood as a multivariate distribution function with all marginals being uniformly distributed C(u 1,..., u n ) = P(U 1 u 1,..., U n u n ). (2) Sklar s Theorem: If F is a multivariate distribution function with marginals F 1,..., F n then there exists a (unique) copula C such that F (x 1,..., x n ) = C(F (x 1 ),..., F (x n )). (3)

9 Copula: Modeling and Estimation 2-3 Copula Classes 1. independence 2. perfect dependence 3. implicit copulae e.g. Gaussion, Student 4. explicit copulae e.g. Gumbel, Clayton, Frank

10 Copula: Modeling and Estimation 2-4 Simplest Copulae 1. independence Π (u 1, u 2 ) = u 1 u 2 (4) 2. perfect dependence M (u 1, u 2 ) = min (u 1, u 2) W (u 1, u 2 ) = max (u 1 + u 2 1, 0) (5) 3. implicit copulas e.g. Gaussian, Student

11 Copula: Modeling and Estimation 2-5 Archimedean Copulae 4. explicit copulas Multivariate Archimedean Copula C : [0, 1] d [0, 1] defined as C (u 1,..., u d ) = φ{φ 1 (u 1 ) + + φ 1 (u d )} (6) where φ (0) = 1, φ (inf) = 0 and φ 1 its pseudo-inverse. Example: φ Clayton (u, θ) = (θu + 1) 1 θ ( ), where θ [ 1, ) \ {0} φ Gumbel (u, θ) = exp u 1 θ, where 1 θ < Disadvantages: too restrictive, single parameter, exchangeable

12 Copula: Modeling and Estimation 2-6 Copulae Product Copula Gaussian Copula Gumbel Copula Clayton Copula Gaussian margins t-margins Figure 3: Different meta distributions

13 Copula: Modeling and Estimation 2-7 Estimation of Copulae I (parametric) a) exact maximum likelihood Λ = ( θ, α 1,..., α n) = arg max {l(θ)} Λ T = ln [c{f 1 (x 1t ; α 1 ),..., F n(x nt; α n)}] + t=1 T t=1 j=1 n ln {F j (x jt ; α j )}. (7) b) IFM estimate parameters for marginals F j (x jt, α j ) estimate θ with ML conditional on α j

14 Copula: Modeling and Estimation 2-8 Estimation of Copulae II (semiparametric) Knowledge about Margins F k (x) = n F k (x) = n n I{X ik x}, (8) i=1 n ( x Xik K h i=1 ), (9) with K (x) = x κ (t) dt and κ : R R, κ = 1, h > 0 F k (x, α) - parametric distribution F k (x) - known distribution F k { F k (x), F k (x), F k (x, α), F k (x)}

15 Copula: Modeling and Estimation 2-9 Estimation of Copulae III (nonparametric) Ĉ (u 1,..., u d ) = 1 n C (u 1,..., u d ) = 1 n n i=1 k=1 n i=1 k=1 d I{ F (X ik ) u k }, (10) d K k { u k F k (X ik ) h k }. (11) If d = 2,one uses other local linear Kernel to avoid bias, see Chen and Huang, C(u 1,..., u d ; θ) - parametric copula

16 Copula: Modeling and Estimation 2-10 Estimation of Copulae IV (nonparametric) if F k (x) = F k (x, α) l (θ, α 1... α d ) = θ, α 1,, α d = arg n log c{f 1 (X i1 α 1 ),..., F d (X id ; α d ) ; θ} i=1 n d + log f k (X ik ; α k ) (12) i=1 k=1 if F k { F k (x), F k (x), F k (x, α)}) l (θ) = max l (θ, α 1,..., α d ) θ,α 1,...,α d n log c{ F 1 (X i1 ),..., F d (X id ) ; θ} (13) i=1 θ = arg max l (θ) θ

17 Copula: Modeling and Estimation 2-11 Copula: Goodness-of-Fit Tests Hypothesis H 0 : C θ C 0 ; θ Θ vs H 1 : C θ C 0 ; θ Θ, (14) Cramér von Mises S = n {Ĉ(u 1,..., u d ) C(u 1,..., u d ; θ)} 2 dĉ(u 1,..., u d) (15) [0,1] d Kolmogorov-Smirnov T = n sup u 1,...,u d [0,1] Ĉ(u 1,..., u d ) C(u 1,..., u d ; θ) (16) in practice p-values are calculated using the bootstrap methods described in Genest and Remillard (2008)

18 Copula: Modeling and Estimation 2-12 Simulation with Copulas marginal distribution marginal distribution of index for region 1 Copula of index for region 2 joint distribution of index distribution of sum of losses VaR

19 Application 3-1 Application Scen. S1 S2 S3 Description Brandenburg (4 contracts, 4 single stations) Germany (4 contracts, 4 single stations, underlined) Germany (4 contracts, average of station (colored)) Figure 4: Location of Weather Stations Table 2: Definition of Scenarios

20 Application 3-2 Weather Indices Cumulative Rainfall Index (CRI) Potential Flood Index (PFI) Growing Degree Days (GDD) Two alternative strike levels: 50% and 15% quantiles of the index distribution

21 Application 3-3 Cumulative Rainfall Index (CRI) CRI t = τ E where τ B is the first of April and τ E is June 30 Loss function for drought risk j=τ B P j,t, (17) L t = max (0, K CRI CRI t ) V, (18) where P j,t is the daily precipitation at day j in year t, and τ B and τ E are the beginning and the end of the vegetation period, respectively.

22 Application 3-4 Potential Flood Indicator (PFI) with s = 5; PFI t = max τ {1,..., }+1(t 1) 365 s+τ 1 P j j=τ Loss function for the risk of excessive rainfall, (19) L t = max (0, PFI t K PFI ) V, (20) where PFI is the rainfallsum of the wettest s- day - period within a year;

23 Application 3-5 Growing Degree Days (GDD) GDD t = τ E,t j=τ B,t max ( 0, T j T ), (21) where τ B,t is the first of March, τ E,t is October 31, L GDD is 5 C; Loss function for the risk of insufficient temperature L t = max (0, K GDD GDD t ) V, (22) where T is the triggering temperature.

24 Application 3-6 Marginal Distributions of Index 1 Contract Scenario A B C D S1 Lognormal Gamma Lognormal Weibull CRI S2 Lognormal Beta Gamma Gamma S3 Normal Beta Logistic Gamma S1 Lognormal Lognormal Lognormal Lognormal PFI S2 Lognormal Lognormal Lognormal Gamma S3 Lognormall Lognormal Beta Beta S1 Weibull Beta Weibull Weibull GDD S2 Weibull Gamma Weibull Gamma S3 Weibull Weibull Weibull Weibull Table 3: Trading Area 1 According to KS test, Chi-square test Anderson-Darling test

25 Application 3-7 Estimation Results for Parametric Copulas I Sce- Test P- nario Copula θ t-value BIC statistics value Rank Gumbel S1 Clayton Frank Gumbel S2 Clayton Frank Frank S3 Frank Frank Table 4: Cumulative Rainfall Index (CRI)

26 Application 3-8 Estimation Results for Parametric Copulas II Sce- Test P- nario Copula θ t-value BIC statistics value Rank Gumbel S1 Clayton Frank Gumbel ,05 NA 1 S2 Clayton ,22 NA 2 Frank ,62 NA 3 Gumbel S2 Clayton Frank Table 5: Potential Flood Indicator (PFI)

27 Application 3-9 Estimation Results for Parametric Copulas III Sce- Test P- nario Copula θ t-value BIC statistics value Rank Gumbel S1 Clayton Frank ,12 NA 3 Gumbel S2 Clayton Frank Gumbel S3 Clayton Frank Table 6: Growing Degree Days (GDD)

28 Application 3-10 Expected Payoff for Different Indices and Regions I Trading Area Contract Scenario A B C D S CRI S S S PFI S S S GDD S S Table 7: Indemnity Trigger = 50% Quantile

29 Application 3-11 Expected Payoff for Different Indices and Regions II Trading Area Contract Scenario A B C D S CRI S S S PFI S S S GDD S S Table 8: Indemnity Trigger = 15% Quantile

30 Application 3-12 Buffer Load and Diversification Effect I Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 9: Contract based on CRI, Trigger = 50% Quantile

31 Application 3-13 Buffer Load and Diversification Effect II Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 10: Contract based on PFI, Trigger = 50% Quantile

32 Application 3-14 Buffer Load and Diversification Effect III Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 11: Contract based on GDD, Trigger = 50% Quantile

33 Conclusions 4-1 Conclusions I Weather risk in Germany has a systemic component on a state level as well as on a national level The possibility of regional diversification depends on the type of weather index (temperature < drought < flooding) Weather risks should be globally diversified of transferred to the capital market (e.g. weather bonds)

34 Conclusions 4-2 Conclusions II Linear correlation may under- or overestimate systemic weather risks Copulas allow a flexible modeling of the dependence structure of joint weather risks But: problem of misspecification

35 Conclusions 4-3 References Wang, H. H. and Zhang, H. On the Possibility of Private Crop Insurance Market: A Spatial Statistics Approach Journal of Risk and Insurance 70: , 2003 Goodwin, B.K. Problems with Market Insurance in Agriculture American Journal of Agricultural Economics 83(3): , 2001

36 Appendix 5-1 Buffer Load I Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 12: Contract based on CRI, Trigger = 15% Quantile

37 Appendix 5-2 Buffer Load II Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 13: Contract based on PFI, Trigger = 15% Quantile

38 Appendix 5-3 Buffer Load III Trading Area Effect of Di- Scenario Method A A+B A+B+C A+B+C+D versification Lin. Corr S1 Gumbel Clayton Frank Lin. Corr S2 Gumbel Clayton Frank Lin. Corr S3 Gumbel Clayton Frank Table 14: Contract based on GDD, Trigger = 15% Quantile

39 Appendix 5-4 Buffer Load IV Contract Trading S1 S2 S3 based on Area LC 2 Copula 3 LC Copula LC Copula A CRI A+B+C+D Change (%) A PFI A+B+C+D Change (%) A GDD A+B+C+D Change (%) Linear Correlation 3 best ranked copula Table 15: Trigger = 15% Quantile

40 Appendix 5-5 Measurement of Dependency Fundamental task of economists Portfolio selection (CAMP) Insurance (VaR; hedging effectiveness) Measurement Multivariate distribution function Linear correlation coefficient (Pearson) Rank correlation coefficient (Spearman, Kendall) Copulas

41 Appendix 5-6 Pitfalls of Linear Correlation Measures only linear dependency Invariant only under linear transformations Perfect dependence does not always imply a linear correlation of 1; Zero correlation does not necessarily imply stochastic independence Marginal distributions and linear correlation of two random variables do not determine their joint distribution The interval [ 1, 1] is not attainable for the linear correlation coefficient for arbitrary distributions

42 Appendix 5-7 Buffer Load Contract Trading S1 S2 S3 based on Area LC 4 Copula 5 LC Copula LC Copula A CRI A+B+C+D Change (%) A PFI A+B+C+D Change (%) A GDD A+B+C+D Change (%) Linear Correlation 5 best ranked copula Table 16: Trigger = 50% Quantile

43 Appendix 5-8 Diversification Effect Contract Trading S1 S2 S3 based on Area LC 6 Copula 7 LC Copula LC Copula 50% quantile CRI 15% quantile Change (%) % quantile CRI 15% quantile Change (%) % quantile CRI 15% quantile Change (%) Linear Correlation 7 best ranked copula