Systemic Weather Risk and Crop Insurance: The Case of China

Transcription

1 and Crop Insurance: The Case of China Ostap Okhrin 1 Martin Odening 2 Wei Xu 3 Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics 1 Department of Agricultural Economics 2 HumboldtUniversität zu Berlin SCOR SE 3 Zürich

2 Motivation 1-1 Motivation High weather sensitivity of agricultural production Increase of extreme weather events Problems with traditional (re)insurance Emergence of weather markets Potential demand for weather derivatives in agriculture

3 Motivation 1-2 Agricultural Insurance 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, ooding) comprehensive Greece 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. Luxembourg comprehensive 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. for extreme Canada multiple peril 50% and uninsurable 50% pri. and crop ins. disasters pub. ins. USA multiple peril crop ins. 35 up to 100% only for uninsurable disasters 80% pri. and pub. ins. Table 1: Agricultural Insurances Systems

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

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

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

7 Motivation 1-6 Outline 1. Motivation 2. Model and Methods 3. Application 4. Conclusion

8 Motivation 1-7 Flow Chart of the Computational Procedure Daily temperature records (1 ) Temperature models Standardize d residuals (2 ) (3 ) Copula (5 ) Simulated daily temperature (6 ) (7 Simulated weather index ) for each weather station Net total losses several for aggregation levels (4 ) Simulated dependent standardized residuals of daily temperature (8 ) Buffer fund, Buffer load and Spatial diversification effect

9 Model and Methods 2-1 Buer Fund I i = I i (T i ), L i = f (I i, K i ) V, Π i = E(L i ), n w i (L i Π i ), NTL = i=1 BF = VaR α (NTL), BL n = BF /n n DE = nbl n / j=1 BL j BF buer fund, NTL net total loss, L loss, Π fair premium, w weight, I weather index, K trigger level, V tick size, α condence level, i region.

10 Model and Methods 2-2 Indices: Growing Degree Days (GDD) GDD i,t = τ E,t j=τ B,t max ( ) 0, T i,t,j T, where τ B,t is the rst of March, τ E,t is October 31, where T is the triggering temperature and is 5 C ; Loss function for the risk of insucient temperature ( ) L GDi,t = max 0, Ki GDD GDD t V, is the strike level being equal to 50% and the 15% quantile of the index distribution. K GDD i

11 Model and Methods 2-3 Indices: Frost Index (FI) τ M FI i,t = I j=τ N ( L FIi,t = max ( ) T i,t,j < T, 0, FI i,t K FI i ) V, where τ N and τ M denote November 1 and March 31, T = 0 C and K FI i is the strike level be equal to 50% and 85%.

12 Model and Methods 2-4 Daily average temperature T i,t = i,t + Ψ i,t, i,t = a 1,i + a 2,i t + a 3,i cos J i Ψ i,t = b j,i Ψ t j,i + σ i,t ε i,t j=1 time-varying variance: K i [ ( σi,t 2 = d 1,i + d 2,i t + d 3,k,i cos 2πk k=1 ( 2π t ) a 4,i, 365 t 365 ) ( + d 4,k,i sin 2πk t 365 )]

13 Model and Methods 2-5 Correlation Gaussian t Gumbel Figure 3: Scatterplots for two distributions with ρ = 0.4 same linear correlation coecient (ρ = 0.4) same marginal distributions rather big dierence

14 Model and Methods 2-6 Copula For a distribution function F with marginals F X1,..., F Xd, there exists a copula C : [0, 1] d [0, 1], such that F (x 1,..., x d ) = C{F X1 (x 1 ),..., F Xd (x d )}.

15 Model and Methods 2-7 Recall Archimedean Copula Multivariate Archimedean copula C : [0, 1] d [0, 1] dened as C(u 1,..., u d ) = φ{φ 1 (u 1 ) + + φ 1 (u d )}, (1) where φ : [0, ) [0, 1] is continuous and strictly decreasing with φ(0) = 1, φ( ) = 0 and φ 1 its pseudo-inverse. Example 1 φ Gumbel (u, θ) = exp{ u 1/θ }, where 1 θ < φ Clayton (u, θ) = (θu + 1) 1/θ, where θ [ 1, )\{0} Disadvantages: too restrictive: single parameter, exchangeable

16 Model and Methods 2-8 Hierarchical Archimedean Copulas Simple AC with s=(1234) C(u 1, u 2, u 3, u 4 ) = C 1 (u 1, u 2, u 3, u 4 ) x 1 x 2 x 3 x 4 z (1234)

17 Model and Methods 2-9 Hierarchical Archimedean Copulas Simple AC with s=(1234) C(u 1, u 2, u 3, u 4 ) = C 1 (u 1, u 2, u 3, u 4 ) x 1 x 2 x 3 x 4 z (1234) Fully nested AC with s=(((12)3)4) C(u 1, u 2, u 3, u 4 ) = C 1 [C 2 {C 3 (u 1, u 2 ), u 3 }, u 4 ] x 1 x 2 x 3 x 4 z 12 z (12)3 z ((12)3)4

18 Model and Methods 2-10 Hierarchical Archimedean Copulas Simple AC with s=(1234) AC with s=((123)4) C(u 1, u 2, u 3, u 4 ) = C 1 (u 1, u 2, u 3, u 4 ) C(u 1, u 2, u 3, u 4 ) = C 1 {C 2 (u 1, u 2, u 3 ), u 4 } x 1 x 2 x 3 x 4 z (1234) Fully nested AC with s=(((12)3)4) C(u 1, u 2, u 3, u 4 ) = C 1 [C 2 {C 3 (u 1, u 2 ), u 3 }, u 4 ] x 1 x 2 x 3 x 4 z 12 z (12)3 z ((12)3)4 x 1 x 2 x 3 x 4 z (123) z ((12)3)4 Partially Nested AC with s=((12)(34)) C(u 1, u 2, u 3, u 4 ) = C 1 {C 2 (u 1, u 2 ), C 3 (u 3, u 4 )} x 1 x 2 x 3 x 4 z 12 z 34 z (12)(34)

19 Model and Methods 2-11 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34

20 Model and Methods 2-12 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13

21 Model and Methods 2-13 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24

22 Model and Methods 2-14 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ˆθ (13)2, ˆθ (13)4, ˆθ 24 } = ˆθ (13)4

23 Model and Methods 2-15 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ˆθ (13)2, ˆθ (13)4, ˆθ 24 } = ˆθ (13)4 x 1 x 3 x 4 x 2 z 13 z (13)4 z ((13)4)2

24 Model and Methods 2-16 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ˆθ (13)2, ˆθ (13)4, ˆθ 24 } = ˆθ (13)4 x 1 x 3 x 4 x 2 z 13 ˆθ (13) ˆθ ((13)4) z (13)4 z ((13)4)2

25 Model and Methods 2-17 Recovering the structure (easy practice) x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ˆθ 12, ˆθ 13, ˆθ 14, ˆθ 23, ˆθ 24, ˆθ 34 } = ˆθ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ˆθ (13)2, ˆθ (13)4, ˆθ 24 } = ˆθ (13)4 x 1 x 3 x 4 x 2 x 1 x 3 x 4 x 2 z 13 ˆθ (13) ˆθ ((13)4) z (13)4 z (134) z ((13)4)2 z ((134)2)

26 Model and Methods 2-18 Estimation Issues - Margins { } n F j (x; ˆα j ) = F j x; arg max α log f j (X ji, α), ˆF j (x) = F j (x) = 1 n n + 1 i=1 n I(X ji x), i=1 n i=1 K ( ) x Xji for j = 1,..., k, where κ : R R, κ = 1, K(x) = x κ(t)dt and h > 0 is the bandwidth. ˇF j (x) {ˆFj (x), Fj (x), F j (x; ˆα j )} h

27 Model and Methods 2-19 Estimation Issues - Multistage Estimation where L j = l j (X i ) ( L 1 θ 1 n l j (X i ) i=1,..., L p θ p ) = 0, = log (c({φ l, θ l } l=1,...,j ; s j ) [ ] ) {ˇFm (x mi )} m sj for j = 1,..., p. Theorem Under regularity conditions, estimator ˆθ is consistent and n 1 2 (ˆθ θ) a IN(0, B 1 ΣB 1 )

28 Model and Methods 2-20 Copula: Goodness-of-Fit Tests Hypothesis H 0 : C θ C 0 ; θ Θ vs H 1 : C θ C 0 ; θ Θ, Cramér von Mises S = n [Ĉ(u1,..., u d ) C(u 1,..., u d ; θ) ] 2 d Ĉ(u 1,..., u d) [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) in practice p-values are calculated using the bootstrap methods described in Genest and Remillard (2008)

29 Simulation 3-1 Simulation Frees and Valdez, (1998, NAAJ), Whelan, (2004, QF), Marshal and Olkin, (1988, JASA) Conditional inversion method: Let C = C(u 1,..., u k ), C i = C(u 1,..., u i, 1,..., 1) and C k = C(u 1,..., u k ). Conditional distribution of U i is given by C i (u i u 1,..., u i 1 ) = P{U i u i U 1 = u 1... U i 1 = u i 1 } Generate i.r.v. v 1,..., v k U(0, 1) Set u 1 = v 1 u i = C 1 k (v i u 1,..., u i 1 ) i = 2, k = i 1 C i (u 1,..., u i ) u 1... u i 1 / i 1 C i 1 (u 1,..., u i 1 ) u 1... u i 1

30 Application 4-1 Location of selected weather stations

31 Application 4-2 Flow Chart of the Computational Procedure Daily temperature records (1 ) Temperature models Standardize d residuals (2 ) (3 ) Copula (5 ) Simulated daily temperature (6 ) (7 Simulated weather index ) for each weather station Net total losses several for aggregation levels (4 ) Simulated dependent standardized residuals of daily temperature (8 ) Buffer fund, Buffer load and Spatial diversification effect

32 Application 4-3 Descriptive Statistics st. GDD FI (198.13) 6.26 ( 6.07) (148.25) (10.29) (146.95) (12.23) (186.12) (12.32) (144.03) 5.86 ( 5.18) (140.68) (11.64) (172.30) (12.07) (141.53) (9.24) (103.70) 0.20 (0.64) (156.99) 0.26 (0.60) (105.20) 0.00 (0.00) Table 2: Descriptives

33 Application 4-4 HAC Structure θ = 1.05 θ = 1.11 θ = θ = 1.3 θ = 1.2 θ = 1.4 θ = 1.35 θ = 1.34 θ = θ = 1.41 θ = 1.49 θ = θ = θ = θ = θ =

34 Application 4-5 Illustration of Dependence Cluster

35 Application 4-6 BL for Dierent Aggregation: GDD Aggregation: (2) (2, 3) (1-3) (1-6, 8) (1-8) (1-8, 15-17) (1-17) Gauss Gumbel Gumbel Rotated monetary units Strike level: 50% Strike level: 15% aggregation level

36 Application 4-7 BL for Dierent Aggregation: FI Aggregation: (2) (2, 3) (1-3) (1-6, 8) (1-8) (1-8, 11-14) monetary units Gauss Gumbel Gumbel Rotated Strike level: 50% Strike level: 15% aggregation level

37 Application 4-8 Fair Prices, Buer Loads and Diversication Eects I Type of Copula Gaussian Gumbel Rotated-Gumbel GDD Strike Level 50% Fair Price Buer Load Diversication Eect GDD Strike Level 15% Fair Price Buer Load Diversication Eect

38 Application 4-9 Fair Prices, Buer Loads and Diversication Eects II Type of Copula Gaussian Gumbel Rotated-Gumbel FI Strike Level 50% Fair Price Buer Load Diversication Eect FI Strike Level 15% Fair Price Buer Load Diversication Eect

39 Conclusion 5-1 Conclusions Weather risk in China has a systemic component on a state level as well as on a national level The possibility of regional diversication depends on the type of weather index (temperature < drought < ooding) Weather risks should be globally diversied or transferred to the capital market (e.g. weather bonds) Linear correlation may under- or overestimate systemic weather risk Copulas allow a exible modeling of the dependence structure of joint weather risks But: risk of misspecication

40 and Crop Insurance: The Case of China Ostap Okhrin 1 Martin Odening 2 Wei Xu 3 Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics 1 Department of Agricultural Economics 2 HumboldtUniversität zu Berlin SCOR SE 3 Zürich