Modeling multivariate time series with copulas: Implications for pricing revenue insurance

Duarte, Gislaine Vieira; Ozaki, Vitor A.

doi:10.5935/0034-7140.20230010

Gislaine Vieira Duarte Gislaine Vieira Duarte: gislainevduarte@gmail.com

Escola Superior de Agricultura Luiz de Queiroz, Universidade de São Paulo, Brasil

http://orcid.org/0000-0003-4372-7499

Vitor A. Ozaki Vitor A. Ozaki: vitorozaki@usp.br

Escola Superior de Agricultura Luiz de Queiroz, Universidade de São Paulo, Brasil

http://orcid.org/0000-0002-0781-4475

Gislaine Vieira Duarte: gislainevduarte@gmail.com

Vitor A. Ozaki: vitorozaki@usp.br

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Figure 1
Plots of the Skew-t density function for different parameter values.

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Figure 2
Plots of the OLLN density function for different values of the parameters.

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Figure 3
Original Price in Reais, Adjusted Price (R$), Histogram and Normal QQ-Plot for the adjustment price series in Reais.

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Figure 4
Price, Histogram, and Normal QQ-Plot for the CME Daily Adjustment Price Series.

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Figure 5
Price, Histogram, and Normal QQ-Plot for the EXG rate series.

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Figure 6
Chi-Plot for variable pairs (Yield, CME price)

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Figure 7
Dependogram.

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Table 1
Descriptive Statistics for the yield series adjusted for Cascavel and Toledo (n = 37 observations)

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Table 2
Estimated parameters of the OLLN(μ,σ,α) distribution simulation, with standard errors in parentheses

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Table 3
Descriptive Statistics for simulated yield series for Cascavel, Toledo and price

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Table 4
AIC and BIC criteria and log-likelihood maximum values for models adjusted to the price series in Reais

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Table 5
AIC and BIC Criteria Values for Copula Selection

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Table 6
Descriptive Statistics for Adjusted Price Series (US$) and EXG rate (Reais)

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Table 7
Ljung-Box test results for squared CME prices (rcme) and EXG rate (r dolar) adjusted price series.

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Table 8
Values of the AIC and BIC criteria and maximum log-likelihood(LLM) for GARCH models (m, n) with t-Student errors to the CME prices and EXG rate series

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Table 9
AIC, BIC, and maximum log-likelihood (MLL) criteria for Copula Selection.

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Table 10
Selected models for revenue insurance pricing with dependency parameters.

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Table 11
Pure and Commercial Rate in % for the two-dimensional revenue insurance premium for the municipalities of Toledo and Cascavel using the Frank copula (δ = −0.3358)

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Table 12
Average commercial rate for revenue insurance using copula and insurer A rate for Toledo and in parenthesis for Cascavel

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Table 13
Pure and Commercial Charging Rates (20%) for revenue Insurance using t-Student copula for Toledo (Yield, CME Price, EXG rate)

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Table 14
Commercial Average Rates for Revenue Insurance using t-Student Copula and Insurer A.

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Series	Mean	Median	Std Dev	Asymmetry	Kurtosis	Max.Value	Min.Value	CV(%)
Cascavel	3405.6	3447.4	329.9	−0.3	−0.7	4021.8	2707.0	9.7
Toledo	3290.0	3442.9	520.7	−1.3	1.9	4005.1	1505.8	15.8

Municipalities	μ	σ	α
Toledo	3289.6	637.9	1.04
	(117.1)	(338.4)	(0.68)
Cascavel	3415.3	680.9	1.63
	(72.6)	(797.1)	(2.09)

Series	Mean	Median	Std Dev	Asymmetry	Kurtosis Max.Value	Min.Value	CV(%)
Cascavel	3402.6	3406.7	425.0 0.002	−0.20	4752.3	2085.2	12.5
Toledo	3279.7	3285.4	593.1	0.001	−0.19	5167.5	1437.1	18.1
Price(R$)	73.5	73.8	5.0	−0.127	−0.76	84.9	62.8	6.8

	Criteria	GARCH(1,1)	GARCH(2,1)	GARCH(2,2)
Price (R$)	AIC	2.85	2.86	2.86
	BIC	2.90	2.91	2.92
	LLV	642.24	642.42	641.74

	Criteria	Frank	AMH	Joe	Gaussiana	t	Gumbel
Cascavel	AIC	0.55	0.66	2.00	1.11	3.11	2.00
Cascavel	BIC	4.67	4.78	6.12	5.23	11.35	6.12
Toledo	AIC	0.55	0.66	2.00	1.11	3.11	2.00
Toledo	BIC	4.67	4.78	6.12	5.23	11.35	6.12

Series	Mean	Median	Std Dev	Asymmetry	Kurtosis	Max.Value	Min.Value	CV (%)
Price	21.25	21.15	1.36	0.34	2.04	24.65	19.19	6.40
EXG rate	3.47	3.39	0.31	0.46	1.91	4.19	2.99	9.05

Series	rcme	rdolar	rcme×rcme	rdolar×rdolar
p-valor	0.54	0.37	0.05	1.31×10⁻⁴

	Criteria	GARCH(1,1)	GARCH(2,1)	GARCH(2,2)
CME price	AIC	−0.18	−0.18	−0.18
	BIC	−0.13	−0.13	−0.12
	LLM	−45.42	−46.99	−47.95
EXG rate	AIC	−3.74	−3.73	−3.74
	BIC	−3.69	−3.68	−3.68
	LLM	−853.24	−853.13	−855.90

	Criteria	Clayton	Frank	Gumbel Joe	Gaussian	t
Toledo	AIC	−116.40	−115.66	−101.40	−67.12	−106.27	−450.55
	BIC	−112.28	−111.55	−97.28	−63.00	−102.15	−438.20
	MLL	59.20	58.83	51.70	34.56	54.14	228.28
Cascavel	AIC	−116.40	−115.66	−101.40	−67.12	−106.27	−450.55
	BIC	−112.28	−111.55	−97.28	−63.00	−102.15	−438.20
	MLL	59.20	58.83	51.70	34.56	54.14	228.28

Data	Copula	Parameters
(Toledo,CME price,EXG rate)	t-Student	(ρ₁,ρ₂,df) = (0.5458;0.0896;1.84)
(Cascavel,CME price,EXG rate)	t-Student	(ρ₁,ρ₂,df) = (0.5458;0.0896;1.84)

N		60%	65%	70%	75%	80%
454	Pure rate	4.50	5.39	6.45	7.65	8.95
	Commercial Rate	9.00	10.77	12.89	15.31	17.90
20 000	Pure rate	4.68	5.84	7.12	8.48	9.97
	Commercial Rate	9.37	11.68 14.23	16.97	19.93

Model	60 − 69%	70 − 79%
Copula	9.89	14.10
insurer A	6.26	5.68 (5.52)

	N	60%	65%	70% 75%	80%
N=454	Pure rate	0.955	1.410	1.937	2.589	3.378
	Commercial rate	1.911	2.820	3.874	5.178	6.755
N=20000	Pure rate 0.947	1.360	1.869	2.486	3.230
	Commercial rate	1.894	2.719	3.738	4.973	6.459

CL	Copula	Insurer A
60 − 69%	2.36	6.26
70 − 79%	4.52	5.68

F (x; μ, σ, α) = \int_{0}^{\frac{Φ (x; μ, σ)}{\bar{Φ} (x; μ, σ)}} \frac{α t^{α - 1}}{{(1 + t^{α})}^{2}} d t = \frac{Φ^{α} (\frac{x - μ}{σ})}{Φ^{α} (\frac{x - μ}{σ}) + {[1 - Φ (\frac{x - μ}{σ})]}^{α}},

f (x; μ, σ, α) = \frac{α ϕ (\frac{x - μ}{σ}) Φ (\frac{x - μ}{σ}) {[1 - Φ (\frac{x - μ}{σ})]}^{α - 1}}{σ {Φ^{α} (\frac{x - μ}{σ}) + {[1 - Φ (\frac{x - μ}{σ})]}^{α}}^{2}},

[1] Gislaine Vieira Duarte: gislainevduarte@gmail.com