Bayesian Credibility

This page contains the R code used for generating the applied example of the article Bayesian Credibility. The published version can be found in this link and a preprint here It is first necessary to install the package rstan. If you have not installed it run first the command install.packages("rstan") before running what follows.

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1. Downloading and preparing the data

The following code downloads the data and modifies the variables as indicated in the article. Note that the code above downloads the file car.csv to your working directory.

download.file("http://www.businessandeconomics.mq.edu.au/our_departments/Applied_Finance_and_Actuarial_Studies/acst_docs/glms_for_insurance_data/data/car.csv",destfile="car.csv")

cars.data <- read.csv("car.csv",header=TRUE)
cars.data$veh_body <- factor(cars.data$veh_body)
cars.data$veh_age <- factor(cars.data$veh_age)
cars.data$gender <- factor(cars.data$gender)
cars.data$area <- factor(cars.data$area)
cars.data$agecat <- factor(cars.data$agecat)

cars.data$veh_value <- cut(cars.data$veh_value,breaks=c(0,1.2,1.86,Inf),labels=c("P1","P2","P3"),include.lowest=T)

 ## Put together area A,B,C,D
cars.data$area <- factor(cars.data$area,levels=c(levels(cars.data$area),"ABCD") )
cars.data$area[cars.data$area %in% c("A","B","C","D")] <- "ABCD"
cars.data$area <- factor(cars.data$area)
cars.data$area <- relevel(cars.data$area,"ABCD")

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2. Using Stan

In order to use Stan we first load the library and detect the cores of the computer.

library(rstan)

rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

Now we create a string with the stan code of the model and store in a variable.

stan.severity.code="
data {
  int M; // Number of rows of the design matrix
  int P; // Number of columns of the design matrix
  matrix[M,P] X; // Design matrix
  vector[M] y; // response variable
  vector[M] ws; // durations
}

parameters {
  vector[P] beta;
  real phi;
}

transformed parameters{
  vector[M] mu;
  mu = exp(X * beta);
}

model{
  beta ~ uniform(-20,20);
  phi ~ uniform(0,1000);
  for (n in 1:M){
     y[n] ~ gamma(ws[n]/phi,ws[n]/(phi*mu[n]));
 }
}"

Now we aggregate the data, get the corresponding design matrix create a list with the data for Stan, and fit the stan model.

agg.data <- aggregate(cbind(claimcst0,numclaims) ~  agecat + gender + area + veh_value,
                        subset=clm==1,
                        data=cars.data,
                        FUN=sum)

X <- model.matrix( ~ agecat + gender + area + veh_value,
                  data=agg.data)


model.data <- list(M=dim(X)[1],
                       P=dim(X)[2],
                       y=agg.data$claimcst0/agg.data$numclaims,
                       X=X,
                       ws=agg.data$numclaims)

sev.stanfit <- stan(model_code=stan.severity.code,
                    data=model.data,
                    iter=30000,
                    warmup=2000,
                    chains=3)

---

3. Entropic Estimates

The following computes the posterior mean, entropic beta and entropic mean respectively.

## Posterior mean
post.mu <- colMeans(extract(sev.stanfit,pars="mu" )$mu)

## Entropic betas
entropic.beta <- coef(glm(post.mu ~ agecat + gender + area + veh_value,
                              family=Gamma(link="log"),
                              weights=numclaims,
                          data=agg.data ))

## Entropic credibility
		 entropic.mu <-exp( X %*% entropic.beta)

---

4. Non Credibility Estimates

Now we compute the premium using frequentist GLMs.

agg.data$obs.premium <- agg.data$claimcst0 / agg.data$numclaims

non.cred.model <- glm(obs.premium ~ agecat + gender + area + veh_value,
                      family=Gamma(link="log"),
                      weights=numclaims,
                      data=agg.data )

non.cred.premium <- predict(non.cred.model ,type="response")

---

4. Classes table

agecat	gender	area	veh_value	claimcst0	numclaims	Credibility Premium	Premium
5	F	ABCD	P3	143286.00	103.00	1373.24	1362.23
5	F	ABCD	P2	190286.65	99.00	1428.23	1416.31
6	F	ABCD	P3	49604.06	33.00	1475.97	1458.01
3	F	ABCD	P3	342153.16	210.00	1517.42	1508.93
4	F	ABCD	P3	175800.70	157.00	1520.57	1512.69
6	F	ABCD	P2	90709.05	74.00	1535.08	1515.90
3	F	ABCD	P2	300949.94	202.00	1578.18	1568.84
4	F	ABCD	P2	427309.04	226.00	1581.46	1572.75
5	F	E	P3	14694.00	16.00	1598.67	1572.53
5	F	ABCD	P1	166165.91	95.00	1604.94	1590.78
5	M	ABCD	P3	107384.24	112.00	1649.01	1635.28
5	F	E	P2	5494.55	9.00	1662.68	1634.97
2	F	ABCD	P3	260872.70	151.00	1670.48	1660.47
5	M	ABCD	P2	93941.60	76.00	1715.05	1700.21
6	F	E	P3	1985.14	3.00	1718.27	1683.11
6	F	ABCD	P1	91259.62	67.00	1725.01	1702.64
2	F	ABCD	P2	243282.20	164.00	1737.38	1726.39
3	F	E	P3	47338.40	25.00	1766.52	1741.88
4	F	E	P3	28446.99	20.00	1770.19	1746.22
6	M	ABCD	P3	150576.26	71.00	1772.38	1750.27
3	F	ABCD	P1	416263.74	204.00	1773.45	1762.09
4	F	ABCD	P1	382979.87	221.00	1777.14	1766.48
6	F	E	P2	33129.19	7.00	1787.07	1749.93
3	M	ABCD	P3	271928.40	162.00	1822.15	1811.38
4	M	ABCD	P3	268648.71	165.00	1825.94	1815.90
3	F	E	P2	43056.17	25.00	1837.26	1811.04
4	F	E	P2	28805.81	18.00	1841.08	1815.55
6	M	ABCD	P2	108135.99	63.00	1843.35	1819.76
5	F	E	P1	14554.23	8.00	1868.41	1836.37
3	M	ABCD	P2	198017.50	112.00	1895.11	1883.30
4	M	ABCD	P2	260961.26	131.00	1899.06	1888.00
5	M	E	P3	39418.93	15.00	1919.71	1887.74
5	M	ABCD	P1	184712.55	75.00	1927.25	1909.65
2	F	E	P3	18717.61	13.00	1944.71	1916.81
2	F	ABCD	P1	341669.44	166.00	1952.34	1939.06
5	M	E	P2	8078.37	7.00	1996.59	1962.69
5	F	F	P3	16626.50	7.00	2002.05	1962.81
2	M	ABCD	P3	415682.61	142.00	2005.95	1993.30
6	F	E	P1	9685.72	6.00	2008.19	1965.50
2	F	E	P2	27660.32	17.00	2022.59	1992.92
1	F	ABCD	P3	127635.32	55.00	2054.96	2033.95
6	M	E	P3	988.43	2.00	2063.33	2020.48
3	F	E	P1	31886.61	16.00	2064.58	2034.13
4	F	E	P1	46814.15	19.00	2068.87	2039.20
6	M	ABCD	P1	117751.87	48.00	2071.43	2043.93
5	F	F	P2	1210.95	1.00	2082.22	2040.74
2	M	ABCD	P2	216822.03	114.00	2086.28	2072.44
3	M	E	P3	56767.03	24.00	2121.27	2091.03
4	M	E	P3	49861.02	14.00	2125.68	2096.24
3	M	ABCD	P1	204610.22	108.00	2129.60	2115.30
4	M	ABCD	P1	273757.03	144.00	2134.03	2120.57
1	F	ABCD	P2	229982.48	122.00	2137.25	2114.71
6	M	E	P2	20723.49	4.00	2145.96	2100.70
6	F	F	P3	4993.37	3.00	2151.82	2100.83
3	M	E	P2	5918.78	5.00	2206.22	2174.05
4	M	E	P2	52690.12	11.00	2210.80	2179.47
3	F	F	P3	46057.12	34.00	2212.25	2174.19
4	F	F	P3	34487.43	20.00	2216.85	2179.61
5	M	E	P1	7157.38	7.00	2243.62	2204.46
2	F	E	P1	13886.76	9.00	2272.84	2238.42
3	F	F	P2	56496.29	19.00	2300.84	2260.52
4	F	F	P2	32559.16	7.00	2305.62	2266.15
2	M	E	P3	44224.71	32.00	2335.25	2301.03
5	F	F	P1	2783.49	2.00	2339.85	2292.13
2	M	ABCD	P1	168426.88	80.00	2344.42	2327.74
1	F	E	P3	8017.28	5.00	2392.30	2347.96
1	F	ABCD	P1	163074.98	76.00	2401.69	2375.21
5	M	F	P3	46239.91	9.00	2404.10	2356.25
6	M	E	P1	4026.32	9.00	2411.47	2359.48
2	M	E	P2	7568.42	11.00	2428.76	2392.39
2	F	F	P3	101384.81	43.00	2435.40	2392.54
1	M	ABCD	P3	213348.43	78.00	2467.64	2441.65
3	M	E	P1	2177.30	2.00	2479.19	2441.86
4	M	E	P1	18164.05	9.00	2484.35	2447.95
1	F	E	P2	5835.85	5.00	2488.10	2441.19
5	M	F	P2	12066.20	5.00	2500.37	2449.80
2	F	F	P2	35546.70	13.00	2532.93	2487.54
1	M	ABCD	P2	151417.25	62.00	2566.46	2538.60
3	F	F	P1	1792.19	3.00	2585.52	2538.98
3	M	F	P3	36945.01	26.00	2656.51	2610.00
4	M	F	P3	55052.36	15.00	2662.04	2616.50
2	M	E	P1	38062.92	6.00	2729.27	2687.10
3	M	F	P2	21796.48	7.00	2762.89	2713.63
4	M	F	P2	8965.32	8.00	2768.64	2720.39
1	F	E	P1	16903.95	9.00	2795.95	2741.90
5	M	F	P1	7310.71	2.00	2809.74	2751.58
2	F	F	P1	1401.77	3.00	2846.33	2793.96
1	M	E	P3	87116.26	17.00	2872.73	2818.60
1	M	ABCD	P1	94418.41	51.00	2884.00	2851.31
2	M	F	P3	24323.16	26.00	2924.49	2872.12
1	M	E	P2	28476.67	7.00	2987.76	2930.51
1	F	F	P3	8258.59	9.00	2995.93	2930.70
2	M	F	P2	18357.16	6.00	3041.60	2986.15
3	M	F	P1	47952.73	5.00	3104.75	3047.90
1	F	F	P2	10839.49	7.00	3115.90	3047.06
1	M	E	P1	490.00	1.00	3357.44	3291.51
2	M	F	P1	6950.54	4.00	3417.93	3354.00
1	F	F	P1	14113.60	6.00	3501.43	3422.41
1	M	F	P3	127855.12	13.00	3597.58	3518.14
1	M	F	P2	11469.09	1.00	3741.64	3657.83
1	M	F	P1	8120.13	1.00	4204.60	4108.42