Modelling-Claim-Severities-Using-the-Pareto-Lomax-Distribution-in-R

The analysis demonstrates a peaks-over-threshold (POT) approach for modeling heavy-tailed insurance claim severities using the Pareto Type II (Lomax) distribution — a standard tool in actuarial risk modeling and extreme value theory. The claim data is simulated, select a high threshold using the mean excess function, fit a Lomax distribution via maximum likelihood and validate tail fit using log–log survival and Pareto-scale QQ diagnostics. Finally, key risk measures are computed which are; Value at Risk (VaR) and Tail Value at Risk (TVaR) — at the 99% confidence level.

#Modelling Claim Severities using Pareto distribution #package for actuarial and risk modelling library(actuar) # Pareto/Lomax functions library(fitdistrplus) # fitting univariate + diagnostics library(evir) # Extreme value theorem library(ggplot2) # Visualization

Sample claims

set.seed(1) # reproducibility

Type I (actuar::rpareto) distribution

claims <- rpareto(3000, shape = 1.7, scale = 200)

Pick threshold using mean excess plot

meplot(claims) # choose u where curve looks ~linear in tail u <- quantile(claims, 0.80) # 80th percentile of claims data

Excesses over u (Lomax fit on excesses)

excess <- claims[claims > u] - u

Fit Lomax (Pareto II) via MLE

lomax_loglik <- function(par, y){

• alpha <- par[1]; beta <- par[2]
• if (alpha <= 0 || beta <= 0) return(Inf)
• -sum(log(alpha/beta) - (alpha+1)*log1p(y/beta))
• }

#Evaluates the negative log-likelihood of the Lomax distribution #fit Lomax via optim fit <- optim(c(alpha=1.5, beta=median(excess)), lomax_loglik, y=excess, method="L-BFGS-B",

•          lower=c(1e-6, 1e-6))
  

#maximum likelihood estimates of Lomax distribution’s parameters alpha_hat <- fit$par["alpha"]; beta_hat <- fit$par["beta"]

Diagnostics: log-log survival plot

library(ismev) # Extreme value modelling #Emphirical tail sample of exceedance over threshold emp_tail <- sort(excess, decreasing=TRUE) # Descending order #Empirical survival probabilities S_emp <- seq_along(emp_tail)/length(emp_tail) #Log-survival plot and Adjusted claim size plot(log(emp_tail+beta_hat), log(S_emp), main="Log-Log Tail", xlab="log(x+beta)", ylab="log S(x)")

QQ plot on Pareto scale

z_emp <- sort(log1p(excess/beta_hat))

since log(1 + X/beta) ~ Exp(alpha)

z_the <- qexp(ppoints(length(z_emp)), rate=alpha_hat) qqplot(z_the, z_emp, xlab="Theoretical Exp Quantiles", ylab="Observed log(1+x/beta)")

straight Line passin through intercept 0 and slope of 1

abline(0,1)

Risk metrics

#Computed fitted VaR at probaility level p VaR <- function(p) beta_hat * ((1-p)^(-1/alpha_hat) - 1) #Computes TVaR for Pareto II distribution TVaR <- function(p) (alpha_hat*beta_hat)/(alpha_hat-1) * (1-p)^(-1/alpha_hat) - beta_hat VaR_99 <- VaR(0.99) # 99% VaR TVaR_99 <- TVaR(0.99) #99% TVaR #Creating the summary of results c(alpha_hat=alpha_hat, beta_hat=beta_hat, VaR_99=VaR_99, TVaR_99=TVaR_99) alpha_hat.alpha beta_hat.beta VaR_99.beta TVaR_99.alpha 1.6954 504.5015 7125.5091 18097.6274