NAIC Schedule P Dataset

## Observations: 77,900
## Variables: 14
## $ grcode          <chr> "266", "266", "266", "266", "266", "266", "266", "2...
## $ grname          <chr> "Public Underwriters Grp", "Public Underwriters Grp...
## $ accidentyear    <int> 1988, 1988, 1988, 1988, 1988, 1988, 1988, 1988, 198...
## $ developmentyear <int> 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 199...
## $ developmentlag  <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7,...
## $ incurloss       <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 24, 21, 24, 25, 2...
## $ cumpaidloss     <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 20, 21, 23, 24, 24...
## $ bulkloss        <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, ...
## $ earnedpremdir   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 25, 25, 25, 25, 2...
## $ earnedpremceded <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ earnedpremnet   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 25, 25, 25, 25, 2...
## $ single          <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ postedreserve97 <int> 932, 932, 932, 932, 932, 932, 932, 932, 932, 932, 9...
## $ lob             <chr> "comauto", "comauto", "comauto", "comauto", "comaut...

## Observations: 77,900
## Variables: 10
## $ grcode     <chr> "266", "266", "266", "266", "266", "266", "266", "266", ...
## $ grname     <chr> "Public Underwriters Grp", "Public Underwriters Grp", "P...
## $ lob        <chr> "comauto", "comauto", "comauto", "comauto", "comauto", "...
## $ acc_year   <chr> "1988", "1988", "1988", "1988", "1988", "1988", "1988", ...
## $ dev_year   <int> 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 19...
## $ dev_lag    <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9...
## $ premium    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 25, 25, 25, 25, 25, 25...
## $ cum_loss   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 20, 21, 23, 24, 24, 24,...
## $ loss_ratio <dbl> NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, 0.2400...
## $ dev_factor <dbl> NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, 0.2500...

Grid Approximation

2D Grid

Core Concept

$$\text{Loss}(t) = \text{Premium} \times \text{ULR} \times \text{GF}(t)$$

Full Specification

$$\text{Loss}(Y, t) \sim \text{Normal}(\mu(Y, t), \, \sigma_Y)$$

where

Importance of the Functional Form

Does the choice of function matter?

Not much difference…

(probably)

Getting Started

functions {
    real growth_factor_weibull(real t, real omega, real theta) {
        return 1 - exp(-(t/theta)^omega);
    }

    real growth_factor_loglogistic(real t, real omega, real theta) {
        real pow_t_omega = t^omega;
        return pow_t_omega / (pow_t_omega + theta^omega);
    }
}

data {
    int<lower=0,upper=1> growthmodel_id;

    int n_data;
    int n_time;
    int n_cohort;

    int cohort_id[n_data];
    int t_idx[n_data];

    int cohort_maxtime[n_cohort];

    vector<lower=0>[n_time] t_value;

    vector[n_cohort] premium;
    vector[n_data]   loss;
}

parameters {
    real<lower=0> omega;
    real<lower=0> theta;

    vector<lower=0>[n_cohort] LR;

    real mu_LR;
    real<lower=0> sd_LR;

    real<lower=0> loss_sd;
}

transformed parameters {
    vector[n_time] gf;
    vector[n_data] lm;

    for(i in 1:n_time) {
        gf[i] = growthmodel_id == 1 ?
            growth_factor_weibull    (t_value[i], omega, theta) :
            growth_factor_loglogistic(t_value[i], omega, theta);
    }

    for (i in 1:n_data) {
        lm[i] = LR[cohort_id[i]] * premium[cohort_id[i]] * gf[t_idx[i]];
    }
}

model {
    mu_LR ~ normal(0, 0.5);
    sd_LR ~ lognormal(0, 0.5);

    LR ~ lognormal(mu_LR, sd_LR);

    loss_sd ~ lognormal(0, 0.7);

    omega ~ lognormal(0, 0.5);
    theta ~ lognormal(0, 0.5);

    loss ~ normal(lm, (loss_sd * premium)[cohort_id]);
}

Model Convergence

Parameter Values

Range of Loss Ratios

Project out ULR values

Find smallest and largest value for each iteration

Compare to data

Total Reserves

Project out ultimate loss ratios

Calculate difference to current loss ratio

Add up across accounting years

Further Iterations

Non-constant variance

Other functional forms (CDFs)

Multiple lines of business

Multiple coverholders

On that note…

https://magesblog.com/post/2018-07-15-hierarchical-loss-reserving-with-growth-cruves-using-brms/

Questions

Thank You!!!

mickcooney@gmail.com

http://mc-stan.org/users/documentation/case-studies/losscurves_casestudy.html