Chain Ladder Model (`ChainLadder`)¶

The chain ladder method is a simple loss development technique that assumes that the ratio of ultimate losses to current losses is the same for all accident periods. Our chain ladder model is based on the chain ladder method, and is implemented by the ChainLadder model type. Mathematically, the base ChainLadder model is expressed as:

\[\begin{split}\begin{align} \begin{split} y_{ij} &\sim \mathrm{Gamma(\mu_{ij}, \sigma_{ij}^2)}, \quad{\forall j \in (1, \tau]}\\ \mu_{ij} &= ATA_{j - 1} y_{ij-1}\\ \sigma_{\text{noise},j} &\sim \mathrm{Normal}(\sigma_{\text{noise},\text{loc}}, \sigma_{\text{noise},\text{scale}})\\ \log \bf{ATA}_{1:M - 1} &\sim \mathrm{Normal}(ATA_{\text{loc}}, ATA_{\text{scale}})\\ \sigma_{\text{int}} &\sim \mathrm{Normal}(\sigma_{\text{int}, \text{loc}}, \sigma_{\text{int}, \text{scale}})\\ \sigma_{\text{slope}} &\sim \mathrm{Normal}(\sigma_{\text{slope}, \text{loc}}, \sigma_{\text{slope}, \text{scale}})\\ ATA_{\text{loc}} &= 0\\ ATA_{\text{scale}} &= 5\\ \sigma_{\text{int}, \text{loc}} &= 0\\ \sigma_{\text{int}, \text{scale}} &= 3\\ \sigma_{\text{slope}, \text{loc}} &= -0.6\\ \sigma_{\text{slope}, \text{scale}} &= 0.3\\ \sigma_{\text{noise},\text{loc}} &= 0\\ \sigma_{\text{noise},\text{scale}} &= 0.5 \end{split} \end{align}\end{split}\]

where \(\bf{ATA}\) is a vector of age-to-age factors that capture how losses change across development lags, \(\tau \in {2,...,M}\) is an integer chosen by an analyst that indicates how many development lags should be used to fit the model to, and \(\mathrm{Gamma(\mu, \sigma^2)}\) is the mean-variance parameterization of the Gamma distribution. In practice, \(\tau\) is determined by preprocessing (i.e. clipping) the triangle before fitting.

Model Fit Configuration¶

The ChainLadder model is fit using the following API call:

model = client.development_model.create(
    triangle=...,
    name="example_name",
    model_type="ChainLadder",
    config={ # default model_config
        "loss_definition": "paid",
        "loss_family": "gamma",
        "use_linear_noise": True,
        "use_multivariate": False,
        "line_of_business": None,
        "informed_priors_version": None,
        "priors": None, # see defaults below
        "recency_decay": 1.0,
        "seed": None
    }
)

The ChainLadder model accepts the following configuration parameters in config:

loss_definition: Name of loss field to model in the underlying triangle (e.g., "reported", "paid", or "incurred"). Defaults to "paid".
loss_family: Outcome distribution family (e.g., "gamma", "lognormal", or ""normal"). Defaults to "gamma".
use_linear_noise: Whether to use the linear noise variance function as specified in the ChainLadder equation above. Defaults to False. If set to True, random intercepts are dropped for each development lag such that the variance function becomes:

\[\begin{align} \sigma_{ij}^2 &= \exp(\sigma_{\text{int}} + \sigma_{\text{slope}} j + \log(y_{ij-1})) \end{align}\]

use_multivariate: Whether to use a industry-informed multivariate normal prior distribution on the age-to-age factors to leverage industry ATA means and covariances across development lags when fitting to the given triangle. Defaults to False. If set to True, line_of_business and informed_priors_version must also be specified. Cannot be used with use_linear_noise=False.
line_of_business: Line of business that the input triangle belongs to. Supported lines include: ["CA", "MC", "MO", "OO", "PC", "PO", "PP", "SL", "WC"]. Abbreviations map to the following lines:

{
    "CA": "Commercial Auto Liability",
    "MC": "Medical Liability: Claims Made",
    "MO": "Medical Liability: Occurrence",
    "OO": "Other Liability: Occurrence",
    "PC": "Product Liability: Claims Made",
    "PO": "Product Liability: Occurrence",
    "PP": "Private Passenger Auto",
    "SL": "Special Liability",
    "WC": "Workers' Compensation"
}

informed_priors_version: Version of the industry-informed priors to use when fitting the model (when use_multivariate=True). Supported versions currently only include: "2022". Specify as "latest" to always use the most up-to-date priors available. Defaults to None.
priors: Dictionary of prior distributions to use for model fitting. Default priors are:

{
    "ata__loc": 0.0,
    "ata__scale": 5.0,
    "sigma_slope__loc": -0.6,
    "sigma_slope__scale": 0.3,
    "sigma_intercept__loc": 0.0,
    "sigma_intercept__scale": 3.0,
    "sigma_noise__sigma_scale": 0.5, # for use_linear_noise=False
}

recency_decay: Likelihood weight decay to down-weight data from older evaluation dates. Defaults to 1.0, which means no decay. If set to a value between 0.0 and 1.0, the likelihood of older evaluation dates will be downweighted by a geometric decay function with factor recency_decay. See Geometric decay weighting for more information.
seed: Random seed for model fitting.

Model Predict Configuration¶

The ChainLadder model is used to predict future losses using the following API call:

predictions = model.development_model.predict(
    triangle=...,
    config={ # default config
        "max_dev_lag": None,
        "include_process_noise": True,
    }
    target_triangle=None,
)

Above, triangle is the triangle to use to start making predictions from and target_triangle is the triangle to make predictions on. For most use-cases, triangle will be the same triangle that was used in model fitting, and setting target_triangle=None will create a squared version of the modeled triangle. However, decoupling triangle and target_triangle means users could train the model on one triangle, and then make predictions starting from and/or on a different triangle. By default, predictions will be made out to the maximum development lag in triangle, but users can also set max_dev_lag in the configuration directly.

The ChainLadder prediction behavior can be further changed with configuration parameters in config:

max_dev_lag: Maximum development lag to predict out to. If not specified, the model will predict out to the maximum development lag in triangle. Note that ChainLadder can only generative predictions out to the maximum development lag in the training triangle, as there is no mechanism in the model to extrapolate out age-to-age beyond the training data.
include_process_noise: Whether to include process noise in the predictions. Defaults to True, which generates posterior predictions from the mathematical model as specified above. If set to False, the model will generate predictions without adding process noise to the predicted losses. Referring to the mathematical expression above, this equates to obtaining the expectation \(\mu_{ij}\) as predictions as oppposed to \(y_{ij}\).

Chain Ladder Model (ChainLadder)¶

Model Fit Configuration¶

Model Predict Configuration¶

Chain Ladder Model (`ChainLadder`)¶