Generalized Bondy Model (`GeneralizedBondy`)¶

Our Generalized Bondy model is a generalization of the Bondy method used by actuaries to estimate ultimate losses given histoic development patterns. Whereas the Bondy method is typically performed given age-to-age factors as input, our Generalized Bondy model is fitted directly to the triangle of interest. The Generalized Bondy model is implemented by the GeneralizedBondy model type, and is expressed mathematically as:

\[\begin{split}\begin{align} \begin{split} \mathrm{LR}_{ij} &\sim \mathrm{Gamma(\mu_{ij}, \sigma_{ij}^2)}\\ \mu_{ij} &= ATA_{j} y_{ij - 1}\\ ATA_{j} &= \exp( ATA_{\text{init}} \beta^{j} )\\ \sigma_{ij}^2 &= \exp(\sigma_{\text{int}} + \sigma_{\text{slope}} j - \log(\mathrm{EP}_{i})), \quad{\forall j \in [\rho_1, \rho_2]}\\ \log ATA_{\text{init}} &\sim \mathrm{Normal}(ATA_{\text{init}, \text{loc}}, ATA_{\text{init}, \text{scale}})\\ \log \frac{\beta}{1 - \beta} &\sim \mathrm{Normal}(\beta_{\text{loc}}, \beta_{\text{scale}})\\ \sigma_{\text{int}} &\sim \mathrm{Normal}(\sigma_{\text{int}, \text{loc}}, \sigma_{\text{int}, \text{scale}})\\ \sigma_{\text{slope}} &\sim \mathrm{Normal}(\sigma_{\text{slope}, \text{loc}}, \sigma_{\text{slope}, \text{scale}})\\ ATA_{\text{init}, \text{loc}} &= 0\\ ATA_{\text{init}, \text{scale}} &= 1\\ \beta_{\text{loc}} &= 0\\ \beta_{\text{scale}} &= .3\\ \sigma_{\text{int}, \text{loc}} &= 0\\ \sigma_{\text{int}, \text{scale}} &= 3\\ \sigma_{\text{slope}, \text{loc}} &= -.6\\ \sigma_{\text{slope}, \text{scale}} &= .3 \end{split} \end{align}\end{split}\]

where \(\bf{ATA}\) is a vector of age-to-age factors that capture how losses change across development and \(\mathrm{EP}_{i}\) is the total earned premium for the given accident period. Instead of being estimated as independent parameters as in the ChainLadder model, here the age-to-age factors are modeled as a function of two parameters, \(ATA_{\text{init}}\) and \(\beta\). The parameter \(ATA_{\text{init}}\) can be interpreted as the initial age-to-age factor, and \(\beta\) as the rate of decay in the age-to-age factors across development. Because \(\log ATA_{\text{init}}\) is constrained to be positive, as development increases, the lowest value an age-to-age factor can take on is 1 (at which point development has reached an asymptote).

Typically, the Generalized Bondy model is fitted to only the window of development lags \(j \in [\rho_1, \rho_2]\), where \((\rho_1, \rho_2) \in {2,...,M}, \rho_1 < \rho_2\), are chosen by an analyst based on where the tail process is assumed to begin and end. In practice, this can be accomplished my mutating/clipping the triangle as a preprocessing step before fitting.

Model Fit Configuration¶

The GeneralizedBondy model above is fit using the following API call:

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

The GeneralizedBondy 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 "reported".
loss_family: Outcome distribution family (e.g., "gamma", "lognormal", or ""normal"). Defaults to "gamma".
line_of_business: Line of business that the input triangle belongs to. If specified, backtest-informed priors leveraging industry data are used to fit the model. Must be preovided if informed_priors_version is specified. Otherwise, defaults to None and the default priors below are used. 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. 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:

{
    "init_log_ata__loc": 0.0,
    "init_log_ata__scale": 1.0,
    "bondy_exp__loc": 0.0,   # beta location above
    "bondy_exp__scale": 0.3, # beta scale above
    "sigma_slope__loc": -0.6,
    "sigma_slope__scale": 0.3,
    "sigma_intercept__loc": 0.0,
    "sigma_intercept__scale": 3.0,
}

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 evaludation 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 GeneralizedBondy model is used to predict future losses using the following API call:

predictions = model.tail_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 GeneralizedBondy 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 GeneralizedBondy can be used to make predictions for development lags beyond the last development lag available in the training triangle, as there is a mechanism in the model to extrapolate out age-to-age beyond the training data.
eval_resolution: the resolution of the evaluation dates in the tail. Defaults to the evaluation date resolution in triangle. If triangle is from a single evaluation date, falls back to the resolution of 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 \(\mathrm{LR}_{ij}\).

Generalized Bondy Model (GeneralizedBondy)¶

Model Fit Configuration¶

Model Predict Configuration¶

Generalized Bondy Model (`GeneralizedBondy`)¶