Meyers Cross Classified Model (MeyersCRC)

This model represents a generalization of the Cross Classified (CRC) Model presented in the 2019 CAS monograph “Stochastic Loss Reserving Using Bayesian MCMC Models, 2nd Edition”. The CRC model breaks losses down into development and experience period components, where the loss for any given accident period at a given development lag is assumed to be a function of the unique factors associated with the period/lag combination.

Our implementation in the MeyersCRC model type is expressed mathematically as:

\[\begin{split}\begin{align} \mathrm{LR}_{ij} &\sim \mathrm{Gamma}(\mu_{ij}, \sigma_{ij}^2) \quad{\forall j \in (1, \tau]}\\ \mu_{ij} &= \exp(\mathrm{LR}_{\text{expected}} + \beta_{\text{lag},j} + \beta_{\text{year},i})\\ \sigma_{ij}^2 &= \exp(\sigma_{\text{int}} + \sigma_{\text{slope}} j - \log(\mathrm{EP}_{i}))\\ \mathrm{LR}_{\text{expected}} &\sim \mathrm{Normal}(\mathrm{LR}_{\text{expected}, \text{loc}}, \mathrm{LR}_{\text{expected}, \text{scale}})\\ \log \boldsymbol{\beta_{\text{lag}}} &\sim \mathrm{Normal}(\beta_{\text{lag}, \text{loc}}, \beta_{\text{lag}, \text{scale}})\\ \log \boldsymbol{\beta_{\text{year}}} &\sim \mathrm{Normal}(\beta_{\text{year}, \text{loc}}, \beta_{\text{year}, \text{scale}})\\ \sigma_{\text{int}} &\sim \mathrm{Normal}(\sigma_{\text{int}, \text{loc}}, \sigma_{\text{int}, \text{scale}})\\ \sigma_{\text{slope}} &\sim \mathrm{Normal}(\sigma_{\text{slope}, \text{loc}}, \sigma_{\text{slope}, \text{scale}})\\ \mathrm{LR}_{\text{expected}, \text{loc}} &= -0.4\\ \mathrm{LR}_{\text{expected}, \text{scale}} &= \sqrt{10}\\ \beta_{\text{lag}, \text{loc}} &= 0.0\\ \beta_{\text{lag}, \text{scale}} &= \sqrt{10}\\ \beta_{\text{year}, \text{loc}} &= 0.0\\ \beta_{\text{year}, \text{scale}} &= \sqrt{10}\\ \sigma_{\text{int}, \text{loc}} &= 0.0\\ \sigma_{\text{int}, \text{scale}} &= 3.0\\ \sigma_{\text{slope}, \text{loc}} &= 0.0\\ \sigma_{\text{slope}, \text{scale}} &= 1.0 \end{align}\end{split}\]

Unlike other loss development models, the MeyersCRC model does not estimate age-to-age factors directly, although implied factors can be derived post-hoc. Instead, expected losses are determined as the product of a general expected losses term, a development lag factor, and an accident period factor. \(\tau \in {2,...,M}\) 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 MeyersCRC model is fit using the following API call:

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

The MeyersCRC 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".

• priors: Dictionary of prior distributions to use for model fitting. Default priors are:

{
   "logelr__loc": -0.4, # expected log loss ratio
    "logelr__scale": np.sqrt(10),
    "lag_factor__loc": 0.0,
    "lag_factor__scale": np.sqrt(10),
    "year_factor__loc": 0.0,
    "year_factor__scale": np.sqrt(10),
    "sigma_intercept__loc": 0.0,
    "sigma_intercept__scale": 3.0,
    "sigma_slope__loc": 1.0,
    "sigma_slope__scale": 1.0
}

• 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 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 MeyersCRC 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 MeyersCRC 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 MeyersCRC 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 \(LR_{ij}\).