Autoregressive Lag 1 Model (AR1)

The autoregressive lag 1 (AR1) model is a standard autogression model that is meant to be used to model changes in (ultimate) loss ratios across accident periods. The AR1 model assumes that the current loss ratio is a function of the previous loss ratio (i.e. auto-regression), the degree to which is controlled by a reversion parameter. Our AR1 model is implemented in the by the AR1 model type, which is expressed mathematically as:

\[\begin{split}\begin{align*} \mathrm{LR}_{i} &\sim \mathrm{Gamma}(\eta_{i}, \sigma_{i}^2)\\ \eta_{i} &= (1 - \phi_{\text{reversion}}) \mathrm{LR}_{\text{target}} + \phi_{\text{reversion}} \mathrm{LR}_{i - 1}\\ \sigma_{i}^2 &= \sigma_{\text{base}} + \sigma_{\text{obs}} / \mathrm{EP}_i\\ \phi_{\text{reversion}} &= \mathrm{logit}^{-1}(\phi_{\text{reversion}}^{*}) \cdot 2 - 1\\ \phi_{\text{reversion}}^{*} &\sim \mathrm{Normal}(\phi_{\text{reversion}, \text{loc}}, \phi_{\text{reversion}, \text{scale}})\\ \log \mathrm{LR}_{\text{target}} &\sim \mathrm{Normal}(\mathrm{LR}_{\text{target}, \text{loc}}, \mathrm{LR}_{\text{target}, \text{scale}})\\ \log \sigma_{\text{base}} &\sim \mathrm{Normal}(\sigma_{\text{base}, \text{loc}}, \sigma_{\text{base}, \text{scale}})\\ \log \sigma_{\text{obs}} &\sim \mathrm{Normal}(\sigma_{\text{obs}, \text{loc}}, \sigma_{\text{obs}, \text{scale}})\\ \phi_{\text{reversion}, \text{loc}} &= 0.0\\ \phi_{\text{reversion}, \text{scale}} &= 1.0\\ \mathrm{LR}_{\text{target}, \text{loc}} &= -0.5\\ \mathrm{LR}_{\text{target}, \text{scale}} &= 1.0\\ \sigma_{\text{base}, \text{loc}} &= -2.0\\ \sigma_{\text{base}, \text{scale}} &= 1.0\\ \sigma_{\text{obs}, \text{loc}} &= -2.0\\ \sigma_{\text{obs}, \text{scale}} &= 1.0 \end{align*}\end{split}\]

where \(\mathrm{LR}_i\) indicates the observed loss ratio for accidenty year \(i\), and \(mathrm{EP}_i\) is the earned premium for the same accident period. The model is specified such that \(\eta_i\) is the expected loss ratio for each accident period, and the observed loss ratios are then assumed to be Gamma distributed where \(\mathrm{Gamma(\eta_i, \sigma_{i}^2)}\) is the mean-variance parameterization of the Gamma distribution.

Model Fit Configuration

The AR1 model is fit using the following API call:

model = client.forecast_model.create(
    triangle=...,
    name="example_name",
    model_type="AR1",
    config={ # default model_config
        "loss_definition": "reported",
        "loss_family": "gamma",
        "priors": None, # see defaults below
        "recency_decay": 1.0,
        "seed": None
    }
)

The AR1 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".

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

{
    "reversion__loc": 0.0,
    "reversion__scale": 1.0,
    "base_sigma__loc": -2.0,
    "base_sigma__scale": 1.0,
    "obs_sigma__loc": -2.0,
    "obs_sigma__scale": 1.0,
    "target_lr__loc": -0.5,
    "target_lr__scale": 1.0,
}

• recency_decay: Likelihood weight decay to down-weight older experience periods. Defaults to 1.0, which means no decay. If set to a value between 0.0 and 1.0, the likelihood of older experience periods 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 AR1 model is used to predict future losses using the following API call:

predictions = model.forecast_model.predict(
    triangle=...,
    config={ # default config
        "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 target_triangle should be specified to include future accident periods (including earned premium values) that forecasts should be made on.

The AR1 prediction behavior can be further changed with configuration parameters in config:

• 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 \(\eta_{i}\) as predictions as oppposed to \(\mathrm{LR}_{i}\).