Sherman Model (Sherman)

Our Sherman model is a Bayesian variant of Sherman’s (1984 [1]) inverse power curve model, which is itself analogous to the Bondy method but with a different functional form. Like our Generalized Bondy model, our Sherman model is fitted directly to the loss triangle (as opposed to being fitted to age-to-age factors). It is implemented by the Sherman model type, which is expressed mathematically as:

\[\begin{split}\begin{align} \begin{split} \mathrm{LR}_{ij} &\sim \mathrm{Gamma(\mu_{ij}, \sigma_{ij}^2)}\\ \mu_{ij} &= ATA_{ij} y_{ij - 1}\\ ATA_{j} &= 1 + \exp( ATA_{\text{int}} - \beta \log(j) )\\ \sigma_{ij}^2 &= \exp(\sigma_{\text{int}} + \sigma_{\text{slope}} j - \log(\mathrm{EP}_{i})), \quad{\forall j \in [\rho_1, \rho_2]}\\ \log ATA_{\text{int}} &\sim \mathrm{Normal}(ATA_{\text{int}, \text{loc}}, ATA_{\text{int}, \text{scale}})\\ \log \frac{\beta}{1 - \beta} &\sim \mathrm{Normal}(\beta_{\text{loc}}, \beta_{\text{scale}})\\ \sigma_{\text{int}} &\sim \mathrm{Normal}(\sigma_{\text{int}, \text{loc}}, \sigma_{\text{int}, \text{scale}})\\ \sigma_{\text{slope}} &\sim \mathrm{Normal}(\sigma_{\text{slope}, \text{loc}}, \sigma_{\text{slope}, \text{scale}})\\ ATA_{\text{int}, \text{loc}} &= 0.0\\ ATA_{\text{int}, \text{scale}} &= 1.0\\ \beta_{\text{loc}} &= 0.0\\ \beta_{\text{scale}} &= 1.0\\ \sigma_{\text{int}, \text{loc}} &= 0.0\\ \sigma_{\text{int}, \text{scale}} &= 3.0\\ \sigma_{\text{slope}, \text{loc}} &= -0.6\\ \sigma_{\text{slope}, \text{scale}} &= 0.3 \end{split} \end{align}\end{split}\]

where \(\bf{ATA}\) is a vector of age-to-age factors that capture how losses change across development and \(EP_{i}\) is the total earned premium for the given accident period. Instead of being estimated as independent parameters as in the, e.g. ChainLadder model, here the age-to-age factors are modeled as a function of two parameters, \(ATA_{\text{int}}\) and \(\beta\). The parameter \(ATA_{\text{int}}\) can be interpreted as the intercept for log age-to-age factors, and \(\beta\) as the rate of decay in the age-to-age factors across development. Because \(\log ATA_{\text{int}}\) is unbounded, as development increases, the lowest value the exponential term can take on is 0 (at which point development has reached an asymptote). The \(1\) is therefore added to ensure the age-to-age factors decay a value of 1 at the lowest.

Typically, tail models like the Sherman model are 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 Sherman model above is fit using the following API call:

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

The Sherman 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:

{
    "dev_intercept__loc": 0.0,
    "dev_intercept__scale": 1.0,
    "sherman_exp__loc": 0.0,
    "sherman_exp__scale": 1.0,
    "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 Sherman 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 Sherman 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 \(\mathrm{LR}_{ij}\).