Motivation

• A unique feature in nonlife insurance risk classification: rating variables are categorical and many have a large number of levels

• The high cardinality in the categorical rating variables imposes challenges in the implementation of the traditional actuarial methods

Motivation

• A unique feature in nonlife insurance risk classification: rating variables are categorical and many have a large number of levels

• The high cardinality in the categorical rating variables imposes challenges in the implementation of the traditional actuarial methods

• In particular, the generalized linear models (GLMs) have some difficulties

  • Unrealistic amount of computational resource due to the high-dimensional design matrix

  • The higher likelihood of insufficient data in some categories of the rating variable

  • The relationship between different levels of the rating variable is usually ignored

Goal

We present several actuarial applications of categorical embedding in the context of nonlife insurance risk classification.

• Single insurance risk

• Dependent insurance risks

• Pricing new risks with sparse data

Based on paper:

P. Shi., K. Shi, 2021, Nonlife Insurance Risk Classification Using Categorical Embedding. Available at SSRN.

What is categorical Embedding?

• The idea is due to Guo and Berkhahn (2016). The method maps each categorical variable into a real-valued representation in the Euclidean space.

• In the embedding space, the categories with similar effects are close to each other. Similar to word embedding in natural languge processing.

What is categorical Embedding?

• The idea is due to Guo and Berkhahn (2016). The method maps each categorical variable into a real-valued representation in the Euclidean space.

• In the embedding space, the categories with similar effects are close to each other. Similar to word embedding in natural languge processing.

For the categorical variable \(x\) with \(K\) levels, the embedding function of \(d\)-dimensional embedding space is given by: \[\begin{align} e: x \mapsto \bf{\Gamma} \times \bf{\delta}, \end{align}\]

The \(k\)th category is represented by the \(k\)th column of \(\bf{\Gamma}\). To see this, for the \(i\)th data point with \(x_i=c_k\), we note: \[\begin{align} e(x_i) = \left( \begin{array}{ccc} \gamma_{11} & \cdots & \gamma_{1K} \\ \vdots & \ddots & \vdots \\ \gamma_{d1} & \cdots & \gamma_{dK} \\ \end{array} \right) \times \left( \begin{array}{c} \delta_{x_i,c_1} \\ \vdots \\ \delta_{x_i,c_K} \\ \end{array} \right) = \left( \begin{array}{c} \gamma_{1k} \\ \vdots \\ \gamma_{dk} \\ \end{array} \right). \end{align}\]

Categorical Embedding and Nueral Networks

The embeddings can be automatically learned by a neural network in the supervised training process.

• Add an embedding layer, an extra layer between the input layer and the hidden layer, in the neural network

• Treat the embedding matrix as the weight parameters of the embedding neurons

Actuarial Literature

We emphasize that categorical embedding is especially useful in two scenarios:

• It mitigates overfitting and thus leads to better prediction for the neural network.
  • Fast growing literautre on applications of neural networks in actuarial applications: Wuthrich and Merz (2019), Wuthrich (2019), Perla et al (2020) among others.
• It is more often that the interest of categorical embedding is the embedding itself rather than the predicted outcome.

Data

The insurance claims dataset is obtained from the local government property insurance fund of Wisconsin

• We examine the building and contents insurance that covers damage to both physical structures and items inside

• There are over one thousand entities observed during years 2006-2013, resulting in 8,880 policy-year observations.

Description of rating variables

Data

We consider a binary outcome that measures the claim frequency by peril

Data

Claim frequency outcomes are dependent:

Rating Classes for A Single Risk

In this case, we consider the context where there is a single insurance risk:

• Treat the open-peril property insurance as an umbrella policy

• Define the claim frequency as a risk measurement for the aggregate claims from all peirls

We fit neural networks:

• One-hot encoding

• Categorical embedding

Rating Classes for A Single Risk

Some results on prediction:

Rating Classes for A Single Risk

We could also use the embeddings to create risk classes:

Rating Classes for A Single Risk

Portfolio Management for Dependent Risks

• In this case, we consider a model for multi-peril risks

• Let \(Z_j\) be the outcome for peril \(j\). We formulate the problem as a multi-output network for the vector \(Y=(Z_1,Z_2,Z_3)\)

• We use the dependence ratio to describe the raltionship among perils

\[\begin{align} \rho(z_1,z_2,z_3) = \frac{{\rm Pr}(Z_1=z_1,Z_2=z_2,Z_3=z_3)}{{\rm Pr}(Z_1=z_1){\rm Pr}(Z_2=z_2){\rm Pr}(Z_3=z_3)} \end{align}\]

Portfolio Management for Dependent Risks

• In this case, we consider a model for multi-peril risks

• Let \(Z_j\) be the outcome for peril \(j\). We formulate the problem as a multi-output network for the vector \(Y=(Z_1,Z_2,Z_3)\)

• We use the dependence ratio to describe the raltionship among perils

\[\begin{align} \rho(z_1,z_2,z_3) = \frac{{\rm Pr}(Z_1=z_1,Z_2=z_2,Z_3=z_3)}{{\rm Pr}(Z_1=z_1){\rm Pr}(Z_2=z_2){\rm Pr}(Z_3=z_3)} \end{align}\]

Portfolio Management for Dependent Risks

We consider two types of insurance coverage, the stop-loss insurance and the excess-of-loss insurance. The insurer’s retained loss can be represented as: \[\begin{align*} {\rm Stop ~loss}:& ~R_1 = \min\{S,d_1\}\\ {\rm Excess~ of~ loss}:& ~R_2 = \max\{S-d_2,0\} \end{align*}\]

Pricing New Risks

• Suppose that the insurer has only provided coverage for water and other perils during years 2006-2011. Starting from year 2012, the insurer plans to offer fire coverage as well.

• We demonstrate the idea of transfer learning using the categorical variable county.

  • Learn the embeddings from single peril: water or other
  • Learn the embeedings from the joint bi-peril model: water and other

Pricing New Risks

• Suppose that the insurer has only provided coverage for water and other perils during years 2006-2011. Starting from year 2012, the insurer plans to offer fire coverage as well.

• We demonstrate the idea of transfer learning using the categorical variable county.

  • Learn the embeddings from single peril: water or other
  • Learn the embeedings from the joint bi-peril model: water and other

• Comparison of similarity matrix