# Loss Reserving Models: Granular and Machine Learning Forms

## Abstract

**:**

## 1. Background

## 2. Notation and Terminology

## 3. The Jurassic Period

- (a)
- Each row of the triangle is a Markov chain.
- (b)
- Distinct rows of the triangle are stochastically independent.
- (c)
- ${X}_{i,j+1}|{X}_{ij}$ is subject to some defined distribution for which $E\left[{X}_{i,j+1}|{X}_{ij}\right]={f}_{j}{X}_{ij}$, where ${f}_{j}$ is a parameter to be estimated from data.

## 4. The Cretaceous Period—Seed-Bearing Organisms Appear

_{ij}:

_{ij}, (3) becomes a Generalised Linear Model (GLM) (McCullagh and Nelder 1989), specified as follows:

- (a)
- ${Y}_{ij}~F\left({\mu}_{ij},\phi /{w}_{ij}\right)\mathrm{where}{\mu}_{ij}=E\left[{Y}_{ij}\right]$ and $F$ is a distribution contained in the exponential dispersion family (EDF) (Nelder and Wedderburn 1972) with dispersion parameter $\phi $ and weights ${w}_{ij}$;
- (b)
- ${\mu}_{ij}$ takes the parametric form $h\left({\mu}_{ij}\right)={x}_{ij}^{T}\beta $ for some one–one function $h$ (called the link function), and where ${x}_{ij}$ is a vector of covariates associated with the $\left(i,j\right)$ cell and $\beta $ the corresponding parameter vector.

- in 1972, the concept was introduced by Nelder and Wedderburn;
- in 1977, modelling software called GLIM was introduced;
- in 1984, the Tweedie family of distributions was introduced (Tweedie 1984), simplifying the modelling software;
- in 1990 and later, seminal actuarial papers (Wright 1990; Brockman and Wright 1992) appeared.

- analysis of an Auto Liability (relatively long-tailed) portfolio (Taylor and McGuire 2004) with:
- ○
- rates of claim settlement that varied over time;
- ○
- superimposed inflation (SI) (a diagonal effect) that varied dramatically over time and also over OT (defined in Section 2);
- ○
- a change of legislation affecting claim sizes (a row effect);

- analysis of a mortgage insurance portfolio (Taylor and Mulquiney 2007), using a cascade of GLM sub-models of experience in different policy states, viz.
- ○
- healthy policies;
- ○
- policies in arrears;
- ○
- policies in respect of properties that have been taken into possession; and
- ○
- policies in respect of which claims have been submitted;

- analysis of a medical malpractice portfolio (Taylor et al. 2008), modelling the development of individual claims, both payments and case reserves, taking account of a number of claim covariates, such as medical specialty and geographic area of practice; and
- a monograph on GLM reserving (Taylor and McGuire 2016).

## 5. The Paleogene—Increased Diversity in the Higher Forms

#### 5.1. Adaptation of Species—Evolutionary Models

_{1}, …, f

_{I−}

_{1}varying stochastically from one row to the next. This type of modelling can be achieved by a simple extension of the GLM framework defined in Section 4. The resulting model is the following.

- (a)
- ${Y}_{ij}~F\left({\mu}_{ij}^{\left(t\right)},\phi /{w}_{ij}\right)\mathrm{where}\text{}{\mu}_{ij}^{\left(t\right)}=E\left[{Y}_{ij}\right]$;
- (b)
- ${\mu}_{ij}^{\left(t\right)}$ takes the parametric form $h\left({\mu}_{ij}^{\left(t\right)}\right)={x}_{ij}^{T}{\beta}^{\left(t\right)}$, where the parameter vector is now ${\beta}^{\left(t\right)}$ in payment period $t$; and
- (c)
- The vector ${\beta}^{\left(t\right)}$ is now random: ${\beta}^{\left(t\right)}~P\left(.;{\beta}^{\left(t-1\right)},\psi \right)$, which is a distribution that is a natural conjugate of $F\left(.,.\right)$ with its own dispersion parameter $\psi $.

- all parameters have been superscripted with a time index;
- the fundamental parameter vector ${\beta}^{\left(t\right)}$ is now randomised, with a prior distribution that is conditioned by ${\beta}^{\left(t-1\right)}$, the parameter vector at the preceding epoch.

^{(t)}and subject to $E\left[{\eta}^{\left(t\right)}\right]=0$.

- the Kalman filter requires a linear relation between observation means and parameter vectors, whereas the present model admits nonlinearity through the link function;
- the Kalman filter requires Gaussian error terms in respect of both observations and priors, whereas the present model admits non-Gaussian within the EDF.

#### 5.2. Miniaturisation of Species—Parameter Reduction

- $\lambda \to 0$: no elimination of covariates (ordinary GLM—see also Table 1);
- $\lambda \to \infty $: elimination of all covariates (trivial regression).

- (a)
- Randomly delete one $n$-th of the data set, as a test sample;
- (b)
- Fit the model to the remainder of the data set (the training set);
- (c)
- Generate fitted values for the test sample;
- (d)
- Compute a defined measure of error (e.g., the sum of squared differences) between the test sample and the values fitted to it;
- (e)
- Repeat steps (a) to (d) a large number of times, and take the average of the error measures, calling this the cross-validation error (CV error).

#### 5.3. Granular (or Micro-) Models

## 6. The Anthropocene—Intelligent Beings Intervene

#### 6.1. Artificial Neural Networks in General

- an accident quarter effect corresponding to the legislative change that occurred in the midst of the data; and
- SI that varied with both finalisation quarter and OT.

#### 6.2. The Interpretability Problem

_{1}, …, γ

_{K}, vector constants β

_{1}, …, β

_{K}, and real-valued functions ${f}_{k}$.

## 7. Model Assessment

- the model’s predictive efficiency; and
- its fitness for purpose.

#### 7.1. Adaptation of Species—Evolutionary Models

- parameter error;
- process error; and
- model error.

- the bootstrap (Taylor and McGuire 2016, sec. 5.3); and
- (in the case of Bayesian models) Markov Chain Monte Carlo (MCMC) (Meyers 2015).

#### 7.2. Fitness for Purpose

## 8. Predictive Efficiency

#### 8.1. Cascaded Models

- claim notification counts;
- claim finalisation counts; and
- claim finalisation amounts.

#### 8.2. Granular Models

**Example**

**1.**

**Example**

**2.**

#### 8.3. Artificial Neural Networks

- for the legislative effect, interaction between accident quarter and OT;
- for SI, interaction between finalisation quarter and OT.

## 9. The Watchmaker and the Oracle

## 10. Conclusions

## Funding

## Conflicts of Interest

## References

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$\mathit{\lambda}$ | $\mathit{p}$ | Special Case |
---|---|---|

0 | - | GLM |

>0 | 1 | Lasso |

>0 | 2 | Ridge regression |

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Taylor, G.
Loss Reserving Models: Granular and Machine Learning Forms. *Risks* **2019**, *7*, 82.
https://doi.org/10.3390/risks7030082

**AMA Style**

Taylor G.
Loss Reserving Models: Granular and Machine Learning Forms. *Risks*. 2019; 7(3):82.
https://doi.org/10.3390/risks7030082

**Chicago/Turabian Style**

Taylor, Greg.
2019. "Loss Reserving Models: Granular and Machine Learning Forms" *Risks* 7, no. 3: 82.
https://doi.org/10.3390/risks7030082