Stochastic loss reserving: A new perspective from a Dirichlet model

• Published date: 01 March 2021
• DOI: http://doi.org/10.1111/jori.12311
• Author: Peng Shi,Karthik Sriram
• Date: 01 March 2021

J Risk Insur. 2021;88:195–230. wileyonlinelibrary.com/journal/JORI

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195

Received: 7 June 2019

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Accepted: 16 March 2020

DOI: 10.1111/jori.12311

ORIGINAL ARTICLE

Stochastic loss reserving: A new perspective

from a Dirichlet model

Karthik Sriram

1

|Peng Shi

2

1

Production and Quantitative Methods

Area, Indian Institute of Management

Ahmedabad, Ahmedabad, Gujarat, India

2

Risk and Insurance Department,

Wisconsin School of Business, University of

Wisconsin‐Madison, Madison, Wisconsin

Correspondence

Peng Shi, 975 University Ave,

Madison, WI 53706.

Email: pshi@bus.wisc.edu

Abstract

Forecasting the outstanding claim liabilities to set

adequate reserves is critical for a nonlife insurer's

solvency. Chain–Ladder and Bornhuetter–Ferguson

are two prominent actuarial approaches used for this

task. The selection between the two approaches is

often ad hoc due to different underlying assumptions.

We introduce a Dirichlet model that provides a

common statistical framework for the two ap-

proaches, with some appealing properties. Depending

on the type of information available, the model in-

ference naturally leads to either Chain–Ladder or

Bornhuetter–Ferguson prediction. Using claims data

on Worker's compensation insurance from several

U.S. insurers, we discuss both frequentist and Baye-

sian inference.

KEYWORDS

Bayesian, Bornhuetter–Ferguson, Chain–Ladder, Dirichlet

distribution, loss reserve

1|INTRODUCTION

Claims reserving is a classical actuarial problem where actuaries estimate the outstanding

liabilities of an insurer and quantify the associated variability. To emphasize its importance,

first, as the largest liability item on an insurer's balance sheet, claims reserve is required to be

opined by qualified actuaries to meet regulatory requirements (Friedland, 2013); second, since

claims for a given insurance portfolio can evolve over time, developing the incurred claims to

the ultimate level is a critical component in ratemaking—another classical actuarial function

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© 2020 American Risk and Insurance Association

for pricing the insurance contracts (Brown & Gottlieb, 2015). In addition, reserving practice is

closely related to the solvency risk. Inadequacy of reserves has been reported as the most

contributing factor to a nonlife insurer's failure (Coyne, 2008).

In the reserving context, insurance claims data, usually referred to as losses, are often aggregated

by lines of business and organized in a triangular format, known as run‐off triangles, to reflect the fact

that losses are incurred and developed over time. For all claims with incidents in a particular year,

known as accident year, the run‐off triangle shows the losses paid every year until the current

calendar year. The data structure (see lower section of Table 1) is triangular because, by the end of the

evaluation year, only 1 year of losses would have been observed for the current accident year, while

10 years of development could have been observed for an accident year that is 10 years prior.

However, it is possible that more payments relating to existing claims from an accident year can arise

in the future, and also new claims corresponding to an accident year can be reported in the future.

The objective is to consider the known losses so far for every accident year and obtain a forecast of

incremental as well as cumulative losses for the subsequent years. The total cumulative losses

resulting from any given accident year is referred to as the ultimate loss. Regulatory reporting requires

that such an exercise consider the recent 10 accident years, and the forecast be obtained for the

10 years, referred to as development years, following each accident year.

Over the years, a large variety of stochastic claims reserving methods based on run‐off

triangles have been proposed by practitioners and academics (see England & Verrall, 2002 for

comparison and Wüthrich & Merz, 2008 for a book‐long review of alternative approaches).

Among them, the most prominent and most venerable are the Chain–Ladder method

and the Bornhuetter–Ferguson method. The original ideas of the Chain–Ladder and

Bornhuetter–Ferguson algorithms trace back to Tarbell (1934) and Bornhuetter and Ferguson

(1972), respectively. Later, stochastic models are proposed to reproduce the prediction from the

two algorithms and to quantify the associated reserving variability. For example, see the

distribution‐free method by Mack (1993), the bootstrap method by England and Verrall (1999),

Peters, Wüthrich, and Shevchenko (2010), and Pinheiro, Andrade e Silva, and de Lourdes

Centeno (2003), and the Bayesian approach by England and Verrall (2006) for the

Chain–Ladder method; and see Verrall (2004), Mack (2008), and Alai, Merz, and Wüthrich

(2009,2011), and Saluz, Gisler, and Wüthrich (2011) for the stochastic models that support the

Bornhuetter–Ferguson method.

The Chain–Ladder and Bornhuetter–Ferguson algorithms are different yet related. Speci-

fically, the former predicts the future cumulative losses by multiplying the current cumulative

TABLE 1 Exhibit of a run‐off triangle of loss ratios

Development year

Accident year 1 2



n−1

n

1

Y

1

1

Y

1

2

⋯

Yn1−

1

Yn1

Fully developed

⋮

⋮

⋮

mn−Y

mn−,

1

Y

mn−,

2

⋯

Ymnn−,−

1

Y

mnn−,

mn−+

1

Y

mn−+1,

1

Y

mn−+1,

2

⋯

Ymn n−+1, −

1

Y

mn

n

−+1,

mn−+

2

Y

mn−+2,

1

Y

mn−+2,

2

⋯

Ymn n−+2, −

1

Run‐off triangle

⋮

⋮

m−

1

Y

m−1,

1

Ym−1,

2

m

Ym,

1

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losses by suitable “development factors”estimated from the triangle data. The latter predicts the

outstanding losses by multiplying the expected ultimate losses by the percentage of (also re-

ferred to as “quota”) unpaid losses. While the percentage of unpaid losses is estimated from the

triangle data, the expected ultimate losses are usually obtained from external information, such

as expert actuarial input or based on industry benchmarks. A common approach to calculate

the expected ultimate loss is by taking the product of earned premiums for a given accident year

and expected loss ratio (i.e., ratio of loss to premium) obtained from external sources. The link

between the two algorithms is the mapping between the development factors and the devel-

opment percentages (or quotas). Because of this link, the Bornhuetter–Ferguson prediction of

the ultimate losses can be viewed as a credibility weighted average of the Chain–Ladder pre-

diction based on the run‐off triangle and the expected ultimate losses based on external sources.

Despite the direct relationship between the Chain–Ladder and Bornhuetter–Ferguson al-

gorithms through the loss development pattern, there is little connection between the asso-

ciated stochastic claims reserving models. Due to the need for assessing the prediction

uncertainty for claims reserves, stochastic methods are independently developed to reproduce

the predictions from the Chain–Ladder and Bornhuetter–Ferguson algorithms. Because these

models are based on different assumptions, the problem is typically framed in the model

selection context and the selection between the two mainstream methods in practice are often

ad hoc and subjective, and depending on the actuary's preference.

Motivated by the above observation, we propose a new stochastic loss reserving model based

on a Dirichlet distribution (see Frigyik, Kapila, & Gupta, 2010 for an introduction to the

Dirichlet distribution). The central idea is to treat the loss development quotas in a run‐off

triangle as compositional data and then formulate them using a Dirichlet distribution. In

Section 3we highlight a mathematical characterization (due to Darroch & Ratcliff, 1971) that

makes the Dirichlet distribution relevant for loss development data. We check the usefulness of

this distributional assumption empirically by using run‐off triangle data from a group of in-

surance companies. The Dirichlet distribution is often used as a conjugate prior for the mul-

tinomial distribution in Bayesian analysis. An example of such application in loss reserving is

Clark (2016). In contrast, our work, to the best of our knowledge, is the first one to employ the

Dirichlet model for the sampling distribution of claims data.

More important, the proposed Dirichlet model offers a new perspective to view the relation

between two mainstream industry methods, namely, the Chain–Ladder and Bornhuetter–Ferguson

methods. Interestingly, we show that the maximum likelihood estimation (MLE) of the model leads

to a reserve prediction that nests the Chain–Ladder prediction. In contrast, a Bayesian inference that

incorporates additional external information or expert knowledge provides the Bornhuetter–Ferguson

type prediction. Therefore, the choice between the Chain–Ladder method and the

Bornhuetter–Ferguson method essentially depends on the types of information available for model

estimation. This is a crucial point. Because both methods can now be derived from a common

stochastic reserving model, the selection of reserving methods becomes an inference problem rather

than a model selection problem.

We emphasize that the proposed Dirichlet framework leads to predictions with an im-

portant desirable property in the loss reserving context. Similar to the Bornhuetter–Ferguson

method, the prediction for accident‐year cumulative losses is shown to be a credibility weighted

average of the Chain–Ladder prediction and the expected (loss ratio) method. It is interesting

that the credibility weight is determined by the coefficient of variation, as opposed to the

expected value of the current cumulative losses that is used in the Bornhuetter–Ferguson

method. So, the weight assigned by the Dirichlet model not only considers the expected value

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