A Portfolio Optimization Approach Using Combinatorics With a Genetic Algorithm for Developing a Reinsurance Model

• Author: Lysa Porth,Jeffrey Pai,Milton Boyd
• Published date: 01 September 2015
• Date: 01 September 2015
• DOI: http://doi.org/10.1111/jori.12037

APORTFOLIO OPTIMIZATION APPROACH USING

COMBINATORICS WITH A GENETIC ALGORITHM FOR

DEVELOPING A REINSURANCE MODEL

Lysa Porth

Jeffrey Pai

Milton Boyd

ABSTRACT

Some insurance firms challenged with a portfolio of high-variance risks face

the classic trade-off between risk spreading and risk retaining. Using crop

insurance as an example, a new solution to this problem is undertaken to

uncover an improved reinsurance design. Joint self-managed reinsurance

pooling and private reinsurance are combined in a portfolio approach

utilizing combinatorial optimization with a genetic algorithm (Model C),

achieving high surplus, high survival probability, and low deficit at ruin.

This portfolio model may also be useful for other large natural disaster and

weather-related insurance portfolios, and other portfolio applications.

INTRODUCTION

Risk management is an integral part of organizational processes, and there is a

potential for many insurance organizations to realize meaningful gains by optimizing

their current risk management policies. Most insurance organizations are often faced

with individual risks that are relatively small, frequent, and uncorrelated, for which

the variance of aggregate risks in the portfolio is low. But conversely, some firms are

a

Lysa Porth is Assistant Professor and Guy Carpenter Professor in Agricultural Risk

Management and Insurance, Warren Centre for Actuarial Studies and Research, Asper

School of Business, University of Manitoba; Assistant Professor, Department of Agribusiness

and Agricultural Economics, Faculty of Agricultural and Food Sciences, University of

Manitoba; Adjunct Professor, Department of Statistics and Actuarial Science, Faculty of

Mathematics, University of Waterloo, Canada. Porth can be contacted via e-mail:

lysaporth@cc.umanitoba.ca. Jeffrey Pai is Professor and Director, Warren Centre for Actuarial

Studies and Research, Asper School of Business, University of Manitoba, Winnipeg, Canada.

Pai can be contacted via e-mail: jpai@cc.umanitoba.ca. Milton Boyd is Professor, Department

of Agribusiness and Agricultural Economics, Faculty of Agricultural and Food

Sciences, University of Manitoba, Winnipeg, Canada. Boyd can be contacted via e-mail:

boyd@cc.umanitoba.ca. Additional supplemental material from this article can be accessed at:

http://umanitoba.ca/faculties/management/faculty_staff/academic_professors/porth_lysa.

html.

© 2014 The Journal of Risk and Insurance. 82, No. 3, 687–713 (2015).

DOI: 10.1111/jori.12037

687

faced with a more difficult challenge of managing risks that are large, infrequent, and

potentially highly correlated within geographic regions and/or product lines. This

leads to a portfolio of aggregate risks with high variance.

Therefore, the objective of this article is to address the common problem of high

variance in insurance portfolios. A solution is provided using a combination of joint

self-managed reinsurance pooling, and private reinsurance, in a portfolio approach

that utilizes combinatorial optimization with a genetic algorithm (Model C), in order

to improve efficiency. This approach takes advantage of the natural offsetting of risks

across regions in order to reduce risk in a cost-effective manner. In addition, the

reinsurance pool is supplemented with private reinsurance for a select group of

correlated risks that do not sufficiently offset, further reducing risk.

Traditionally, insurance firms seek to reduce the risk associated with a portfolio of

high-variance risks by purchasing private reinsurance. Through transferring risk to a

reinsurer, the insurer incurs additional cost in the form of reinsurance premium. The

higher the expected risk transferred to a reinsurer, the more costly the reinsurance

premiums. However, on the other hand, the insurer can lower the cost of reinsuring

by maintaining higher exposure to expected retained risk. This is the classic trade-off

between risk spreading and risk retaining, which demonstrates the importance of

optimal reinsurance design (Borch, 1960; Denuit and Vermandele, 1998; Cai and

Tan, 2007; Cai et al., 2008; Tan, Weng, and Zhang, 2009, 2011; Tan and Weng, 2012;

Porth, Tan, and Weng, 2013).

Crop insurance is a particularly suitable example to consider for reinsurance pooling

and private reinsurance. While crop insurance has generally been successful in a

number of countries when supported by government, without a subsidy it has often

been considered unsuccessful (Miranda and Glauber, 1997). Commonly, asymmetric

information has been cited for crop insurance market difficulties, including adverse

selection, and moral hazard (Holstrom, 1979; Skees and Reed, 1986; Chambers, 1989;

Quiggin, 1994; Nelson and Loehman, 1997; Cohen and Siegelman, 2010; Woodard

et al., 2012).

In addition to asymmetric information, another explanation for crop insurance

market difficulties has been that crop risks are substantially larger (Miranda and

Glauber, 1997) than the portfolio risk faced when risk exposures are independent.

Further, these losses tend to be very large, highly variable from year to year, and

correlated within geographical areas, making these different than other insurable

risks (Miranda and Glauber, 1997; Skees and Barnett, 1999; Pai, Boyd, and

Porth, forthcoming). However, Wang and Zhang (2003) show that while losses are

often correlated within regions, they are less correlated across regions in the United

States. Similarly, Okhrin, Odening, and Xu (2013) show significant nonlinear spatial

diversification effects for crop insurance in China. They find that indemnities are

found to be dependent on the distance between regions, but as the geographic region

is increased, indemnities exhibit less dependence.

In Canada, each province operates its own crop insurance program independently of

the other nine provinces, and it is cost-shared between farmers (40 percent) and

government (60 percent). Currently, the provincial crop insurance firms in Canada do

688 THE JOURNAL OF RISK AND INSURANCE