The×Effectiveness of Gap Insurance With Respect to Basis Risk in a Shareholder Value Maximization Setting

• Author: Nadine Gatzert,Ralf Kellner
• Date: 01 December 2014
• Published date: 01 December 2014
• DOI: http://doi.org/10.1111/j.1539-6975.2013.01523.x

THE EFFECTIVENESS OF GAP INSURANCE WITH RESPECT TO

BASIS RISK IN A SHAREHOLDER VALUE MAXIMIZATION

SETTING

Nadine Gatzert

Ralf Kellner

ABSTRACT

The purchase of index-linked alternative risk transfer instruments can lead

to basis risk, if the insurer’s loss is not fully dependent on the index. One way

to reduce basis risk is to additionally purchase gap insurance, which fills the

gap between an insurer’s actual loss and the index-linked instrument’s

payout. The previous literature detects gains in the effectiveness of this

hedging strategy in a mean–variance framework. The aim of this article is to

extend this analysis and to examine the effectiveness of gap insurance in a

shareholder value maximization framework under solvency constraints.

Our results show that purchasing gap insurance can generally increase the

hedging effectiveness in multiple ways by reducing basis risk, thus

increasing shareholder value and, at the same time, lowering shortfall risk.

INTRODUCTION

The increasing number and magnitude of catastrophic events in recent years

emphasized the potential stress on insurance and traditional reinsurance markets’

capacities. To overcome these capacity constraints, alternative risk transfer (ART)

instruments such as cat bonds, cat options, or industry loss warranties (ILWs) have

been introduced in the past decades. These instruments often feature a contract

design that links their payoff to the development of an index. Thus, they come along

with benefits such as higher transparency; lower transaction costs than, for example,

traditional reinsurance; and a reduction of moral hazard (see, e.g., Gatzert and

Schmeiser, 2011). However, at the same time, basis risk can occur, as the insurer’s

exposure is usually not fully dependent on the index (see, e.g., Harrington and

Niehaus, 1999; Zeng, 2000), thus implying that the index-linked product does not pay

off, even though the buying insurer has a high loss. A potential strategy to overcome

Nadine Gatzert and Ralf Kellner are at the Department of Insurance Economics and Risk

Management, Friedrich-Alexander-University (FAU) Erlangen-Nu

¨rnberg, Lange Gasse 20,

D-90403 Nuremberg, Germany. The authors can be contacted via e-mail: nadine.gatzert@fau.de

and ralf.kellner@fau.de. The authors would like to thank two anonymous referees for valuable

comments and suggestions on an earlier version of the paper.

DOI: 10.1111/j.1539-6975.2013.01523.x

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© The Journal of Risk and Insurance, 2013, Vol. 81, No. 4, 831–859

basis risk is to additionally purchase so-called gap insurance, thus filling the gap

between an insurer’s actual loss and the index-linked instrument’s payout. In

previous work, Doherty and Richter (2002) demonstrate potential gains in the

effectiveness in a mean–variance framework if the index-based hedge is replenished

through a fraction of an indemnity-based instrument. The aim of this article is to take

this analysis further and to analyze whether gains in the effectiveness from gap

insurance can also be observed in a comprehensive shareholder value maximization

framework under solvency constraints.

There has been steady growth in research on index-linked cat instruments. Even

though ILWs were the first traded index-linked instruments in the 1980s (see Swiss

Re, 2009), the literature only began to focus on these instruments with the

implementation of insurance futures based on catastrophic loss indices in 1992 by

the Chicago Board of Trade. Early articles analyze the usage of these instruments (see,

e.g., D’Arcy and France, 1992; Harrington, Mann, and Niehaus, 1995) and discuss

possible impediments for their success (see, e.g., Cox and Schwebach, 1992; Hoyt and

Williams, 1995). The impact of basis risk is measured in several articles by means of

the hedging effectiveness. Major (1999), for instance, determines an insurer’s loss

volatility reduction through a linear hedge in a simulation analysis, comparing

attained volatilities through hedging strategies using statewide and zip-based

indices. Harrington and Niehaus (1999) and Cummins, Lalonde, and Phillips (2004)

empirically analyze basis risk of insurance derivatives and find basis risk not to be a

significant impediment for hedging strategies that are based on state-specific indices.

However, they detect that using state-wide indices leads to substantial basis risk,

especially for insurance companies whose underwriting business is not diversified

across the country. An extension of existing basis risk definitions is introduced by

Zeng (2000, 2003, 2005), who compares the hedging effectiveness of index-linked

instruments to traditional reinsurance. Lee and Yu (2002) develop a model to price cat

bonds, incorporating moral hazard and basis risk, while Gatzert, Schmeiser, and

Toplek (2011) simultaneously examine basis risk and pricing of ILWs based on

various measures of basis risk and a comparison of different actuarial and financial

pricing approaches. An analysis of contract features, pricing, and central demand

factors of ILWs is conducted in Gatzert and Schmeiser (2011), and a comparison of

existing basis risk definitions and the impact of nonlinear dependencies on basis risk

in the context of ILWs are studied in Gatzert and Kellner (2011).

Besides the examination of basis risk, interaction effects between traditional

reinsurance and ART instruments are analyzed in several articles. Doherty and

Richter (2002) examine a hedging strategy that combines an index-linked instrument

and indemnity-based (gap) reinsurance and find potential gains in the effectiveness in

a mean–variance framework, while Nell and Richter (2004) analyze interactions

between reinsurance and cat bonds in an expected utility approach, thereby

identifying substitution effects between cat bonds and the demand for reinsurance for

large losses. Lee and Yu (2007) study how the value of a reinsurer’s contract can be

increased by means of issuing cat bonds, whereas Song and Cummins (2008) observe

substitution effects between derivative hedging and traditional reinsurance. One

example of a transaction that combines a reinsurance structure (i.e., comparable to

gap insurance) and a parametric cat bond is a transaction sponsored by the Mexican

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