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

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 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. …