Portfolio optimization in the catastrophe space

• Published date: 01 November 2020
• Author: Carolyn W. Chang,Jack S. K. Chang,Min‐Teh Yu,Yang Zhao
• Date: 01 November 2020
• DOI: http://doi.org/10.1111/eufm.12265

Eur Financ Manag. 2020;26:1414–1448.wileyonlinelibrary.com/journal/eufm1414

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© 2020 John Wiley & Sons Ltd.

DOI: 10.1111/eufm.12265

ORIGINAL ARTICLE

Portfolio optimization in the catastrophe space

Carolyn W. Chang

1

|Jack S. K. Chang

2

|Min‐Teh Yu

3

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Yang Zhao

4

1

California State University, Fullerton,

California

2

California State Polytechnic University,

Pomona, California

3

NCCU‐RIRC and NCTU‐PAIR Lab,

Providence University, Taichung, Taiwan

4

School of Finance, Nankai University,

Tianjin, China

Correspondence

Min‐Teh Yu, Department of Finance,

NCCU‐RIRC and NCTU‐PAIR Lab,

Providence University, #200, Sec. 7,

Taiwan Ave., Taichung 43301, Taiwan.

Email: mtyu@nctu.edu.tw

Abstract

In today's global catastrophe space, the role of

insurance‐linked securities has evolved from that

of a threatened reinsurance substitute to now

being a viable complementary reinsurance pro-

duct, underpinning the convergence of the two

markets. This study constructs a two‐agent se-

quential optimization framework to mimic the

economics of the reinsurance/insurance markets

and shows how NPV‐maximizing reinsurers and

hedging cost‐minimizing insurers can optimally

allocate default‐risky catastrophe reinsurance and

default‐free catastrophe bonds at the interface of

these two markets. We analyze parametric impacts

considering interest rate risk, financial leverage,

basis risk, differential markup, catastrophe arrival

intensity, and severity, as well as other relevant

characteristics.

KEYWORDS

catastrophe bonds, catastrophe space, default risk, optimum

allocation, traditional reinsurance

JEL CLASSIFICATION

G14; G22

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We are indebted to the Editor, John Doukas, and two anonymous referees for their constructive comments and

suggestions. We would also like to thank seminar participants at Temple University, the National University of

Singapore, National Taiwan University, and the ICW Group of Insurance Companies for valuable comments, and Owen

Jeng and Don Mango at Guy Carpenter, and Kimberly Anthony and Shawn Adams at the ICW Group of Insurance

Companies for industry insights.

1|INTRODUCTION

Insurance‐linked securities (ILS), as a form of alternative reinsurance capital, continue to

expand their presence and influence within the global reinsurance landscape. According to a

report by McKinsey and Company (2014), about 18% of the approximately $420 billion in global

catastrophe (CAT) reinsurance (CAT reinsurance hereafter) capacity (i.e., about $75.6 billion) is

provided by third‐party capital in the form of ILS, up from 2% to 3% in the late 1990s.

1

Artemis

(2018)

2

indicates this ILS capacity will grow to about $100 billion by the end of 2018, following

the substantial catastrophe losses in 2017 from Hurricanes Harvey, Irma, Jose, and Maria and

rampant California wildfires. Among the three main types of ILS –CAT bonds, industry loss

warranties (ILW), and collateralized reinsurance –CAT bonds are the dominant one and are

truly securitized, in which a secondary market exists for them with daily trading available on an

OTC market organized by a number of brokers, including Deutsche Bank, BNP Paribus, Willis,

Swiss Re, etc. The ILS marketplace now features a solid, expanding core of experienced and

dedicated third‐party fund investors, rating agencies, and modeling firms, while issuers have

also become far more comfortable with using ILS strategies. To wit, the role of ILS has evolved

from that of a threatened reinsurance substitute to one that is a complementary product,

underpinning the ultimate convergence of the two markets.

In the insurance industry a cedent's top‐down catastrophe risk financing decision starts first

with deciding on how much internal capital to hold versus how much external risk transfer

programs to purchase. Next, the cedent decides how to optimally allocate between traditional

reinsurance and ILS (in particular, CAT bonds) across ascending loss layers to achieve the most

cost‐effective risk transfer solution. While academic research has been lagging in the optimal

allocation of these two inherently different assets, the insurance industry regularly faces such

decisions. For example, as demonstrated in Figure 1, NCJUA/NCIUA's (North Carolina Joint

Underwriting Association/North Carolina Insurance Underwriting Association) funding structure

as of May 2010 was composed of four excess‐of‐loss reinsurance layers and three CAT bond

issuances in addition to other sources. For another example, for the fourth layer to cover a 135‐year

event with attachment point $3.34bn and detachment point $4bn and a funding limit of $700mn,

the allocation is composed of 50.27% of excess‐of‐loss reinsurance purchased, 43.57% of CAT bonds

issued with respective sizes $200mn and $105mn, and 6.16% of member company assessments. For

both the first and second layers, however, the decisions were not to issue CAT bonds.

The purpose of this research is to develop a novel theoretic model of portfolio optimization

in the catastrophe space between traditional reinsurance and CAT bonds for both insurers and

reinsurers. This optimal allocation decision though is not as straightforward as de facto

asset allocation in traditional portfolio analysis for two reasons. As Cummins and Trainar

(2009) succinctly suggest, optimum allocation models have been lacking in the insurance and

risk management literature, because of the inherently different trading mechanisms, risk

structures, and contract designs of these two products. First, standardized CAT bonds are being

fully collateralized, default‐free, and traded on capital markets, whereas customized traditional

1

Entitled ‘Could Third‐Party Capital Transform the Reinsurance Markets?’The market for ILS emerged in the mid‐

1990s following the aftermath of Hurricane Andrew (1992) and the Northridge earthquake (1994). Third parties have

often worked with reinsurers to bring capital to insurance risks, for example, pension funds working together with

reinsurers to window out default‐free collateralized reinsurance by providing cash collaterals. A new batch of CAT bond

funds has also emerged in recent years to heed the demand of institutional investors for higher returns than comparable

corporate bonds with the same ratings, due to the US Federal Reserve's ultra‐low interest rate policy.

2

Entitled ‘Nephila Capital Adds 11% to Reach $12.2bn of ILS Assets Managed,’17 July 2018.

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reinsurance is only partially collateralized and thus default‐risky. Second, one must consider a

large number of attributes, such as a reinsurer's markup, layering, capital/debt positions, and

interest rate/credit risk exposures; a CAT bond's markup, trigger point and size, and basis risk;

and the catastrophe arrival frequency and loss volatility.

As a novel endeavor to fill this gap we contribute to the literature by providing optimization and

computational methods for portfolio optimization in the catastrophe space in a threefold manner.

First, we propose a new two‐step integrated reinsurance and CAT bond pricing approach by

straddling the actuarial science and the mathematical finance approaches. We apply an incomplete‐

market version of the no‐arbitrage martingale pricing paradigm of Harrison and Kreps (1979)and

Harrison and Pliska (1981) to factor in the modeled catastrophe risk premiums by means of a

measure change from the physical measure to the risk‐neutralized measure Qand then top up the

risk premiums by factoring in the respective non‐modeled factors as markups using the actuarial

science approach. Second, we construct a two‐agent sequential optimization framework to mimic

the economics of the reinsurance/insurance markets and show how NPV‐maximizing reinsurers

and hedging cost‐minimizing insurers would optimally allocate these two distinct assets at the

interface of the two markets. Third and finally, we conduct an extensive sensitivity analysis via

simulation to study the parametric impact on our optimum allocation results.

This proposed optimal allocation method is rooted in the structure of a CAT bond, as

illustrated in Figure 2,

3

and issued by a reinsurer‐sponsored SPV (Special Purpose Vehicle).

FIGURE 1 NCJUA/NCIUA's (North Carolina Joint Underwriting Association/North Carolina

Insurance Underwriting Association) funding structure as of May 2010. Source: Aon Benfield (2013)

3

See section 3.2 for more details about the allocation method as nested in the CAT bond structure.

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CHANG ET AL.