CONVERGENCE OF CAPITAL AND INSURANCE MARKETS: CONSISTENT PRICING OF INDEX‐LINKED CATASTROPHE LOSS INSTRUMENTS

• Author: Nadine Gatzert,Nikolai Vogl,Sebastian Pokutta
• Date: 01 March 2019
• DOI: http://doi.org/10.1111/jori.12191
• Published date: 01 March 2019

CONVERGENCE OF CAPITAL AND INSURANCE MARKETS:

CONSISTENT PRICING OF INDEX-LINKED CATASTROPHE

LOSS INSTRUMENTS

Nadine Gatzert

Sebastian Pokutta

Nikolai Vogl

ABSTRACT

Index-linked catastrophe loss instruments have become increasingly

attractive for investors and play an important role in risk management.

Their payout is tied to the development of an underlying industry loss index

(reflecting losses from natural catastrophes) and may additionally depend

on the ceding company’s loss. Depending on the instrument, pricing is

currently not entirely transparent and does not assume a liquid market. We

show how arbitrage-free and market-consistent prices for such instruments

can be derived by overcoming the crucial point of tradability of the

underlying processes. We develop suitable approximation and replication

techniques and—based on these—provide explicit pricing formulas using

cat bond prices. Finally, we use empirical examples to illustrate the

suggested approximations.

Nadine Gatzert and Nikolai Vogl are at the Department of Insurance Economics and Risk

Management, Friedrich-Alexander University Erlangen-N€

urnber g (FAU) , N€

urnber g,

Germany. Gatzert and Vogl can be contacted via e-mail: nadine.gatzert@fau.de,

nikolai.vogl@fau.de. Sebastian Pokutta is at the Department of Industrial and Systems

Engineering (ISyE), Georgia Institute of Technology, Atlanta, Georgia. Pokutta can be

contacted via e-mail: sebastian.pokutta@isye.gatech.edu. The authors would like to

thank two anonymous referees as well as Semir Ben Ammar, David Cummins, Helmut

Gr€

undl,StevenKou,AlexanderM

€

urmann, Mary Weiss, and the participants of the Risk

Theory Society Annual Meeting 2014 in Munich, the Annual Seminar of the European

Group of Risk and Insurance Economists 2014 in St. Gallen, the International AFIR/ERM

Colloquium 2013 in Lyon, the German Finance Association Annual Meeting 2013 in

Wuppertal, the Annual Meeting of the German Insurance Science Association 2014 in

Stuttgart, and the 8th Conference in Actuarial Science & Finance 2014 on Samos for

valuable comments and suggestions on earlier versions of the article.

© 2017 The Journal of Risk and Insurance. Vol. 86, No. 1, 39–72 (2019).

DOI: 10.1111/jori.12191

39

INTRODUCTION

Alternative risk transfer (ART) has become increasingly relevant in recent decades for

insurers and investors,

1

especially due to a considerably growing risk of extreme

losses from natural catastrophes caused by value concentration and climate change,

as well as the limited (and volatile) capacity of traditional reinsurance markets in the

past (Cummins, Doherty, and Lo, 2002). Among the most commonly used ART

instruments are index-linked catastrophe loss instruments such as index-based cat

bonds

2

or industry loss warranties (ILWs), for instance, whose defining feature is

their dependence on an industry loss index and that may also depend on the

company-specific loss resulting from a natural catastrophe.

3

However, the current

degree of liquidity of the various index-linked instruments considerably differs.

Although the market for cat bonds is fairly well developed with an increasingly

relevant secondary market (Albertini, 2009), for instance, the market for ILWs is less

liquid and limited (Elementum Advisors, 2010).

In this article, we focus on how these products can be priced in a consistent way and

discuss under which assumptions (e.g., regarding a liquid underlying market) risk-

neutral valuation can be used. This procedure can considerably simplify pricing and

enhance transparency, making the market as a whole more efficient. In addition, risk-

neutral valuation is of great relevance for the inclusion of such instruments in

enterprise risk management strategies as it provides a mark-to-market valuation

approach, allowing for (partial) hedging, versus the traditional mark-to-model

approaches with the associated model risk (which is very hard to quantify). We

develop a new pricing approach by means of approximations and replication

techniques and apply it to ILWs as a representative of index-linked catastrophe loss

instruments under the assumption of a liquid cat bond market, while addressing the

necessary prerequisites and limitations, and we also illustrate the approach by

consistently pricing different cat bonds. We study binary contracts in detail, whose

payout depends on the industry index only, and discuss indemnity-based contracts,

where the payout depends on both the industry index and the individual company

losses, thus representing a double-trigger product. The approach derived in this

article can also be transferred to the consistent pricing of other index-linked

catastrophe loss instruments.

In the literature, several articles examine the actuarial and financial pricing of index-

linked catastrophe loss instruments such as ILWs, for instance (e.g., Ishaq, 2005;

Braun, 2011; Gatzert and Schmeiser, 2012), and discuss the underlying assumption

briefly (see Braun, 2011). However, the tradability of the underlying processes as well

1

The volume of outstanding cat bonds, for instance, reached $17.5 bn. in 2013 (see AON, 2013).

Investors include, for example, specialized funds, institutional investors, mutual funds, and

hedge funds (see AON, 2013).

2

There are various versions of cat bonds with different types of triggers, including indemnity-

based and nonindemnity-based triggers with parametric, modelled loss, and industry loss

triggers, for instance (see, e.g., Hagedorn et al., 2009).

3

See, for example, Barrieu and Albertini (2009); Cummins and Weiss (2009) for an overview of

the ART market.

40 THE JOURNAL OF RISK AND INSURANCE

as direct replication and consistent pricing has not been discussed in detail so far in

this context. Several articles have dealt with risk-neutral valuation in the context of cat

bonds (see, e.g., Haslip and Kaishev, 2010; Nowak and Romaniuk, 2013) and compute

explicit pricing formulas, while other authors have focused on the consistent pricing

of double-trigger contracts (e.g., Lane, 2004) or empirical aspects using econometric

pricing approaches (e.g., Jaeger, M€

uller, and Scherling, 2010; Galeotti, G€

urtler, and

Winkelvos, 2012; Braun, 2015; G€

urtler, Hibbeln, and Winkelvos, 2016).

In general, the main assumption when using risk-neutral valuation is the tradability

of an underlying process. As the underlying process can usually not be traded directly

like a stock, one has to assume a liquid market for (certain) derivatives. We derive a

general approach for dealing with this issue, describe the underlying assumptions,

and apply this approach to binary ILWs as well as cat bonds. This is done by means of

direct or approximate replication with traded derivatives using available cat bonds,

which leads to explicit and consistent prices.

4

In particular, using ILWs as an example,

we assume the existence of a liquid cat bond market to handle the tradability of the

industry loss index and to apply arbitrage-free valuation. As there is a growing

secondary market for cat bonds, this assumption appears to be appropriate, at least in

the foreseeable future (see, e.g., Albertini, 2009, for a description of the secondary

market). Moreover, we show that liquidity assumptions are not needed to the same

extent as in classical option pricing theory because continuous trading is not

necessary to replicate ILWs when using cat bonds; that is, a static hedging approach is

sufficient, which also reduces transaction costs and possible tracking errors.

Therefore, the liquidity requirement is reduced to the availability of suitable cat

bonds at the time of replication. We derive prices for binary/nonindemnity-based

ILWs, where the payout only depends on the industry loss index exceeding a

contractually defined trigger level during the contract term. If a suitable cat bond is

not available for deriving ILW prices, we provide proper approximations under some

additional assumptions. To illustrate and test the proposed approximations in case of

index-linked instruments, we conduct simulation analyses and additionally derive

the price of an ILW using secondary market cat bond prices and compare the resulting

price with the available real-world ILW prices, finding a high degree of consistency

with both robustness tests, which support our suggested approximations for

replicating portfolios. Moreover, as a further application, we approximate prices of

cat bonds using empirical data and compare them with real secondary market data in

order to examine whether the market prices consistently.

The presented approach f or pricing index-linked catastrophe loss i nstruments is of

high relevance today and esp ecially for the future (for bo th practical and academi c

endeavors), when index -linked catastrophe l oss hedging instrument s will become

even more widespread than today and when some mar kets for derivatives li ke

4

The prices of available index-linked catastrophe loss instruments such as ILWs should

generally be consistent with the prices of other derivatives traded on an already liquid market

such as in the case of cat bonds. To ensure this consistency, the prices of ILWs should equate

the prices of replicating portfolios consisting of tradable derivatives (cat bonds). This is also

generally in line with the findings in Jaeger, M€

uller, and Scherling (2010).

PRICING INDEX-LINKED CATASTROPHE LOSS INSTRUMENTS 41