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

INTRODUCTION

Alternative risk transfer (ART) has become increasingly relevant inrecent 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 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, Muller, and Scherling, 2010; Galeotti, Gurtler, and Winkelvos, 2012; Braun, 2015; Gurtler, 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, astatic 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 for pricing index-linked catastrophe loss instruments is of high relevance today and especially for the future (for both practical and academic endeavors), when index-linked catastrophe loss hedging instruments will become even more widespread than today and when some markets for derivatives like the cat bond market are truly liquid. One main contribution of our work is to overcome the crucial point of the tradability of the loss index through suitable approximations and to provide explicit pricing formulas using replication techniques. Although the focus of the article is primarily theoretical, we use empirical examples and simulation analyses to illustrate the suggested approach by comparing ILW and cat bond prices with the ones derived based on the replication and approximation techniques.

The remainder of the article is structured as follows. The "Further Related Literature" section gives an overview of related literature with a focus on the underlying theory and assumptions. The "Modeling Index-Linked Catastrophe Loss Instruments" section introduces index-linked catastrophe loss instruments and ILWs as representatives of index-linked catastrophe loss hedging instruments, their basic properties, and the underlying industry loss indices. In the "Pricing Index-Linked Catastrophe Loss Instruments" section, we present the pricing approach and as an example apply the approach to consistently price ILWs and cat bonds in the "Empirical Examples for Pricing Index-Linked Catastrophe Loss Instruments by Replication Using Cat Bonds" section. The "Outlook: Pricing Indemnity-Based (Double-Trigger) Contracts" section gives an outlook on the pricing of indemnity-based products, and the "Summary and Conclusions" section concludes.

FURTHER RELATED LITERATURE

There are several articles that deal with the pricing of index-linked catastrophe loss instruments such as ILWs or strongly related products. Actuarial pricing principles are applied to ILWs in Gatzert and Schmeiser (2012), for instance, while Grundl and Schmeiser (2002) compare actuarial pricing approaches with the capital asset pricing model to calculate prices for double-trigger reinsurance contracts, which are similar to indemnity-based ILWs. Furthermore, there are articles that combine financial and actuarial pricing approaches such as Moller (2002, 2003), who discusses the valuation and hedging of insurance products that depend on both the financial market and insurance claims. Regarding the arbitrage-free pricing of ART instruments related to index-linked catastrophe loss instruments, there is a wide literature, which is outlined in what follows.

First, Cummins and Geman (1995) develop a model to price cat futures and call spreads written on the aggregated claims process using an arbitrage approach based on a jump diffusion model. In contrast, Bakshi and Madan (2002) price options written on the average level of a Markov process using a mean-reverting process and derive closed-form solutions for cat option prices. Haslip and Kaishev (2010) price reinsurance contracts with a specific focus on catastrophe losses. They assume a liquid market of indemnity-based cat bonds and that the aggregated loss process of a company follows a compound Poisson process, and then calculate arbitrage-free prices for the excess of loss reinsurance contracts using Fourier transformations. In contrast to their setting, we only assume a liquid cat bond market with comparable index-based cat bonds and weaken the distributional assumptions.

Integral to our pricing approach for ILWs is the valuation of cat bonds. Due to jumps induced by natural catastrophes, this market is generally incomplete. To deal with incompleteness, one approach is offered by Merton (1976), who argues that the risk of jumps can be completely neutralized by diversification, as jumps are of an idiosyncratic nature. Under this "risk-neutralized" assumption, one can employ arbitrage-free pricing. (5) The consistent pricing of double-trigger cat bonds based on single-trigger bonds is studied in Lane (2004), who points out that in the absence of arbitrage, the sum of the prices of two single-trigger cat bonds should generally be equal to the sum of the prices of two fitting double-trigger cat bonds (one junior and one senior tranche). In the "Empirical Examples for Pricing Index-Linked Catastrophe Loss Instruments by Replication Using Cat Bonds" section, we further develop this idea when applying our approach to the consistent pricing of single-trigger cat bonds and without assuming perfectly fitting cat bonds. An empirical example for a potential security-class arbitrage opportunity is discussed in Jaeger, Muller, and Scherling (2010, p. 27f), who show that the same risk may be priced differently depending on the type of security using ILWs and cat bonds as an example. (6) However, the authors also emphasize the different contract terms as one potential main reason for observed differences in prices. Based on their empirical data they further critically evaluate other types of arbitrage, stating that there may be rather small program replication arbitrage opportunities (mainly arising from other risks or transaction costs), and regarding trigger type arbitrage find no evidence in their historical data.

MODELING INDEX-LINKED CATASTROPHE LOSS INSTRUMENTS

One main property of index-linked instruments such as ILWs is that they depend on an industry loss index, which generally does not instantaneously reflect the exact insured damage. Instead, if a catastrophe occurs, the first value of the index is a preliminary estimation (see, e.g., Kerney, 2013a), which is adjusted if necessary or at predetermined points in time (see Property Claim Service [PCS], 2013; PERILS, 2013). To take this development into account, we follow the approach of Vaugirard (2003a, 2003b) and assume the actual (contractually defined) maturity T' of the index-linked catastrophe loss instruments to exceed the "risk exposure period" T, during which the catastrophe must occur in order to trigger a payoff. Newly occurring catastrophes between the end of the risk exposure period T and the maturity T' are not taken into account, but adjustments of the loss estimates from the catastrophes that occurred during the risk exposure period (between times0 and T)are taken into account and are reflected in the index. Hence, while the time interval from 0 to T represents the "risk exposure period," the time interval from T to T' is referred to as the "development period." Thus, an industry loss index incorporates data from insured losses arising from an occurring catastrophe and each qualifying event is reflected in the index.

In what follows, we consider an aggregated industry loss index [I.sup.T] for the risk exposure period until time T by aggregating estimated losses from all catastrophes (e.g., of the same and contractually defined type) that occurred during the risk exposure period. (7) Hence, following, for example, Biagini, Bregman, and Meyer-Brandis (2008), the aggregated industry loss index [I.sup.T] for the risk exposure period until T, taking into account the development of the estimations until time t [less than or equal to] T', is given by

[mathematical expression not reproducible]

where [N.sub.t[and]T] (with t [and] T = min (t, T)) is the number of catastrophes that occurred up to time t within the risk exposure period until T and [X.sup.i.sub.t] denotes the time-dependent estimation of the insured loss arising due to the ith catastrophe at time t. In the following, we focus on index-linked catastrophe loss instruments, whose payoffs depend on the industry loss index at maturity T' and that can be defined by

[P.sup.T.sub.T'] = h([I.sup.T.sub.T'),

where h is a nonnegative measurable function. The payoff at time T' of an "indemnity-based" index-linked catastrophe loss instrument, which additionally depends on the company loss, is analogously given by

[P.sup.T.sub.T'] = h ([I.sup.T.sub.T'],[L.sup.T.sub.T']), (1)

where [L.sup.T] denotes (analogously to [I.sup.T]) the estimated aggregated company loss for the risk exposure period until T. In Figure 1, we depict the development of an industry loss index for a single catastrophe. After the catastrophe occurred, the loss estimations are adjusted several times, and additionally occurring catastrophes during the risk exposure period would cause additional jumps. The solid dot on the right side corresponds to the value of the industry loss index at maturity and hence the payoff is given by applying the function h to this value. Note that while downward adjustments of the estimation are possible in principle, they are unlikely and adjustments are typically upward. (8)

We assume that [I.sup.T] is given by an industry loss index provider such as, for example, the PCS index for the United States or the PERILS index for Europe. It represents an estimation of the total insured loss caused by a catastrophe. The loss data by PCS, for instance, is updated approximately every 60 days, if the event caused more than $250 million in insured property losses, until PCS believes the estimated loss reflects the insured loss for the industry (see Kerney, 2013a; PCS, 2013). It is important to keep in mind that these indices do not incorporate information instantaneously but with a certain delay; however, under the assumption of sufficient liquidity in the derivatives market, we can assume that information is incorporated almost instantaneously (similar to the CDS market for credit defaults) into the derivatives prices (e.g., cat bonds that play a crucial role here).

One main problem related to the use of an index-linked catastrophe loss instrument in the context of hedging is basis risk, which arises if the dependence between the index and the company's losses (which are to be hedged) is not sufficiently high. In particular, the company loss could be high, but the industry loss index could be too low to trigger a payoff. There are several definitions of basis risk and we refer to Cummins, Lalonde, and Phillips (2004) for a discussion of basis risk associated with index-linked catastrophe loss instruments.

PRICING INDEX-LINKED CATASTROPHE LOSS INSTRUMENTS

Pricing by Replication Under the No-Arbitrage Assumption

In general, actuarial valuation methods for insurance contracts are based on the expected loss plus a specific loading, which depends, for example, on the insurers' risk aversion and its already existing portfolio (see Gatzert and Schmeiser, 2012). Financial pricing approaches generally allow the derivation of market-consistent prices. For arbitrage-free valuation, one has to calculate the expected value under a certain risk-neutral measure Q given the existence of a liquid market and independent of the existing portfolio. If there is a unique risk-neutral measure (i.e., the market is complete), the arbitrage-free price is unique and the price corresponds to the initial value of a self-financing portfolio replicating the cash flow (Harrison and Kreps, 1979).

In the present setting, we focus on the two stochastic processes of importance: the industry loss index [I.sup.T] and the company loss [L.sup.T]. These processes need to be tradable in a more general sense, as there is no possibility to trade [I.sup.T] or [L.sup.T] directly like a stock and it is unrealistic to assume that there is a possibility to buy or sell this index; however, we assume there is a liquid market for certain derivatives on the industry loss index [I.sup.T]. (9) To avoid arbitrage opportunities, the prices of these derivatives should coincide with the expectation of the discounted cash flow under a risk-neutral measure (unique or not, depending on the market's completeness), and the prices of other index-linked catastrophe loss instruments should be consistent with the prices of the derivatives already traded on a liquid market (given a sufficient degree of comparability in regard to transaction costs, for instance). To ensure this consistency, the prices of these other instruments should be equal to the expectations under the same risk-neutral measure or, equivalently, should be equal to the prices of replicating portfolios consisting of tradable derivatives, for instance. In case the market is incomplete, that is, there are not sufficient traded derivatives available (no exact replicating portfolio; only partial hedging is possible), other approaches can be used, such as sub-/super-hedging (e.g., Bertsimas, Kogan, and Lo, 2001), by choosing a self-financing portfolio based on maximizing expected utility (e.g., Henderson, 2002) or by selecting a risk-neutral measure according to certain criteria (e.g., jump risk is not priced; see Merton, 1976; Delbaen and Schachermayer, 1996).

We thus assume that the liquid market for certain index-linked catastrophe loss instruments (e.g., cat bonds) is at least large enough to allow the derivation of an exact replicating portfolio or at least a close approximation. For instance, in case suitable cat bonds are available for perfectly replicating the respective instrument's cash flows (e.g., an ILW or another cat bond), a unique arbitrage-free price can be derived. (10) Alternatively, in case the required cat bonds are not available (e.g., mismatching trigger level or time to maturity), we derive suitable approximations for the replicating portfolio. Using direct replication also has the significant advantage that a risk-neutral measure does not have to be specified and that model risk is significantly reduced for the hedger.

In case of "indemnity-based" index-linked catastrophe loss instruments, one crucial point and major problem in addition to the treatment of the industry loss index is the tradability of the company loss [L.sup.T]. As potential remedies, we briefly discuss three preliminary approaches with varying degrees of assumptions in the "Outlook: Pricing Indemnity-Based (Double-Trigger) Contracts" section.

Pricing Binary ILWs by Replication Using Cat Bonds

We apply the proposed approach to price binary ILWs, whose payoff depends on whether the industry loss index exceeds the trigger level Y at the end of the development period T (see McDonnell, 2002). The payoff at maturity T of a "binary ILW" with trigger level Y and risk exposure period until T is given by

[mathematical expression not reproducible] (2)

where D represents the possible payout and 1[mathematical expression not reproducible] denotes the indicator function, which is equal to 1 if [I.sup.T.sub.T'], > Y, that is, if the industry loss index at maturity [I.sup.T.sub.T'], exceeds the trigger level Y (due to catastrophes that occurred during the risk exposure time T), and 0 otherwise. (11)

To illustrate our approach more specifically, we assume that there is a liquid cat bond market consisting of cat bonds with comparable maturities, strikes, and so on that can be used to derive consistent ILW prices by means of replicating the ILWs' cash flows. (12) In case of a liquid cat bond market, the whole market is represented by the filtered probability space ([OMEGA], F, [mathematical expression not reproducible], P), where [mathematical expression not reproducible] is a filtration satisfying the usual assumptions and [F.sub.t] represents all information up to time t. Following the fundamental theorem of asset pricing, the price of every contingent claim is given by the expectation of the discounted payoff under an equivalent martingale measure Q. The discounted payoff (at time 0) of an index-based cat bond with binary payoff (denoted b), (13) trigger level Y, risk exposure period until T, coupon payment c, and maturing at time T, and without loss of generality an assumed nominal of 1 is generally given by

[mathematical expression not reproducible] (3)

where r is the constant risk-free interest rate (this assumption can be weakened) and n is the number of coupon payments (coupon time intervals). In what follows, we first assume that there is a perfectly fitting cat bond, that is, a zero coupon (c = 0) cat bond with the same trigger level Y, the same risk exposure period until T, and maturing at the same time T', and then extend our formula to imperfectly fitting cat bonds under certain additional assumptions. For the remainder of the article, we calculate prices at time t = 0 to simplify the notation.

Matching Maturity and Trigger Level. First, if there are binary zero coupon cat bonds with the same maturity and the same trigger level as the ILW, according to Equation (3), the risk-neutral price at time t = 0 of a zero coupon cat bond with binary payoff, trigger level Y, risk exposure period until T, maturing at time T' is (see Haslip and Kaishev, 2010)

[mathematical expression not reproducible] (4)

Hence, together with (2) (and D = 1), Equation (4) turns into

[mathematical expression not reproducible]

where [V.sup.B,Y,T.sub.ILW,T'] is the price of a binary ILW with risk exposure period until T, trigger level Y, and maturing at time T'. (14) Thus, in the presence of a zero coupon cat bond with the same trigger level Y and same maturity, the price is given by

[mathematical expression not reproducible] (5)

where [e.sup.-rT'] is the price of the related zero coupon bond without catastrophe (or any other default) risk under the assumption of a constant interest rate. In case of a nonconstant interest rate this could be replaced with the price of a zero coupon bond maturing at time T' without default risk. Note that as direct replication is used, independence between the risk-free rate and the industry loss index is not necessary in this case. Equation (5), thus, also yields a replicating portfolio for a binary ILW. In particular, the cash flow of a binary ILW with risk exposure period until T, trigger level Y maturing at time T' can be perfectly replicated by buying a zero coupon bond without default risk and selling a binary zero coupon cat bond with risk exposure period until T, trigger level Y, and maturing at time T'.

Although these considerations are straightforward, they serve as a starting point, as one has to take into account that the chance of finding a perfectly fitting cat bond in the market might be low, such that an approximation in spite of a maturity mismatch might be easier. Therefore, in what follows we provide approximations if there are only cat bonds with different maturity, different trigger level, nonbinary payoff, or nonzero coupons available. As described before, using direct replication thereby has the significant advantage that a risk-neutral measure does not have to be specified and that model risk is significantly reduced, which represents one major benefit.

Mismatching Trigger Level or Maturity: General Approach. We next assume that there is a liquid market for binary zero coupon cat bonds with different trigger levels, different maturities, and different risk exposure periods, thus implying an incomplete market setting. In this case, sub- and super-hedging can be applied, where the arbitrage-free price of an ILW with maturity T lies between the prices of ILWs with shorter and longer times to maturity, both given by Equation (5), which results in upper and lower bounds for the ILW price. The obtained range can be rather large, which is why we choose the zero coupon cat bond with trigger level Y the maturity T' and risk exposure period until T that is closest to the parameters of the ILW we intend to price (everything else equal, i.e., trigger level and index). The aim is thus to approximate the binary ILW price [V.sup.B,Y,T.sub.ILW,T'] with the price of this cat bond, which can be observed in the market. First, the price of a binary ILW with risk exposure period until T, trigger level Y, and maturing at time T is given through Equation (5) by

[mathematical expression not reproducible] (6)

which can then be used to price the ILW with different maturity and different risk exposure period by

[mathematical expression not reproducible] (7)

where

[mathematical expression not reproducible]

is the residual difference in (7) arising from the approximation error due to using an observable traded cat bond with mismatching maturity (closest to the one of the ILW). To calculate [D.sup.Y,T,T'.sub.Y,T,T'] additional assumptions regarding the distribution of I are needed. However, if the trigger levels, maturities, and risk exposure periods are close, the approximation difference [D.sup.Y,T,T'.sub.Y,T,T'] is small and only marginally depends on the underlying distribution assumptions, as shown in the "Quality of Approximations and Reduction of Model Risk: Numerical Analysis" section using simulation analyses. Hence, combining market data (i.e., available cat bond prices) with model assumptions regarding the difference term [D.sup.Y,T,T'.sub.Y,T,T'] in (7) will generally reduce potential model risk involved in pricing index-linked catastrophe instruments, which would be considerably higher when applying the distributional assumptions for pricing the ILW without using any market data.

Matching Trigger Levels but Mismatching Maturity: An Approximation. To calculate [D.sup.Y,T,T'.sub.Y,T,T'], in practice, where the loss distribution under Q is usually unknown, we propose the following approximations using only available data, that is, the price of the traded cat bond [V.sup.B,0,Y,T.sub.CAT,T'] and the trigger probabilities, that is, P([I.sup.T.sub.T'] > Y) and P([I.sup.T.sub.T'] > Y), which are usually published (see, e.g., Table 1 in the "Empirical Examples for Pricing Index-Linked Catastrophe Loss Instruments by Replication Using Cat Bonds" section).

First, we assume the trigger levels of the ILW and the traded cat bond are equal. Let the risk exposure period be equal to the maturity (T =T', T = T') and assume that the industry loss index reflects the real catastrophe losses instantaneously. Furthermore, let [I.sup.T] follow a compound Poisson process under Q, which is not too restrictive if [I.sup.T] follows a compound Poisson process under the physical measure P. Delbaen and Haezendonck (1989) show under some reasonable assumptions, which should be fulfilled in most nonlife insurance cases, that [I.sup.T]remains a compound Poisson process under any equivalent risk-neutral measure. For a further treatment, we refer to Aase (1992) and Murmann (2008). These assumptions lead to

[mathematical expression not reproducible] (8)

where [N.sub.t] and [X.sup.i] are independent, [N.sub.t] denotes the claims arrival process (i.e., a Poisson process with intensity [lambda]) and [X.sup.i] the i.i.d. claims (see, e.g, Levi and Partrat, 1991). Using these assumptions and a resulting [D.sup.Y,T,T'.sub.Y,T,T'] as shown in the Appendix, we can approximate the ILW price through

[mathematical expression not reproducible] (9)

for small intensity values [lambda]. We analyze the quality of this approximation in Quality of Approximations and Reduction of Model Risk: Numerical Analysis section.

Matching Maturity but Mismatching Trigger Level: An Approximation. If the maturities of the cat bond and the ILW are the same, but the trigger level differ, prices can be obtained by analogously applying the approach in the previous section (Equation (7)). We similarly obtain

[mathematical expression not reproducible] (10)

The value of

Q([I.sup.T.sub.T'] > Y)/Q([I.sup.T.sub.T'] > Y)

depends on the change from the physical measure to the risk-neutral measure. One approach for the change of measure is the Wang transform (see Wang, 2000), which is also suitable from an empirical perspective, as, for example, Wang (2004) shows that this probability transform can well explain empirical cat bond prices, whereby the Wang transform extends the Sharpe ratio concept for risk adjustment to skewed distributions (see Wang, 2004). The exceedance probability under Q is thereby defined by

Q([I.sup.T.sub.T'] > Y) = [PHI]([[PHI].sup.1] (P([I.sup.T.sub.T'] > Y)) + [alpha]), (11)

where [PHI] denotes the standard normal cumulative distribution function and the parameter [alpha], also called market price of risk, is a direct extension of the Sharpe ratio, which can be calculated by solving

[mathematical expression not reproducible]

for [alpha] using the observable price of a traded cat bond [V.sup.b,0,Y,T.sub.cat,T'].

Nonzero Coupon Cat Bonds. If there is no market for zero coupon cat bonds (as is currently the case), nonzero coupon cat bonds need to be used for approximating ILW prices. The decomposition follows the standard bond stripping approach, where each coupon itself is understood as a zero coupon bond with adjusted nominal, interest rate, and maturity. To calculate the price of an ILW involving only the nonzero coupon cat bond price [V.sup.b,Y,T.sub.cat,T'], we use Equation (6). Recall Equation (3) for the payoff of such a cat bond and Equation (7) for adjusting a mismatching maturity to obtain the price of a binary ILW [V.sup.b,Y,T.sub.ILW,T'] with trigger level Y, risk exposure period until T, and maturity T', that is,

[mathematical expression not reproducible] (12)

Cat Bonds With Nonbinary Payoff. In practice, cat bonds often have a nonbinary payoff; that is, the payment depends on the extent to which the industry loss index exceeds the trigger. The payoff of such a cat bond with nonbinary payoff, risk exposure period until T, trigger level Y, limit M, and maturing at time T can be defined by

[mathematical expression not reproducible]

where without loss of generality the maximum payoff is normalized to 1. The price of this cat bond is given by

[mathematical expression not reproducible] (13)

In order to price binary ILWs, the prices of binary cat bonds are required, which in turn can be approximated by prices of cat bonds with nonbinary payoff by using the midpoint rectangle rule, which leads to

[mathematical expression not reproducible] (14)

Combining Equations (13) and (14) results in

[V.sup.0,Y,M,T.sub.cat,T'] [approximately equal to] [V.sup.b,0,Y+M/2,T.sub.cat,T']

Therefore,

[mathematical expression not reproducible] (15)

for the price of a binary ILW with trigger level Y + 1/2M, risk exposure period until T, and maturing at time T'. The approximation generally works well for small layers (see also simulation study in the "Quality of Approximations and Reduction of Model Risk: Numerical Analysis" section).

Quality of Approximations and Reduction of Model Risk: Numerical Analysis

Input Parameters. In what follows, we use simulation analyses to study the quality of the approximations and the reduction of model risk in more detail. Toward this end, we follow Braun (2011) and assume that [I.sup.T.sub.T'] follows a compound Poisson process with either log-normal, Burr, or Pareto distributed losses (under Q). (15) The cumulative distribution function of the Burr distribution with parameters a, b, c is thereby given by

[mathematical expression not reproducible]

with the Pareto distribution being a special case with b = 1. We use the parameters estimated by Braun (2011) for U.S. earthquakes losses during the period 1900-2005, that is, [micro] = -1.3778 and [sigma] = 2.5835 for the log-normal distribution; a = 0.4027, b = 1.1018, c = 0.0426 for the Burr distribution; a = 0.4602, c = 0.0503 for the Pareto distribution; and the intensity of the Poisson process is given by [lambda] = 0.76. Furthermore, we assume a constant interest rate of r = 0.01.

We now consider a tradable zero coupon cat bond with trigger level Y = 30 and a maturity of 1 year. Assuming that the parameters estimated in Braun (2011) imply "real" prices, the price of the cat bond is [mathematical expression not reproducible], which in case of a log-normal distribution, for instance, is given by [V.sup.b,0,30,1.sub.cat,1] = 0:9653.

Mismatching Maturity or Trigger Level. We study the impact of model risk by varying the input parameters as well as the distributional assumptions. Figure 2 shows the pricing error for an ILW with varying trigger level Y' and maturity T' when using incorrect parameters [lambda]*, [micro]*, [sigma]* (for the intensity of the Poisson process and the log-normally distributed losses) for directly calculating the price using the pricing formula [mathematical expression not reproducible],as compared to calculating the price using Equation (7), that is, [mathematical expression not reproducible], with [mathematical expression not reproducible] being calculated using simulation approaches as well (i.e., we first study the case without the developed approximations). The incorrect parameters are in a range of [+ or -]10 percent of the "true" values.

The results show that the model risk is strongly reduced if the ILW's characteristics (Y, T') are close to the ones of the available cat bond and that for mismatching maturities (T' = 0.9), higher trigger levels imply a stronger pricing error. Overall, the pricing error due to the difference term in (7) is considerably lower as compared to the direct pricing approach that entirely relies on input parameters instead of making use of available market data. In particular, a misestimation in parameters of [+ or -]10 percent implies a mispricing of at most 13 percent in most cases, typically considerably lower at around 5 percent in comparison to an error of up to 48 percent using direct calculation. Similar results can be found for the Burr and Pareto distributions (see Figures A1 and A2 in Appendix).

Approximation Quality in Case of Mismatching Maturity. We next consider the developed approximation approaches that only rely on market data, and first focus on the quality of the approximation in Equation (9) for mismatching maturities as illustrated in Figure 3. We again assume there is a tradable zero coupon cat bond with trigger level Y = 30 and a maturity of 1 year and that the industry loss process is a compound Poisson process with log-normally distributed losses under Q with input parameters from Braun (2011). The relative pricing errors for ILWs with maturity T' and different intensity values [lambda] are exhibited in Figure 3, showing that the quality of the approximations crucially depends on the intensity of the Poisson process [lambda] as already formally shown in the Appendix, as the approximation error converges to 0 for [lambda] [right arrow] 0.

Approximation Quality In Case of Mismatching Trigger Level. To illustrate the quality of the proposed approximation for mismatching trigger levels stated in Equation (10), we also assume that there is a tradable zero coupon cat bond with trigger level Y = 30 and a maturity of 1 year and that the industry loss process is a compound Poisson process with log-normally distributed losses (parameter: [[lambda].sup.P] = 0.76, [[micro].sup.P] = -1.3778, [[sigma].sup.P] = 2.5853) under the physical measure P. Figure 4 shows the relative pricing error for an ILW with various trigger levels Y' using the approximation in Equation (10), which makes use of the empirically observed trigger probability under the physical measure P. In particular, we assume in the left graph of Figure 4 that the industry loss process follows a compound Poisson process with intensity [lambda] and log-normally distributed losses (parameter: [micro], [sigma]) under Q, and then in Equation (10) use the Wang transform for the change of measure, that is, Q([I.sup.T.sub.T] > Y) = [PHI]([[PHI].sup.-1](P([I.sup.T.sub.T] > Y)) + [alpha]). The right graph of Figure 4 illustrates the quality of the approximation for a different loss distribution, where we assume that the industry loss process is a compound Poisson process with Burr distributed losses with parameters [[lambda].sup.P] = 0.76, [a.sup.P] = 0.4027, [b.sup.P] = 1.1018, [c.sup.P] = 0.0426 under P and parameters [lambda], a, b, c under Q. Figure 4 shows the relative pricing error for an ILW with different trigger levels Y' using the Wang transform in Equation (10).

Our results show that the quality of the approximation in Equation (10) strongly depends on how the loss distribution is affected by the change of measure (from P to Q). The quality of the approximation also depends on distributional assumptions as can be seen in Figure 4. However, in all cases considered, the pricing error remains below 4 percent.

Quality of Approximation in Case of Nonbinary Cat Bonds. The quality of the approximation in Equation (15) using the midpoint rectangle rule is studied in Figure 5. For all considered loss distributions (log-normal, Burr, and Pareto) and various limit values (M = 5, 10, 15), the pricing error caused by the approximation is below 5percent. One can observe that the approximation works better for high trigger levels and low limit values. Furthermore, the skewness of the distribution only has a minor influence on the quality of the approximation. Even when increasing [sigma] of the log-normal distribution from 2.5835 to 4, the approximation error is not increasing and stays below 5 percent for all tested values.

Discussion of the Method

This section presented an approach for the valuation of index-linked catastrophe loss instruments under the assumption of no arbitrage, which we applied to the pricing of ILWs. In the presence of a liquid cat bond market, we obtain Equation (5) independent of the respective distributional assumption. As there may not be a sufficient variety of cat bonds to match the parameters of every ILW, we further provide approximations for different trigger levels and maturities.

As shown in the "Quality of Approximations and Reduction of Model Risk: Numerical Analysis" section our approach can considerably decrease model risk. However, one major limitation is that the quality of the approximations and the reduction of model risk depend on the underlying distributions. Therefore, while our simulation analysis suggests that the approximations are very stable, the exact distributions are usually unknown, implying that the exact reduction of model risk is also unknown. However, for the considered distributions, the approximations work well for a wide range of trigger levels and maturities, and the reduction of model risk is considerable. When using nonbinary cat bonds to calculate prices of binary ILWs, we further make use of the midpoint rectangle rule. The quality of this approximation also depends on the underlying distribution, but additionally on the limit value of the nonbinary cat bond, whereby the approximation error is below 5 percent for all considered loss distributions and limit values.

Our approach extends the work of Haslip and Kaishev (2010), who also assume a liquid cat bond market but--in contrast to our analysis--focus on indemnity-based cat bonds, and then evaluate excess of loss reinsurance contracts under additional assumptions concerning the risk-neutral measure Q. The assumption of a liquid cat bond market can be considered reasonably realistic given the high volume of cat bonds (see Figure 6) and a growing secondary market with a considerable volume of cat bond transactions (see Moody, 2013). Furthermore, sponsors are becoming more and more familiar with this kind of risk transfer, and service providers involved in structuring and marketing exhibit increasing experience. In addition, frictional costs of cat bond transactions are decreasing (less than 20 basis points; see Kerney, 2013b). Time to market and frictional costs are further improved by the use of different securitization structures involving a Special Purpose Insurer (SPI) instead of a Special Purpose Vehicle (SPV), which can be set up in less than 3 to 4 weeks overall (see Garrod, 2014), thus also improving the degree of liquidity in the cat bond market. In addition, Braun (2015) points out that sponsors recently started to increasingly use shelf-offerings (e.g., Swiss Re Successor Series), where additional classes of notes can be issued repeatedly out of the same SPV, which eases access to capacity and considerably reduces transaction costs. These developments also contribute to reducing differences in transaction costs and transaction times between cat bonds and ILWs, as the latter are generally more easily available and less costly. (16) However, liquidity is not needed to the same extent as in classical option pricing as continuous replication is not necessary to replicate ILWs when using cat bonds; that is, a static hedging approach is sufficient. Moreover, the Bermuda Stock Exchange, among others (see Artemis, 2013), has started to list cat bonds, which is a first step to trading them on the exchange, indicating that the secondary market is growing, the transaction time on the primary market and the frictional costs are decreasing, and even if the market is not yet enough liquid, it appears reasonable to assume that it will become liquid in the foreseeable future in view of the current developments.

The approach using available cat bonds in order to (at least approximately) replicate cash flows can also be used for pricing other index-linked catastrophe loss instruments covering a wide variety of instruments. Hence, once the replicating portfolio is known, prices can easily be calculated as they can directly be observed in the market. In addition, this approach can be used to test the degree of liquidity in the market. Consistent prices would encourage the assumption of a liquid market, while inconsistent prices would contradict this assumption and lead to an arbitrage opportunity or indicate the existence of an external risk that might not have been incorporated into market prices.

EMPIRICAL EXAMPLES FOR PRICING INDEX-LINKED CATASTROPHE LOSS INSTRUMENTS BY REPLICATION USING CAT BONDS

Consistent Pricing of Binary ILWs Using Cat Bonds

To illustrate and test the previously developed approximations, we use secondary market cat bond prices provided by Lane (2002) and compare the theoretically derived ILW prices obtained by means of our approximations (see the "Pricing Binary ILWs by Replication Using Cat Bonds" section) with the actual ILW prices dated from April 1,2002 as provided by McDonnell (2002). (17) To approximate the price of a binary ILW related to earthquake risk in California with a maturity of 1 year, the secondary market price of the cat bond "Western Capital" (18) is used, which features the same underlying risk, but mismatching maturity (10 months instead of 1 year) and a nonbinary payoff (layer between $22.5 and $31.5 bn.). The secondary market price is given as the spread (spreads published by Goldman Sachs at March 31, 2002: bid: 642 bp, ask: 554 bp; Lane, 2002) over LIBOR (2.03 percent at (19) March 31,2002). We use the developed approximations to derive a bid-ask interval for ILW prices and in the following exhibit the calculation for the ask price and define March 31, 2002 as t = 0. As our approximations use zero coupon cat bond prices, we transform the secondary market spread using the LIBOR rate to a nonbinary zero coupon price through

[mathematical expression not reproducible]

where 10 months (from March 31,2002 to January 31,2003) are left until the cat bond matures. According to Equation (14), we can use the midpoint rectangle rule to approximate the price of a zero coupon cat bond with binary payoff and trigger level 1/2(22:5 + 31:5) = 27 (middle of the layer of the nonbinary cat bond) by

[mathematical expression not reproducible]

which, as illustrated in the "Quality of Approximations and Reduction of Model Risk: Numerical Analysis" section, should generally work well for smaller layers and higher trigger level. (20) The maturity of the ILWs, whose prices are stated in McDonnell (2002), is 1 year. For the adjustment of the mismatching maturity (from 10 months to 1 year) we use Equation (9), resulting in

[mathematical expression not reproducible]

The ask price is thus given by 5.05 percent and the same calculation for the bid price yields 5.80 percent. (21) The actual prices stated by McDonnell (2002) are 5.25 percent for an ILW with a trigger level of $25 bn. and 4.25 percent for trigger level of $30 bn., which already shows a high degree of consistency with our approximation (despite the use of only partial information for the ILW price, for instance).

Consistent Pricing of Cat Bond

Another application of our proposed approximation formulas is the consistent pricing of cat bonds. The available characteristics and prices of four selected cat bonds written on U.S. hurricane risk using the PCS industry loss index are summarized in Table 1.

As in the "Summary and Conclusions" section, prices are given as the (annual) spread over LIBOR and the quarterly secondary market spreads are displayed in Figure 7. U.S. hurricanes can only occur during the hurricane season from June to November, such that these cat bonds cover the same seasons and we can assume that they have the same maturity and risk exposure period. Due to sub- and super-hedging considerations, prices at every point in time are ordered according to the related expected loss; that is, a cat bond with higher expected loss generally has a higher spread.

Analogously to the ILW pricing formulas in Equation (7), the price of cat bonds (in terms of the spread premium) is given by

Data Source: Lane (2013, 2014).

[mathematical expression not reproducible] (16)

where we observe the price of a binary cat bond with trigger level Y ([V.sup.b,Y,T.sub.cat,T'] in the market and, analogously to the "Quality of Approximations and Reduction of Model Risk: Numerical Analysis" section, use an empirically calibrated model to calculate the difference term

[D.sup.Y,T,T'.sub.Y,T,T'] = [V.sup.B,Y,T.sub.cat,T'] - [V.sup.b,Y,T.sub.cat,T']

We follow Galeotti, Gurtler, and Winkelvos (2012) and apply the model of Wang (2004) (see also the "Pricing Binary ILWs by Replication Using Cat Bonds" section) to

calculate [D.sup.Y,T,T'.sub.Y,T,T'], that is,

[mathematical expression not reproducible]

where PFL is the probability of the first loss, PLL is the probability of the last loss, and the market price of risk a is obtained by solving the nonlinear regression

[mathematical expression not reproducible]

With [alpha] and the parameters stated in Table 1, [D.sup.Y,T,T'.sub.Y,T,T'], and [V.sup.b,Y,T.sub.cat,T'] can be calculated using Equation (16) and be compared with the empirical prices. As an example, the real and approximated prices of Ibis Re II Ltd. are shown in Figure 8 (where a is determined by using the empirical input parameters of the other three cat bonds in Table 1), where we can again observe a high degree of consistency. As mentioned before, by combining empirically observed prices and using theoretical models only for the difference term (as displayed by Equation (16)), model risk can be considerably reduced.

OUTLOOK: PRICING INDEMNITY-BASED (DOUBLE-TRIGGER) CONTRACTS

For pricing indemnity-based contracts by means of replication, the tradability of the company loss [L.sup.T] must be examined. Toward this end, we provide first thoughts on how the company loss could be treated and suggest three approaches, which can be explored in future research. The payoff of an indemnity-based ILW with attachment point A, maximum payoff M, trigger level Y, risk exposure period until T, and maturing at time T is given by

[mathematical expression not reproducible] (17)

where [(x).sub.+] =max(x,0). From Equations (2) and (17), one can see the option-like structure of ILWs. A binary ILW is a binary call on the industry loss index [I.sup.T], and an indemnity-based ILW corresponds to a call spread option on the company loss [L.sup.T] with an additional trigger for the industry loss.

First, in case [I.sup.T] and [L.sup.T] are independent, the price of an indemnity-based ILW (Equation (17)) can be described by

[mathematical expression not reproducible]

where [mathematical expression not reproducible] is the price of a call option on the company loss [L.sup.T] with strike price A maturing at time T'. For the calculation of [V.sup.b,Y,T.sub.ILW,T'] using tradable cat bonds, we refer to the previous sections, and [mathematical expression not reproducible] can be calculated using actuarial pricing principles, for instance (see also Moller, 2003). Still, even though independence between the index and the company loss would imply a substantial degree of basis risk and would thus be problematic regarding the risk management purpose of an indemnity-based, index-linked catastrophe loss instrument, empirical results by Cummins, Lalonde, and Phillips (2004) for the U.S. market show that especially small insurers exhibit very low correlations between their own losses and industry loss indices. In addition, this calculation could provide a lower bound for an efficient price.

Second, one could approximate the company loss [L.sup.T] with the loss of the stock price, which in turn is assumed to be traded, thus allowing arbitrage-free valuation. Doing this results in a structured risk management product, which can be treated as in Cox, Fairchild, and Pedersen (2004) and by using arbitrage-free valuation. However, due to catastrophe losses, the market is generally incomplete and choosing the right risk-neutral measure Q is a nontrivial task.

Third, one could assume a functional relationship between the industry loss index and the company loss motivated by the typically high degree of dependence between these processes. (22) This approach reduces indemnity-based index-linked catastrophe loss instruments to nonindemnity-based instruments and allows applying the replication techniques developed previously. A high degree of dependence is realistic, since basis risk arises if the company loss and the industry loss are not fully dependent (see, e.g., Harrington and Niehaus, 1999; Gatzert and Kellner, 2011). For instance, Cummins, Lalonde, and Phillips (2004) show that 36 percent of Florida hurricane insurers could effectively use instruments based on a statewide index without being exposed to a high degree of basis risk, while smaller insurer are generally exposed to high basis risk (see also Harrington and Niehaus, 1999).

SUMMARY AND CONCLUSIONS

This article presents a new approach and techniques of how prices for index-linked catastrophe loss instrument can be derived using arbitrage-free pricing by proposing replication techniques and approximations that aim to overcome the requirement of direct tradability of the underlying loss indices. In particular, contrary to traditional option pricing theory, one cannot necessarily assume that the underlying industry loss index is tradable itself; however, there may be a liquid market for certain derivatives, including cat bonds. This is of great relevance today in the academic literature, but it will be even more relevant in the future for the insurance industry and financial investors, when index-linked catastrophe loss instruments become even more widespread than today and when there are truly liquid markets for derivatives (e.g., cat bonds) on the industry loss index.

We apply the proposed approach and considerations to the arbitrage-free pricing of ILWs, where we assume a liquid cat bond market to ensure the tradability of the underlying industry loss index. We hence consider a liquid index-linked cat bond market (equivalent to a liquid option market) for various trigger levels and maturities, and first derive prices for binary ILWs. We thereby show that we do not need any assumption concerning the distribution of the underlying industry loss index, which represents a major advantage as compared to other pricing approaches. If a suitable cat bond is not available for approximating ILW prices, we provide approximations under some reasonable assumptions. Thus, one main contribution of the article is to propose an approach to overcome the crucial point of tradability of the underlying loss processes in case of index-linked catastrophe loss instruments through suitable approximations and by deriving explicit replicating portfolios or close approximations using traded derivatives providing explicit pricing formulas. We test these approximations by studying empirical examples and by conducting comprehensive simulation analyses, which suggest that the approximations are very stable and that model risk can be considerably reduced as compared to standard pricing that makes use of distributional assumptions.

When calculating prices for indemnity-based double-trigger catastrophe loss instruments, which in addition to the index depend on the company loss, one major problem is the behavior of the company loss, which should be tradable in some sense when using risk-neutral valuation. We provide first brief considerations regarding three potential approaches for solving this issue and address problems associated with them, leaving further theoretical and empirical analyses for future research. In general, such considerations regarding indemnity triggers will be increasingly relevant in the future, especially against the background of an increasing share of indemnity-based cat bond transactions.

APPENDIX

[I.sup.T.sub.t] is a compound Poisson process defined through

[mathematical expression not reproducible]

where [N.sub.t] and [X.sup.I] are independent, [N.sub.t] denotes the claims arrival process (i.e., a Poisson process with intensity [lambda]), and X are the i.i.d. claims. The probability for exceeding the trigger level is given through

[mathematical expression not reproducible] (A1)

Using the Landau symbol O leads to

[mathematical expression not reproducible] (A2)

for [lambda] [right arrow] 0, where the approximation in the second line is inspired by the principle of a single big jump (see, e.g., Foss, Konstantopoulos, and Zachary, 2007). We use Equation (A2), and

[mathematical expression not reproducible] (A3)

to calculate [D.sup.Y,T,T.sub.Y,T,T]:

[mathematical expression not reproducible]

[mathematical expression not reproducible]

Hence,

[mathematical expression not reproducible] (A4)

As Q([X.sup.1] > Y) and X are usually unknown, we use Equation (A3) to rewrite Equation (A4) as follows

[mathematical expression not reproducible]

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Nadine Gatzert and Nikolai Vogl are at the Department of Insurance Economics and Risk Management, Friedrich-Alexander University Erlangen-Nurnberg (FAU), Nurnberg, 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 Grundl, Steven Kou, Alexander Murmann, 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.

DOI: 10.1111/jori.12191

(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.

(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, Muller, and Scherling (2010).

(5) See, for example, Lee and Yu (2002) for an application of this approach to pricing cat bonds using risk-neutral valuation under default risk, basis risk, and moral hazard. Alternatively, an equilibrium model can be employed (see, e.g., Cox and Pedersen, 2000; Zhu, 2011, for a brief discussion on the relation of the two frameworks).

(6) The example (referring to March 2007) suggested buying the (3-year) Mystic Re cat bond on Northeast U.S. wind with an industry loss trigger of $30-40 billion and linear payout with a spread of LIBOR + 700 bps and buying a (12-month) binary ILW on Northeast U.S. wind with a $30 billion trigger level for 615 bps (p. 27).

(7) This loss information is provided by specialized industry loss index providers. Note that this also depends on the type of index-linked catastrophe loss instruments. For example, if dealing with occurrence ILWs (see Ishaq, 2005), whose payoff does not depend on the sum of insured damage but only on the loss caused by the first event), one needs a different approach, but our techniques are applicable as well.

(8) See, for example, McDonnell (2002) for the development of the PCS estimations of the 10 U.S. natural catastrophes with the highest insured damage.

(9) These derivatives also represent index-linked catastrophe loss instruments. Note, however, that the assumption of liquidity only holds for certain types of derivatives, as standard call and put options on an industry loss index are not traded on a liquid market. Call and put options were introduced in 1995 by Chicago Board of Trade, but due to limited trading these options were delisted in 2000 (see, e.g., Cummins and Weiss, 2009). Hence, the derivatives we focus on in the following are cat bonds, which exhibit a considerable market volume with a relevant secondary market. In case that there is a liquid market for call and put options on the underlying industry loss index at all strikes, the Litzenberg formula could be applied to for static replication.

(10) The instruments we focus on are not path dependent and there could be various measures resulting in the same prices for instruments with a payoff depending only on [I.sup.T.sub.T']. As there is no difference for the prices of the instruments we consider, it is not relevant which of these measures we take.

(11) Note that an alternative but less common representation of an ILW would be that the payoff is triggered if the index exceeds the trigger limit during the contract term (knock-in barrier option). The following analysis can be extended to this case as well.

(12) In general, in case the market is incomplete and ILWs cannot be fully replicated, under suitable assumptions it may still be possible to at least approximately (dynamically) replicate the ILWs (see, e.g., Bertsimas, Kogan, and Lo, 2001; Xu, 2006, where arbitrage-free price windows are derived by means of sub- and super-hedges). With the derivation of sub- and super-hedges, one obtains a price interval in which any arbitrage-free price of the ILW has to be contained. Within the interval, the chosen price will depend on the risk preference of the investor and might still be subject to inefficiencies.

(13) Even though proportional payouts are generally more common in practice, the Mexican cat bond issued in 2009, for instance, featured a binary trigger (see Cummins, 2008).

(14) A related observation is made by Vaugirard (2003a), who observes that the buyer of a catastrophe bond holds a short position on a binary option on the index.

(15) Note that the reduction of model risk also works for more complex loss-frequency models (see, e.g., Lin, Chang, and Powers, 2009; Chang, Lin, and Yu, 2011).

(16) Note that in case the approach presented here is applied to consistently price cat bonds, transaction costs would be even more comparable.

(17) Note that we need information regarding the attachment point and the layer of the cat bond, which in case of secondary (and primary) market data is typically not provided. Also, ILW prices are very difficult to obtain due to OTC transactions, which is why we use an empirical example from 2002 where this information is available.

(18) Date of issue: February 2001; maturity: January 2003; coupon: 510; attachment point: $22.5 bn.; exhaustion point: $31.5 bn.; probability of first loss: 0.0082; probability of exhaustion: 0.0034; expected loss: 0.0055; index: PCS; risk: California earthquake (see Lane and Beckwith, 2001; Michel-Kerjan et al., 2011, for the attachment and exhaustion points). As we have no information regarding the risk exposure period, we assume that it equals the maturity.

(19) See http://de.global-rates.com

(20) Taking Figure 5 (U.S. earthquake data instead of California earthquake data) as a rough indication for the pricing error, the latter would amount below 2.5 percent.

(21) We additionally calculated the outcome including coupon payments with resulting ask and bid prices of 5.07 and 5.83 percent.

(22) Alternatively, one can assume stochastic dependence to reflect the case where the company is not affected by a catastrophe.
TABLE 1
Characteristics of Successor X Ltd 2012-1; Ibis Re II Ltd. 2012-1 A;
Ibis Re II Ltd. 2012-1 B; Mythen Re Ltd. 2012-1

                                                         Conditional
                                    Spread     Expected  Expected
                 Date of            (At        Loss      Loss
Name             Issue    Maturity  Issuance)  (%)       (%)

Successor X      01/2012  01/2015    1,100     2.59      83
  Ltd. 2012-1
Ibis Re II Ltd.  01/2012  02/2015      835     1.38      59.2
  2012-1 A
Ibis Re II Ltd.  01/2012  02/2015    1,350     3.38      67.9
  2012-1 B
Mythen Ltd.      05/2012  05/2015      850     1.09      73.6
  2012-1

                 Prob. of  Prob. of
                 1st Loss  Last
Name             (%)       Loss (%)

Successor X      3.12      2.24
  Ltd. 2012-1
Ibis Re II Ltd.  2.33      0.89
  2012-1 A
Ibis Re II Ltd.  4.98      2.36
  2012-1 B
Mythen Ltd.      1.48      0.82
  2012-1

Notes: See Lane (2013) for Mythen Ltd. 2012-1 and Lane (2012) for Ibis
Re II Ltd. and Successor X Ltd.
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Author:Gatzert, Nadine; Pokutta, Sebastian; Vogl, Nikolai
Publication:Journal of Risk and Insurance
Date:Mar 1, 2019
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