Market-Consistent Valuation of Natural Catastrophe Risk

Simone Beer∗ Alexander Braun†

February 2019

Abstract Natural catastrophe risk is increasingly covered through alternative capital instead of classical reinsurance. Due to the absence of a secondary market for almost all of these instruments, their ongoing valuation poses a challenge to investors. We suggest extracting information contained in regularly observed catastrophe bond prices by means of a reduced-form model. The Poisson intensity of the latter is found to materially depend on time to maturity and initial probability of ﬁrst loss. Along these two dimensions, we estimate smooth implied intensity surfaces for peak and non-peak perils that allow us to mark any catastrophe risk contract to market.

Key words: Natural Catastrophe Risk · Market-Consistent Valuation · Reduced-Form Model · Ran- dom Eﬀects Model · Intensity Surface JEL Classiﬁcation: C23 · C58 · G12 · G22

∗Simone Beer ([email protected]) is with the Institute of Insurance Economics, University of St. Gallen, Tannen- strasse 19, CH-9000 St. Gallen.

†Alexander Braun ([email protected]) is with the Institute of Insurance Economics, University of St. Gallen, Tannenstrasse 19, CH-9000 St. Gallen. 1 Introduction

Natural catastrophes are among the most severe risks faced by households and businesses in the modern age (see Allianz, 2019). The record-breaking year 2017 alone saw USD 330 billion of worldwide economic losses from tropical cyclones, severe windstorms, floods, wildfires and other weather-related perils, which is around half a percent of global GDP. Due to persisting protection gaps, only USD 138 billion were recovered through insurance claims (see Swiss Re, 2018).1 In comparison to that, the re/insurance industry commanded USD 605 billion of risk-bearing capital at year-end 2017 (see Aon Benfield, 2019).

Hence, its capacity to provide further coverage is limited. Against this backdrop, direct risk transfer to the ﬁnancial markets has become a state-of-the-art solution. The rapidly growing alternative capital available to absorb reinsurance risks amounted to about 100 billion in 2018.

At least 33 open-end insurance-linked securities (ILS) funds worldwide currently deploy investor money in the market for natural catastrophe (NatCat) risk (see Eurekahedge ILS Advisers Index, Ben Ammar et al., 2016). However, the majority of instruments oﬀering this type of exposure, such as collateralized reinsurance or industry loss warranties, lack an active secondary market. Therefore, their ongoing valuation poses a major challenge. An exception are catastrophe (cat) bonds, which oﬀer daily liquidity based on indicative quotes from broker-dealers (see www.artemis.bm). In this paper, we suggest exploiting information embedded in prices of traded cat bonds to mark NatCat risk positions to market. To this end, we draw on a reduced-from model, which is deeply-rooted the credit derivatives literature (see, e.g.,

Jarrow and Turnbull, 1995). Jarrow(2010) proposes to adopt this methodology for cat bonds and derives a closed-form pricing equation. The essential input parameter is a Poisson intensity (or hazard rate) that drives defaults. For indemnity and industry-index triggers, the latter occur when natural disasters strike and cause losses in excess of the bond’s attachment point. In contrast, a cat bond with parametric trigger defaults when a physical parameter value such as wind speed crosses a preset threshold. The collateral is then liquidated to compensate the cedent (sponsor) and investors lose all or part of the notional.2 Since the intensity is not directly observable, Jarrow(2010) recommends computing it from historical NatCat event data. We follow a diﬀerent logic. Being able to regularly observe indicative cat

1Amounting to approximately USD 30 billion each, hurricanes Harvey, Irma and Maria in North America and the Caribbean turned out to be the most costly events for the insurance industry in 2017.

2A detailed discussion of the anatomy of cat bonds can be found in Braun(2016).

2 bond prices in the secondary market allows us to extract implied intensities that can subsequently be employed for the pricing of non-liquid ILS. Similar to implied volatilities for options, we expect the implied intensities to be a better predictor of future realized intensities than estimators based on past data.3

A major hurdle is the low degree of standardization of cat bonds. More specifically, transactions may differ by trigger type, covered territory, reference peril, attachment and detachment point, maturity, and rating. These factors are known to affect primary and secondary market spreads (see Braun, 2016;

Gürtleret al., 2016). Thus, they might also have an impact on implied intensities and turn the derivation of suitable estimates into a high-dimensional problem with only few data points for each case. To explore this possibility, we run a random effects model on the time-series-cross-section of implied intensity estimates. Our results reveal that the latter differ between peak and non-peak perils, but otherwise only materially depend on the time to maturity and the initial probability of first loss (PFL). Hence, we may construct smooth surfaces that combine term structure and loss distribution information and allow us to mark any catastrophe risk contract to market.4 When investigating the accuracy of our pricing approach, we obtain an excellent in-sample fit statistics.

The rest of the paper is structured as follows: In Section2, we revisit the reduced-form cat bond pricing model of Jarrow(2010) and explain its numerical implementation. Moreover, in Section3, we introduce our data set and show some descriptive statistics. The methodology of deriving implied intensities along with the empirical results are illustrated in Section4. Section5 states the hypotheses of potential intensity rate determinants. In addition, we present our panel data model. Finally, in Section6, we compute smooth implied intensity rate surfaces and evaluate their pricing ﬁt. The last section, Section7, concludes.

3An extensive list of studies has proven this point for option-implied volatilites (see, e.g., Flemming et al., 1995; Jorion, 1995; Christensen and Prabhala, 1998). Similarly, hazard rates derived from observed CDS spreads are richer in information than their historical counterparts, as they additionally incorporate non-default factors, such as premia for liquidity risk and spread risk, and eﬀect for the current market supply-demand situation (see, e.g., O’Kane and Turnbull, 2003).

4Our approach works analogous to volatility surfaces estimated from options and hazard rate term structures estimated from credit default swap (CDS) spreads, which are the key valuation reference for exotic derivatives.

3 2 Reduced-form pricing model

2.1 Modelling assumptions and cat bond terms and conditions

We follow Jarrow(2010) and apply the reduced-form pricing framework in continuous time. In doing so, we rely on certain modelling assumptions which shall read as follows. Firstly, a trigger event (default) can occur at any point in time denoted by τ before the bond’s maturity. Given indemnity, industry index and modelled loss bonds, the event itself is deﬁned as [event losses > attachment point], whereas for parametric bonds it holds that [parameter values > threshold]. Secondly, future ﬁxings of the reference rate equal the current forward rates such that no term structure model is required. Thirdly, both sponsor and collateral are default-free. As a result, there is no need to assume credit risk that requires the application of a credit risk model.

With respect to the cat bond terms, the following holds. There is a ﬁxed maturity T and a notional

N that is due at T unless a trigger event occurs at time τ before T . Additionally, the cat bond terminates directly after an event with a recovery rate α. The valuation dates (i.e. the dates on which price quotes are available) are denoted by tν ∈ [0,T ]. The preceding coupon (or issuance) dates of the cat bond are given by t0 ≤ tν , while future coupon times refer to tk > tν , where (k = 1, ..., K). The ﬁnal coupon is paid at maturity such that tk = T . The length of the coupon periods are indicated by ∆tk = tk − tk−1 for ∀k.

Floating coupons are paid in arrears based on the ∆tk-year reference rate Ltk−1 (p.a.) that is ﬁxed on the last coupon date tk−1. Hence, we require both spot and forward yield curves of the reference rate

(i.e. the current interest rate term structure). The cat bond spread scat (p.a.) is ﬁxed at issuance of the cat bond, and we assume accrued interest for the time between default and preceding coupon date

(tk−1 < τ ≤ tk).

4 2.2 Valuation formula

The cat bond price CBtν at time tν can be computed by summing (10), (13) and (14) (see appendix), and rearranging:

" 1 CBt = N DF (tν , t1) Lt + scat ∆t1 1 + SP (tν , t1) ν 2 0

K # 1 X + DF (tν , tk) Lt + scat ∆tk SP (tν , tk−1) + SP (tν , tk) 2 k−1 k=2 (1) " M·T # X Q + N DF (tν ,T ) + DF (tν , tm) Etν (α)(SP (tν , tm−1) − SP (tν , tm)) ; m=1

where

DF (t, s) = elt,s(s−t)

is the discount factor (i.e. the time t value of a rate’s deposit paying one dollar at time s, with lt,s being the spot rate (p.a.) for the period from t to s),

Q Etν (α) = [.]

is the distribution for the recovery rate α (or the percentage loss in case the bond is triggered),5

Z s Q λt,s = Et (Ns − Nt) = λ(u)du (2) t

is the expected number of events from time t to time s (or the likelihood of an event (per unit of time)

that triggers the bond)6, and

0 e−λt,s λ ( t,s) −λt,s SP (t, s) = P r (Ns − Nt = 0) = = e (3) 0!

is the probability of survival from time t to time s, conditional on having survived to time t.

5We will illustrate the speciﬁc form in Section 2.3.

6The likelihood of an event, henceforth intensity λ(t), can be estimated with historical data (e.g. Jarrow(2010)) or retrieved from observed cat bond market prices. We will focus on the latter approach (see Section4 for details).

5 2.3 Numerical implementation

The data set is comprised of cat bonds whose coupons are paid either monthly, quarterly, or semi-annually.

In addition, there are zero-coupon bonds. In the following, we assume the length of each coupon period to be constant, i.e. ∆tk = tk − tk−1 = ∆t for ∀k. In line with the above mentioned order of diﬀerent coupon payment schedules, ∆t ∈ [0.083, 0.250, 0.500, not applicable]. Please note that the cat bond pricing framework illustrated in Equation (1) requires ﬂoating coupons to be considered at time tk that are

based on the ∆t-year reference rate Ltk−1 (p.a.) ﬁxed on the preceding coupon date tk−1; where L ∈ [3-month Euribor rate, 3-month Libor rate, 6-month Libor rate, U.S. money market fund yield]. For the latter, we construct a synthetic money market fund consisting of 1-month, 3-month, and 6-month U.S.

7 Treasury bills, being assigned equal portfolio weights. For each valuation date tν , where tν implies the availability of a cat bond market price, the reference rate for the ﬁrst coupon is the current rate

(yield) and for the subsequent coupons, forward rates (yields) are used. Coupon payments and the notional repayment are discounted using spot rates. In case of a trigger event, we assume a ﬁxed recovery

Q value α; where, given the cat bond’s conditional expected loss (CEL), Etν (α) in Equation (1) reduces to α = (1 − CEL (%)). Finally, the assumption is made that an event may occur at M = 252 discrete times per year, provided that there are 252 trading days a year. Precisely speaking, we assume a cat bond to be triggered once a day.8

Table1 shows the required model parameters to evaluate Equation (1), along with the corresponding variable names and the ﬁnancial instruments in case real data is used. To discount coupon payments and the principal repayment both in the non-default and in the default case, we apply the EUR swap zero curve for bonds whose ﬂoating coupons are based on Euribor rates, the USD swap zero curve in case of Libor-based coupons, and the U.S. Treasury zero-coupon yield curve where the bond’s reference rate is a U.S. money market fund. Here, we follow Ron(2000) and use the subsequently listed instruments

(subject to the instruments’ tradability) to derive a complete swap zero curve term structure for the required currencies EUR and USD, respectively, starting at overnight rates. The short end of the swap

7For the fund’s composition, we use the 2017 money market fund statistics provided by the U.S. Securities and Exchange Commission. It includes, amongst others, the weighted average maturity of assets held by money market funds in the preceding year (see U.S. Securities and Exchange Commission(2017) for more information).

8Unreported results show that an increase in the number of events per year up to a value of 1,000 does not lead to a diﬀerent result in terms of estimated implied intensity rates.

6 curve, out to three months, results from overnight, 1-week, 2-week, 1-month, 2-month, and 3-month interbank deposit rates. The middle area of the swap curve up to two years is based on interest rate futures contracts. Finally, we derive the long end of the swap curve out to five years directly from observable coupon swap rates. In particular, we collect 1-month, 2-month and 3-month Euribor (Libor) rates, Euro Libor (Eurodollar) futures out to two years, and swap rates out to five years to construct a full EUR (USD) swap curve term structure up to five years.9 Treasury zero-coupon yield curves for a complete term structure are, however, entirely obtained from our data source (see Section 3.1).

model parameter variable instrument 1. reference rate L • 3-month Euribor spot rate; 3-month Euribor forward rate (curve) • 3-month Libor spot rate 3-month Libor forward rate (curve) • 6-month Libor spot rate 6-month Libor forward rate (curve) • money market fund yield money market fund forward yield (curve) 2. discount rate l • EUR swap zero curve (for coupon payments, • USD swap zero curve and principal repayment • U.S. Treasury zero-coupon yield curve in (non-)default case) 3. coupon payment schedule • zero-coupon • monthly • quarterly • semi-annual 4. length of coupon period ∆t • not applicable • 0.083 • 0.250 • 0.500 5. number of events per year M 252 (default time steps)

Table 1: Description of pricing model parameters. Table1 shows the required model parameters to evaluate Equation (1), along with the corresponding variable names and the ﬁnancial instruments in case real data is used.

3 Data

3.1 Data set and sample selection

Cat bond data

The initial data set comprises a total of 580 cat bonds that are traded on secondary markets. The bonds were issued between November 1997 and May 2017 and provide insurance against natural catastrophe

9The cat bond with the longest maturity has a tenor of ﬁve years.

7 perils. This data set is compiled from two diﬀerent sources. On the one hand, end-of-month secondary

market bid quotes are provided by Aon Benﬁeld, along with issue price, per annum spread and per annum spread discount. In addition, we obtain information about the issue date as well as the ﬁnal

maturity date, payment schedule, reference rate, issue size (in USD million), tenor, trigger type, covered

territory, reference peril, and bond rating.10 On the other hand, we also consider trade notes by Lane

Financial LLC, the Artemis Deal Directory, as well as market research of Swiss Re, Aon Benﬁeld, and

Guy Carpenter that cover cat bond issues in primary markets between November 1997 and May 2017. In doing so, we are able to not only do a cross-check of all the static cat bond parameters that are obtained from Aon Benﬁeld, but also to collect additional parameters, that is to say expected loss (EL), probability

of first loss (PFL), conditional expected loss (CEL) and beneficiary. Finally, given each cat bond, we manually gather information on whether its protection (or coverage) is of type ‘annual aggregate’ or of type ‘per-occurrence’ using the Artemis Deal Directory.11 Unfortunately, we find only 387 of the 580 cat bonds to make its type of protection publicly available. In particular, a large majority of bonds issued prior to 2010 report missing values. We will address this issue in more detail in Section 5.3.

Exclusions

Several cat bonds are deleted from the initial data set. First, we exclude cat bonds for which the static parameter values probability of ﬁrst loss and conditional expected loss are missing since these factors are

essential determinants required for computing implied intensities. Second, we remove cat bonds whose

reference rate is based on a yen-dominated money market fund. This is due to the lack of available data

on short-term historical spot and forward yields for Japanese treasuries that are the usual constitutions

of a representative money market fund and are required for computing coupon payments (as well as

for discounting these cash ﬂows and the bond’s notional). Third, we eliminate all second- as well as

multi-event bonds.12 The reasons for these bonds’ elimination are twofold: i) their corresponding implied

10Data on bond rating is available from the following four rating agencies: Standard & Poor’s, Moody’s, Fitch Ratings and A.M. Best. We decide to choose Standard & Poor’s as the main provider since it has the most comprehensive data base, i.e. ratings available for most of the bonds. In case of missing rating information (which applies to 32 bonds), we opt for Moody’s. However, to ensure consistency, all bond ratings of Moody’s are translated into an equivalent rating of Standard & Poor’s.

11The Artemis Catastrophe Bond & Insurance-Linked Securities Deal Directory is accessable via www.artemis.bm.

12Information on whether a bond has a second- or multi-event feature is manually collected using the Artemis Catastrophe Bond & Insurance-Linked Securities Deal Directory.

8 intensity rates are not directly comparable to those of single-event bonds. As the notation already implies, these bonds’ intensities refer to a series of consecutive events, thus making it nearly impossible to extract the intensity of the first event only. ii) due to their multi-event feature, bonds of this type generally report conditional expected loss (CEL) values above 100.00%. Since we assume the recovery value α required in the pricing Equation (1) to be of the form α = 1−CEL(%), using a CEL larger than 100.00% would result in a negative α value. Fourth, concerning the type of peril, we remove cat bond transactions that insure rare perils. Precisely speaking, there is only one cat bond that insures floods. Fifth, we eliminate cat bonds whose maturity exceeds a threshold of five years since we are unable to collect historical interest rate data for certain types of rates beyond a maturity of five years.

Defaulted and distressed bonds

Sixth, we remove cat bonds that either defaulted in light of the Lehman bankruptcy (since Lehman

Brothers was the swap counterparty in the pre-crisis cat bond structure) or that were triggered subsequent to the occurrence of a NatCat event using the Artemis Deal Directory. Here, we consider Hurricane Ivan in 2004, Hurricane Katrina, Rita and Wilma in 2005, Hurricane Ike in 2008, the Tohoku earthquake in 2011, and Hurricane Sandy in 2012. Four bonds are labelled lehman, while there are ten cat bond transactions identified as triggered. We decide to exclude each of these bonds’ full price history. The reason is simple: we thus avoid any potentially arising conflicts when determining at which precise level a single market prices is considered to be affected by default. Please be aware of the fact that defaulted bonds are not to be confused with distressed bonds, the latter are still included in the data set (see

Section 5.2 for more details). To be more precise, a bond is classiﬁed as being defaulted if its market prices reaches a value of zero and does not recover again. In contrast, a distressed bond exhibits a price decline that is only temporarily, reaching its prior price level again and additionally never has a zero price quote. The remaining data set consists of 539 cat bonds with 15,807 observations of secondary market prices.

Interest rate data

We gather end-of-month interest rates data in line with Table1 for the time period from December, 31st

2000 to May, 31st 2017.13 Since the cat bond with the longest maturity in the ﬁnal data set has a tenor of

13The reason for the selection of end-of-month rates only will be given in Section 4.1.

9 ﬁve years, we consider a term structure for every interest rate curve of up to ﬁve years at each given month.

The data provider is Bloomberg. Firstly, as illustrated in Section 2.3, we collect all required instruments to build EUR swap zero curves and USD swap zero curves, respectively, that are used for discounting.

Secondly, for most of the required curves listed in Table1 that act as reference rate, entire historical rate curves are directly available on Bloomberg, given the desired tenors (i.e. monthly intervals). However, an exception are 6-month forward Libor rate curves which cannot be retrieved. Here, we compute arbitrage- free 6-month Libor forward rates using U.S. swap curves. Thirdly, we apply a logarithmic transformation to all spot rates that are used for discounting only. It ensures the conversion of discrete spot rates into continuous spot rates which are required for continuous discounting. Finally, it has to be mentioned that U.S. Treasury spot rate, U.S. Treasury yield and U.S. Treasury forward yield curves have rates retrievable only at certain tenors.14 However, at each month, we require a term structure of monthly intervals, ranging from one month up to ﬁve years, for both yield and forward yield curves, and of daily intervals for spot rate curves. The latter are used for discounting both in the default and non-default case, and hence require daily maturity dates. Thus, given each month, we ﬁt a Nelson-Siegel-Svensson model to both observed spot rates (already stated in log terms) and yields, and a Nelson-Siegel model to observed forward yields, which we then use to estimate unobserved spot rates, yields and forward yields, respectively, whose maturities then are of the requested intervals.15 Furthermore, we are also unable to collect rates for constructing EUR swap zero curves, and USD swap zero curves, in the desired term structure format. Thus, in line with the methodology to build interest rate swap curves that is used by

Bloomberg, we apply the smooth interpolation method to model the term structure of both EUR and

USD swap zero curves, such that these curves are then available in daily intervals, up to a maturity of 5 years (see Bloomberg(2016) for more information on the interpolation method).

14Bloomberg oﬀers the following tenors in case spot rates and yields are considered: 1-month, 3-month, 6-month, 1-year, 2-year, 3-year, 4-year, and 5-year. In addition, to construct the short end of spot rate curves up to a maturity of 1 month, we use overnight Treasury repo rates (i.e. rates for overnight Treasury general collateral repurchase agreement (repo) transactions), as well as 1-week, 2-week and 3-week Fed Fund rates. Regarding forward yields, the available tenors are as follows: 1-month, 3-month, 6-month, 1-year, 2-year, 3-year, 4-year, and 5-year.

15We justify the model choice as follows. Firstly, we fit both a Nelson-Siegel model and a Nelson-Siegel-Svensson model to observed spot rates, yields, and forward yields. Secondly, the two models’ goodness of fit for each rates type are gauged with respect to minimizing the root mean square error (RMSE), calculated as the difference between observed and fitted rates, subject to the tenors’ availability. We find the Nelson-Siegel-Svensson model to have a superior fit, i.e. a smaller RMSE, for spot rate curves and yield curves, while the Nelson-Siegel model better fits forward yield curves.

10 3.2 Descriptive statistics

In this section, we provide a variety of descriptive statistics with regard to our final sample. Table2 presents summary statistics for the key characteristics of the final cross section of 539 cat bonds. The statistics for variables that are bond specific and vary over time are calculated on the observation level.

For variables that are bond speciﬁc and time-invariant, we use the tranche level for computation. The mean market price amounts to USD 100.59 and the mean value for the probability of ﬁrst loss equals

2.82 percent. In addition, the PFL ﬁgure varies considerably across all bonds in the sample, which is indicated by the respective standard deviation as well as the minimum and maximum values. It can, for example, be as low as 1 bp or as high as 21.38 percent. Market prices, in contrast, range between

USD 50.00 and USD 119.36.16 In general, most cat bonds typically insure only a maximum of two perils, while the mean number of covered territories is 1.4. Furthermore, the average issue size of the cat bond tranche in the ﬁnal sample is USD 123.92 million, while issues as small as USD 0.25 million and as large as USD 1,500.00 million have been observed. Finally, the sample comprises transactions with terms from a minimum of 6 months up to a maximum of 5.08 years, and the average cat bond matured after 3.23 years.

Obs. Mean S.D. Min. q25 q50 q75 Max. Cat-bond-specific variables Market Price 15,807 100.59 3.22 50.00 99.71 100.40 101.83 119.36 Probability of First Loss (PFL) (in %) 539 2.82 2.77 0.01 1.13 1.85 3.74 21.38 No. of Covered Territoriesa 539 1.40 0.86 1 1 1 1 10 No. of Perilsb 539 1.99 1.65 1 1 1 1 12 Issue Size (in USD m) 539 123.92 121.99 0.25 50.00 100.00 160.00 1,500.00 Maturity (in years) 539 3.23 0.84 0.50 3.00 3.08 4.00 5.08 a The maximum number of covered territories of 10 results from counting the different single territory specifications in the final sample of 539 bonds, and indicates multiterritory bonds. b The maximum number of perils of 12 results from counting the different single peril specifications in the final sample of 539 bonds, and indicates multiperil bonds.

Table 2: Descriptive statistics of cat-bond-specific variables. Table2 shows the number of observations, the mean, median, and standard deviation (S.D.), as well as the minimum and maximum values, and the 25 and 75 percent quantile for the market price, the probability of first loss (PFL), the number of covered territories and perils, the issue size, and maturity of the full cross section of 539 cat bonds in the final sample. The statistics for variables that are bond specific and vary over time are calculated on the observation level. For variables that are bond specific and time-invariant, we use the tranche level for computation.

Further descriptive statistics for different categories of cat bonds in the final sample can be found in Table3. When focusing on the geographic classification, we observe that the vast majority of issues

16As indicated in Section 3.1, the ﬁnal data sets contains both distressed and non distressed bonds, which explains the considerably wide dispersion in observed bond prices.

11 (59.00 percent) cover the United States. The second-largest fraction is represented by multiterritory bonds (24.86 percent). In contrast, deals referencing a single territory other than the U.S. or Europe are very rare. ‘Japan’ and ‘Other’ represent a mere 5.57 percent and 2.78 percent of the final sample, respectively. Finally, the average issue size (maturity) of transactions on Japanese areas is substantially higher (longer) than for the remaining territories. Turning to the peril-specific figures, we notice that the sample comprises 19.48 percent earthquake, 33.02 percent wind, and 47.50 percent multiperil bonds.

We find both multiterritory and multiperil tranches to exhibit considerably larger average PFL values compared to earthquake bonds. Overall, the differences in the average maturity for the first two categories of Table3 are negligible. With regard to trigger types, indemnity (38.03 percent), industry loss index

(28.20), and parametric index (22.23 percent) clearly dominate the ﬁnal sample. The by far highest average PFL value is associated with bonds that contain more than one trigger mechanism. Furthermore, transactions with indemnity and industry loss index triggers are associated with the largest, and bonds having multiple triggers with the smallest average issue volumes. In terms of rating, we see that only

4.83 percent of the cat bonds in our sample carry an investment-grade rating.17 On average, these tranches have a considerably lower PFL value than high-yield cat bonds. Finally, with respect to the type of protection, we differentiate between annual aggregate and per occurrence. We find the first to report an average probability of first loss of higher magnitude compared to the latter. From a rational perspective, this is not surprising. Given annual aggregate bonds, events are considered over the entire year to determine whether a bond is triggered. On the contrary, the trigger event coincides with the first occurrence of an event in the case of per occurrence bonds. As already mentioned in Section 3.1, only

355 cat bonds make their type of protection publicly available. Thus, we are faced with 184 tranches for which we are unable to report this information. In summary, our ﬁndings are consistent with the observations of Braun(2016).

Figure1 shows the historical development of monthly cat bond prices for U.S. wind contracts between

December 2013 and May 2017, presented as a time series of monthly box plots. In the following, we focus on the mean monthly price quote series, indicated by the bold grey line. In August 2005, prices were quite low, with quotes substantially beneath USD 100.00. Thus, reinsurance was rather cheap. Following

Hurricane Katrina, Rita and Wilma in 2005, prices rose steadily until they peaked in the summer of 2006.

17All rating classes from BBB− upwards by Standard & Poor’s and Fitch(or the Moody’s equivalent Baa3) are termed investment grade (see, e.g., www.standardandpoors.com.)

12 No. of Bonds Percentage PFL Maturity Issue Size (in %) (in years) (USD m) Territory Europe 42 7.79 2.13 3.11 102.73 Japan 30 5.57 1.52 3.85 152.65 Multiterritory 134 24.86 4.40 2.98 99.24 Other 15 2.78 2.44 2.89 107.00 United States 318 59.00 2.90 2.93 138.78 Peril Earthquake 105 19.48 1.63 2.81 135.88 Multiperil 256 47.50 3.93 3.10 117.99 Wind 178 33.02 2.84 2.97 131.77 Trigger Indemnity 205 38.03 2.72 3.22 161.66 Industry Loss Index 152 28.20 3.59 3.01 137.89 Modelled Loss 33 6.12 2.17 2.68 91.33 Multiple Trigger 29 5.39 7.06 2.68 61.26 Parametric Index 120 22.23 2.53 2.80 75.33 Rating Class AA 2 0.37 0.01 3.00 194.75 A 5 0.93 0.06 2.60 167.92 BBB 19 3.53 0.27 2.71 59.44 BB 231 42.86 1.49 3.04 130.30 B 119 22.07 4.43 2.74 111.88 NR 163 30.24 4.95 3.19 135.94 Type of Protection Annual Aggregate 103 19.11 3.78 3.53 178.00 Per Occurrence 252 46.75 2.78 3.18 143.61 Not Available 184 34.14 3.22 2.46 72.85

Table 3: Descriptive statistics for different categories of cat bonds in the final sample. Table3 shows the classification of cat bonds in the final sample according to covered territory, reference peril, trigger type, rating class, and type of protection. For each category the number of bonds and percentage of tranches as well as their average probability of first loss (PFL), term, and issue size are provided.

This period may be seen as a classical hard market when premiums generally increased, and reinsurance capacity decreased (Cummins et al., 2006). After a brief recovery phase, when conditions softened again, a sharp rise could be observed around the time of the collapse of Lehman Brothers and Hurricane Ike in

2008. However, markets eased again quickly in the aftermath. The next sharp upheaval occurred in the summer of 2010. During this hurricane season, four category 4 hurricanes were observed: Danielle, Earl,

Igor and Julia (NOAA, 2011). From then on until May 2017 (with Hurricane Sandy occurring in 2012), prices slowly increased. It is interesting to note that risk pricing seemed to have changed. However, this is not surprising, and clearly in line with the ongoing softening of conditions in the reinsurance market that has been observed for several years now. Generally speaking, the reinsurance business is exposed to periods of soft markets with ample coverage and fairly low premiums as well as hard markets with limited risk-bearing capacities and high premiums (see, e.g., Cummins and Weiss, 2009). Since cat bonds are a direct substitute for traditional reinsurance contracts, they should, by no-arbitrage reason, exhibit

13 comparatively similar pricing patterns over time. In addition, the graph clearly indicates the seasonality

eﬀect in bond prices. The time series of price movements of both U.S. earthquake and U.S. multiperil

bonds depicted in Figures7 and8 (see appendix) show very similar price ﬂuctuations. This suggests

that movements may also be explained by structural eﬀects and not merely by events with regard to the

respective reference peril.

110.00

105.00

100.00

95.00

90.00 Market Price Market

85.00

80.00

75.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Blue Coast Ltd. A Carillon B Foundation Re A Palm Capital Ltd. Blue Coast Ltd. B Citrus Re Ltd. 2014 1.A Globecat USW 3.A Queen City Re Blue Coast Ltd. C Citrus Re Ltd. 2014 2.1 Long Point Re III 2012 1.A Successor Hurricane Ind. IV.E Carillon 2.A Compass Re II 2015 1 Manatee Re 2016 1.C Successor Hurricane Ind. V.E

Figure 1: Time series of monthly cat bond prices for U.S. wind contracts (12/2000–05/2017). Figure1 shows the historical development of monthly cat bond prices for U.S. wind contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of bond prices within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Outliers are depicted using a range of different symbols which are labelled by the respective bonds’ names. These names are listed in the table below the graph. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly bond prices. In addition, the red line indicates the cut-off price level at which bonds may be qualified as distressed.

4 Derivation of implied intensity rates

4.1 Modelling assumptions and motivation

To decrease complexity in the bonds’ various (and presumably diﬀerent) event dates we are faced with in each month and given the fact that market prices are available at month-end only, we assume all cat bonds to be issued, to mature, and to have coupon payments exclusively at each month’s last trading

14 day. As a consequence, both spot and forward rate curves and yield curves may only be collected at month-end dates which considerable reduces the amount of data to be retrieved from Bloomberg. Simul- taneously, we naturally simplify the degree of complexity of the algorithm to derive implied intensities since we consider less input parameters, making it a more robust and stable estimation technique. In a

ﬁrst step, we thus convert all inception, maturity and coupon payment dates of each single contract (or transaction) into month-end dates. To be more precise, there are two approaches possible. On the one hand, if the original date in a given month is after the 15th of the respective month, one may convert it to a month-end date using the last trading day of the current month; otherwise, the month-end date of the preceding month may be used. On the other hand, there is the possibility of converting each original date to the month-end date of the current month only. Since, in the ﬁrst case, we would be faced with the issue of missing market prices if the bond’s inception is before the 15th of a given month (and a simul- taneously common discrepancy between a bond’s maturity and its last quoted price), we opt for the latter.

The motivation for deriving implied intensity rates is based on O’Kane and Turnbull(2003). The paper addresses the estimation of single hazard rates (and the simultaneous ﬁtting of a term structure) from credit default swap (CDS) spreads using the reduced-form model proposed by Jarrow and Turnbull

(1995). These hazard rates may then be directly translated into corresponding default probabilities of single credit names. The remarkable similarity between hazard rates and intensity rates thus becomes obvious. The computation of single implied hazard rates is based on an iterative process that is generally known as bootstrapping. In detail, the contract with the shortest maturity is used to compute the first hazard rate by applying an one-dimensional root-searching algorithm. The implied hazard rate then is the value for which the model spread of a CDS contract (i.e. the spread that is in line with the valuation formula to price the contract) and the market spread are equal. Subsequently, the technique is repeated to solve for the next hazard rate and so forth until the final maturity CDS contract is reached. Please note that the standard modelling assumption is to presume that the hazard rate is a piecewise flat function of maturity. This is due to the fact that, given merely one price quote, it is not possible to extract more than one piece of information about the term structure of hazard rates (see, e.g., O’Kane and Turnbull,

2003). Figure2 illustrates a hazard rate term structure.

15 0.018

0.016

0.014 Hazard Rate

0.012

0.010

0.00 2.00 4.00 6.00 8.00 10.00 Time to Maturity (in Years)

Figure 2: Hazard rate term structure.

Figure2 illustrates a hazard rate term structure with five sections indicated by λ0,1, λ1,3, λ3,5, λ5,7 and λ7,10, given 1-year, 3-year, 5-year, 7-year and 10-year credit default swap spreads. A piecewise flat hazard rate term structure is assumed. Starting with the contract that has the shortest maturity, the first hazard rate is computed. In detail, the 1-year default swap is used to estimate the value of λ0,1 by applying a technique called bootstrapping. Subsequently, the methodology is repeated to solve for the next hazard rate λ1,3 and so forth until the final maturity CDS contract is reached. Source: O’Kane and Turnbull(2003).

4.2 Model speciﬁcation

Unfortunately, the aforementioned approach of computing implied hazard rates from CDS spreads using bootstrapping cannot be directly carried over to the realm of cat bonds. The main reason for the non- applicability of bootstrapping is that usually not more than two tranches are issued for each cat bond contract. Unlike in the credit world, we thus fail to obtain instruments with different time to maturities for a single contract. To solve this issue, one could assume cat bonds of similar terms and conditions (i.e. reference peril, covered territory, trigger type, pool size of the underlying portfolio of insurance policies, risk parameters) to form a syndicate. Provided that such a syndicate is comprised of bonds of different time to maturities at a given date in time, bootstrapping may be applied. However, the wide variety of terms and conditions leads to a high-dimensional setting with only few observations for each case. Thus, the data scarcity and consequently the reason of practicability led us refrain from proceeding in such a way. We therefore adopt a slightly different approach. At first, in line with implied volatilities in option markets, we define the implied intensity as the fixed annual value λt,t+∆t = λ∆t (∀t ∈ (0,T − ∆t]). In other words, the annual λt,t+∆t corresponds to an intensity of λ∆t per trading day. We then embed the valuation framework into a one-dimensional optimization algorithm which searches for λt,t+∆t by recalculating the model price until it matches the observed monthly market price. To be more precise, a root-fining algorithm is applied to solve for the root (i.e. the implied intensity rate) of an equation

16 f (λt,t+∆t) deﬁned as

f (λt,t+∆t) = CBtν [λt,t+∆t,.] − observed market price = 0, (4)

where CBtν [λt,t+∆t,.] is deﬁned according to Equation (1) and λ ∈ [0.00, 100.00]. In doing so, we extract time series of annualized implied intensities for each cat bond tranche at month-end dates tν (tν ∈ [0,T ]).

4.3 Empirical results

To begin with, we remove all implied intensity values that relate to the bonds’ ﬁnal maturity, i.e. whose associated time to maturity is zero. The reason is simple: these values must be zero according to the valuation framework of Equation (1), and thus lack any economic interpretability. Next, we provide a detailed investigation of the remaining intensity rates in the following.

Figure3(a) illustrates the distribution of cat bond market prices in relation to the corresponding estimated implied intensity rates. The time period under investigation ranges from December 2000 to

May 2017. For comparison purposes, we group the data according to distressed bonds and non distressed bonds. We ﬁnd the majoritarian part of rates to be closely related, clustering around a market price of between USD 95.00 and USD 110.00 and an intensity value being located in an interval of [0.01; 0.30].

However, a handful of rates are beyond this range, with some outliers even located far outside in the upper left corner. Thus, there is clear evidence of lower market prices to be generally associated with higher intensities. In contrast, Figure3(b) displays the distribution of market prices of bonds that are labelled ‘distressed’ in relation to ‘non distressed’ bonds using a density plot. While the shape for the latter category may be considered to be of type Gaussian, the ﬁrst is characterized by a long left tail and thus heavily left-skewed. In addition, the mean market price of distressed bonds is substantially below the mean price of non distressed bonds. To conclude, these ﬁndings clearly support the partition of bond prices into the previously mentioned two categories. In summary, controlling for distressed bonds is inevitable in the subsequent panel data analysis of implied intensity rates to avoid distorting its results

(see Section 5.2 for more information on the dummy variable distressed bonds).

Figure4(a) shows the distribution of cat bonds’ probability of ﬁrst loss values in relation to the corresponding estimated implied intensity rates. Here as well, for comparison purposes, data is grouped

17 0.40 1.00

0.30 0.75

0.50 0.20

Density

Implied Intensity 0.25 0.10

0.00 0.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00 120.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00 120.00 Market Price Market Price

Distressed Bond Non distressed Bond Distressed Bond Non distressed Bond

(a) Scatter plot of cat bond prices in relation to im- (b) Density plot of distribution of distressed cat plied intensity rates, separated according to dis- bonds’ prices in relation to non distressed cat tressed bonds and non distressed bonds. bonds’ prices.

Figure 3: Graphical illustration of the distribution of cat bond prices. Figure3(a) illustrates the distribution of cat bond prices in relation to corresponding estimated implied intensity rates between December 2000 and May 2017. For comparison purposes, data is grouped according to distressed bonds and non distressed bonds. Figure3(b) displays the distribution of market prices of bonds that are labelled ‘distressed’ in relation to ‘non distressed’ bonds using a density plot. according to distressed bonds and non distressed bonds. As already assumed, we detect strong evidence of a positive relationship between probability of first loss and implied intensity, indicated by the red trend line. This finding serves as a first lead for the subsequent panel data analysis in which we include the bonds’ PFL values as the main explanatory variable (see Section5). Additionally, Figure4(b) depicts the distribution of estimated implied intensity rates of bonds that are labelled distressed in relation to

non distressed bonds using a density plot. In line with Figure3(b), the distressed bonds’ density again

is heavily skewed; in the present case, however, to the right. These bonds’ estimated intensities are of

much larger magnitude and more dispersed compared to non distressed bonds.

Figure5 shows the historical development of estimated monthly implied intensity values for U.S. wind

contracts between December 2013 and May 2017, presented as a time series of monthly box plots. In the

following, we focus on the mean monthly rates series, indicated by the bold grey line. We ﬁnd most of the

events being described for the corresponding contract’s market prices in Figure1 to inﬂuence intensities

as well (see Section 3.2 for details). We also observe both a seasonality eﬀect in estimated rates and a

decrease in the level of overall intensities from the year 2013 onwards; the latter indicating a changing

risk assessment with respect to the probability of triggering payments on these instruments. It seems

as if industry experts, and overall markets, assess their occurrence to be less likely in recent times. To

conclude, there is strong evidence of a close co-movement between implied intensity rates and observed

18 1.00 10.00

0.75 7.50

0.50 5.00

Density

Implied Intensity 0.25 2.50

0.00 0.00 0.00 0.05 0.10 0.15 0.20 0.00 0.25 0.50 0.75 1.00 Probability of First Loss (PFL) Implied Intensity

Distressed Bond Non distressed Bond Distressed Bond Non distressed Bond

(a) Scatter plot of PFL values in relation to implied (b) Density plot of distribution of distressed bonds’ intensity rates, separated according to distressed implied intensity rates in relation to non dis- bonds and non distressed bonds. tressed bonds’ intensities.

Figure 4: Graphical illustration of the distribution of probability of ﬁrst loss (PFL), and estimated implied intensity rates. Figure4(a) illustrates the distribution of cat bonds’ probability of ﬁrst loss values in relation to corresponding estimated implied intensity rates between December 2000 and May 2017. For comparison purposes, data is grouped according to distressed bonds and non distressed bonds. A trend line (in red) has been added to facilitate the interpretation. Figure4(b) displays the distribution of estimated implied intensity rates of bonds that are labelled ‘distressed’ in relation to ‘non distressed’ bonds using a density plot. cat bond quotes. To put it another way, the conclusion may be drawn that market prices adjust mainly because of changes in implied intensities. Looking at the time series of intensity rates movements of both

U.S. earthquake and U.S. multiperil bonds depicted in Figures9 and 10 (see appendix), we are able to reach similar conclusions.

5 Determinants of implied intensity rates

5.1 Testable hypotheses and model speciﬁcation

A multitude of various studies have investigated the factors that influence both premiums and prices of cat bonds (see, e.g. Dieckmann, 2010; Braun, 2016; Gürtleret al., 2016). Since we observe a close co-movement between implied intensity rates and cat bond secondary market quotes (see Section 4.3), these studies serve as a starting point. Thus, building on them, but having our attention focused on intensity values as the dependent variable instead, Table4 shows the different testable hypotheses. Each of them refers to one potential determinant of cat bond implied intensity, stating our postulated direction of influence based on the motivation below and adjusted to the situation of intensities.

19 0.40

0.30

0.20 Implied Intensity

0.10

0.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Figure 5: Time series of monthly implied intensity rates for U.S. wind contracts (12/2000– 05/2017). Figure5 shows the historical development of monthly implied intensity rates for U.S. wind contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of these rates within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Distressed bonds are removed. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly implied intensity values. In addition, the red box indicates the time period that is used for computing smooth implied intensity rate surfaces and subsequently investigating the these rates’ pricing ﬁt.

First of all, we consider the probability of ﬁrst loss as the main measure of intensity determinant.

Consequently, it is also considered to be the single most important driver of the intensity rate. We justify this assumption as follows: implied intensity may be interpreted as the time evolution of the static parameter probability of first loss, thus having a more direct link to the intensity rate than, for example, the expected loss (see Section 5.2 for more details). Nevertheless, since both expected loss and probability of first loss relates to a bond’s risk parameters, we find them to be directly comparable. We therefore rely on several earlier studies on the empirical evidence for the importance of expected loss (see, e.g.,

Dieckmann, 2010; Braun, 2016; G¨urtleret al., 2016) and carry these ﬁndings over to the realm of the

PFL value.

Second, we account for the bond-specific characteristics issue size, term, trigger type, and type of protection. We base our presumption concerning the first three of these four variables on the findings of

20 Edwards et al.(2007), Cummins and Weiss(2009), Dieckmann(2010), and Braun(2016).

Third, we turn to the covered territory and the reference peril, which characterize the underlying catastrophe risk of a transaction. Braun(2016) shows that cat bonds covering peak territories, such as the U.S., carry larger spreads than transactions exposing investors to similar natural hazards in the rest of the world, i.e. non peak zones. He ﬁnds a similar eﬀect with regard to peak perils (e.g., windstorms).

Furthermore, there are also combined impacts of territory and peril on spread reported. These ﬁndings are in line with industry experts’ opinions (see, e.g., Aon Benﬁeld, 2011, 2012).

A forth batch of potential explanatory factors relates to the investors’ perception of credit risk inherent in cat bond structures. Previous studies on determinants of cat bond spreads note that experienced investors also assess the speciﬁcs of the sponsor (see, e.g., Spry, 2009; Braun, 2016). Moreover, Braun

(2016) and G¨urtler et al.(2016) ﬁnd a link between the spread and market price, respectively, of a cat bond and its rating.

The subsequent empirical analysis is comprised of diﬀerent linear approaches in which the implied intensity rate is always the dependent variable. The period under investigation ranges from January

2001 to May 2017. Firstly, we identify how much of the variance may be explained by time-ﬁxed bond- dependent or by time-dependent variables. Secondly, we test a range of bond-dependent hypotheses by including the corresponding variables in the model.

To begin with, we apply a least squares dummy variable (LSDV) estimation as follows:

0 0 implied intensityit = αi + β Xi + γ Yt + uit, (5)

where i = 1, ..., I cat bonds and t = 1, ..., T different points in time. The variable Xi denotes bond fixed effects. Precisely speaking, we include dummy variables for every cat bond resulting in an intercept for every bond, αi. Time fixed effects are indicated by the variable Yt. Here, we test monthly, quarterly, as well as yearly time fixed effects, thus generating a dummy variable for every unit of time. Lastly, the error term that varies over bond and time is given by uit. The model coefficients are estimated employing pooled OLS regressions.

Generally speaking, Equation (5) allows us to investigate the maximal proportion of variance that is explainable by time-ﬁxed bond-dependent and, if applicable, time-dependent variables. The results of the least squares dummy variable (LSDV) estimation are presented in Table5. We measure the goodness

21 Hypothesis Label Description

Trigger probability hypothesis [H1]: A positive correlation exists between the intensity implied by a cat bond’s observed market price and its probability of ﬁrst loss.

Liquidity hypothesis [H2(a)]: The intensity implied by a cat bond’s observed market price is uncorrelated with the bond’s issue size.

Term structure hypothesis [H2(b)]: Implied intensities increase with the term of the security.

Protection hypothesis [H2(c)]: There is no diﬀerence in implied intensities between annual aggregate and per-occurrence protection.

Trigger mechanism hypothesis [H2(d)]: There is no diﬀerence in implied intensities between the various types of trigger mechanisms.

Covered territory hypothesis [H3(a)]: Implied intensities for diﬀerent territories vary in their levels.

Reference peril hypothesis [H3(b)]: Implied intensities for diﬀerent perils vary in their levels.

Interaction hypothesis [H3(c)]: Given the most common peril-territory-combinations, interaction eﬀects between territories and perils result in diﬀerent cat bond implied intensity.

Beneﬁciary hypothesis [H4(a)]: There is no relation between a beneﬁciary and implied intensity values extracted from its issued bonds.

Rating hypothesis [H4(b)]: The rating of a cat bond is uncorrelated with its implied intensity.

Table 4: Overview of testable hypotheses for implied intensity rate determinants. Table4 shows the diﬀerent testable hypotheses. Each of them refers to one potential determinant of cat bond implied intensity value, and is termed accordingly. The title is depicted in the left part of the table. The right part provides a short description of the hypotheses, specifying the assumed direction of inﬂuence of the explanatory variable of interest on estimated rates.

of fit of the models using the adjusted R2. In model (I.1), only bond fixed effects are considered. Hence,

2 Yt = 0. We ﬁnd the adjusted R to be of 67.4 percent, implying that a considerable proportion of variation in implied intensity is explained by cat-bond-speciﬁc characteristics. Model (I.2) incorporates

2 both bond fixed effects Xi and month fixed effects. The associated adjusted R is 68.3 percent. Next, we replace month fixed effects by quarter fixed effects, which produces an R2 of 67.7 percent (see model

(I.3)). In the final model, model (I.4), year fixed effects are employed, reporting the highest adjusted

R2 value. Hence, the following conclusions may be drawn. Firstly, bond fixed effects are by far more important than time fixed effects when explaining implied intensity variation. Secondly, time effects are best measured by year fixed effects. Thirdly, the remaining percentage share in intensity variation may be explained by bond-independent time-variant factors.

For the subsequent main analysis, we draw the following three conclusions from Table5: (i) as

22 (I.1) (I.2) (I.3) (I.4)

Bond ﬁxed eﬀects Yes Yes Yes Yes

Month ﬁxed eﬀects No Yes No No

Quarter ﬁxed eﬀects No No Yes No

Year ﬁxed eﬀects No No No Yes

Observations 15,807 15,807 15,807 15,807 Bonds 539 539 539 539 R2 0.685 0.694 0.689 0.710 Adjusted R2 0.674 0.683 0.677 0.698

Table 5: Preliminary analysis: determinants of implied intensity rate. Table5 shows least squares dummy variable (LSDV) estimates of cat-bond-dependent variables and, if applicable, different categories of time fixed effects (monthly, quarterly, and yearly) on implied intensity. The model coefficients are estimated employing pooled OLS regressions. Standard errors are clustered by bonds and robust to heteroskedasticity. Both the R2 and the adjusted R2 are reported. Each of the four presented models allows to investigate the maximal proportion of variance that is explainable by bond-dependent and, if any, time-variant variables. mentioned above, we opt for year fixed effects only since the corresponding adjusted R2 is the highest and the number of variables to be estimated is the smallest; and (ii) we employ random effects models to capture bond-dependent time-invariant characteristics as they explain the largest share in intensity variation. Precisely speaking, we consider the random effects model to be of the following form

0 0 0 0 implied intensityit = α + β Xi + γ Yt + δ Zit + ai + uit, (6) where i = 1, ..., I cat bonds and t = 1, ..., T different points in time. Cat-bond-independent intercept are denoted by αi, Xi are cat-bond-dependent time-invariant variables, Yt is comprised of time-variant variables, Zit measures cat-bond-dependent time-variant variables, ai capture unobservable time-invariant individual effect and uit is the time-variant error term. For each of the fixed effects model under investigation, both the Within-R2 and the adjusted Within-R2 are reported. They refer to the percentage of an explanatory variable’s variation around the cat-bond-dependent means being explainable by the dependent variables of interest.18

For each of the performed random eﬀects models, we employ the Breusch-Pagan Lagrange multiplier

(LM) test to substantiate whether our assumption of a random eﬀects model is valid or whether we should apply a pooled OLS regression instead. The null hypothesis is that the variances of the individual intercept

σa across all entities is zero, thus assuming that there are no signiﬁcant bond-dependent disparities across

18For a more comprehensive description of ﬁxed eﬀects models as well as panel data analyses in general, the reader is referred to Woolridge(2013).

23 bonds (i.e. there is no panel eﬀect). We ﬁnd the hypothesis to be rejected for all subsequent models.19

5.2 Variables

A crucial determinant of estimated implied intensities is the probability of first loss (PFL). Recall that the intensity rate is the combined likelihood of the occurrence of a catastrophe event, per unit time, that causes losses at a sufficiently high level, or that are of sufficient magnitude, such that a payment on the cat bond is triggered. Hence, PFL and intensity rate have the same interpretation, with the difference however that implied intensity values may be considered as the PFL’s evolution over time after issuance of the bond. In contrast, PFL is a static parameter set at bond inception and constitutes a time-invariant risk component. Since this value remains constant following its computation, it does not incorporate an expectation update of the market on the probability of prospective trigger events that may arise while the bond is traded. Nevertheless, the logical consequence is to consider PFL as one of the main factors influencing the intensity rate.

Next, we measure the impact of a bond’s trigger mechanism on the intensity rate by accounting for the precise trigger types using dummy variables. More speciﬁcally, INDEMi is a dummy variable that equals one if transaction i refers to an indemnity-based trigger, INDUSi is set to one for industry loss bonds, MODELi takes on a value of one if the contract has a modelled loss trigger mechanism, and

MULT Ii equals one in case of multiple triggers. If all of these variables are zero, transaction i has a parametric trigger.

We also employ a range of variables that are related to the geographic area in which a catastrophe event has to occur (territory) and to the underlying type of disaster (peril) in order to be relevant under the bond indenture. Firstly, we use covered territory and reference peril as speciﬁed in the bonds’ terms and conditions. We again employ dummy variables, such that USi equals one if transaction i securitizes perils in the United States. The variable EUi takes on a value of one for all perils being securitized in Europe, while JPi and MTi are each set to one in case of a cat bond that reference Japan and Multiterritories, respectively. If none of these variables equals one, then the bond refers to another geographic area (i.e. Mexico, Taiwan, or Turkey). Relating to the diﬀerent peril types, the dummy

19We show the Lagrange multiplier (LM) statistics and their statistical signiﬁcance for each of the estimated random eﬀects model.

24 variable WINDi assumes a value of one, if cyclones, hurricanes, tornadoes, typhoon or windstorms are

referenced and EQi takes on a value of one for earthquake contracts. In case of both variables being

zero, transaction i constitutes a multiperil bond. Secondly, we generate the variables NUMPERILi

and NUMTERRITORYi that measure how many diﬀerent perils are securitized in how many diﬀering regions of transaction i. This allows us to quantify each transaction’s complexity. Thirdly, we also

consider interaction eﬀects for the most common combinations of territory and peril. Precisely speaking,

we use U.S. W ind, U.S. Earthquake, Europe W ind, and Japan Earthquake.

Furthermore, referring to the bond’s rating, AAi is a dummy variable that equals one, if cat bond i has a AA rating, Ai is set to one in case of an A-rated transaction i, BBBi takes on a value of one,

if the contract has a BBB rating, and BBi and Bi each equal one in case of a BB- and B-rated bond, respectively. If all of these variables are set to zero, transaction i represents a non-rated bond.

The variable BENEF ICIARYi relates to the precise beneﬁciary of transaction i, while the dummy

variable PROTECTIONi equals one, if the protection of contract i is of type ‘annual aggregate’, and zero otherwise.

In addition, we also employ the dummy variable DISTRESSEDi that takes on a value of one, if bond price i is classiﬁed as being distressed. Since literature does not provide a precise deﬁnition of when

a cat bond transaction may be labelled as ‘distressed’, we follow G¨urtleret al.(2016) and adopt their

classiﬁcation. Based on a private conversation with a vice president of Lane Financial LLC, they identify

eleven bonds to be distressed prior to the year 2012. When looking at these bonds’ full price history in

detail, we ﬁnd all bonds to report at least one transaction price i below USD 95.00. Hence, the variable

DISTRESSEDi equals one, if a bond price i falls below this threshold. Detecting 24 additional bonds to have prices below USD 95.00 by ourselves from the year 2012, we have a total of 35 bonds that have

at least one distressed bond price.

Finally, to measure liquidity of each cat bond transaction, we employ the following two variables. The

variable TTMi relates to the maturity that remains at the precise time we observe a market price. Hence, it is time-variant. The volume of transaction i is quantiﬁed by the natural logarithm of the issue volume

(in USD million) and denoted by the variable LOG(SIZE)i. The ﬁrst is of particular importance for the following reason. In line with the term structure of hazard rates (see Section 4.1), we are interested

in whether there also is a term structure with respect to implied intensity rates. It may subsequently

be used as a reference point for pricing any type of cat bond. Thus, we investigate the possibility of

25 intensities being depended on time to maturity.

5.3 Empirical results

5.3.1 Main analysis

We begin with model (I.1), whose results are shown in column one of Table6. Of little surprise is the high significance of the coefficient for the probability of first loss, thus confirming hypothesis [H1]. Apart from that, we find evidence for a positive impact of bonds being classified as distressed on our dependent variable, justifying the approach of controlling for them in the analysis. We also include year fixed effects to account for the differences in intensities over time, and find them significant at a 0.1 percent level.20

Having demonstrated that the year fixed effects are important determinants, we use these variables in the subsequent models along with the PFL and other potential influencing factors.

Next, we turn our attention to model (I.2) and model (I.3), respectively. When examining the corresponding results in Table6, hypothesis [ H2 (b)] can be confirmed, implying that the term structure of cat bond implied intensities in the secondary market is non flat but increasing with the length of the transactions’ term. Hypothesis [H2 (a)], in contrast, has to be rejected. We observe a negative liquidity impact of larger cat bond issues on their implied intensity values. Investigating the coefficients’ magnitude, this effect remains negligible in economic terms though. Looking at the coefficient for the trigger type, we make the observation that all of them are insignificant. Thus, we are able to approve hypothesis [H2 (d)].

Model (I.3) additionally includes a variable to test for hypothesis [H2 (c)]. Please note that the number of observation reduces from 15,807 to 10,450 since the type of protection is not publicly available for each transaction (see Section 3.2). We ﬁnd no evidence for an impact of the protection type on intensity rates,

20In detail, we observe that compared to 2001 the intensity rate is significantly lower in the years after 2013, particularly in 2016 and 2017, which could be a result of markets assigning trigger events, i.e. natural catastrophe events that trigger bonds, a lower probability of occurrence in recent times (see Figure5 in Section 4.3 for more information). In a separate model, we also include the interaction effect of PFL and year, taking into account any differences in relative intensities over time. The variables of the interaction effects are significant, and in line with the previously described pattern of decreasing risk assessment. Nevertheless, the adjusted R2 hardly changes compared to the previous model specification without interactions. In addition, all single year coefficients have now become insignificant, while both significance and magnitude of all other variables remain virtually the same. Given these findings, and to reduce the number of variables to be estimated, we opt for year fixed effects only. Please note, in order to preserve space, we do not separately report the coefficients for each year, and each interaction effect between PFL and year, respectively. However, results can be made available upon request from the authors.

26 (I.1) (I.2) (I.3)

coeff. s.e. coeff. s.e. coeff. s.e.

Intercept 0.049 (0.029) 0.060 (0.050) 0.153∗ (0.070) PFL 3.288∗∗∗ (0.608) 3.198∗∗∗ (0.690) 1.143∗∗∗ (0.094) Distressed Bond 0.081∗∗∗ (0.015) 0.080∗∗∗ (0.015) 0.089∗∗∗ (0.021) Trigger Indemnity Trigger 0.013 (0.007) 0.016 (0.009) Industry Loss Trigger 0.011 (0.006) 0.013 (0.008) Modelled Loss Trigger 0.010 (0.007) 0.014 (0.011) Multiple Trigger 0.004 (0.008) 0.013 (0.011) Number of Perils -0.000 (0.001) -0.001 (0.002) Number of Territories -0.001 (0.003) -0.003 (0.003) Peril Earthquake -0.011 (0.007) -0.007 (0.009) Wind 0.007 (0.007) 0.001 (0.009) Territory Europe 0.006 (0.006) -0.011 (0.022) Japan -0.015 (0.009) -0.009 (0.013) Multiterritory 0.029∗∗∗ (0.007) 0.028∗∗ (0.010) United States 0.027∗∗∗ (0.008) 0.022 (0.011) Territory-Region EU × Wind -0.017∗ (0.008) -0.010 (0.021) JP × EQ 0.025∗∗ (0.009) 0.016 (0.012) US × Wind - -0.012 (0.008) -0.009 (0.010) US × EQ -0.010 (0.007) -0.016 (0.010) Rating AA -0.016 (0.018) -0.030 (0.017) A -0.002 (0.019) -0.001 (0.018) BBB 0.003 (0.013) 0.005 (0.018) BB -0.003 (0.008) -0.007 (0.006) B -0.006 (0.007) 0.002 (0.006) Time to Maturity 0.004∗∗ (0.001) 0.003∗∗ (0.001) log(Size) -0.005∗ (0.002) -0.007∗ (0.003) Beneficiary -0.000 (0.000) -0.000 (0.000) Type of Protection 0.004 (0.004) Year fixed effects Yes Yes Yes

Observations 15,807 15,807 10,450 Bonds 539 539 355

σa 0.032 0.029 0.025 LM statistics 22,790.24∗∗∗ 9,879.17∗∗∗ 9,633.42 ∗∗∗

σu 0.038 0.038 0.035 R2 0.573 0.616 0.592 Adjusted R2 0.566 0.599 0.578

Table 6: Main analysis: impact of cat-bond-specific variables on implied intensity rate. Table6 shows random effects estimates of cat-bond-dependent variables on implied intensity. The base variables are ‘Parametric Index’ for trigger, ‘Multiperil’ for peril, ‘Other’ for territory, and ‘NR’ for rating. Standard errors are clustered by bonds and are robust to heteroskedasticity. The symbols *, **, and *** indicate statistical significance at the 5%, 1%, and 0.1% levels, respectively. In addition, the black boxes indicate both the coefficients and standard errors that will be discussed in detail with respect to various robustness checks.

27 and consequently do not reject the hypothesis.

We now address hypotheses [H3 (a)], [H3 (b)], and [H3 (c)]. Examining the respective results in Table6, we notice that for both earthquake and wind bonds, there is no significantly different intensity by contrast with multiperil securities. However, models (II.2) and (III.3) do not permit us to directly compare earthquake and wind risk, owing to the way the peril-related dummy variables are specified. To address this issue, we conduct a separate analysis in which ‘Earthquake’ is used as the base category. The results, which can be make available upon request, reveal that no significant distinction can be make between implied intensities of the three peril types. In addition, we also discover the following. The geographic characteristics ‘Multiterritory’ and ‘U.S.’ report markups on their respective intensity rates in comparison to transactions that refer to other regions of the world. Employing ‘Multiterritory’ as the base category in a comparative analysis, we identify it to be indistinguishable from ‘U.S.’. Thus, we group these two variables to form the category peak territories. Analogously, we do not find any sign of significant intensity differences between ‘Europe’, ‘Japan’, and ‘Other’, presuming that those three specifications may be merged to non peak territories. Eventually, two significant interaction effects between the different peril and territory variables are observed. In particular, we find the effects to be prevalent for EU wind and Japan earthquake bonds only. We will discuss the implications of these results in more detail in the robustness checks below. To sum up, hypotheses [H3 (a)] and [H3 (c)] can be confirmed, but hypothesis

[H3 (b)] needs to be rejected. It appears as if risks which occur more frequently, and (or) with higher severity are indeed associated with higher intensities. Of particular interest is the fact that Braun(2016) who investigates the determinants of cat bond spreads at issuance reports similar ﬁndings pertaining to the territory.

Finally, we consider the results for hypotheses [H4 (a)], and [H4 (b)], i.e., the effects of the beneficial entity, and the bond rating class. As postulated, we find all of the tested variables to be insignificant intensity drivers. To conclude, accounting for other potential influencing factors along with the bond’s probability of first loss does not substantially enhance the model’s explanatory power. The adjusted

R2 increases only moderately when moving from model (I.1) to model (I.2) and (I.3), respectively. As discussed in Section 5.1, the PFL of a single tranche thus is undoubtedly the most important driver of its corresponding implied intensities.

28 5.3.2 Robustness check

A reliable model requires distinct economic relationships for its overall ﬁt to show stability across both diﬀerent samples and time periods drawn from the same population. We thus decide to reestimate model

(I.2) on four subsamples of similar size for different time periods, each of which cover a particular cat bond market environment.21 Here, we follow Braun(2016) in parts. The first subperiod (01/2001 −05/2007) represents the phase in which the cat bond asset class has been launched, covering the Sumatra-Andaman earthquake and Hurricane Katrina, the second one (06/2007−12/2010) involves the recent financial crisis, the third one (01/2011−12/2012) refers to Hurricane Sandy, thus being a single event period, and the fourth one (01/2013−05/2017) represents the current benign loss environment for natural catastrophes.

Beyond that, three peril-related subsamples are analysed, being solely comprised of earthquake, multiperil, or wind transactions. Lastly, we also investigate two territory-specific subsamples, and consider the partition into peak and non peak territory. The results for these robustness checks are illustrated in Tables8,9 and 10 (see appendix). With regard to the adjusted R2, we find the model fit to remain strong for each of the nine subsamples. Yet, we must admit that some regression coefficients vary, both in magnitude and direction. In addition, we observe certain factors to be priced now while others are not any longer. To further confirm our chosen model’s validity we will briefly explore these puzzles below.

We start with the subsamples for different time periods. Here, the following observations are especially worth mentioning. First, consider the ‘rating’ coefficients for the first subperiod (01/2001−05/2007) as shown in the second column of Table8. The coefficients for each single rating class turn out to be highly significant, whereas these effects are no longer observable in subsequent periods. This suggests that credit-related issues instead of catastrophe risk were the dominant concern in the market during the takeoff phase of the cat bond asset class. However, the market’s focus soon shifted from credit risk to

(re-)insurance prices inherent in cat bonds from mid 2007 on. Second, we ﬁnd a markup associated with indemnity, industry loss, and modelled loss trigger mechanisms on implied intensity rates in comparison to securities referring to a parametric index. It thus follows that these three variables may form the group of monetary loss triggers.22 In contrast, bonds based on a parametric index trigger constitute the

21We opt for model (I.2) simply because the additional variable, controlling for the type of protection, that is included in model (I.3) and being the only diﬀerence between the two models proves to be insigniﬁcant.

22Unreported results with diﬀerent base categories support this proposal.

29 correspondent group which we term parametric loss triggers.23 Third, while we do not ﬁnd evidence for an impact of the covered territory on implied intensities in the period from June 2006 to December

2012, there is a reversed eﬀect for subperiod four when compared with the full model, i.e. model (I.2).

Precisely speaking, the regions ‘Europe’ and ‘Japan’ are now associated with a discount relative to transactions that cover other territories. Nevertheless, the main implication remains the same: we still opt for a categorization into peak and non peak territories. Forth, we observe significant interaction effects between the peril and territory variables in model (II.4), i.e. the effects are prevalent for Japan earthquake, US wind, and US earthquake bonds. Yet, we decide to neglect them in future applications

(see Section6) for the following two reasons. On the one hand, we already account for any impacts the region may have on intensities by considering peak and non peak territory groups. On the other hand, the eﬀect of both US wind and US earthquake is of weak signiﬁcance only, while the number of

Japan earthquake bonds is very small as of January 2013. In the latter case, less than ﬁve percent of all outstanding bonds in subperiod four refer to this particular peril-territory-combination.

The most striking observations with respect to the results for the peril-specific subsamples in Table9, and the region-specific subsamples in Table 10 can be shortly described as follows. To begin with, we are unable to document any significant impact of the covered territory on intensities of multiperil bonds.

This is hardly surprising, however, since there are merely two bonds that securitize perils in Europe, while

Japan-based securities are nonexistent at all. In the first case, the observed variation is simply too low to yield significant estimates. In addition, we find the trigger effect observed in model (II.4) to most likely result from multiperil bonds (see column four of Table9 in comparison to columns two and six of the same table). Lastly, we notice that, unlike wind bonds, earthquake bonds show significant lower implied intensity rates in comparison to multiperil transactions for the subgroup of peak territories. In contrast, in case of non peak territories, securities insuring wind and earthquake risk alike exhibit higher intensities compared to multiperil bonds. Unreported results using ‘Earthquake’ as the base category additionally confirm the existence of peak (wind and multiple disasters) and non peak (earthquake) perils, however, for peak regions only. Braun(2016) also proposes such a breakdown of reference perils when investigating the determinants of cat bond spreads in the primary market. It should be noted though, that he does not

23Multiple trigger securities report intensity values that are indistinguishably from those of parametric-index-based bonds. Therefore, one may also assign them to the group of parametric loss triggers. In the present case, however, we do not explicitly take them into account since they constitute a minuscule share of all outstanding bonds in subperiod four (less than 2.00 percent, or ﬁve bonds). The observed variation may thus simply be too low to yield signiﬁcant estimates.

30 separate data according to regions, and thus merely justiﬁes the two peril groups in case of the full data set.

To conclude, the final results may be summarized as follows. We find the bond’s probability of first loss to have the most prevailing influence on implied intensities. Apart from that, there is evidence for an impact of time to maturity. These effects remain strong both across different samples and time periods. In addition, another important explanatory variable is the territory. Precisely speaking, our results show that we may cluster intensities according to whether their associated bonds are either of type peak territory or of non peak territory. The remaining factors, i.e. trigger, peril, rating, size, and beneficiary, are of subordinated importance only. Either, the associated coefficients are insignificant or the reported effects are relatively small and may thus be ignored. In summary, these findings support our approach of constructing implied intensity rate surfaces that depend on both PFL and time to maturity.

Here, it may be suﬃcient to group the data in peak and non peak territories.

6 Computation of implied intensity rate surfaces

6.1 Surface construction

To begin with, we construct monthly raw implied intensity rate surfaces for the time period from January

2013 to May 2017. The reason for our choice of time frame is twofold. First, we consider this period to have overcome the cat bonds’ takeoff phase. Pricing anomalies due to investors’ inexperience regarding this asset class, also affecting estimated implied intensities, should thus no longer be observed on a large scale. Second, trading volume has increased substantially, particularly during the last several years, leading to a reduction in the bonds’ relatively high illiquidity. As a result, bond price data availability, and thus intensities as well, is just sufficient to allow us to construct entire surfaces. In doing so, we remove distressed bonds, for the simple reason that our aim is not to propose a pricing approach in times of single bonds’ upheaval. Our previous analyses have shown that markets put a markup on such intensity rates. Based on the results from Section 5.3, we then separate the intensity data according to different territories. Precisely speaking, we opt for the two groups peak territory and non peak territory. For the subsequent investigation of their surfaces’ pricing fit, we decide to consider a control group termed

‘complete’ that does not diﬀerentiate between regions. Thus, we generate three diﬀerent types of surfaces.

Here, again in line with our previous ﬁndings, probability of ﬁrst loss is depicted on the x-axis, and time

31 to maturity on the y-axis.

Since there are not enough quoted cat bond prices and thus implied intensity rates, we are faced with

monthly surfaces that are highly fragmentary. However, both the construction of entire surfaces and the

subsequent proper application of a smoothing estimator requires more data; there should be virtually

continuous surfaces at best. To solve this issue, we linearly interpolate between single rates. It is stressed

that, due to data constraints, we are faced with the sole possibility of applying the weak assumption of

a linear relationship. Precisely speaking, for each surface, we ﬁrst interpolate along the range of PFL

values for a particular time to maturity, provided that there are at least two rates available for this

maturity. We repeat this procedure for all of the diﬀerent available time to maturities. Then, we apply

the interpolation to the sequence of time to maturity for a ﬁxed PFL value, and move along the various

PFL values. As a result, we obtain an extended grid of intensity values which we again convert into

corresponding surfaces. To prevent loss of data, we eventually reinsert all single rates that were ignored

during interpolation.

Next, we build a monthly time series of smooth implied intensity rate surfaces for all three types of

pricing surfaces depicted in Table7, each deﬁned on a ﬁxed grid of PFL and time to maturity values. We

denote these series by {IIt(P F L, T T M)}t=1,...,T = 53. In detail, we follow Cont and da Fonseca(2002) and employ a non-parametric Nadaraya-Watson estimator with a Gaussian kernel, i.e.

I J P P λt,ij(PFLt,i,TTMt,j)g(PFL − PFLt,i,TTM − TTMt,j) i=1 j=1 II (P F L, T T M) = , (7) t I J P P g(PFL − PFLt,i,TTM − TTMt,j) i=1 j=1 2 2 −1 x y g(x, y) = (2π) exp − − , x, y ∈ R, (8) 2ht,1 2ht,2

where PFL ∈ [0.0004, 0.2138],TTM ∈ [0.08, 5.08], and with bandwidth parameters ht,1 and ht,2, respectively, that determine the degree of smoothing.24 It further holds that t = 1, ..., T = 53, i = 1, ..., I

and j = 1, ..., J; the latter two being the indices of observations of implied intensity rates at month t.

Please note that PFL and TTM may vary for each single surface. In addition, both PFL and TTM

are consistent with the respective surfaces’ original axis values, i.e. time to maturity is kept at monthly

24For a more comprehensive review of the Nadaraya-Watson estimator, the reader is referred to H¨ardle(1990) as well as A¨ıt-Sahaliaand Lo(1998).

32 intervals and PFL equals the range of available parameter values. Strictly speaking, we thus simply

smooth the surfaces, and only interpolate between single time to maturities if there is a disruption in

monthly time steps due to missing quotes.

Within the ﬁeld of non-parametric modelling, choosing the proper smoothing parameters is essential.

If these values are too small, the resulting surface is undesirably bumpy, whereas too large ones have

the disadvantage of smoothing away crucial details. In literature, there exists a variety of methods

that can be applied to determine the optimal bandwidth parameters h1 and h2, e.g. the application of a cross-validation criterion, penalization and bootstrapping techniques, the employment of an adaptive

bandwidth estimator, or plug-in methods. For a detailed review on the many ways of bandwidth selection,

the reader is referred to H¨ardle(1990), Gasser et al.(1991), as well as Brockmann et al.(1993). We decide

to apply the leave-one-out cross validation technique (see, e.g., Green and Silverman(1994)). Here, the

cross validation term is deﬁned as

I J 2 1 X X (−k) CV (h , h ) = λ − λˆ (h , h ) , (9) t,1 t,2 I × J t,ij t,ij t,1 t,2 i=1 j=1

ˆ(−k) for t = 1, ..., T = 53, i = 1, ..., I and j = 1, ..., J. The leave-one-out estimator is given by λt,ij (ht,1, ht,2). In essence, the technique follows a re-substitution procedure where the kth observation is left out. To

put it diﬀerently, Equation (9) thus is the sum of squared residuals resulting from the application of the

Nadaraya-Watson estimator when the kth observation is unknown. The optimal smoothing parameters

ht,1,opt and ht,2,opt, respectively, are then chosen to minimize the term CV (ht,1, ht,2).

In line with the previously deﬁned three types of surfaces, Figure6 (as well as Figures 11 and 12 in the appendix) each illustrate two selected monthly smooth implied intensity rate surfaces. It shall be noted that one could have also chosen a smaller grid of time to maturities, leading to a more precise, i.e. smoother, evaluation of these surfaces. However, we decide to opt for monthly steps. The reasons for our choice are threefold. Firstly, the data set is comprised of monthly cat bond prices such that the computation of implied intensity rates follows these time steps as well. For the subsequent pricing ﬁt investigation, we thus have a perfect match of time to maturity and available cat bond price quotes. At this stage, there is no need to price bonds on a weekly or even daily basis, simply because the market is not as liquid as it is for other asset classes that generally have a higher trading frequency. Secondly,

33 having a rougher grid allows us to speed up computation. This particularly is a major advantage when dealing with the cross-validation technique, where one needs to solve a considerably large number of separate smoothing problems. Thirdly, any other selection of maturities would neither change the pricing

ﬁt nor the implications drawn from investigating the plotted surfaces.

Overall, we find the term structure of cat bond implied intensities to be non flat but to generally increase with the length of the transactions’ term. This effect is particularly apparent for the case of non peak territory surfaces. The positive impact of PFL on intensity rates is clearly observable as well, though less pronounced for smaller values of the bonds’ respective probability of first loss. In addition, peak territory and all territory pricing surfaces show striking similarities with respect to their shapes, and thus strongly distinguish themselves from non peak territory surfaces. It may therefore be concluded that there are at least two different risk factors, leading to a heterogeneous fluctuation of these three surface types over time.

0.30 0.30

Implied Intensity 0.25 Implied Intensity 0.25

0.20 0.20

0.15 0.15

0.10 0.10

0.05 0.05

0.00 0.00 0.10 0.10 4.5 0.04 4.0 0.05 4.0 0.03 3.5 3.5 3.0 0.04 3.0 2.5 0.02 0.03 2.5 2.0 PFL PFL 2.0 1.5 0.02 1.5 1.0 1.0 0.01 0.5 0.01 0.5 0.00 0.0 Time to Maturity 0.00 0.0 Time to Maturity

(a) Trading date: November 30th, 2015. (b) Trading date: May 31st, 2017.

Figure 6: Peak territory: selected end-of-month smooth implied intensity rate surfaces. Figure 11 shows two selected end-of-month smooth implied intensity rate surfaces for the case in which peak territory only is considered. Probability of ﬁrst loss is depicted on the x-axis, and time to maturity on the y-axis. A non- parametric Nadarayan-Watson estimator with a Gaussian kernel is employed to smooth the surfaces. The application of cross-validation determines the optimal bandwidth parameters.

6.2 Investigation of the pricing ﬁt

The goodness of the implied intensity rate surfaces’ pricing ﬁt is gauged with respect to an error function.

In the following we draw upon a minimization of the mean absolute error (MAE). However, we will also report the mean error (ME), the mean absolute percentage error (MAPE), and the root mean square

34 error (RMSE) associated with the generated cat bond prices (deﬁnition see appendix).

Table7 shows the pricing ﬁt. Note that rows generally indicate the type of surface used for pricing, while the particular columns refer to the respective aforementioned error functions, starting from column three. In addition, the time-series-cross-section, i.e. the monthly cross-section, minimum and maximum of the MAE for each of the three pricing surfaces is reported in the ﬁrst two columns of the table.

Pricing Surface Min. Max. ME MAE RMSE MAPE Peak Territory 0.731 1.377 -0.062 0.984 1.384 0.010 Non Peak Territory 0.979 2.789 -1.749 1.827 2.211 0.018 Complete 0.741 1.517 -0.068 1.181 1.606 0.012

Table 7: Fit statistics for the entire time-series-cross-section sample of cat bond prices. Table7 shows the goodness of the peak territory, the non peak territory and the complete pricing surfaces, respectively. Results are based on a minimization of four error functions. These are the mean error (ME), the mean absolute error (MAE), the root mean squared error (RMSE) and the mean absolute percentage error (MAPE), respectively. Based on these measures, loss statistics for the entire time-series-cross-section sample of cat bond prices are calculated. In addition, the time-series-cross-section, i.e. the monthly cross-section, minimum and maximum of the MAE for each of the three pricing surfaces is reported in the ﬁrst two columns of the table.

Generally speaking, we obtain very good ﬁt statistics, with the worst performing month reporting an average maximum MAE of 1.377 for peak territory bonds and of 2.789 for non peak territory bonds, respectively. Thus, the maximum monthly cross-section absolute pricing error in dollar terms is USD

1.377 (USD 2.789). With respect to the average minimum MAE for the full time period, we ﬁnd it to be within USD 0.984 and USD 1.827.

A closer look into the pricing performance of single transactions implies the following. The deviation between observed cat bond price and model price is the largest for those bonds whose corresponding raw implied intensity rate can be seen as an outlier (in terms of the level) in the respective unsmoothed intensity surface for a particular month. Due to its mode of operation, the non-parametric smoothing methodology along with the cross-validation technique thus leads to an over-smoothing of this single outlier rate and eventually to this bond’s overpricing. This is especially the case if only a comparatively small number of outliers in intensities for a given surface are found. Please bear in mind the inverse relationship between implied intensities and associated cat bond prices. The motivation for some bonds reporting low prices is twofold. Firstly, they may actually be traded at a discount to all other bonds for some reason.25 However, the following second explanation probably is far more reasonable in clarifying mispriced bonds. We assume large pricing errors to be due to trading inactivity. Unfortunately, infor-

25It is only mentioned in passing that such a situation may indicate an arbitrage opportunity, in particular if all bonds involved are of similar terms and conditions and are thus directly comparable.

35 mation on the trading activity is not publicly available. But a private conversation with an ILS portfolio manager of a large German reinsurance company revealed that, depending on the particular month, non-traded bonds which are generally associated with indicative prices set by brokers may account for a considerable share in a given month. This would explain a range of observed high monthly cross-section pricing errors since such non-traded quotes distort the smoothing of implied intensity rate surfaces to a certain extent. It is particularly problematic if brokers’ indicative prices are set too low. Compared to traded bonds with similar terms and conditions in a given month, the corresponding estimated implied intensity rate of non-traded bonds are then too high. The smoothing algorithm leads to a considerable penalization of this rate, such that the resulting smoothed rate is far below the raw rate, and the bond becomes highly overpriced. In addition, we were told that a handful of bonds are not traded for several months in a row which is consistent with our observation that we are unable to market-consistently price them during these months. Again, we do not have information on whether the obtained quotes refer to prices being actually traded or if they are merely indicative ones, making it impossible for us to exclude them prior to surface computation.

Another important observation with regard to the different types of pricing surfaces in Table7 is that we find non peak territory pricing surfaces to have a worse pricing fit across all error functions compared to both peak territory surfaces and the complete case. This is hardly surprising in view of the aforementioned issue of some bonds’ trading inactivity and the resulting mispricing of these transactions.

The assumption that non peak territory bonds have a lower trading frequency, simply because there is an evidentially increased demand for peak territory bonds, would support these ﬁndings.

To conclude, we would like to shortly illustrate two possibles approaches for improving the pricing

ﬁt. First of all, we currently consider a partition of the estimated implied intensity rates into peak and non peak territory only. This mainly is due to data scarcity since any further separation would refrain us from constructing entire surfaces. However, there are other determinants of intensity rates as well, most notably the bond’s reference peril and the trigger type (see Section 5.3). If the market for cat bonds should become more liquid in the future, one could think of intensity surfaces that depend on covered territory, type of peril and the particular trigger mechanism. A more liquid market also has the advantage of refraining us from applying a linear interpolation ﬁrst to smooth the surfaces in a subsequent step.

Secondly, as stated above, excluding non traded bonds from the surfaces’ construction would certainly enhance the pricing ﬁt as well. Dedicated brokers might have knowledge about each bonds’ trading

36 frequency that can be exploited for such purposes. Nevertheless, one should always bear in mind to develop a framework that has a superior pricing ﬁt with just a few parameters to consider for a broad range of diﬀerent instruments. We already achieve this very well at the present stage.

7 Conclusion

This paper contributes to the literature through a comprehensive analysis of possible practical applications of an existing reduced-form pricing model that is known form the structured credit literature but tailored to cat bonds. One of the essential input factors to the valuation formula is the combined likelihood of the occurrence of a catastrophic event, per unit time, that causes losses at a suﬃciently high level, or that are of suﬃcient magnitude, such that a payment on the cat bond is triggered; hereinafter referred to as intensity rate. It is the only quantity that is not directly observable. Prior research suggests computing it using historical event data on natural catastrophes.

Based on a large data set, encompassing more than one and a half decades of monthly cat bond secondary market prices from December 2000 to May 2017, we propose to estimate the rate by directly extracting it from these price quotes and call this quantity implied intensity rate. In line with the common approach that is used to build hazard rate term structures from observed CDS spreads, though slightly modified to make it applicable to cat bonds, we use an one-dimensional root-finding optimization algorithm, and obtain monthly time series of implied intensity rates for 539 bonds that exhibit both seasonality effects and changes in overall level over time.

Next, we analyse which both time-variant and time-invariant bond-specific characteristics that are given in the instruments’ terms and conditions influence cat bond implied intensities. For this purpose, a series of random effect models with year fixed effects and standard errors that are robust to heteroskedasticity and clustered by bonds is run. Our findings indicate that the probability of first loss, time to maturity, and the covered territory are major drivers of implied intensity values. For the latter, it is sufficient to distinguish between peak and non peak territory. In contrast to that, issue volume, trigger type, rating as well as reference peril are much less influential and the complexity of the transaction, beneficiary, and the type of protection do not seem to be priced at all. In addition, we exhibit a stable

ﬁt across diﬀerent calibration subsamples.

37 Common to all these models is that implied intensity rates depend on both time to maturity and probability of ﬁrst loss, separable according to peak and non peak territory. This allows us to construct implied intensity rate surfaces for the two region types, where the x- and y-axis refer to the ﬁrst two above mentioned variables, while the intensity rate itself is depicted on the z-axis. To subsequently obtain smooth surfaces, a non-parametric Nadaraya-Watson estimator with a Gaussian kernel is employed, where the optimal bandwidth parameters are determined by means of leave-one-out cross validation.

These surfaces may then be used as a reference point for pricing any type of cat bond. Lastly, we thus investigate their pricing ﬁt and obtain excellent in-sample ﬁt statistics.

An important implication of our constructed smooth implied intensity rate surfaces is that they do not only allow us to market-consistently price a broad range of various secondary-market-traded cat bonds but also any other instruments that refer to catastrophe risk. In the latter case, this is particularly relevant if these are less liquid or have no market at all. Thus, our ﬁndings should be relevant to investors and sponsors alike, helping to promote a better understanding of the cat bond asset class that certainly further enhance the growth of this relatively small segment of the capital markets.

38 8 Appendix

8.1 Main price components

(1) Present value of the coupon stream at time tν

K ! X −l (t −t ) Q tν ,tk k ν 1 ctν = Etν e Ltk−1 + scat ∆tkN τ>tk k=1 K X = DF (t , t ) L + s ∆t N · SP (t , t ) ν k tk−1 cat k ν k (10) k=1 | {z } | {z } coupon at tk based on probability reference rate from of survival last reset date from tv to tk

Intuition: Discount back from coupon date tk to valuation date tν by multiplying with DF (tν , tk).

Sum over all K future coupon dates tk, ..., tK . For each coupon date tk the reference rate Ltk−1 is

ﬁxed at time tk−1. Use forward rates for the unknown future reference rate ﬁxings Lt1 , ..., Ltk−1 .

(2) Present value of the accrued interest at time tν

Q −ltν ,τ ,(τ−tν ) 1 atν = Etν e Ltk−1 + scat (τ − tk−1)N tk−1<τ≤tk Z tk = DF (tν , τ) Lt0 + scat (τ − t0)N · SP (tν , τ) λ(τ)dτ tν | {z } ∗ n X Z tk (11) + DF (tν , τ) Ltk + scat (τ − tk−1)N · SP (tν , τ) λ(τ)dτ k=2 tk−1 | {z } | {z } accrued interest from probability of survival last coupon date to time τ to τ and default in the next instance

Intuition: Discount back from default time τ to valuation date tν by multiplying with DF (tν , τ). Integrate over all times in the coupon period since trigger event τ may occur at any time. Sum over

running coupon periods [tν , t1] and all k − 1 remaining full periods.

∗ The trigger probability from time t to time s, conditional on having survived to time t, is deﬁned as

TP (t, s) = P r (Ns − Nt > 0) = 1 − SP (t, s) ,

39 (3) Present value of the principal repayment at time tν

Q −ltν ,T ,(T −tν ) 1 Q −ltν ,τ ,(τ−tν ) 1 Ptν = Etν e N τ>T + Etν e αN τ≤T | {z } | {z } no trigger case trigger case Z T Q = DF (tν ,T ) N · SP (tν ,T ) + DF (tν , τ) (α) N · SP (tν , τ) λ(τ)dτ Etν (12) | {z } tν | {z } | {z } probability expected probability of survival of survival residual to τ and default in the from tν to T notional next instance

Intuition: For the no trigger case, discount back from ﬁnal maturity date T to valuation date tν by

multiplying with DF (tν ,T ). For the trigger case, discount back from default time τ to valuation date

tν by multiplying with DF (tν , τ). Integrate from tν to maturity T since trigger event τ may occur at any time.

In order to simplify the evaluation of (11), we approximate the integrals as follows

1 at = DF (tν , t1) Lt + scat ∆t1N 1 − SP (tν , t1) ν 2 0 K X 1 + DF (tν , tk) Lt + scat ∆tkN (1 − SP (tν , tk)) − (1 − SP (tν , tk−1)) . (13) 2 k−1 k=2

Assuming the event can only occur at M discrete times per year, (12) can be approximated according to the following form

Ptν = DF (tν ,T ) N · SP (tν ,T ) M·T X Q + DF (tν , tm) Etν (α)N (1 − SP (tν , tm)) − (1 − SP (tν , tm−1)) . (14) m=1

whereas the trigger probability for the inﬁnitesimal instant between time t and time t + dt, conditional on having survived to time t, is of the form Z t+dt Q TP (t, s) ≈ Et (Nt+dt − Nt) = λ(u)du ≈ λ(t)dt. t

40 8.2 Time series of monthly cat bond prices (U.S. earthquake contracts)

110.00

105.00

100.00

95.00

90.00 Market Price Market

85.00

80.00

75.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Calabash Re II 1.D Golden State Re Newton Re 2007 1.A Redwood Capital VI Globecat CAQ 1.A Golden State Re II 2014 1 Redwood Capital V Ursa Re 2014 1.A

Figure 7: Time series of monthly cat bond prices for U.S. earthquake contracts (12/2000–05/2017). Figure7 shows the historical development of monthly cat bond prices for U.S. earthquake contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of bond prices within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Outliers are depicted using a range of different symbols which are labelled by the respective bonds’ names. These names are listed in the table below the graph. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly bond prices. In addition, the red line indicates the cut-off price level at which bonds may be qualified as distressed.

41 8.3 Time series of monthly cat bond prices (U.S. multiperil contracts)

110.00

105.00

100.00

% n & n 95.00 $ nn %% % % % % & 90.00 % % & && Market Price Market % & & & 85.00 & &

80.00

75.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Blue Halo Re 2016 1.B Laetere Re 2016 1.B Mystic Re III 2012 1.B Residential Re 2011 CL.2 Residential Re 2016 II.2 East Lane Re IV 2011 1.A Laetere Re 2016 1.C Residential Re 2005 A Residential Re 2011 II CL.1 Residential Re 2016 II.4

East Lane Re IV 2011 1.B Merna Re A Residential Re 2005 B n Residential Re 2011 II CL.2 $ Successor X IV AL.3

Foundation Re D Merna Re B Residential Re 2010 II 3 Residential Re 2012 1.7 % Successor X Ltd. 1.U1

Laetere Re 2016 1.A Montana Re Ltd. B Residential Re 2010 4 Residential Re 2016 I.11 & Successor X Ltd. 1.X1

Figure 8: Time series of monthly cat bond prices for U.S. multiperil contracts (12/2000–05/2017). Figure8 shows the historical development of monthly cat bond prices for U.S. mutliperil contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of bond prices within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Outliers are depicted using a range of different symbols which are labelled by the respective bonds’ names. These names are listed in the table below the graph. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly bond prices. In addition, the red line indicates the cut-off price level at which bonds may be qualified as distressed.

42 8.4 Time series of monthly cat bond implied intensity rates (U.S. earthquake contracts)

0.40

0.30

0.20 Implied Intensity

0.10

0.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Figure 9: Time series of monthly implied intensity rates for U.S. earthquake contracts (12/2000– 05/2017). Figure9 shows the historical development of monthly implied intensity rates for U.S. earthquake contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of these rates within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Distressed bonds are removed. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly implied intensity values. In addition, the red box indicates the time period that is used for computing smooth implied intensity rate surfaces and subsequently investigating the these rates’ pricing ﬁt.

43 8.5 Time series of monthly cat bond implied intensity rates (U.S. multiperil contracts)

0.40

0.30

0.20 Implied Intensity

0.10

0.00

Dec 2003 Dec 2004 Dec 2005 Dec 2006 Dec 2007 Dec 2008 Dec 2009 Dec 2010 Dec 2011 Dec 2012 Dec 2013 Dec 2014 Dec 2015 Dec 2016 Year (in Months)

Figure 10: Time series of monthly implied intensity rates for U.S. multiperil contracts (12/2000– 05/2017). Figure 10 shows the historical development of monthly implied intensity rates for U.S. multiperil contracts between December 2013 and May 2017, presented as a time series of monthly box plots to illustrate the distribution of these rates within each month and across the full time period. The upper and lower whiskers indicate the 2.5 and 97.5 percent quantile, respectively. Distressed bonds are removed. The time period (in months) is depicted on the x-axis and the price (in USD) on the y-axis. Please note that the bold grey line refers to the time series of mean monthly implied intensity values. In addition, the red box indicates the time period that is used for computing smooth implied intensity rate surfaces and subsequently investigating the these rates’ pricing ﬁt.

44 8.6 Robustness check for implied intensity rate determinants: time periods

01/2001-05/2007 06/2007-12/2010 01/2011-12/2012 01/2013-05/2017

(II.1) (II.2) (II.3) (II.4)

coeff. s.e. coeff. s.e. coeff. s.e. coeff. s.e.

Intercept 0.144 (0.078) 0.117∗∗∗ (0.029) 0.137∗∗ (0.044) 0.031 (0.016) PFL 2.517∗∗∗ (0.676) 1.087∗∗∗ (0.208) 1.092∗∗∗ (0.283) 1.235∗∗∗ (0.284) Distressed Bond 0.097∗ (0.043) 0.052∗∗∗ (0.007) 0.270∗∗ (0.090) 0.300∗∗∗ (0.007) Trigger Indemnity Trigger 0.010 (0.011) 0.015 (0.012) -0.006 (0.019) 0.022∗∗ (0.007) Industry Loss Trigger -0.005 (0.010) 0.011 (0.010) -0.012 (0.015) 0.024∗∗∗ (0.007) Modelled Loss Trigger -0.009 (0.009) 0.004 (0.011) -0.025 (0.017) 0.038∗∗∗ (0.008) Multiple Trigger -0.002 (0.021) 0.010 (0.014) -0.024 (0.018) 0.016 (0.008) Number of Perils 0.010 (0.018) -0.003∗∗ (0.001) -0.004 (0.002) -0.001 (0.001) Number of Territories -0.028∗∗ (0.010) 0.006 (0.004) 0.022 (0.011) 0.003 (0.002) Peril Earthquake -0.041 (0.026) -0.016∗ (0.008) -0.045 (0.027) -0.005 (0.005) Wind -0.018 (0.014) -0.003 (0.009) -0.023 (0.016) 0.019∗∗ (0.007) Territory Europe 0.053∗ (0.021) -0.005 (0.009) -0.019 (0.017) -0.027∗∗ (0.009) Japan 0.040 (0.026) -0.014 (0.012) 0.031 (0.023) -0.057∗∗∗ (0.013) Multiterritory 0.122∗∗∗ (0.025) 0.003 (0.013) 0.024 (0.028) -0.005 (0.007) United States 0.056∗∗ (0.017) 0.028∗ (0.012) 0.046 (0.030) 0.005 (0.008) Territory-Peril EU × Wind 0.005 (0.023) -0.006 (0.013) 0.003 (0.022) -0.021 (0.011) JP × EQ 0.040 (0.024) 0.017 (0.016) 0.035 (0.029) 0.046∗∗∗ (0.010) US × Wind 0.018 (0.023) -0.007 (0.011) 0.003 (0.017) -0.022∗ (0.009) US × EQ 0.040∗ (0.020) -0.022∗ (0.011) -0.003 (0.027) -0.015∗ (0.006) Rating AA -0.146∗∗ (-0.051) -0.017 (0.026) A -0.137∗∗ (0.049) -0.002 (0.029) -0.031 (0.031) BBB -0.119∗∗ (0.046) 0.003 (0.026) -0.030 (0.026) -0.022 (0.019) BB -0.102∗ (0.042) -0.003 (0.019) -0.038∗ (0.015) -0.001 (0.018) B -0.084∗ (0.034) 0.002 (0.015) -0.016 (0.012) 0.006 (0.004) Time to Maturity 0.006∗∗∗ (0.001) 0.007∗∗∗ (0.001) 0.011∗∗∗ (0.003) 0.006∗∗∗ (0.001) log(Size) -0.003 (0.003) -0.007∗ (0.003) -0.009 (0.005) -0.001 (0.002) Beneficiary -0.000 (0.000) -0.000∗∗ (0.000) -0.000∗ (0.000) -0.000 (0.000) Year fixed effects Yes Yes Yes Yes

Observations 3,586 3,790 2,320 6,111 Bonds 172 197 159 283

σa 0.031 0.023 0.035 0.019 LM statistics 1,112.91∗∗∗ 1,401.22∗∗∗ 1,297.41∗∗∗ 2,801.38∗∗∗

σu 0.034 0.047 0.040 0.027 R2 0.719 0.580 0.567 0.654 Adjusted R2 0.700 0.499 0.521 0.588

Table 8: Robustness check for cat-bond-dependent variables: time periods. Table8 shows random effects estimates of cat-bond-dependent variables on implied intensity. The base variables are ‘Parametric Index’ for trigger, ‘Multiperil’ for peril, ‘Other’ for territory, and ‘NR’ for rating. Standard errors are clustered by bonds and are robust to heteroskedasticity. The symbols *, **, and *** indicate statistical significance at the 5%, 1%, and 0.1% levels, respectively. In addition, the black boxes indicate both the coefficients and standard errors that will be discussed in detail with respect to various robustness checks.

45 8.7 Robustness check for implied intensity rate determinants: reference peril types

Earthquake Multiperil Wind

(III.1) (III.2) (III.3)

coeff. s.e. coeff. s.e. coeff. s.e.

Intercept 0.032 (0.023) 0.116 (0.066) 0.100∗∗ (0.038) PFL 1.631∗∗∗ (0.181) 1.312∗∗∗ (0.126) 1.764∗∗∗ (0.238) Distressed Bond 0.024∗∗∗ (0.004) 0.094∗∗∗ (0.025) 0.064∗∗∗ (0.015) Trigger Indemnity Trigger -0.002 (0.005) 0.037∗∗ (0.012) -0.015 (0.015) Industry Loss Trigger -0.003 (0.006) 0.031∗∗ (0.011) -0.002 (0.011) Modelled Loss Trigger 0.002 (0.006) 0.023 (0.014) -0.004 (0.018) Multiple Trigger 0.035∗∗ (0.011) -0.012 (0.019) Number of Perils -0.002 (0.195) Number of Territories 0.002 (0.016) 0.004 (0.003) -0.023 (0.020) Territory Europe 0.004 (0.006) -0.022 (0.022) 0.004 (0.009) Japan 0.017∗∗ (0.006) 0.018 (0.012) Multiterritory 0.022∗ (0.011) 0.013 (0.010) 0.064∗∗ (0.022) United States 0.023∗∗∗ (0.006) 0.011 (0.012) 0.040∗∗∗ (0.012) Rating AA -0.046∗∗ (0.015) A -0.039∗ (0.019) BBB -0.009 (0.005) -0.033 (0.024) BB 0.006 (0.005) -0.010 (0.010) 0.000 (0.013) B 0.004 (0.010) 0.001 (0.010) -0.006 (0.012) Time to Maturity 0.003∗∗ (0.001) 0.004∗∗ (0.002) 0.003∗∗ (0.001) log(Size) -0.001 (0.001) -0.005 (0.004) -0.005 (0.003) Beneficiary 0.000 (0.000) -0.000 (0.000) -0.000 (0.000) Year fixed effects Yes Yes Yes

Observations 2,985 7,271 5,551 Bonds 105 256 178

σa 0.011 0.031 0.033 LM statistics 512.12∗∗∗ 5,112.85∗∗∗ 3,035.36∗∗∗

σu 0.026 0.044 0.036 R2 0.496 0.580 0.572 Adjusted R2 0.488 0.438 0.539

Table 9: Robustness check for cat-bond-dependent variables: reference peril types. Table9 shows random effects estimates of cat-bond-dependent variables on implied intensity. The base variables are ‘Parametric Index’ for trigger, ‘Multiperil’ for peril, ‘Other’ for territory, and ‘NR’ for rating. Standard errors are clustered by bonds and are robust to heteroskedasticity. The symbols *, **, and *** indicate statistical significance at the 5%, 1%, and 0.1% levels, respectively. In addition, the black boxes indicate both the coefficients and standard errors that will be discussed in detail with respect to various robustness checks.

46 8.8 Robustness check for implied intensity rate determinants: covered territory

Peak Territory Non peak Territory

(IV.1) (IV.2)

coeﬀ. s.e. coeﬀ. s.e.

Intercept 0.129∗∗ (0.048) 0.079 (0.030) PFL 1.368∗∗∗ (0.117) 1.747∗∗∗ (0.416) Distressed Bond 0.087∗∗∗ (0.018) 0.036∗∗∗ (0.006) Trigger Indemnity Trigger 0.017∗ (0.008) 0.013∗ (0.006) Industry Loss Trigger 0.013 (0.007) 0.002 (0.008) Modelled Loss Trigger 0.012 (0.010) 0.008 (0.007) Multiple Trigger 0.015 (0.009) Number of Perils -0.001 (0.001) 0.024 (0.013) Number of Territories 0.003 (0.002) 0.014 (0.010) Peril Earthquake -0.024∗∗∗ (0.006) 0.026∗ (0.010) Wind -0.002 (0.005) 0.030∗∗ (0.009) Rating AA -0.033∗ (0.013) A -0.029 (0.019) BBB -0.023 (0.015) -0.005 (0.013) BB -0.007 (0.009) -0.001 (0.006) B 0.002 (0.007) -0.012 (0.007) Time to Maturity 0.004∗∗ (0.001) 0.003∗∗ (0.001) log(Size) -0.007∗∗ (0.002) -0.000 (0.002) Beneficiary -0.000 (0.000) 0.000 (0.000) Year fixed effects Yes Yes

Observations 12,780 3,027 Bonds 452 87

σa 0.032 0.014 LM statistics 9,464.09∗∗∗ 8,051.53∗∗∗

σu 0.041 0.024 R2 0.581 0.448 Adjusted R2 0.537 0.416

Table 10: Robustness check for cat-bond-dependent variables: covered territory. Table 10 shows random effects estimates of cat-bond-dependent variables on implied intensity. The base variables are ‘Parametric Index’ for trigger, ‘Multiperil’ for peril, ‘Other’ for territory, and ‘NR’ for rating. Standard errors are clustered by bonds and are robust to heteroskedasticity. The symbols *, **, and *** indicate statistical significance at the 5%, 1%, and 0.1% levels, respectively. In addition, the black boxes indicate both the coefficients and standard errors that will be discussed in detail with respect to various robustness checks.

47 8.9 Smooth implied intensity rate surfaces (all territories)

0.30 0.30

Implied Intensity 0.25 Implied Intensity 0.25

0.20 0.20

0.15 0.15

0.10 0.10

0.05 0.05

0.00 0.00 0.08 0.10 0.05 3.5 0.05 4.5 3.0 0.04 4.0 0.03 2.5 3.5 0.03 3.0 2.0 2.5 0.02 PFL 1.5 PFL 0.02 2.0 1.0 1.5 1.0 0.01 0.01 0.5 0.5 0.00 0.0 Time to Maturity 0.00 0.0 Time to Maturity

(a) Trading date: January 31st, 2013. (b) Trading date: May 31st, 2017.

Figure 11: All territories: selected end-of-month smooth implied intensity rate surfaces. Figure 11 shows two selected end-of-month smooth implied intensity rate surfaces for the case in which there is no diﬀerentiation between territories. Probability of ﬁrst loss is depicted on the x-axis, and time to maturity on the y-axis. A non-parametric Nadarayan-Watson estimator with a Gaussian kernel is employed to smooth the surfaces. The application of cross-validation determines the optimal bandwidth parameters.

48 8.10 Smooth implied intensity rate surfaces (non peak territory)

0.05 0.05

Implied Intensity Implied Intensity 0.04 0.04

0.03 0.03

0.02 0.02

0.01 0.01

0.00 0.00 0.040 0.020 2.0 0.016 3.5 0.025 3.0 1.5 0.010 2.5 2.0 1.0 PFL 0.020 PFL 0.008 1.5 0.006 1.0 0.015 0.5 0.5 0.000 0.0 Time to Maturity 0.000 0.0 Time to Maturity

(a) Trading date: March, 31th 2013. (b) Trading date: May 31st, 2017.

Figure 12: Non peak territory: selected end-of-month smooth implied intensity rate surfaces. Figure 11 shows two selected end-of-month smooth implied intensity rate surfaces for the case in which non peak territory only is considered. Probability of ﬁrst loss is depicted on the x-axis, and time to maturity on the y-axis. A non-parametric Nadarayan-Watson estimator with a Gaussian kernel is employed to smooth the surfaces. The application of cross-validation determines the optimal bandwidth parameters.

49 8.11 Error functions

The four measures are deﬁned as follows (see, e.g., Armstrong and Collopy, 1992; Xu and Taylor, 1995):

• Mean error (ME): N 0 T 0 1 X X ME = pCAT − pˆCAT, N 0T 0 i,t i,t i=1 t=1

• Mean absolute error (MAE):

N 0 T 0 1 X X CAT CAT MAE = p − pˆ , N 0T 0 i,t i,t i=1 t=1

• Mean absolute percentage error (MAPE):

N 0 T 0 CAT CAT 1 X X pi,t − pˆi,t MAPE = N 0T 0 pCAT i=1 t=1 i,t

• Root mean square error (RMSE):

v u N 0 T 0 u 1 X X 2 RMSE = t pCAT − pˆCAT , N 0T 0 i,t i,t i=1 t=1

where N 0 denotes the number of transactions in each month, T 0 equals the number of months in the time

CAT CAT series, pi is the observed bond price of transaction i, andp ˆi represents the bond price generated by the pricing model. Based on these measures, we calculate loss statistics for the entire time-series-cross-

section sample of quoted prices.

50 References

A¨ıt-Sahalia, Y. and Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in

ﬁnancial asset prices. The Journal of Finance, 53(2):499–547.

Allianz (2019). Allianz Risk Barometer - Top Business Risks for 2019. Report.

Aon Benﬁeld (2011). Insurance-linked securities: Consistency and conﬁdence 2011. Research report.

Aon Benﬁeld (2012). Insurance-linked securities: Evolving strength 2012. Research report.

Aon Benﬁeld (2019). Reinsurance Market Outlook January 2019. Report.

Armstrong, S. J. and Collopy, F. (1992). Error measures for generalizing about forecasting methods:

Empirical comparisons. International Journal of Forecasting, 08(1):69–80.

Ben Ammar, S., Braun, A., and Eling, M. (2016). Asset Pricing and Extreme Event Risk: Common

Factors in ILS Fund Returns. Working Paper.

Bloomberg (2016). Building the Bloomberg interest rate curve: Deﬁnitions and methodology. Quantita-

tive Research and Development FX Team.

Braun, A. (2016). Pricing in the primary market for cat bonds: New empirical evidence. Journal of Risk

and Insurance, 83(4):811–847.

Brockmann, M., Gasser, T., and Hermmann, E. (1993). Locally adaptive bandwidth choice for kernel

regression estimators. Journal of the American Statistical Association, 88(424):1302–1309.

Christensen, B. J. and Prabhala, N. R. (1998). The relation between implied and realized volatility. The

Journal of Financial Economics, 50(2):125–150.

Cont, R. and da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quantitative Finance,

2(1):45–60.

Cummins, J. D. et al. (2006). Should the government provide insurance for catastrophes? Federal Reserve

Bank of St. Louis Review, 88(4):337–379.

Cummins, J. D. and Weiss, M. A. (2009). Convergence of insurance and ﬁnancial markets: hybrid and

securitized risk-transfer solutions. Journal of Risk and Insurance, 3(76):493–545.

51 Dieckmann, S. (2010). By force of nature: Explaining the yield spread on catastrophe bonds. Working

Paper.

Edwards, A. K., Harris, L. E., and Piwowar, M. S. (2007). Corporate bond market transaction costs and

transparency. The Journal of Finance, 62(3):1421–1451.

Flemming, J., Ostdiek, B., and Whaley, R. E. (1995). Predicting stock market volatility: A new measure.

The Journal of Futures Markets, 15(3):265–302.

Gasser, T., Kneip, A., and K¨ohler,W. (1991). A ﬂexible and fast method for automatic smoothing.

Journal of the American Statistical Association, 86(415):643–652.

Green, P. J. and Silverman, B. W. (1994). Nonparametric regression and generalized linear models. In

Monographs on Statistics and Applied Probability, volume 58. Chapman & Hall, London.

G¨urtler,M., Hibbeln, M., and Winkelvos, C. (2016). The impact of the ﬁnancial crisis and natural

catastrophes on cat bonds. The Journal of Risk and Insurance, 83(3):579–612.

H¨ardle,W. (1990). Applied nonparametric regression. Cambridge, England: Cambridge University Press.

Jarrow, R. A. (2010). A simple robust model for cat bond valuation. Finance Research Letters, 2(7):72–79.

Jarrow, R. A. and Turnbull, S. (1995). Pricing derivatives on ﬁnancial securities subject to credit risk.

The Journal of Finance, 50(1):53–85.

Jorion, P. (1995). Predicting volatility in the foreign exchange market. The Journal of Finance, 50(2):507–

528.

NOAA (2011). Hurricanes and tropical storms for annual 2010. Technical report, National Oceanic and

Athmospheric Administration.

O’Kane, D. and Turnbull, S. (2003). Valuation of credit default swaps. Lehman Brothers: Fixed Income

Quantitative Credit Research.

Ron, U. (2000). A practical guide to swap curve construction. Working Paper.

Spry, J. (2009). Non-life insurance securitization: Market overview, background and evolution. In Barrieu,

P. and Albertini, L., editors, The Handbook of Insurance-Linked Securities, chapter 2, pages 9–18. John

Wiley & Sons, Inc., West Sussex, 1 edition.

52 Swiss Re (2018). Natural Catastrophes and Man-Made Disasters in 2017: A Year of Record-Breaking

Losses. Sigma 1/2018.

U.S. Securities and Exchange Commission (2017). Money market fund statistics. Division of Investment

Management.

Woolridge, J. (2013). Introductory econometrics: A modern approach. Mason, OH: South-Western

Publications, 5th edition.

Xu, X. and Taylor, S. J. (1995). Conditional volatility and the information eﬃciency of the PHLX

currency options market. Journal of Banking and Finance, 19(5):803–821.