Journal ArticleDOI

# Accuracy of Premium Calculation Models for CAT Bonds - An Empirical Analysis

TL;DR: In this article, the authors apply different premium calculation models in order to compare them with regard to their predictive power, without taking the financial crisis into account, a version of the Wang transformation model and the linear model are the most accurate ones.
Abstract: CAT bonds are of significant importance in the field of alternative risk transfer. Since the market of CAT bonds is not complete, the application of an appropriate pricing model is of high relevance. We apply different premium calculation models in order to compare them with regard to their predictive power. Without taking the financial crisis into account, a version of the Wang transformation model and the linear model are the most accurate ones. In contrast, under consideration of the financial crisis, all analyzed models are approximately equivalent. Furthermore, we find that CAT bond specific information does not improve out-of-sample results.

### Introduction

• Since both the trend of insured losses and the trend of numbers of catastrophes are positive, (re-)insurance companies have to consider new ways of coping with the risk.
• Therefore, the authors compare different selected premium calculation models and include pricing determining factors.
• In order to guarantee insurance coverage for the sponsor, the SPV in turn issues CAT bonds to an investor5 who pays the par amount h at issue date.
• 3 a basis and models Λ by applying a Cobb-Douglas production function on the probability of first loss PFL and the conditional expected loss CEL.
• The empirical analysis is established on the basis of CAT bond premiums within the period from 1999 to 2006.

• Premium calculation models determine the premiums ρ, which need to be paid by the sponsor in order to receive protection against predefined losses.
• In order to introduce some relevant variables and to understand the models under consideration, the authors start with a short introduction to insurance pricing.
• The determination of the expected loss of an insurance product corresponds to the calculation of expected loss of the associated layer.

### Empirical Methodology

• In the literature, the premium calculation models as described above have only been analyzed in isolation.
• Hence, the authors will analyze three types of models.
• Analogous to the linear models, the authors implement the loglinear1 model (ρLogL1M) and the loglinear2 model (ρLogL2M).
• Afterwards, the authors apply the parameterized models on the CAT bond contracts that started between June 2006 and June 2008 (out-of-sample period 1)22 in order to evaluate the deviations between the model-predicted and the real CAT bond premiums ρL1M , ρL2M , ρLogL1M , ρLogL2M , ρW1T , and ρW2T .
• In both studies the prediction accuracy of the respective (in-sample) model is evaluated by comparing the coefficients of determination in the out-of-sample analysis on the basis of the premiums.

### Predicting the CAT Bond Premium – the Test Environment

• The authors present the regression equations that are applied in the subsequent study.
• Since the linear models and the loglinear model use different additional premium determining factors, the authors standardize the factor 24See (Campbell and Thompson, 2008).
• The precise descriptions of all factors are presented in the next section.
• In contrast to the linear and the loglinear model, the two versions of the Wang transformation establish a relationship between ρ(X) and the above described transformation EL+ of the expected loss.

### Description of the Data

• The empirical analysis uses original26 data sets provided by Lane Financial LLC and Standard & Poor’s (S&P), where 176 CAT bond transactions between the years 1999 and mid 2009 are specified.
• The data include, in particular, values of the above mentioned PFL, PLL, CEL, and EL.
• Furthermore, CAT bond specific information is available for all CAT bonds regarding maturity, rating, trigger mechanism, peril, and issue date.
• The authors only take into account CAT bonds rated by S&P27 since only for these CAT bonds they received information on the peril and the applied trigger mechanism.
• The CAT bond specific information is described in the following.

### Trigger Mechanisms

• Basically, there are five different trigger mechanisms.
• The indemnity trigger uses the height of actual losses of the sponsor, the parametric trigger uses a physical measure like the Richter scale, the index trigger uses a specified index, the modeled loss trigger uses catastrophe modeling software, and the hybrid trigger uses combinations of different triggers in order to define the payout in case of catastrophe.
• (Lane and Mahul, 2008) as well as (Dieckmann, 2008) establish different analyses using both original and secondary market data.
• (Cummins and Weiss, 2009) and (Dubinsky and Laster, 2003) suppose that prices for CAT bonds with an indemnity trigger might be higher compared to CAT bond prices with different trigger mechanisms due to basis risk.

### Rating of the bond

• The purpose of a CAT bond rating is to provide independent and professional 29(Cummins and Weiss, 2009) find that the CAT bond volume for 1997-2007 was distributed as follows: 25.9% for parametric triggered bonds, 30% for indemnity triggered bonds, 21.5% for industry index triggered bonds, 8.5% for modeled triggered bonds, and 14% for hybrid triggered bonds.
• Therefore, rating agencies evaluate the catastrophe risk analysis as established by specialized firms (e.g. AIR, RMS, EQE).
• (Anders, 2005) objects that rating agencies only have little knowledge in the field of catastrophe risk assessment.
• Against this background, the authors want to examine whether the CAT bond rating has an impact on the premiums.
• Since the data set only contains CAT bonds with rating classes A, BBB, BB, and B, the included dummy variables are yBB = 1, if the bond is rated BB,0, else, (25) for BB rated CAT bonds and yA,BBB for the aggregated class of A or BBB rated CAT bonds.

### Perils

• Most of the perils are hurricane and earthquake perils in the US.
• (Cummins and Weiss, 2009) find that 61.4% of the CAT bonds issued between 1997 and 2007 covered perils in the US.30 Apart from that, European windstorms and earthquakes and typhoons in Japan are securitized quite often.
• The authors data show that 25% of the CAT bonds issued between 1999 and 2006 (and rated by S&P) covered earthquake perils, 23% covered hurricane perils, 13% covered the combination hurricane/earthquake, 25% covered combinations of any perils, 12% covered European windstorms, and 2% covered industrial 30Thereby, 29.6% covered US earthquakes and about 31.8% covered hurricanes.
• The authors include dummy variables yH = 1, if hurricane is the peril,0, else, (26) for hurricane perils, yH,EQ for hurricane and earthquake perils, yEurwind for European windstorms, yComb for combinations of any perils, and yInd.
• The base variable is yEQ for earthquake perils.

### Maturity

• There are different maturities of CAT bonds in the markets.
• In the literature, on the one hand it is stated that the CAT bond market is less affected by insurance cyclic effects than the reinsurance market.
• Instead, price changes from one year to another have been used.
• The authors include the cyclic index in their regression analysis .

### Capital Markets

• The purpose is to verify the statement from the literature that developments on capital markets are independent of developments on CAT bond markets.
• For the assumption of independence, the authors refer to (Litzenberger et al., 1996) and 32For the analysis, (Lamm-Tennant and Weiss, 1997) used the average loss ratio.

### Empirical Findings

• The following empirical study is based on two different situations.
• On the one hand, the authors consider a reduced data set which does not comprise data after June 2008 since the smooth functioning of markets cannot be taken for granted during the financial crisis.
• On the other hand, the authors consider the complete data set including the financial crisis in order to test the predictive power of the models in an arbitrary situation.
• Within the framework of the linear2 and the loglinear2 model the authors apply the stepwise linear regression method (by using the PASW Statistics 18 procedure) to obtain the factors that should be included in the model.33.

### Reduced Data Set – Stable Market Environment

• The models are evaluated on the basis of the coefficients of determination referring to the out-of sample analyses.
• The premium influencing factors in the linear2 and loglinear2 models can be explained economically and are discussed briefly.
• Thus, the authors can assume that multi-peril CAT bonds are imposed by the market with an additional risk load compared to earthquake perils.
• In the literature it is stated that a parametric trigger significantly reduces moral hazard for investors compared to indemnity triggers.

### Complete Data Set – Consideration of the Financial Crisis

• Analogous to the preceding section, the authors proceed by separating the whole data set into an in-sample and an out-of-sample data set.
• The out-of-sample data set is extended compared to the first analysis and covers the period from June 2006 to March 2009 (approximately 1/3 of the data).
• This issue and the influence of the model factors on the CAT bond premiums are presented in table 2.
• The results referring to the premium influencing factors in the linear2 and the loglinear2 model are also slightly different from the corresponding results of the first study.
• Advantages for the sponsor when 28 29 using industry index triggers are that the transaction is simple to execute and that the sponsor does not need to provide confidential information.

### Conclusion

• Due to an incomplete market for catastrophe risks and the lack of transparency on the CAT bond market, it is difficult to determine an accurate pricing model for CAT bonds.
• The models suggest a relationship between the CAT bond premium ρ and the expected loss EL.
• The models are compared on the basis of an out-of-sample analysis which has been carried out as follows.
• Moreover, the authors have found that the Wang2 transformation model always leads to better in-sample and out-of-sample results than the Wang1 trans- 32 formation.
• This results from the fact that the Student’s t-distribution is able to fit the data better than the normal distribution.

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Galeotti, Marcello; Gürtler, Marc; Winkelvos, Christine
Working Paper
Accuracy of premium calculation models for CAT
bonds: An empirical analysis
Working Paper Series, No. IF29V4
Provided in Cooperation with:
Technische Universität Braunschweig, Institute of Finance
Suggested Citation: Galeotti, Marcello; Gürtler, Marc; Winkelvos, Christine (2009) : Accuracy of
premium calculation models for CAT bonds: An empirical analysis, Working Paper Series, No.
IF29V4, Technische Universität Braunschweig, Institut für Finanzwirtschaft, Braunschweig
This Version is available at:
http://hdl.handle.net/10419/55239
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Working Paper Series
Accuracy of Premium Calculation Models for CAT Bonds
an Empirical Analysis
by Marcello Galeotti, Marc Gürtler, and Christine Winkelvos
No.: IF29V4/09
First Draft: 2009-03-23
This Version: 2011-10-13
University of Braunschweig Institute of Technology
Department of Finance
Abt-Jerusalem-Str. 7
D-38106 Braunschweig

Accuracy of Premium Calculation Models for CAT Bonds
an Empirical Analysis
+
by Marcello Galeotti*, Marc Gürtler**, and Christine Winkelvos***
Abstract. CAT bonds are of significant importance in the field of alternative risk transfer. Since the
market of CAT bonds is not complete, the application of an appropriate pricing model is of high rele-
vance. We apply different premium calculation models in order to compare them with regard to their
predictive power. Without taking the financial crisis into account, a version of the Wang transformation
model and the linear model are the most accurate ones. In contrast, under consideration of the finan-
cial crisis, all analyzed models are approximately equivalent. Furthermore, we find that CAT bond
specific information does not improve out-of-sample results.
Keywords: CAT Bonds, Alternative Risk Transfer, Premium Calculation Models, Empirical Analysis
JEL classification: G13, G22
+
Earlier versions of the paper are known under the title “Accuracy (and Determining Factors) of Pric-
ing Models for CAT bonds - an Empirical Analysis” and “Determining Pricing Factors of CAT bonds”.
We would like to thank Andre Liebenberg, Tristan Nguyen, and Barbara Klimaszewski-Blettner for their
discussions. We would also like to thank J. David Cummins, Richard D. Phillips, Kim B. Staking, Pierre
Picard, and the participants of the Annual Meeting 2009 of the American Risk and Insurance Associa-
tion, the Annual Congress of the German Insurance Association 2009 and 2010, and the Annual Meet-
ing 2010 of the Western Risk and Insurance Association for interesting and helpful comments.
*
Professor Dr. Marcello Galeotti
University of Florence
Department of Mathematics
for Decisions
Via Lombroso 6/17
50134 Firenze
Italy
Fon: +39-554796821
Fax: +39-554796800
E-Mail: marcello.galeotti@dmd.unifi.it
**
Professor Dr. Marc Gürtler
Braunschweig Institute of Technology
Department of Finance
Abt-Jerusalem-Str. 7
D-38106 Braunschweig
Germany
Fon: +49 531 3912895
Fax: +49 531 3912899
E-mail: marc.guertler@tu-bs.de
***
Dipl.-Math. Oec. Christine Winkelvos
Braunschweig Institute of Technology
Department of Finance
Abt-Jerusalem-Str. 7
D-38106 Braunschweig
Germany
Fon: +49 531 3912893
Fax: +49 531 3912899
E-mail: c.winkelvos@tu-bs.de

Introduction
Since both the trend of insured losses and the trend of numbers of catas-
trophes are positive, (re-)insurance companies have to consider new ways of
coping with the risk.
1
One possibility is to transfer the risk from reinsurance
markets to ﬁnancial markets. Important ﬁnancial instruments which are used
for the transfer are (CAT-)astrophe bonds.
2
The volume of CAT bond princi-
pal outstanding rose to USD 13.8 billion in 2007.
3
After a collapsing market
has been observed in 2008, the market regained strength in 2009. The main
idea of catastrophe securitization by a CAT bond transaction is that a spon-
sor usually a (re-)insurer enters into an alternative reinsurance contract
with a Special Purpose Vehicle (SPV). Thus, the sponsor is protected against
high losses due to a speciﬁed catastrophe up to a certain limit. In order to
guarantee insurance coverage up to the limit, the SPV issues CAT bonds to
investors. Investors buy the bonds to diversify their portfolios and to receive
high yields resulting from the covered peril.
4
A challenging question for the
trading of CAT bonds is how CAT bond transactions can be priced best.
The objective of this paper is the identiﬁcation of the most accurate pricing
model. Therefore, we compare diﬀerent selected premium calculation models
and include pricing determining factors.
In order to describe these models, ﬁgure 1 presents the basic structure of
1
See (Munich Re, 2010).
2
Generally, bonds whose payment structure is linked to insurance risk are called
Insurance-Linked Securities (ILS). If catastrophe risk is securitized, one refers to CAT
bonds.
3
See (Carpenter, 2008, p. 13).
4
For a more precise description of the functionality of CAT bonds see, for instance,
(Carpenter, 2006).
2

a CAT bond transaction.
SPV
Investor
EL

cr
no triggering event
triggering event
1c ( r )( d )
no triggering event
triggering event
payment
payment = 0
no triggering event
triggering event
h
h
h
Figure 1: CAT Bond Transaction
ρ to the SPV to receive insurance coverage up to the limit h. The premium
ρ consists of the expected value of loss EL plus a load for risk margin and
expenses Λ. In order to guarantee insurance coverage for the sponsor, the
SPV in turn issues CAT bonds to an investor
5
who pays the par amount h
at issue date. If no triggering event occurs, the investor receives at maturity
the par amount h and a coupon c consisting of the risk-free interest rate r
and the premium ρ. In case of a triggering event, the coupon to the investor
is reduced by d, 0 d 1. Furthermore, the par amount at maturity h
might be reduced by f, 0 f 1. However, the sponsor receives insurance
coverage according to the reinsurance contract between the sponsor and the
SPV up to the limit h.
Obviously, the key parameter of a CAT bond transaction and thus of the
CAT bond price is the premium ρ. The premiums are usually determined on
the basis of premium calculation models which use the relationship between
ρ and EL. For instance, (Lane, 2000) takes the relationship of ﬁgure 1 as
5
Typically, there are more than one investors. For simplicity, we assume only one
investor within the basic transaction scheme.
3

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