Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaréand … · 2017. 9. 8. · Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaréand Michel Gendron Hedging Flood

Featured Article

Alexandre Têtu, Van Son Lai, Issouf Soumaré* and Michel Gendron

Hedging Flood Losses Using Cat Bonds

DOI 10.1515/apjri-2014-0024

Abstract: In this paper, we develop a methodology to model the risk of lossesresulting from a natural disaster in which the intensity parameter of the non-homogeneous Poisson process has an upward trend and a seasonal component.We apply this model to losses due to floods in the Financial Assistance Programof the Government of Quebec (Canada). We use the historically observed riskpremiums to assess the financial costs for the government if it had issued suchinstruments to hedge risk linked to floods.

Keywords: cat bond, catastrophe bond, catastrophe risk, floods, insurance, riskmanagement, risk transfer, securitization

1 Introduction

With average annual temperatures rising and frequencies of natural disastersincreasing, dire environmental impacts, vital sociopolitical and economicproblems commonly linked to global warming are growingly alarming. The2007 report of the Intergovernmental Panel on Climate Change (IPCC) describesthe effects of global warming quite plausibly due to ongoing climate change andthe IPCC expects these effects and related costs to grow with time (IPCC 2007).These issues are of great importance for insurance companies and governmentswhose finances are greatly affected by the occurrence of natural disasters.1

*Corresponding author: Issouf Soumaré, Department of Finance, Insurance and Real Estate,Faculty of Business Administration, Laval University, Quebec City, QC, Canada, E-mail: [email protected] Têtu, Financial Markets Placements, Investments, General Funds, Industrial AllianceInsurance and Financial Services Inc., Quebec City, QC, Canada, E-mail: [email protected] Son Lai: E-mail: [email protected], Michel Gendron: E-mail:[email protected], Department of Finance, Insurance and Real Estate, Faculty ofBusiness Administration, Laval University, Quebec City, QC, Canada

1 For example, the Insurance Bureau of Canada (IBC, 2008) reported that the amount of claimsdue to natural disasters exceeded 900 million Canadian dollars in 2005.

APJRI 2015; 9(2): 149–184

Brought to you by | Bibliotheque de l'Universite LavalAuthenticated

Download Date | 9/4/17 2:42 PM

Traditionally, insurance companies and governments have turned to rein-surance to hedge their exposure to the risk of natural disasters. The last twodecades have seen an increase in the number of disasters and the value of theproperties insured. Reinsurance companies have responded by increasing pre-miums required to hedge against the risk of a natural disaster and also bydecreasing the level of coverage they are willing to offer. Further, in the 1990s,following the huge losses caused by the earthquake in California and byHurricane Andrew on the east coast of North America, insurance companiescame up with the innovation of transferring part of their risk to capital marketsby issuing the first catastrophe bonds (hereafter referred to as cat bonds).

Floods are natural disasters with the highest frequency of occurrence (IBC2008). For instance, European flood damage resulting from the June 2013 floodacross Central Europe – in particular in Austria, the Czech Republic, Germanyand Hungary – caused $22 billion in economic losses; insurance payouts totaled$5 billion according to an Aon Benfield catastrophe report. In Canada, inSeptember 2013, the Insurance Bureau of Canada (IBC) reports that PCS-Canada estimated insured property damage caused by the June 2013 southernAlberta floods to have exceeded $1.7 billion. The September 2013 Coloradofloods are expected to have caused economic damages in excess of $2 billion.2

In Quebec, Canada, as elsewhere, floods are severe natural disasters; sincehomeowners cannot purchase insurance against damages caused by floods,the Government of Quebec created a Financial Assistance Program to helphomeowners after floods.3 Thus, when big disasters occur, this provincial pro-gram, supported in part by the Federal Government, can be very costly. A goodexample is the Saguenay-Lac-Saint-Jean4 flood in 1996. To eliminate theseunexpected expenditures, the government wants flood risk to become insurablefor homeowners. However, as stated by Sandink et al. (2010), insurance compa-nies may refuse to insure properties located in areas that are too risky, oralternatively charge these homeowners very high premiums, making the finan-cial assistance program indispensable. The government would not allow suchadverse selection from the private insurance companies, as it wants all citizens,including those living in the most vulnerable areas of the province as well as the

2 Flooding could cost US$1 trillion a year by 2050, http://www.rtcc.org/2013/08/19/flooding-could-cost-us1-trillion-a-year-by-2050/.3 In the United States, the National Flood Insurance Program (NFIP) enables property ownersin participating communities to purchase insurance protection from the government againstlosses from flooding.4 Saguenay-Lac-Saint-Jean is a northern region of the Province of Quebec in Canada (see mapin Appendix A).

150 A. Têtu et al.



most vulnerable citizens, to be insured against flood losses. Thus, the FinancialAssistance Program is the only flood insurance in effect right now. The programreimburses renovation costs and essential good replacement costs for victims offloods and is entirely funded by the government.

One way the government could control the expenses related to its financialaid program, without altering it, would be to transfer some of the risk to capitalmarkets. This can be done by issuing a catastrophe bond, better known as a “catbond,” on floods. In this paper we will study how the government could havereduced its costs by issuing cat bonds to cover floods. Allianz, in 2007, closedthe first cat bond ever to cover floods to transfer the risk of severe river floods inGreat Britain. Recently, Canada’s banking regulator was urging insurers to sellcatastrophe bonds for the first time to cut the risk linked to natural disasters.Correspondingly, US Congress has called a group of insurance-linked securities(ILS) experts to look into whether US flood risk could be transferred to capitalmarkets via cat bonds. To get the most from this risk sharing between thegovernment, investors and insurance companies, it is important to perform anassessment of the underlying risks relating to floods.

The goal of this paper is to examine the hedging of losses due to floods inQuebec by issuing cat bonds by addressing the following four research ques-tions: (1) How to model the occurrence of floods in Quebec? (2) How to value catbonds issued to transfer flood risk under the Financial Assistance Program of theGovernment of Quebec? (3) What is the impact of the Federal Government aid onthe value of the cat bonds? (4) What are the risk-adjusted returns of suchinstruments for investors for given risk premiums?

First, we propose a cat bond valuation model, which includes an upwardtrend and a seasonal effect in the intensity parameter of the non-homogenousPoisson process (NHPP) used to model the occurrence of disasters. This valua-tion model has an analytical solution when the losses distribution allows for aclosed-form solution. Second, using 1992–2011 historical data of flood lossesunder the Financial Assistance Program of the Government of Quebec, ourcalibration exercises confirm the presence of an upward trend in the numberof flood events and of a seasonal cyclical effect. Finally, we use historicallyobserved risk premiums to assess the financial costs for the government if it hadissued such instruments to hedge risk linked to floods.

The rest of the paper is structured as follows. Section 2 provides the defini-tion and the characteristics of a typical catastrophe bond. Section 3 gives anoverview of the literature review related to our work. Section 4 presents the dataused for our analyses. Section 5 develops the proposed valuation model for catbonds. Section 6 discusses the empirical results obtained with Monte Carlosimulations. We conclude in Section 7.

Hedging Flood Losses Using Cat Bonds 151



2 Overview on Catastrophe Bonds

Cat bonds are issued by insurance companies, reinsurers or by governments whowish to transfer some of their exposure to the risk of occurrence of a naturaldisaster, also known as securitization of disaster risks, to investors. At issuance ofcat bonds, the notional amount paid by investors to acquire the bonds is placedin a trust fund called an SPV (special purpose vehicle), which is an entitycompletely separated from the issuer. This protects investors against the risk ofdefault of the issuer. If no natural disaster covered by the bonds occurs during thelife of the instrument, investors receive periodic coupons, and at maturity, theyreceive payment of the principal. If a natural disaster occurs (thereby triggeringpayout of the cat bond), investors lose part or all of their investment.5 Thisstructure is employed in this paper for an analysis of a cat bond, albeit fictive,but potentially may be used to study the costs of covering flood losses in Quebec.The flowchart in Figure 1 illustrates a typical cat bond securitization.

There are many reasons for an investor to want to hold these bonds. Oneimportant reason is diversification. Indeed, cat bonds have little or zero correla-tion with the market (Litzenberger, Beaglehole, and Reynolds 1996; Constantin

Figure 1: Securitization of a catastrophe bond.Note: This figure summarizes the trail of cash flows that happens at issuance, during the life ofand at maturity of a cat bond.

5 Note that there are different types of trigger events upon the occurrence of a disaster coveredby a catastrophe bond. The main types are indemnity trigger, industry loss index trigger,modeled loss trigger, pure parametric trigger and parametric index trigger.

152 A. Têtu et al.



2011). However, for the asset to be totally independent from market movements,the market should not be affected at all by the occurrence of the natural disasterunderlying the cat bond. We know that this is not always the case, especiallywith large-scale natural disasters such as the 2011 earthquake in Japan,Hurricane Katrina in New Orleans in the United States in 2005.6

3 Literature Review

In pricing cat bonds, two key aspects need to be considered: the structuring of thebond issue and the valuation model used to price the bond. The accuracy of thevaluation model mainly depends on the modeling of the occurrence of the under-lying disaster. Accordingly, we review the three segments of the literature. We firstreview cat bond valuation models. We then focus on disaster risk occurrencemodeling, and finally, give careful consideration to the risk premium issue.

3.1 Cat Bond Valuation Models

The literature offers different modeling approaches to price cat bonds. Mostprevious works use Merton (1976)’s contingent claims or no-arbitrage valuationmodel and assume the reinsurance market to be sufficiently efficient to diver-sify the risk of occurrence of disasters (e.g., Dassios and Jang 2003; Jarrow2010; Ma and Ma 2013; Nowak and Romaniuk 2013, among many others). Themain weakness of these no-arbitrage pricing models is that it is almost impos-sible to find financial instruments in the marketplace that can replicate exactlythe cash flows of a cat bond. In addition to this weakness, some researchers(such as Duan and Yu 2005) have argued that this risk diversification or no-arbitrage argument is not applicable in all situations, particularly when thedisaster is large enough to affect the whole economy. Even if a disaster wouldnot affect the whole economy, the risk may still be non-diversifiable if theprobability distribution of occurrence has a thick left tail. For disaster risk tobecome diversifiable, there is a need for coordination among a large number ofinsurers/reinsurers so that a single insurer limits his exposure to the occur-rence of a given catastrophe in a given region (e.g., Ibragimov, Jaffee, andWalden 2009).

6 Although interesting, to improve understanding of how disasters and markets are connectedin more extreme events is out of the scope of this paper.




From the above literature review, cat bond markets are incomplete, so theexistence of several equivalent martingale measures rules out the existence of asingle price. Based on this argument, some researchers use the actuarialapproach by relaxing the assumption of complete markets (e.g., Wang 2004;Lane and Mahul 2008, among many others). The method consists of evaluatingthe required premium on cat bond issues by taking into account the expectedloss for the issuer and the reinsurance cost. The advantage of these actuarialapproaches is that they are valid even if markets are incomplete.

Our work integrates and further develops on previous closed-form models toinclude seasonal and global warming effects as buttressed by the ability of ourmodel to capture the observed upward trend in floods occurrences. The pro-posed extended model is used to model for the first time the exposure to floodlosses in the Province of Quebec in Canada.

3.2 Modeling Disaster Risk

Many of the existing works on cat bond valuation use the Poisson process as theappropriate stochastic process to model disaster occurrence. Although othermodels can be used for the arrival process, the Poisson process is the most usedin the industry for modeling occurrence of rare events such as disasters. However,since there are different types of disasters, the Poisson process must be adapted tothe disaster class being studied. Hence, different variations of the Poisson process,ranging from constant jump amplitude (e.g., Cummins and Geman 1995) to mixedor double stochastic process (e.g., Baryshnikov, Mayo, and Taylor 1999; Dassiosand Jang 2003; Embrechts and Meister 1997; Louberge, Kellezi, and Gilli 1999), areused depending on the type of catastrophe being studied.

More recently, Lin, Chang, and Powers (2009) proposed a double stochasticPoisson process to reflect the likelihood of an increase in the frequency ofoccurrence of natural disasters due to global warming. Hainaut (2012) proposeda process that takes into account the seasonal effect of some disasters, likefloods, hurricanes and tornadoes. Our work integrates this feature of seasonaleffect with a time dependent variable to account for global warming, sincefloods are seasonal events amplified by recent global warming.

3.3 Risk Premium Puzzle

As mentioned above, since the market for catastrophic risk is not complete, catbonds bear a risk premium at issuance which is very difficult to gauge. Existingworks, such as Bantwal and Kunreuther (1999), showed that cat bonds have higher

154 A. Têtu et al.



risk premiums than equivalently rated bonds, in some cases preventing institu-tional investors from entering this market. This is the result of several factors,including the ignorance of investors regarding this relatively new type of securitizedinstrument and investor risk aversion. Braun (2012), for instance, using a multi-factor pricing model, found that, apart from the expected loss, the covered territory,the sponsor, the reinsurance cycle, and the spreads on comparably rated corporatebonds exhibit a significant impact on cat bond spread at issuance.

Cummins (2008) showed that the risk premium demanded by investors for catbonds between 2001 and 2007 ranges between 2 and 6 times expected losses.According to Froot (2008), the risk premium has even exceeded 10 times expectedlosses in the past. In particular, the risk premium associated with any particulartype of disaster rises following the occurrence of some other type of disaster. Forexample, prices of cat bonds prices decreased (or risk premiums increased)following the terrorist attacks of September 11, 2001, and risk premiums increasedfollowing Hurricane Andrew in 1992. These risk premium patterns can beexplained by the fact that there is much less capital available on risk transfermarkets after a major disaster event, leading to higher premiums. We account forthis risk premium behavior in our calibration to better evaluate the costs of floodslosses coverage using cat bonds in the province of Quebec.

4 Data

We use data from the disaster database of the Financial Assistance Program ofthe Government of Quebec in Canada. The data shown in Table 1 contains theamounts of losses resulting from floods for which the defrayed costs exceeded$300,000 for the provincial government. It is important to differentiate betweenthe amounts provided by the financial aid program and the level of coveragepotentially provided by an insurance company. Indeed, an insurance companycould offer complete coverage to individuals, whereas this governmental aidprogram provides only coverage for essential goods.7 In short, the loss amountsgiven in Table 1 are compensation paid to individuals for damages to essentialgoods, to uninsured companies, to municipalities and to organizations. Theyalso include the costs of municipality emergency measures and temporaryrelocation costs of people who had to leave their homes. Since catastrophicflooding is not an insurable loss for the majority of homeowners in Quebec, it iscurrently impossible to obtain accurate historical data on total losses incurred

7 More detailed information can be found in the guideline of the Financial Assistance Program.




Table 1: Floods that cause more than $300,000 of damages in Quebec since 1992 and their2011 dollars equivalent costs.

Event date Disaster type Costs for the financial aidprogram of the PublicSafety Department of

Quebec

Administrativeareas affected

(*)

Equivalentcosts in

dollars

Mar Floods $, , $,,Jul Floods $,, , , $,,Apr Floods $,, , , $,,Jan Floods $,, , $,,Jul Heavy rainfall $, , $,,Jul Floods and heavy

rainfall$,, $,,

Aug Floods $,, , , $,,Nov Heavy rainfall $,, , $,,Apr Floods $,, , , $,,Jul Heavy rainfall $,, , , $,,Mar Floods (winter

and spring)$,, , $,,

Oct Heavy rainfall $,, , $,,Jan Floods $, $,Jul Heavy rainfall $, , , , ,

$,

Jun Heavy rainfall $,, , $,,Aug Heavy rainfall $,, , $,,Apr Floods $,, , , , ,

, $,,

Jul Heavy rainfall $,, , $,,Jul Heavy rainfall $, , $,Apr Floods $, , , $,Aug Floods $,, , $,,Dec Floods $,, , , $,,May Floods $, . $,,Sept Heavy rainfall $, , , $,Dec Floods $, , , $,Apr Floods $,, , , , ,

, , $,,

Jun Floods $,, $,,Jul Heavy rainfall $, , $,Aug Heavy rainfall $,, , $,,Sept Heavy rainfall $,, , $,,Oct Heavy rainfall $,, , , , $,,Dec Floods $, $,May Floods $, $,

(continued )

156 A. Têtu et al.



by individuals. For an insurance company to accurately assess flood losses atthe individual level, it would need to have an estimate of losses incurred byindividual floods in comparatively recent history in Quebec. Unfortunately, inthe absence of this detailed losses data, we model the risk of losses under thefinancial aid program administered by the Department of Public Safety of theGovernment of Quebec. The database provides monthly data from March 1992 toJune 2011.

Table 1: (continued )

Event date Disaster type Costs for the financial aidprogram of the PublicSafety Department of

Quebec

Administrativeareas affected

(*)

Equivalentcosts in

dollars

May Heavy rainfall $,, $,,Oct Heavy rainfall $,, , , $,,Mar Floods $, $,Aug Heavy rainfall $,, $,,Nov Heavy rainfall $, , $,Nov Heavy rainfall $,, $,,Jan Floods $,, , , , $,,Apr Floods $,, , , , ,

,, ,,, , ,

$,,

Jul Heavy rainfall $, , $,Aug Heavy rainfall $,, , , , ,

, $,,

Aug Heavy rainfall $, , , , $,Dec Floods $, , $,Apr Floods $, , , , ,

$,,

Jul Heavy rainfall $, , , $,Jul Heavy rainfall

and heavy winds$, $,

Dec Floods $, $,Oct Heavy rainfall $,, , , , $,,Dec Heavy rainfall $,, , , $,,Apr Floods $,, $,,Jun Heavy rainfall $,, $,,

Notes: *Appendix A contains a map detailing the different administrative areas.This table contains the loss amounts supplied by the Financial Aid Program of Quebec duringfloods for which the defrayed costs exceeded $300,000.




Loss amounts in this database provide information on historical costs of theoccurrence of disasters. The number of properties at risk is also included in thedatabase. If a disaster of the same magnitude occurs today, the losses wouldreflect (i) the price level today (including inflation and the increase in the valueof houses) and (ii) the increase in the number of properties in the affected area.To convert historical costs into their 2011 equivalent, we use a method similar toCollins and Lowe (2001). The formula used to convert the costs is as follows:

Dr;2011 ¼ Dr;y � 0:25 � MLSr;2011MLSr;y

� �þ 0:75 � CPI2011

CPIy

� �� NPr;2011

NPr;y

� �½1�

where Dr,y is the amount of losses arising from a disaster in area r in year y;MLSr,y is the average value of houses in area r in year y; CPIy is the value of theconsumer price index in year y; and NPr,y is the number of houses in area r inyear y.

The rationale behind this equation is that the amount of damages is relatedto the number of houses affected, the price of these houses and inflation.Appendix B contains a table which presents the data on increases in the valueand prices and the number of properties, by area. The proxy used for inflation isthe consumer price index and the average value of homes in the affected area.Collins and Lowe (2001) do not use the average value of homes in the affectedarea; we decide to include this in our equation since some people must move orrebuild part of their home following the flood. The 75/25 ratio was arbitrarilychosen to give more weight to inflation (proxied by the consumer price index).

The last column of Table 1 gives the 2011 equivalent of the historical losses.8

If a disaster has occurred outside a region containing administrative metropoli-tan areas for which we have data on the value and number of properties, we usethe mean values of the entire province for the adjustment. If the affected areacontains administrative metropolitan areas for which we have data, we use the

8 Note that we do not have data on events where losses were less than $300,000 because theseare not recorded in our dataset. Moreover, as we have adjusted the loss amounts to reflect thelosses as though these events were occurring today, there is a risk of introducing an over-estimation bias relative to the actual number of flood events. We could adjust the data byexcluding any disaster having caused less than $300,000 in damages (1992 dollars); as our datasample is already small, we assume that the impact of this distortion is negligible. Anotherpotential weakness of the data concerns the various changes that have been made to theguidelines of the financial assistance program, such as restrictions to building new houses inrisky areas, increases in the maximum compensation and the redefinition of items covered bythe program. We presume that these changes were made in order to keep up with the increasingcosts and the changing nature of the household goods in each home; therefore, these changesdo not invalidate our data.

158 A. Têtu et al.



corresponding index. In addition, if two administrative regions have beenaffected by a single flood, we use the average of two indices (one of themetropolitan areas included in the administrative regions affected if available;otherwise, for the region that does not contain any metropolitan area, we use theindex of the entire province). In cases where more than three administrativeregions were affected, we use the index of the entire province.

5 Methodology

As in the extant literature, we assume that the occurrence of disasters follows aPoisson process. We assume that the severity of losses is independent from thePoisson process and follows a random variable whose distribution is determinedbased on historical data.

5.1 Modeling the Disaster Occurrence Process

We use a Poisson process to model the occurrence of floods, because as noted inthe literature review above, the Poisson process is now widely accepted as theindustry standard and has a good fit for the probability of rare events occur-rence.9 To calibrate the events occurrence probabilities, we need to determinethe intensity λ of the Poisson process. To do this, we must choose between aconstant intensity λ (homogeneous Poisson process, HPP) and a time-dependentintensity λ(t) (NHPP). The intensity λ in the case of an HPP is determined bydividing the number of historical catastrophes by the number of periods(months). We thus obtain an intensity of λ ¼ 53

20�12 ¼ 0:2208333. For the NHPP,we innovate from previous works by modeling both the effects of seasonalityand global warming. Floods occurrence is seasonal. Indeed, historically, therehas been 4 times more flooding in the summer than in the winter. To include thiseffect in our intensity function λ(t), we introduce a periodic component whichwill increase or decrease our intensity, depending on the month. To do this, weuse a sinusoidal function with a period of 12 months. Relying on historical data,we determine that our minimum must be during the months of January andFebruary, while the maximum should be during the months of July and August.Our periodic component is of the form

9 Future works could explore other types of processes used in the literature to model theoccurrence of floods.




per tð Þ ¼ a � sinðbt þ cÞ: ½2�We set the initial date, t ¼ 0, at January 1, 1992, and t ¼ 1 corresponds toFebruary 1, 1992, etc. We use monthly intervals, implying 12 periods per year,yielding 2π

b ¼ 12 ⇒ b ¼ π6. We want the minimum at t ¼ 1, which gives c ¼ 4π

3 .Equation [2] then becomes

per tð Þ ¼ a � sin πt6þ 4π

3

� �: ½3�

Once the seasonal effect is identified, we need to determine the function λ(t) thatminimizes the sum of the squared errors when calibrated with historical data.The calibration is done with an optimization algorithm. Using the Nelder–Meadsimplex method, after several iterations, we find that a first-degree polynomial10

plus the seasonal component given in eq. [3] is more appropriate. We obtain thefollowing function:

λ tð Þ ¼ 0:064105þ 0:001417 � t þ 0:063782 � sin πt6þ 4π

3

� �; ½4�

where t is in months (with t ¼ 60 after 5 years).11 We can decompose eq. [4] intothree components: a constant, an increasing trend (global warming) and aperiodic variation (seasonality). The long-term trend predicted by this functionis an average increase in the intensity by 0.001417 each month.

Figure 2 compares the goodness of fits of the two types of Poisson processes:HPP and NHPP using historical data. It clearly shows that the NHPP fits betterwith our historical data. Using a two-sample Kolmogorov–Smirnov (KS) test, wecompare both the HPP and NHPP with the historical data. The statistic is lowerfor the NHPP (0.073 for NHPP versus 0.200 for HPP) and thus its fit is better withthe historical data. Hence, we have successfully integrated both the seasonaland the global warming effects in our occurrence process.

5.2 Loss Distribution

We assume that the loss amounts are independent random variables as seen inCabrera and Hardle (2010) and others. In this section, our aim is to characterizethe distribution of losses by identifying the distribution function that best fits the

10 Even increasing the degree of the polynomial beyond one does not significantly reduce thesum of the squared residuals.11 Note that this function assumes that λ(t) is deterministic. In future research, it would beinteresting to investigate the effect of a stochastic function λ(t) on our results as well.

160 A. Têtu et al.



historical data. To achieve our goal, we first study the shape of the mean excessfunction in our historical data. Burnecki, Janczura, and Weron (2011) stated thatif the mean excess function increases (decreases), it implies that the distributionhas a thicker (thinner) tail than the exponential distribution. In Figure 3, weobserve that the mean excess function is increasing, thus our distribution seemsto have a bigger tail than the exponential distribution.

Figure 2: Comparison of the Poisson processes and the historical data.Note: The figure compares the number of catastrophic floods according to different fitted occurrenceprocesses with the historical occurrence of floods. We fit the historical data to the homogeneousPoisson process ðHPP : λ ¼ 0:2208333) and to the non-homogeneous Poisson Process with seaso-nal effect and upward trend ðNHPP: λ tð Þ ¼ 0:064105þ 0:001417 � t þ 0:063782 � sin πt

6 þ 4π3

� �). K-S2

designates the two-sample Kolmogorov–Smirnov test statistic used to compare with historical data.

0 2 4 6 8 10 12x 107

0

0.5

1

1.5

Mea

n ex

cess

val

ue

2

2.5x 108

Historical data

Mean excess function threshold

Figure 3: Historical data, mean excess function.Note: The figure shows the mean excess function of the historical loss data. The mean excessvalue is determined by averaging all loss amounts that are greater than the mean excessfunction threshold.




Second, we study the distribution of losses. According to Burnecki, Janczura,and Weron (2011), log-normal, gamma, exponential, Weibull, Burr and Paretodistributions are typical candidates for a loss distribution. We will thereforestudy these distributions and the log-logistic distribution to determine whichbest fits our data. Because our data contains only floods that caused more than$300,000 of damage, we subtract this amount from our historical costs. Thefinal model will therefore be the chosen distribution plus $300,000. Accordingto D’Agostino and Stephens (1986), when the fitted distribution deviates fromthe true distribution in the tails, it is best to use the Anderson–Darling (A2) test.This statistic developed by Anderson and Darling (1952) is a quadratic statisticmeasuring the difference between the empirical distribution function of thesample and the distribution under study. To calibrate the parameters that fiteach of the distributions to the historical data, we minimize the A2 statisticsusing the Nelder–Mead simplex method. We reject the null hypothesis that theempirical data comes from the specified distribution if the test statistic value isgreater than the specified critical value; the appropriate distribution is thereforethe one with the smallest value for the test statistic. As shown in Table 2, threedistributions have the lowest values for the test statistics: log-normal, general-ized Pareto and Burr distributions. For these three distributions, the values forthe A2 statistic are more or less the same. We therefore use other statistics todifferentiate between these three distributions: the KS and the Cramer–vonMises quadratic (W2) statistics, as did D’Agostino and Stephens (1986). Usingthese additional statistics, we can see that the generalized Pareto distributionhas values slightly lower than the values for the two other distributions (log-normal and Burr).

Further, we compare the mean excess functions of the distributions withthe historical data. We find that, toward the higher end of the distribution,the generalized Pareto distribution fits the data rather well whereas towardthe lower end of the distribution, the log-normal distribution seems moreappropriate. However, since the mean excess function of the log-normaldistribution is below the historical data, we retain the generalized Paretodistribution so as to not underestimate the losses incurred. Indeed, this latterdistribution is a relatively good fit to the data for the mean excess functionfor the majority of the distribution and it does not underestimate the functiontoward the tails of the distribution. Our choice of the generalized Paretodistribution confirms earlier results, where this distribution had the smallestvalues for the KS and W2 statistics among the three distributions (log-normal,generalized Pareto and Burr distributions) with more or less similar A2 valuesas reported in Table 2.

162 A. Têtu et al.



Table2:

Loss

distribu

tion

s’test

statistics.

Distribution

Statistic

Expo

nential

β¼,,

Weibu

lla¼,,

b¼.

Gam

ma

a¼.

b¼,,,

Log-logistic

µ¼.

σ¼.

Log-no

rmal

µ¼.

σ¼.

Gen

eralized

Pareto

k¼.

σ¼,,θ¼

Burrα¼.

δ¼,,

τ¼.

KS

.

.

.

.

.

.

.

W

.

.

.

.

.

.

.

A.

.

.

.

.

.

.




5.3 Cat Bonds Valuation Model

We use the no-arbitrage argument (e.g., Jarrow 2010) for our actuarial modelto value cat bonds because it is relatively easy to understand and to imple-ment. The value of the cat bond is obtained as the present value of theexpected future payments. Our proposed model, however, innovates fromJarrow (2010) in that we adjust for a fixed coupon rate and allow the catbond to undergo several triggering events before the remaining capital isliquidated. In addition, as we mentioned above, the proposed model accountsfor seasonal and global warming effects in the occurrence process. We alsoassume zero correlation between the occurrence process, the loss variableand the interest rates (e.g., Burnecki, Janczura, and Weron 2011).12 We definethe following variables:– p t; sð Þ: the value at time t of $1 paid at time s;– r tð Þ: the risk-free rate at time t;– c: the percentage coupon rate paid in each period;– CFcoupons: cash flows from coupons;– CFmaturity: the principal to be paid at maturity;– Li: the amount of losses resulting from disaster i;– A: the principal value of the bond at issuance;– τi: time of occurrence of disaster i;– T: time to maturity of the bond;– Ns�t: the number of disasters between t and s (with s > t);– 1τi > s > τi�1 : equals 1 if time s is between the occurrence of disaster i–1 and

disaster i, otherwise it is 0.

5.3.1 Present Value of Expected Coupon Payments

The present value of expected coupons received before the first disaster isdefined as

Et¼0 A � cXTs¼1

1τ1 > s > 0 � pð0; sÞ" #

: ½5�

When a catastrophe occurs, we assume that the notional value of the bond isreduced by the amount of losses incurred, and thus the remaining coupons are

12 For future works, it would be interesting to investigate if historical data confirms thisabsence of correlation.

164 A. Têtu et al.



paid on this adjusted notional value. The present value of expected couponsreceived after the occurrence of a first catastrophe is obtained as follows:

Et¼0 maxðA� L1Þ � cXTs¼1

1τ2 > s > τ1 � pð0; sÞ" #

: ½6�

We can generalize this equation across i disasters. Thus, the present value ofexpected coupons received after the occurrence of i disasters is

Et¼0 maxðA�Xij¼1

LjÞ � cXTs¼1

1τiþ1 > s > τi � pð0; sÞ" #

: ½7�

To obtain the analytical form of this equation, we need to compute the prob-ability that i disasters occurred before time s. Since we defined our arrivalprocess for disasters as a Poisson process with a deterministic intensity functionλðtÞ, the probability of having n disasters between time t and time s is given by

ProbðNs�t ¼ nÞ ¼ exp � Ð st λðuÞdu� � � Ð st λðuÞdu� �nn!

: ½8�

The probability of having zero disasters between time 0 and time s is

ProbðNs�0 ¼ 0Þ ¼ exp �ðs0

λðuÞdu0@

1A: ½9�

Hence, before the first disaster, the present value of expected coupon payments,given in eq. [5], becomes

Et¼0 A � cXTs¼1

1τ1 > s > 0 � pð0; sÞ" #

¼ A � cXTs¼1

pð0; sÞ � exp �ðs0

λðuÞdu0@

1A: ½10�

For i disasters, the present value of expected coupons received, given in eq. [7],becomes

Et¼0 max A�Xij¼1

Lj;0

!" #� cXTs¼1

p 0; sð Þ � exp � Ð s0 λ uð Þdu� � � Ð s0 λ uð Þdu� �ii!

:

½11�The present value of expected coupons to be received is the sum of the expectedpresent values defined above for all occurrences of possible disasters. It then follows:




Et¼0 CFcoupons� � ¼ A � c

XTs¼1

p 0; sð Þ � exp��ðs0

λ uð Þdu�

þX1i¼1


Lj;0

!" #� cXTs¼1

p 0; sð Þ�exp � Ðs

0λ uð Þdu

� �� Ðs

0λ uð Þdu

� �i

i!

0BBB@

1CCCA:

½12�We poseP

i ¼Pij¼1

Lj andP

0 ¼ 0, thus


Lj;0

!" #

¼ Et¼0 max A�X

i;0

� h i¼ðA0

A�X

i

� þ� g

Xi

� � dX

i;

½13�

where g(∑i) is the density function of the sum of losses for i occurrences.We assume that losses due to the disasters are iid, i.e. independent from

each other and identically distributed, with density function f(Li), which yields

gX

i

� ¼ f L1ð Þ � f L2ð Þ � � � � � f ðLiÞ;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

i times

½14�

where “�” is the convolution product operator, i.e. f xð Þ � f xð Þ ¼ Ð10f x � tð Þ

f tð Þdt: Equation [12] then becomes

Et¼0½CFcoupons� ¼X1i¼0

Et¼0 max A�X

i;0

� h i�

� cXTs¼1

pð0; sÞ �exp � Ðs

0λðuÞdu

� �� Ðs

0λðuÞdu

� �i

i!

1CCCA:

¼X1i¼0

ðA0

A�X

i

� þ� g

Xi

� � dX

i

0@

1A

0@

�cXTs¼1


0λðuÞdu

� �� Ðs

0λðuÞdu

� �i

i!

1CCCA:

½15�

5.3.2 Present Value of Expected Principal at Maturity

The present value of the expected principal at maturity when no disaster occurs is

Et¼0�A � 1τ1 > T � p 0; Tð Þ� ¼ A � p 0; Tð Þ � exp �

ðT0

λ uð Þdu0@

1A: ½16�

166 A. Têtu et al.



The present value of the expected principal at maturity when i disasters occur is


Lj;0

!� 1τi < T � p 0;Tð Þ

" #

¼ Et¼0 max A�Xij¼1

Lj;0

!" #� p 0;Tð Þ �

exp � ÐT0λ uð Þdu

� �� ÐT

0λ uð Þdu

� �i

i!

½17�

The present value of the expected principal to be received at maturity is the sumof the present value for each disaster occurrence possible:

Et¼0 CFmaturity� � ¼X1

i¼0

Et¼0 max A�

Xi;0

� h i

� pð0; TÞ �exp � ÐT

0λ uð Þdu

� �� ÐT

0λ uð Þdu

� �i

i!

!

¼X1i¼0

ðA0

A�X

i

� þ� g

Xi

� � dX

i

!:

� pð0; TÞ �exp � ÐT

0λ uð Þdu

� �� ÐT

0λ uð Þdu

� �i

i!

!½18�

5.3.3 Formula for the Present Value of the Cat Bond

Observing eqs [15] and [18], we can quickly note that the most significant termsare the first ones in each of the equations. In fact, as the number of disastersincreases, the last terms in the equations will become very small and negligible.To implement the model for a catastrophe bond, one needs to consider themagnitude of the terms as i increases and decide at which critical value K tostop, as the inclusion of the terms beyond K disasters has no significant impacton the value of the bond. Therefore, eqs [15] and [18] can be rewritten as follows:

Et¼0 CFcoupons� � ¼ XK

i¼0

ðA0

A�X

i

� þ � gX

i

� � dX

i

!0@

� c �XTs¼1


0λðuÞdu

� �� Ðs

0λðuÞdu

� �i

i!

!þ O :ð Þ;

½19�




Et¼0 CFmaturity� � ¼ XK

i¼0

ðA0

A�X

i

� þ � gX

i

� � dX

i

!:

�pð0; TÞ �exp � ÐT

0λ uð Þdu

� �� ÐT

0λ uð Þdu

� �i

i!

!þ O :ð Þ;

½20�

where O :ð Þ is the error term that can be ignored. The formula for the cat bondvalue is obtained by summing these two expressions [19] and [20] as follows:

VCat�Bond ¼XKi¼0

ðA0

A�X

i

� þ� g

Xi

� � dX

i

!0@

�c �XTs¼1

p 0; sð Þ �exp � Ðs

0λ uð Þdu

� �� Ðs

0λ uð Þdu

� �i

i!

!

þXKi¼0

ðA0

A�X

i

� þ� g

Xi

� � dX

i

0@

1A

�pð0;TÞ �exp � ÐT

0λ uð Þdu

� �� ÐT

0λ uð Þdu

� �ii!

!½21�

To solve this equation analytically, we need to know the density function gP

i

� �,

which is obtained as the convolution product of i identical density functions f(x).Here, since our loss amounts are distributed according to the generalized Paretodistribution, there is no analytical solution. Indeed, Ramsay (2006) stated thatthe analytical solution for the convolution product of n density functions existsonly for specific cases with Pareto distributions, which is not our case. Nor dothe analytical solutions exist using log-normal distributions (Li 2007) or Burrdistributions (Kortschak and Albrecher 2010). Since we are unable to obtainanalytical solutions with the Pareto distributions for the losses, we will resort toMonte Carlo simulations to obtain our empirical results.

6 Simulation Results

Above we found the generalized Pareto distribution to be a better fit to our lossdata. Unfortunately, with this distribution, known closed-form analytical solu-tions for cat bond values are not available. Therefore, we use Monte Carlo

168 A. Têtu et al.



simulations. We first define the characteristics of the cat bond that the govern-ment could have issued in the market to cover its losses from floods. And then,before conducting the Monte Carlo simulations for our empirical analyses, weprovide an example with the gamma distribution for the loss amounts for whichwe obtain an analytical form. We then compare the results obtained with thisanalytical solution to the results of the Monte Carlo simulations to check if ourproposed formulas are correct. More importantly, we use the analytical formulaas a control variate and follow the regression method described in Boyle,Broadie, and Glasserman (1997) in subsequent Monte Carlo simulations forvariance reduction. We then compare the losses that have been incurred histori-cally to the cost of the hedge using a cat bond.

6.1 Cat Bonds Characteristics and Assumptions

We assume that the government issues one type of cat bonds to hedge againstflooding losses. The bonds have a fixed coupon rate. At each occurrence of adisaster, the principal is reduced by the loss amount. We assume the followingadditional parameters at issuance:– The coupons are paid monthly and the annual coupon rate is c.– The principal amount at issuance is A0¼ $500 million.– The maturity of the bond is 1 year (to allow a comparison with the historical

losses on an annual basis).– The bonds are issued on December 1 of each year. For example, when the

hedging cost is shown for year 1993, it is for a hedge that covers the periodfrom December 1, 1992, to December 1, 1993.

– We use the zero-coupon yield curves data from the Bank of Canada. For thezero-coupon yield curve methodology, we refer the interested reader toBolder, Johnson, and Metzler (2004).

We also use the following assumptions:– Losses (Li) follow a generalized Pareto distribution and are iid. The distribu-

tion is calibrated to historical data.– The occurrences of catastrophe floods follow an NHPP with deterministic

intensity function λ(t) given in eq. [4] and calibrated to the historicaldata.

– When a catastrophic event occurs during a period, we assume that the lossamount can be immediately assessed and the principal is reduced at the endof the period by the equivalent amount.




Under the risk-neutral probability, the value of the catastrophe bond at issuanceis given by

VCat�Bond ¼XNn¼1

c12

:An:e�n:rn12 þ AN � e�N�rN

12 ; ½22�

where n is the period; N is the number of periods up to the maturity of the bond;An is the adjusted principal amount at the end of period n; rn is the risk-free ratewith a term that corresponds to the end of period n; cis the annual percentagecoupon paid.

The coupon rate that gives a bond value at par at its issuance (based on aninitial principal amount of 100) is obtained as follows:

c ¼ 12100� AN � e�N�rN

12PNn¼1 Ane

�n�rn12

: ½23�

We use Monte Carlo simulations to obtain the principal amounts An. Then, usingthe zero-coupon yield curve, we determine the coupon rate which allows us toissue the cat bond at par. The cost of an issue is the spread relative to anequivalent risk-free bond and is given by

Costs ¼XNn¼1

An � c� b12

� e�n�rn12 ; ½24�

where b is the annual coupon rate of an otherwise equivalent risk-freebond issued at par with the same maturity and same coupon frequency as

the catastrophe bond; An ¼ max A0 �Pij¼1

Lj;0

!is the adjusted principal with

Lj representing the amount of losses incurred following the occurrence of dis-aster j, and i is the number of disasters that have happened between 0 and n.

We generate the paths of An over the life of the catastrophe bond in whichthe dynamics of Lj follows the generalized Pareto distribution and the arrival ofdisasters follows the NHPP as specified above.

6.2 Validation with a Gamma Distribution for the Losses

In this section we validate our valuation model and since the generalized Paretodistribution doesn’t allow us to obtain an analytical solution to our model, weuse a gamma distribution to describe the losses here. With the gamma distribu-tion, it is possible to obtain the solution to formula [21] which is the value of thecat bond. We will compare the results obtained with this analytical solution tothose obtained from the Monte Carlo simulations.

170 A. Têtu et al.



For the analytical solution, we assume the loss amounts to have the follow-ing gamma distribution: Li , gamma ðα; θÞ, with α¼0.438 and θ ¼ 2.609eþ 7.With these distributions, we obtain an analytical solution to eq. [21].13

We compute the value of a catastrophe bond on December 1, 2011 (t¼ 239 inthe λ(t) equation), with a principal of $500 million at the issue, a monthlycoupon rate of 0.666% (8% per annum) and maturities of 1, 2, 5, 10 and 20years. We used a critical value of K¼ 100 obtained by trial and error with ourdata. The results from the analytical solution and Monte Carlo simulations aregiven in Table 3. We observe that the valuation made with the analytical modelis very close to the one obtained by Monte Carlo simulations. In fact, themaximum difference is only $0.025 for a principal of $100. Since this differenceis minimal, we can conclude that our model is valid. As the analytical modelcannot be used with the distribution chosen for the losses caused by floods inQuebec, we proceed by Monte Carlo simulations for the rest of this paper.However, we have developed a simple and efficient model to evaluate catbonds when the losses distribution has a closed-form solution for the convolu-tion product of n density functions.

6.3 Comparing the Actual Historical Lossesand the Hedging Costs

Here we compare the annual losses that have occurred with the costs that thegovernment would have borne if there was a hedging strategy via a cat bondissue of one year maturity. Figure 4 plots the historical losses and the costs of

Table 3: Comparison of the analytical solution and the Monte Carlo simulation results.

Maturity (years) Price per $ nominal

Analytical solution Monte Carlo Difference

. . . . . . . . . . . . . . .

Note: This table compares the results of using our analytical model to price a cat bond versususing Monte Carlo simulations. We can see that for a distribution that has an analytical solutionsuch as the gamma distribution, the results are close to the Monte Carlo simulation results.

13 We did not report the analytical solution because the formula is too long.




issuing a bond offering complete disaster coverage over the course of one year.When hedging at actuarial cost would have been possible, the issuance ofcatastrophe bonds would have greatly reduced flood losses uncertainty since1993. As we can observe, the objective of hedging with a cat bond is justly toprotect the provincial government against catastrophic events similar to the 1996Saguenay-Lac-Saint-Jean floods, which cost more than $100 million Canadian toQuebec taxpayers.

6.4 Analyzing the Effect of the Risk Premium

The development of an actuarial valuation model is the first step in valuing thecost of issuance of this type of risk transfer solution. As discussed in ourliterature review, the risk premium required by investors to issue a cat bonddepends on the prevailing market conditions. The risk premium is measured asexpected returns demanded by investors divided by expected losses. For exam-ple, a risk premium ratio of 1 corresponds to the fair actuarial value, i.e., thetotal expected return required by investors is equal to the total expected loss;meanwhile, a ratio of 3 means that the issuer must provide a return that coversthree times expected losses. Following Cummins (2008) and Lane and Mahul(2008), we examine the impact of a risk premium varying between 1 and 6 timesexpected losses. Indeed Lane and Mahul (2008) found an average long-term

1994 1996 1998 2000 2002 2004 2006 2008 20100

0.5

1

1.5

2

2.5

3x 108

Historical lossesHedging costs

Costs

Year

Figure 4: Historical losses caused by floods and annual hedging costs (100% hedge) between1993 and 2011.Note: This figure compares the annual losses that have occurred with the costs that thegovernment would have borne if there was a hedging strategy via a cat bond issue of 1-yearmaturity. The hedging by cat bond on this figure is done at actuarial costs.

172 A. Têtu et al.



trend risk premium ratio of 2.69. Figure 5 shows the historical losses as well asthe costs that would have been incurred from issuing catastrophe bonds whichoffer full flood risk coverage during one year for various risk premiums. Ofcourse, the cost of the hedge increases with the level of risk premium.

Table 4 presents the average historical flood losses and hedging costs overthe study period (1992–2011) for various risk premium ratios. Complete coveragewould have been beneficial if the risk premium ratio was less than 2. However,one must be aware that the objective of insurance or hedging is not necessarilyto lower costs, but rather to stabilize annual costs, i.e., reduce the volatility of

Figure 5: Historical costs of a complete hedge for various risk premiums.Note: This figure shows the historical losses as well as the costs that would have been incurredfrom issuing catastrophe bonds offering full flood risk coverage during the year for various riskpremiums.

Table 4: Historical costs of a complete coverage for various risk premium ratios.

Risk premium ratio Average annual costs Total costs(over the study period)

No cat bond issue (or no hedge) $,, $,,. (actuarially fair value) $,, $,,. $,, $,,. $,, $,,. $,, $,,. $,, $,,. $,, $,,,. $,, $,,,

Note: This table shows the historical average annual costs and the total costs, over the studyperiod, of hedging flood losses using cat bonds for different risk premiums. The deductiblelevel is set to zero, which corresponds to full coverage.




annual cash outlays and to prevent unexpected situations in which a singlecatastrophic event forces the issuer to make a large payment that could threatenits financial viability.

6.5 Analyzing the Impact of the Trigger Level

We rerun our analyses with different trigger levels using the average long-termrisk premium estimated by Lane and Mahul (2008), i.e., 2.69. A $1 milliondeductible means that there will be a cat bond payoff only if losses exceed$1 million, and there is no payout otherwise. The higher the deductible, theless the coverage provided by the hedge. Figure 6 shows the historical annualcosts in 2011 dollars that would have been paid if the government had issuedcatastrophe bonds with a risk premium ratio of 2.69 and for different triggeringlevels. These costs include hedging costs as well as losses in the case of partialor no hedging strategy. All else being equal, we can see that full coverageoffers more stability in the historical annual costs than partial coverage.However, losses must be significantly larger for the insurance hedge to beviable; otherwise it is better for the issuer to choose a higher deductible level.

Figure 7 plots the average annual costs over the study period as a function of thedeductible level with a risk premium ratio of 2.69. We can see that the relation-ship between the trigger level and the total costs is not linear but “bound.”Indeed, setting too low or too high deductible levels would not have beenadvantageous.

Figure 6: Historical costs for different trigger levels and a risk premium of 2.69.Note: This figure compares the annual losses that have occurred with the costs that thegovernment would have borne if there was a hedging strategy via a at bond issue of 1-yearmaturity, with a risk premium of 2.69, for different deductibles.

174 A. Têtu et al.



6.6 Studying the Impact of the Federal GovernmentContribution

Under the current provincial flood financial aid program, the federal govern-ment has to defray part of the costs when the per capita costs of a disaster in theprovince exceed a certain amount. Table 5, taken from the website of PublicSafety Canada, shows the percentage paid by the federal government duringdisasters such as floods in Quebec.

Many scenarios can be explored. On the one hand, the federal governmentcould enter into an agreement to share the issuance costs with the provincialgovernment in cases where the provincial government decides to issue cat bondsto hedge the total exposure (including the part covered by the federal govern-ment). On the other hand, the provincial government could decide to issue catbonds to cover only the portion of expected losses not reimbursed by the federal

Figure 7: Historical average annual costs as a function of the deductible with a risk premium of 2.69.Note: This figure plots the average annual costs as a function of the deductible level with a riskpremium ratio of 2.69.

Table 5: Disaster financial assistance arrangements per capita sharing formula.

Eligible Provincial/territorial expenditures Government of Canada share

First $ per capita NilNext $ per capita %Next $ per capita %Remainder %

Note: This table, taken from the website of Public Safety Canada, shows thepercentage of losses paid by the Federal Government as financial aid duringdisasters in the provinces.




government. Since the risk premium is generally less for small-scale events, it isreasonable to assume that the existence of federal aid would decrease the riskpremium required by the market. Figure 8 compares the historical losses and thehedging costs with and without federal aid, assuming a risk premium ratio of2.69 and full coverage (zero deductible). As expected, the expected losses andhedging costs are lower when federal aid is included.

6.7 Analyzing the Risk-Adjusted Returns to Investors

The principal benefit to an investor that invests a portion of its wealth in acatastrophe bond is diversification. Returns offered by disaster risk markets havelittle or no correlations with traditional securities (stocks and bonds) markets. Toanalyze the risk-adjusted returns to investors, we use the Sharpe ratio because it isa risk-adjusted measure that is widely understood by portfolio managers.14 Sincethe Sharpe ratio is the portfolio excess return adjusted by its risk level (measuredby the portfolio volatility), it shows which investment would have given the higherreturn for each unit of risk taken. To obtain the risk-adjusted performance mea-sure, we compute the Sharpe ratio of the investment as follows:

S ¼ R� rσ

; ½25�

where R is the average annual return of the investment over the period, r is therisk-free rate and σ is the standard deviation of annual returns. For the risk-free

Figure 8: Losses and hedging costs with federal aid and with a risk premium of 2.69.Note: This figure compares the historical losses and the hedging costs with and without federalaid, assuming a risk premium ratio of 2.69 and full coverage.

14 Of course, we could have used other more sophisticated risk-adjusted performance measureswith asymmetric risk measures like value at risk. Since this is not the main focus of this paper,we leave these interesting issues for further studies.

176 A. Têtu et al.



rate, we use historical returns on 3-month government T-bills. Table 6 presentsthe Sharpe ratios for the cat bonds. The Sharpe ratios with federal aid are higherthan those without federal aid.

6.8 Stress Test with the Distributions Parameters

In this section, we stress the parameters of the generalized Pareto distributionand the intensity of the Poisson process to see their impact on the historicalaverage annual costs of hedging. We assume a risk premium of 2.69. The resultsare presented in Table 7. As we can see, the impact of a 10% variation in thevalue of the parameter k of the generalized Pareto distribution has much bigger

Table 6: Sharpe ratios.

Asset Sharpe ratio

Federal aidnot included

Federalaid included

Cat bond with trigger ¼ $ . .Cat bond with trigger ¼ $M . .Cat bond with trigger ¼ $M . .Cat bond with trigger ¼ $M . .Cat bond with trigger ¼ $M . .

Note: This table presents the Sharpe ratios for the cat bonds.

Table 7: Stress test with the distributions’ parameters.

Historical average annual costs

Baseline average annual cost ,,Scenarios (% change of parameters’ value) −% þ %k (pareto distribution)Average annual cost ,, ,,Percentage change from baseline −% %σ (pareto distribution)Average annual cost ,, ,,Percentage change from baseline −% %λ (Poisson process)Average annual cost ,, ,,Percentage change from baseline −% %

Note: This table shows the historical average annual costs and the percentage change in thecost from the baseline value when the distribution parameter values vary by −10% and þ 10%.We assume a risk premium of 2.69.




impact on the historical average annual costs than identical percentage changein the other parameters (σ for the generalized Pareto and λ for the Poissonprocess).

7 Conclusion

In this paper, we have developed a methodology to model the risk of floodlosses. We extended the model proposed by Jarrow (2010) and derived a semi-closed-form formula for cat bond valuation that accounts for coupons andoccurrence of multiple events. The proposed model uses an NHPP to capturethe probability of disaster occurrence and includes at the same time theseasonal and the global warming effects on the occurrence of floods. Wecarefully performed estimation of the intensity function parameters for theNHPP describing the occurrence of disasters as well as the loss distributiondynamics for the occurrence of disasters. We used losses recorded by theGovernment of Quebec flood financial aid program for our calibrations. Ourintensity function calibration results confirm the presence of an upward trendin the number of occurrences of floods and also of a seasonal cyclical effect.15

However, it is possible that the upward trend was overstated because of thedata available to us.

We used our model to price cat bonds that the government couldhave issued to hedge its losses. Using extant empirical studies on the riskpremium of catastrophe bonds, we were able to analyze the effect of differentlevels of risk premiums on the hedging costs. Of course these costs can varydepending on the market sentiment and conditions, so the timing of the issue

15 Nonetheless, the independence assumption between losses from one period to another canbe questioned given the observed upward trend in floods occurrences. We thank an anonymousreferee for this very insightful remark. Indeed, one could argue that with more frequentoccurrences of floods and less time between them, current losses could be dependent onprevious ones. This would mean that our assumption about the independence of lossesamounts could eventually become wrong. We could take this into account by reducing theamounts of estimated losses for further floods occurrences in the same year or in subsequentyears because items would not have been replaced or repaired yet. This would reduce theexpected losses and thus the hedging costs. However, given the big size of the territory coveredby the assistance program, we are far from having material probabilities that multiple floodscould hit the same region during the same year or within a short time period. Moreover, thesmall amount by which we could have reduced the losses amounts means that changing theindependence assumption would have small or even zero impact on our results andconclusions.

178 A. Têtu et al.



will be important. Indeed, issuance following a major natural disaster willprobably be more expensive and should be captured by the risk premium. Weexogenously use historical risk premiums observed in the marketplace, thisconstitutes one main limitation of our study. Nevertheless, with more informa-tion on total historical losses resulting from floods in affected regions, ourmethodology could be used by government mitigation agencies or by an insur-ance company to evaluate the costs of offering flood protection to the inhabi-tants in disaster-prone areas.

After all, the purpose of this paper is to propose a practical framework toshow how one may use cat bonds to hedge flood losses in Quebec and else-where. Our simple modeling approach to model catastrophe risk and price theproposed cat bond evokes some weakness. However, future research couldextend our present work by using a more innovative pricing model and complexprocesses for flood occurrences and losses.

Appendix A

Map of Quebec – administrative areas




Appendix B

Data for indices by area

Weight

. Consumer price index

(increase %)

.% .% −.% .% .% .% .% .% .%

Saguenay

MLS mean value (PM) , , , , , , , , ,

MLS mean value (PMP) ,

. MLS increase, % .% −.% .% −.% .% .% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes

(year’s beginning)

, , , , , , , , ,

New homes (during

the year)

INDEX . . . . . . . . .

Trois-Rivières

MLS mean value (PM) , , , , , , , , ,


. MLS increase, % .% .% −.% −.% .% .% −.% −.% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


, , , , , , , , ,

New homes (during

the year)

INDEX . . . . . . . . .

Sherbrooke

MLS mean value (PM) , , , , , , , , ,


. MLS increase, % −.% .% −.% −.% .% .% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


, , , , , , , , ,

New homes (during

the year)

INDEX . . . . . . . . .

Gatineau

MLS mean value (PM) , , , , , , , , ,


. MLS increase, % .% .% .% −.% .% −.% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


, , , , , , , , ,

New homes (during

the year)

, , , , , , , , ,

INDEX . . . . . . . . .

Québec City

MLS mean value (PM) , , , , , , , , ,


. MLS increase, % −.% .% .% −.% .% −.% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


,, ,, ,, ,, ,, ,, ,, ,, ,,

New homes (during

the year)

, , , , , , , , ,

INDEX . . . . . . . . .

(continued )

180 A. Têtu et al.



.% .% .% .% .% .% .% .% .% .% .%

, , , , ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

, , , , , , , , , , ,

. . . . . . . . . . .

, , , ,, ,, ,, ,, ,, ,, ,, ,,

−.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

, , , , , , , , , , ,

, , , , ,

. . . . . . . . . . .

, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

, , , , , , , , , , ,

, , , , , , , ,

. . . . . . . . . . .

,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

, , , , , ,, ,, ,, ,, ,, ,,

, , , , , , , , , ,

. . . . . . . . . . .

, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

, , , , , , , , , ,

. . . . . . . . . . .

(continued )




Acknowledgments: The authors thank the editor Michael R. Powers and twoanonymous referees for their very constructive comments and suggestions,which have greatly enhanced the quality of the paper. The authors also thankGeorge Dionne, Skander Lazrak and seminar participants at the 2014 CanadianEconomics Association (CEA) Conference in Vancouver (Canada), the 2014Midwest Finance Association (MFA) Conference in Orlando (USA) and the 17thannual conference of the Asia-Pacific Risk and Insurance Association (APRIA) inNew York City (USA), Sylvain Tremblay from the Quebec Public SafetyDepartment, Marie-Claude Beaulieu and Catherine Cournoyer. All errors andomissions are the authors’ sole responsibilities.

Funding: The authors acknowledge the financial support received from theFonds Conrad Leblanc, the Industrielle-Alliance Chair in Insurance andFinancial Services, the Institut de Finance Mathématique of Montréal (IFM2)and the Social Sciences and Humanities Research Council of Canada.

References

Anderson, T. W., and D. A. Darling. 1952. “Asymptotic Theory of Certain ‘Goodness of Fit’ CriteriaBased on Stochastic Processes.” The Annals of Mathematical Statistics 23 (2):193–212.

(continued )

Weight

Montreal

MLS mean value (PM) ,, ,, ,, ,, ,, ,, ,, ,, ,,

MLS mean value (PMP) ,,

. MLS increase, % –.% .% –.% −.% −.% .% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


,, ,, ,, ,, ,, ,, ,, ,, ,,

New homes (during

the year)

, , , , , , , , ,

INDEX . . . . . . . . .

Province of Québec

MLS mean value (PM) ,, ,, ,, , , ,, ,, ,, ,,

MLS mean value (PMP) ,,

. MLS increase, % −.% .% −.% −.% −.% .% .% .% .%

Number of homes

(increase, %)

.% .% .% .% .% .% .% .% .%

Number of homes


,, ,, ,, ,, ,, ,, ,, ,, ,,

New homes (during

the year)

, , , , , , , , ,

INDEX . . . . . . . . .

182 A. Têtu et al.



Bantwal, V., and H. Kunreuther. 1999. “A Cat Bond Premium Puzzle?” Working Papers, WhartonSchool Center for Financial Institutions, University of Pennsylvania.

Baryshnikov, Y., A. Mayo, and D. Taylor. 1999. “Pricing Cat Bonds.” Working Paper.Bolder, D. J., G. Johnson, and A. Metzler. 2004. “An Empirical Analysis of the Canadian Term

Structure of Zero-Coupon Interest Rates.” Working Paper, Bank of Canada.Boyle, P., M. Broadie, and P. Glasserman. 1997. “Monte Carlo Methods for Security Pricing.”

Journal of Economic Dynamics and Control 21 (8–9):1267–321.Braun, A. 2012. “Pricing in the Primary Market for Cat Bonds: New Empirical Evidence.” Working

Papers on Risk Management and Insurance No. 116, University of St. Gallen.Burnecki, K., J. Janczura, and R. Weron. 2011. “Building Loss Models.” In Statistical Tools for

Finance and Insurance, edited by P. Cizek, W. K. Härdle, and R. Weron, 293–328.Springer-Verlag Berlin Heidelberg.

Cabrera, B. L., and W. K. Hardle. 2010. “Calibrating Cat Bonds for Mexican Earthquakes.” TheJournal of Risk and Insurance 77 (3):625–50.

Collins, D. J., and S. Lowe. 2001. “A Macro Validation Dataset for US Hurricane Model.” CasualtyActuarial Society, Winter Forum.

Constantin, L. 2011. “Portfolio Diversification through Structured Catastrophe Bonds Amidst theFinancial Crisis.” Economic Sciences Series 63 (3):75–84.

Cummins, J. D. 2008. “Cat Bonds and Other Risk Linked Securities: State of the Market andRecent Developments.” Risk Management and Insurance Review 11 (1):23–47.

Cummins, J. D., and H. Geman. 1995. “Pricing Catastrophe Insurance Futures and Call Spreads:An Arbitrage Approach.” Journal of Fixed Income 4:46–57.

Dassios, A., and J.-W. Jang. 2003. “Pricing of Catastrophe Reinsurance and Derivatives Usingthe Cox Process with Shot Noise Intensity.” Finance and Stochastics 7:73–95.

(continued )

,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

, , , , , , , , , ,

. . . . . . . . . . .

,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.% .% .% .% .% .% .% .% .% .% .%

.% .% .% .% .% .% .% .% .% .% .%

,, ,, ,, ,, ,, , ,, ,, ,, ,, ,,

, , , , , , , , , ,

. . . . . . . . . . .




Duan, J., and M. Yu. 2005. “Fair Insurance Guaranty Premia in the Presence of Risk-BasedCapital Regulations, Stochastic Interest Rate and Catastrophic Risk.” Journal of Banking &Finance 29 (10):2435–54.

D’agostino, R., and M. Stephens. 1986. Goodness-of-Fit Techniques. New York: Marcel.Embrechts, P., and S. Meister. 1997. “Pricing Insurance Derivatives, the Case of Cat Futures.” In

Proceedings of the 1995 Bowles Symposium on Securitization of Risk, 15–26. Atlanta, GA:Georgia State University Atlanta: Society of Actuaries.

Froot, K. A. 2008. “The Intermediation of Financial Risks: Evolution in the CatastropheReinsurance Market.” Risk Management & Insurance Review 11 (2):281–94.

Hainaut, D. 2012. “Seasonality Modeling for Catastrophe Bond Pricing.” Bulletin FrancaisD’actuariat 12 (23):129–50.

Ibragimov, R., D. Jaffee, and J. Walden. 2009. “Nondiversification Traps in CatastropheInsurance Markets.” The Review of Financial Studies 22 (3):959–93.

Insurance Bureau of Canada (IBC). 2008. Facts of the General Insurance Industry in Canada.Intergovernmental Panel on Climate Change Working Groups (IPCC). 2007. Climate Change:

Synthesis Report.Jarrow, R. A. 2010. “A Simple Robust Model for Cat Bond Valuation.” Finance Research Letters

7:72–9.Kortschak, D., and H. Albrecher. 2010. “An Asymptotic Expansion for the Tail of Compound

Sums of Burr Distributed Random Variables.” Statistics and Probability Letters 80:612–20.Lane, M., and O. Mahul. 2008. “Catastrophe Risk Pricing: An Empirical Analysis.” Policy

Research Working Paper Series, The World Bank.Li, X. 2007. “A Novel Accurate Approximation Method of Lognormal Sum Random Variables.”

M.Sc. thesis, Wright State University.Lin, S.-K., C.-C. Chang, and M. R. Powers. 2009. “The Valuation of Contingent Capital with

Catastrophe Risks.” Insurance: Mathematics and Economics 45:65–73.Litzenberger, R. H., D. R. Beaglehole, and C. E. Reynolds. 1996. “Assessing Catastrophe

Reinsurance-Linked Securities as a New Asset Class.” Journal of Portfolio Management23:76–86.

Louberge, H., E. Kellezi, and M. Gilli. 1999. “Using Catastrophe-Linked Securities to DiversifyInsurance Risk – A Financial Analysis of Cat Bonds.” Journal of Insurance Issues 22 (2):125–46.

Ma, Z.-G., and C.-Q. Ma. 2013. “Pricing Catastrophe Risk Bonds: A Mixed ApproximationMethod.” Insurance: Mathematics and Economics 52:243–54.

Merton, R. 1976. “Option Prices When Underlying Stock Returns Are Discontinuous.” Journal ofFinancial Economics 3:125–44.

Nowak, P., and M. Romaniuk. 2013. “Pricing and Simulations of Catastrophe Bonds.”Insurance: Mathematics and Economics 52:18–28.

Ramsay, C. M. 2006. “The Distribution of Sums of Certain I.I.D. Pareto Variates.”Communications in Statistics – Theory and Methods 35 (3):395–405.

Sandink, D., P. Kovacs, G. Oulahen, and G. Mcgillivray. 2010. “Making Flood Insurable forCanadian Homeowners.” Discussion Paper, Swiss RE.

Wang, S. 2004. “Cat Bond Pricing Using Probability Transforms.”Geneva Papers: Etudes et Dossiers,special issue on “Insurance and the State of the Art Cat Bond Pricing” No. 278, 19–29.

184 A. Têtu et al.



Documents

Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaré*and … · 2017. 9. 8. · Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaré*and Michel Gendron Hedging Flood

Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaréand … · 2017. 9. 8. · Featured Article AlexandreTêtu, Van Son Lai, IssoufSoumaréand Michel Gendron Hedging Flood