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The Impact of Climate Change

on Catastrophe Bonds

Hendrik Rozema

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies

Supervisor: dr. E.L. Kramer Second assessor: dr. L. Dam

Acknowledgements

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The Impact of Climate Change

on Catastrophe Bonds

Hendrik Rozema

Abstract

With increasing global warming, the amount of damage caused by extreme weather is expected to increase. Since most catastrophe (cat) bonds are linked to extreme weather events, this will have impact on the cat bond market. This thesis examines the effects of climate change on risk and return of US wind related cat bonds. A time series linear regression model is used to model US storm related cat bond returns as a function of various variables. It is found that the BB corporate spread and US storm damages have a significant effect on the cat bond returns. Using projections for these variables, scenarios of future returns are estimated and compared for three different climate pathways. From these scenarios it follows that in the absence of extra risk premiums, expected returns are lower on the long term under climate pathways with more global warming. Furthermore, standard deviations are higher under climate pathways with

more global warming. Compared to a world without global warming,

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Contents

1 Introduction 5

2 Theoretical background 7

2.1 Cat bonds and the development and current state of the market 7

2.2 Literature review . . . 9

2.3 The problem . . . 11

2.4 Research question . . . 12

2.5 Hypotheses . . . 12

3 Methodology 14 3.1 Modeling Cat bond returns . . . 14

3.1.1 Time series linear regression model . . . 14

3.1.2 ARIMAX model . . . 17

3.2 Modeling future implications . . . 17

3.2.1 Scenario analysis . . . 18

3.2.2 Modeling monthly storm damages . . . 18

4 Data 20 4.1 Data used for the scenarios . . . 31

5 Results 35 5.1 Model selection . . . 35

5.2 Future implications . . . 47

5.2.1 Simulating monthly storm damages . . . 51

5.2.2 Future implications using scenarios . . . 55

5.2.3 Sensitivity analysis . . . 61

6 Conclusion 73

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1

Introduction

The insurance industry is exposed to enormous risks from natural disasters like hurricanes. In 1992, for example, insured losses from hurricane Andrew were as high as $16 billion. Insured losses caused by hurricane Katrina in 2005 were estimated to range from $40-$60 billion (Muermann, 2008). Even only one event can cause serious problems for an insurance company (Nowak and Romaniuk, 2013). Hurricane Andrew, for example, resulted in more than 60 insurance com-panies becoming insolvent.

Most traditional insurance portfolio models focused on the independence of claims. However, claims arising from natural disasters are obviously not inde-pendent. Therefore, new approaches were needed to build insurance company portfolios (Nowak and Romaniuk, 2013).

After 1992, large insurance and reinsurance companies initially reduced their

exposure to natural catastrophic events. Insurance companies bought more

reinsurance form reinsurance companies that were less affected by losses from hurricane Andrew. Furthermore, increasing demand for insurance related to natural disasters meant that more capital was needed in reinsurance (Polacek et al., 2018).

An approach to raise more capital is the securitization of losses. This means that the risks of natural disaster are ”packaged” into tradable assets that are called catastrophe derivatives (Nowak and Romaniuk, 2013). The most popular catastrophe derivative is the catastrophe (cat) bond, which was introduced in late 1990’s by the insurance industry to increase the available capital (Polacek et al., 2018).

Cat bonds are insurance linked securities that transfer risks arising from disas-ters from the issuer to investors. When investors buy a cat bond, they take over the risk of the occurrence of a specified event, for example, a tropical storm. If this specified event occurs, the investor will lose (part of) his investment and the issuer uses this money to cover the damage. In return, the investors receive a premium.

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in frequency.

Since most cat bonds are linked to extreme weather events, climate change will affect the cat bond market (Morana and Sbrana, 2019). Moreover, climate change may also affect corporate bond prices (Dafermos, Nikolaidi, and Galanis, 2018). As corporate bond spreads are found to be a determinant of cat bond

risk and return characteristics (G¨urtler, Hibbeln, and Winkelvos, 2016), the cat

bond market may also be affected by climate change indirectly.

In this thesis, the future implications of climate change on cat bond risk and return are examined. This is done by first modeling cat bond returns as a func-tion of various explanatory variables. Then, using scenarios and projecfunc-tions of the explanatory variables, cat bond risk and return are estimated for different climate pathways.

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2

Theoretical background

2.1

Cat bonds and the development and current state of

the market

After hurricane Andrew in 1992, the insurance and reinsurance industry started looking for alternative methods to hedge their risks related to natural disasters. The first attempt to introduce catastrophe securities was by the Chicago Board of Trade (CBOT). They introduced catastrophe futures in 1992. Later, they also introduced put and call options. However, investors were not really inter-ested in these contracts for multiple reasons (Cummins, 2008).

In the late 1990’s, the first cat bonds were introduced. As mentioned in the introduction, cat bonds let investors take over the risks of the issuer. Cat bonds are often issued to cover the high layer of reinsurance. This means they protect against events that occur with a probability of 1% or less (Cummins, 2008). These events are often natural disasters. However, also other types of risk, such as cyber attacks and terror risk, are examined to be covered via cat bonds the last years (Polacek et al., 2018).

Nowadays, cat bonds are perceived to be the most successful among the various

alternative risk transfer measures (Braun, M¨uller, and Schmeiser, 2013). The

popularity of cat bonds has grown over the years. Figure 1 shows the risk cap-ital issued and risk capcap-ital outstanding per year. For both an increasing trend over time is observed.

Figure 1: Risk capital issued and outstanding per year.

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the sponsor. In return, the SPV provides the coverage via issuing securities to investors. Moreover, the SPV receives the principal of the investors. The principal is then deposited in a collateral account. From here, it is invested, mostly in safe short term securities, such as government bonds or AAA rated corporate bonds (Cummins, 2008). The coupon payments to the investors are then determined by the premiums the sponsor pays plus the interest payment the SPV receives from the collateral account.

If the specified conditions that activate a payout have not been met before maturity, the SPV liquidates the collateral at maturity and the investors are repaid. However, when the specified trigger conditions are met before maturity, the SPV immediately liquidates the collateral and pays out the amount specified in the agreement, and the investors lose (part of) their investment.

Figure 2: Typical structure of a cat bond.

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Figure 3: Part of risk capital outstanding per trigger type.

A reason that cat bonds are attractive to investors, is that cat bonds are of-ten perceived to have low correlation with investment returns (see Litzenberger, Beaglehole, and Reynolds (1996)). Therefore, cat bonds can be used for diver-sification purposes (Cummins, 2012). Furthermore, since cat bonds are fully collateralized, there is no exposure to credit risk (Cummins, 2008). This is also a feature that attract investors.

2.2

Literature review

Most studies on cat bonds focus on the valuation of the bonds. A study by Litzenberger et al. (1996) is one of the first studies that does this. They price a one year zero-coupon bond and compare prices estimated by a bootstrap

ap-proach with prices estimated from the hypothetical loss distribution. Louberg´e,

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the best. Zimbidis, Frangos, and Pantelous (2007) use extreme value theory to model losses. By using real data on earthquake losses, they value cat bonds for

earthquakes in Greece. H¨ardle and Cabrera (2010) use real data to calibrate a

cat bond for Mexican earthquakes in continuous time. Furthermore, similar to Lee and Yu (2002), they take basis risk and moral hazard into account. Nowak and Romaniuk (2013) propose a pricing formula that can be used for cat bonds with different payoff functions. Moreover, they use different models for the in-terest rate dynamics.

Another part of the existing literature is about whether cat bonds are uncor-related with other financial markets, which is often assumed. For example, Litzenberger et al. (1996) argue that cat bonds have low correlation with the stock and bond market. This is supported by Hoyt and McCullough (1999). They review catastrophe insurance options and found that these options are so-called zero-beta assets. From this, it follows that these options could be used for diversification purposes. Furthermore, Tao (2011) found that cat bonds are uncorrelated with European stock markets and US stock markets. They show their beta can be approximated by zero, even in the financial crisis of 2007 − 2009. They conclude that cat bonds could be used for diversification, since including them in an investment portfolio shifts their efficient frontier to the upper left. By using a multivariate GARCH model, Carayannopoulos and Perez (2015) found that cat bonds are zero beta assets only in non-crisis periods. They found significant correlation between returns from cat bonds and other fi-nancial markets during the fifi-nancial crisis of 2007 − 2009. However, as the effect of the financial crisis was small relative to other asset classes, they argue that cat bonds are still a valuable source of diversification for investors. Clark, Dickson, and Neale (2016) also used a GARCH model. They also found diversification

benefits of including cat bonds in a portfolio. G¨urtler et al. (2016) argue that

cat bonds should not be perceived as zero beta securities, since there exists a positive relation between corporate bond spreads and cat bond premiums. This relation became even stronger during the financial crisis in 2008. Hence, they argue that when diversification is needed the most, investors must be aware that the correlation between cat bonds and other securities becomes bigger.

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spread is found to be significant and positive. G¨urtler et al. (2016) also found that the risk premiums for cat bonds were positively affected by hurricane Kat-rina. They argue that this increase is mainly driven by an increase in expected loss coefficients derived by modeling companies. After hurricane Ike in 2008, the increase in risk premium was smaller. They also found that spreads are affected by the BB corporate spread, their rating and deal complexity in terms of underlying perils and regions. Similar to Lane and Mahul (2008), they also found that spreads are affected by the reinsurance cycle. For the reinsurance cycle they used the Guy Carpenter Global Property Catastrophe Rate-On-Line Index, which is a measure of the change in dollars paid for coverage per year for all major global catastrophe reinsurance markets. Braun (2012) found that the spread is also affected by which company issues the bond.

Lastly, there is one study that focuses on climate change implications for cat bonds. Using the federal fund rate, corporate bond spreads and climate vari-ables related to US hurricanes, Morana and Sbrana (2019) try to explain the dynamics of the average multiple of cat bonds. The average multiple is defined by the average coupon rate divided by the average expected loss in the cat bond market. The expected loss for a certain cat bond is calculated by modeling companies and is expressed as a percentage. Morana and Sbrana (2019) found a negative effect for various climate related variables on the average multiple. This means that the average expected loss grows harder with climate change than the average coupon rate. Hence, they conclude that climate change is undervalued in the cat bond multiple.

2.3

The problem

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this will develop in the future.

Moreover, most studies agree that the cat bond coupon rates depend on the corporate spreads. These corporate spreads might be different depending on the climate pathway. Hence, the impact of the spreads on risk and return char-acteristics might also differ for different climate pathways.

The capital outstanding in the cat bond market has grown over the last years. According to Morana and Sbrana (2019) the cat bond market has a large growth potential. With this growing popularity it becomes more important to look at the future implications of climate change on the cat bond market.

2.4

Research question

The goal of this thesis is to get insight in how the cat bond market will be affected by climate change. Therefore, our research question is the following: ”To what extend will climate change affect the cat bond market? And in how far will this impact differ among different global warming pathways?”

To answer this question, we look at which variables explain cat bond returns. Furthermore, we look at the future dynamics of the cat bond returns under different climate pathways. Therefore, to answer our main research question, we address the following two subquestions:

”How will cat bond risk and return characteristics differ among different cli-mate pathways?”

”Should there be an extra risk premium to compensate for climate change under less favorable climate circumstances? And if so, how much should it be?”

2.5

Hypotheses

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H1: Storm damages have a negative effect on cat bond returns.

As the climate change measures that affect the cat bond multiple become more extreme in less favourable climate pathways, we expect damages to be bigger when there is more global warming. Therefore, we expect that returns will be lower in less favourable climate pathways if extra risk premiums are not intro-duced. This leads to the following hypothesis.

H2: Without extra risk premiums, cat bond returns will be lower in climate pathways with more global warming than in climate pathways with less global warming.

Furthermore, since we expect that more extreme weather leads to more ex-treme losses on the cat bond market, we expect the standard deviation of the returns to be higher in less favorable climate pathways. Hence we have the following hypothesis:

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3

Methodology

3.1

Modeling Cat bond returns

Most studies on cat bonds focus on determinants of spreads and valuation of specific cat bonds. Only a limited amount of studies consider cat bond returns. Carayannopoulos and Perez (2015) and Clark et al. (2016) use a multivariate GARCH model to study correlations with other asset classes like equity and corporate bonds. The only study that looks at returns itself is by Trottier, Godin, et al. (2019). They use a regime switching GARCH model to explain

dynamics of multiple cat bond performance indices. However they did not

include explanatory variables. Since we study the effect of climate change on cat bond returns through explanatory variables, we start by using a linear time series regression. To make sure we have enough data points, we use monthly cat bond returns.

3.1.1 Time series linear regression model

The first model we consider in order to model the cat bond returns, is the time series linear regression model. This model is given by the following equation:

rcb,t= α + β0xt+ t. (1)

Here, rcb,tis the cat bond return at time t, xtis a vector of explanatory variables

at time t and t is the error term at time t.

We want to use a time series regression to estimate future cat bond returns. Unfortunately, we can therefore only use explanatory variables for which we have projections or for which we can make reasonable assumptions. We have projections for US corporate spreads, US treasury bill rates, global temperature anomaly and US storm damages. Below it is explained in more detail which explanatory variables are used.

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US storms that cause big damages are likely to affect cat bond returns, as they can trigger cat bonds. If cat cat bonds are triggered, (part of) the prin-cipal will be used to cover damages. This means that the market value of the bond decreases, or becomes even zero when investors lose their full principal. Hence, the total monthly US storm damage will be included as an explanatory variable. Also, some other specifications of the monthly damage are considered. These will be explained in more detail in the results section.

Morana and Sbrana (2019) found that various climate change related variables negatively affect the cat bond multiple. That variables related to hurricane intensity have negative effects is what one would expect, since higher intensity will lead to higher damages. However, Morana and Sbrana (2019) also found that temperature volatility affected the multiple, which is a less straightfor-ward result. To see whether we can obtain similar results, we include monthly temperature anomalies as well as the 12 month moving average of temperature anomalies. Unfortunately, we can only include variables for which projections are available. Moreover, we do not have access to data on temperature volatil-ity on a monthly basis. Therefore, we can not include temperature volatilvolatil-ity. However, since we want to examine whether there is a direct relation of global warming and cat bond returns, temperature anomalies should also be a good candidate.

As figure 1 showed, the size of the market has increased over time. Braun (2016) found some evidence that issue volume negatively affects spreads at the individual cat bond level. This might also be true for the market as a whole, leading to lower returns. The cat bond market is relatively young. It could be that in the first years a a liquidity premium was paid. As the market became more mature and therewith more liquid, this premium then decreased. To ac-count for this effect, we add the natural logarithm of volume outstanding and the natural logarithm of the amount issued each month to our model. We take natural logarithms since we believe the liquidity premium does not decrease linearly with volume outstanding or amounts issued. Instead, we believe that it decreases faster at the beginning, but as the market becomes more mature, the decrease in liquidity premium will slow down. Unfortunately, we do not have projections for the market volume and future issued amounts. Therefore, we assume that the market volume and the issued amount remain constant at the current level. Since we use the natural logarithm and since the market volume and issued amount in 2019 US dollars did not change much the last year, we believe this assumption will not have large impact on our results.

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risky period for cat bonds. As often, higher risk coincides with higher expected returns. This reasoning is confirmed if we compare the standard deviations of the returns for each month of the year, meaning that standard deviations are also higher during the hurricane season. To account for this seasonality,

we include the categorical variable montht. The levels of this variable equal

the months of the year. The month January is taken as the reference variable. Therefore, the coefficients of the other months are with respect to January.

Fur-thermore we try some other specification for the the variable montht. These

will be explained in more detail in the results section.

Lane and Mahul (2008) and G¨urtler et al. (2016) found that cat bond spreads

are affected by the reinsurance cycle. Therefore, it would have been interesting to include the Guy Carpenter Global Property Catastrophe Rate-On-Line Index as a measure for the reinsurance cycle. Unfortunately, we do not have projec-tions for this index. Since it is hard to make reasonable assumpprojec-tions about this index we do not include it in our model. However, to see whether this index would have improved our final model, a model with the index included is given in the appendix.

Summarizing, the following variables are considered as explanatory variables:

• BBspreadt−1, which is the BB corporate spread at time t − 1;

• ∆BBspreadt, which is the difference between the BB spread at t and t−1;

• ∆BBspread2

t, which is the squared difference between the BB spread at

t and t − 1;

• U S3Mt−1, which is the 3-month US treasury bill rate at t − 1;

• Damaget which is the total damage from US storms during month t;

• T emperatureAnomalytwhich is the temperature anomaly in month t;

• M A12T emperatureAnomalyt, which is the 12-month moving average

tem-perature anomaly over the last 12 month up to month t;

• log(V olumet) which is the natural logarithm of the amount of capital

outstanding in the cat bond market in month t;

• Issuet, which is the natural logarithm of the average monthly amount

issued in month t;

• M ontht, which is a categorical variable , where the reference level is

Jan-uary and the other levels are the other months of the year.

We try different models, all with different explanatory variables. To compare the fit of the models, we make use of the following measures of fit: R2, adjusted R2,

absolute mean error (AME), root mean squared error (RMSE) and corrected

Akaike information criterion (AICc). The AME and RMSE are calculated by

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predictions for the last 24 months and finally comparing them with the observed

returns of the last 24 months. The AICc is similar to AIC, but with a correction

for small sample sizes N . Note that for large N , AICc converges to AIC.

3.1.2 ARIMAX model

As we will see in the next section, there is a significant autocorrelation between the cat bond returns and their first lag. To include lagged values of the cat bond returns, we consider the auto regressive integrated moving average with exogenous variables (ARIMAX) model of order (p,d,q). The ARIMAX(p,d,q) model is given by equation 2.

∇drcb,t= p X j=1 ∇drcb,t−j+ q X j=0 t−j+ β0xt (2)

Here, ∇dis the difference operator with d as the number of first differences, r cb,t

is the cat bond return at time t and xt is a vector of explanatory variables at

time t. Now t is again the error term, but lagged values of this error term are

also included.

In the ARIMAX model, we use the same explanatory variables as for the linear

time series regression model described above. Furthermore, we use AICc values

to compare the fits of different models, and from therewith to choose p and q. An augmented Dickey-fuller (ADF) test will be used to determine how many first differences are needed to make the model stationary, i.e., to select d.

3.2

Modeling future implications

Once we have selected a model to estimate cat bond returns, we can look at the future implication on cat bond returns. We do this by comparing cat bond returns under three different climate pathways. The three climate pathways we consider are the following:

• no global warming: global warming is ignored;

• Paris transition: global warming does not exceed 2 degrees by the year the 2100;

• failed transition: global warming is 4 degrees by the year 2100.

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storms are multiplied by the expected growth under the corresponding climate pathway.

To compare the returns under the different climate pathways, we first compute the expected monthly returns from our model. Then, to keep everything acces-sible, we compute the yearly returns from the monthly returns geometrically, and compare these for the three climate pathways.

3.2.1 Scenario analysis

To obtain more realistic results, we also perform a scenario analysis. For each climate pathway, we simulate scenarios for the future returns. For this, we use scenarios of monthly values for the financial variables. Furthermore we use sce-narios for US storm damages. Once we have the scesce-narios of monthly returns for each climate pathway, we again compute from them the yearly returns geo-metrically. Finally, we compare expected returns, median returns and standard deviations of returns. Furthermore we compute possible extra risk premiums that compensate for negative climate change effects.

3.2.2 Modeling monthly storm damages

Since we do not have monthly scenarios for US storm damages, we have to simulate them ourselves. The first step we take to model monthly US storm damages is to fit a probability distribution to the historical storm data. Since storm damages are not identically distributed among the months of the year, we consider different distributions for different periods of the year.

Similar to for example Louberg´e et al. (1999) and Burnecki and Kukla (2003),

we assume that the frequency of the catastrophes (in our case US storms) is distributed independently from the severity as a Poisson distribution. For the intensity rate of the Poisson distribution, λ, we simply use the historical av-erage number of storms per month. For the damage caused by a storm, we consider different distributions. Loss distributions, especially those related to property losses, are usually heavy-tailed. Following Burnecki and Kukla (2003), we therefore consider the following distributions:

• the lognormal distribution, with the distribution function (d.f.) F (x, µ, σ) = Φ(log(x) − µ σ ) = Z x 0 1 √ 2πσye −1 2( log(y)−µ σ ) 2 dy, where Φ denotes the standard normal distribution, x > 0, σ > 0 and

µ ∈R;

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where x > 0, α > 0 and β > 0; • the pareto distribution, with the d.f.

F (x, α, β) = (

1 − (βx, if x ≥ β

o, if x < β

where x > 0, α > 0 and β > 0.

To select the appropriate distribution we use, similar to Burnecki and Kukla

(2003), the following four Goodness of fit tests: the χ2 test, the

Kolmogorov-Smirnov test, the Andreson-Darling test and the Cramer-von Mises test. To simulate monthly storm damages for the different climate pathways, we make use of the fact the that the monthly storm damage follows a compound Poisson process. This means that if D is the total storm damage in a month,

then D =PN

i=1Xi. Here, N follows a Poisson distribution with intensity rate

λ and the individual damages X1, X2,... are i.i.d. random variables.

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4

Data

For the cat bond returns, we use the Swiss Re US Wind Cat Bond Perfor-mance Index Total Return, obtained from Thomson Reuters Eikon (Thomson Reuters Eikon, 2020c). This index tracks the aggregate performance of USD de-nominated cat bonds, that are exposed to US Atlantic Hurricanes. Cat bonds exposed to hurricanes outside the US are not considered. Swiss Re has multi-ple cat bond indices. Our choice for the US wind index is based on the fact that we want to examine climate change effect. Since, other indices are partly based on cat bonds related to non-weather events, we decide to choose the Swiss Re US wind cat bond performance index. For this index, there are 3 different time series. First, the coupon return series (Thomson Reuters Eikon, 2020a), which accounts for accrued coupon and collateral return. Second, the price return series (Thomson Reuters Eikon, 2020b), which measures the movement of secondary bid indications. Lastly, there is the total return series, which is the composite of the previous series. We will use the total return series in our analysis, since we believe climate change may affect both, coupon and collateral return and price movements. The returns that follow from the other series are given graphically at the end of this section.

Other data we consider are the ICE BofA BBB US Corporate Index Total Return Index Value (FRED, 2020e), the ICE BofA BBB US Corporate Index Option-Adjusted Spread (FRED, 2020d), the ICE BofA BB US High Yield In-dex Total Return InIn-dex Value (FRED, 2020c), ICE BofA BB US High Yield Index Option-Adjusted Spread (FRED, 2020b) and the 3-month US treasury bill rate (FRED, 2020a), all obtained via the Federal Reserve Economic Data

(FRED) website. To account for climate change impact, we have data on

monthly global temperature anomalies and US storm damages, obtained from the National Oceanic and Atmospheric Administration (NOAA) (NOAA, 2020) and EM-DAT (EM-DAT, 2020) respectively. The temperature anomalies are measured with respect to the 20’th century average in degrees Celsius. The storm damages are given per storm event. Note that a storm only enters into the EM-DAT database if at least one of the following criteria in fulfilled: 10 or more deaths, 100 or more people affected/injured/homeless or declaration of a state of emergency and/or an appeal for international assistance by the country where the storm took place. Hence, only big storms are in the EM-DAT database. Lastly, for the cat bond market, we have yearly data on the volume outstanding and the yearly issued volume. This data is obtained from ARTEMIS (ARTEMIS, 2020a).

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damages are assigned to a certain month if the end date of the storm was in that month. We removed all storm events of which no damage costs are known from the dataset. To obtain monthly values for the volume outstanding in the cat bond market, we linearly interpolated the yearly volumes over the 12 months of the year. Since the data was obtained at the end of October, we linearly interpolated the volumes of 2020 over 10 months. Lastly, for the monthly issued amounts, we compute the average amount per month for each year. Again, we take into account that we have 10 months in 2020. Both the volume and issued amounts are in millions of 2019 US dollars

Table 1 shows the summary statistics of all the variables we will use in our models. Furthermore, to compare return characteristics, the returns on the BBB and BB index are also given in table 1. When we compare the return on cat bonds with the returns on BB and BBB corporate bonds, we see that the average return for cat bonds is higher than the return on BBB rated corporate bonds. Compared to BB rated corporate bonds, it is slightly lower. The stan-dard deviation of the cat bond returns is much lower than that of both types of corporate bonds. By looking at the minimum and maximum monthly returns, we see that, for corporate bonds, the downside is more extreme and the upside potential is higher. Hence, compared to the corporate bonds, investing in cat bonds seems to be safe. However, if we look at the median returns, we see that these are higher for corporate bonds. Both, cat bond returns and corporate bond returns are negatively skewed. Though, the corporate bond returns are highly skewed and the cat bond returns only moderately. When we look at the correlations, we see that there is a small positive correlation between the returns on the cat bond index and the corporate bond indices. The cat bond returns seem to be uncorrelated with the corporate spreads and the 3-month US Treasury bill rate. When we look at the climate related variables, we see that the monthly storm damage has a negative correlation with cat bond returns. This is what one would expect. Furthermore, the correlation of the temperature anomaly with cat bond returns is a bit smaller, but also negative. The correla-tion with cat bond returns for the volume outstanding and the issued amounts are also negative.

rcb rBBB BBBspread rBB BBspread US3M Damage Temp. ano. Volume Issued

Mean 0.59 0.53 2.13 0.61 3.85 1.29 3.74 1.13 20060 607 Std. dev. 1.10 1.87 1.13 2.28 1.94 1.50 14.77 0.37 10710 325 Min -5 -11 1.12 -15.13 1.74 0 0 0.22 4276 129 Max 4.46 6.29 7.84 7.59 13.96 5.16 163.64 2.53 43677 1250 Median 0.48 0.57 1.85 0.86 3.43 0.92 0.35 1.11 17473 586 Skewness -0.51 -1.78 2.90 -1.93 2.38 1.25 7.71 0.72 0.45 0.36 Corr with CAT USW 1.00 0.28 -0.07 0.23 -0.01 0.07 -0.24 -0.16 -0.19 -0.13 N 224 224 224 224 224 224 224 224 224 224

Table 1: Summary statistics of cat bond returns, returns on corporate bonds, corporate spreads, 3-month US treasury bill rate, monthly storm damage, temperature anomaly, volume outstanding and amount issued.

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plotted. Moreover, we numbered some of the biggest losses. Overall, we see that before 2008, the monthly returns increased. Around the financial crisis in 2008 and 2009, there was a big decrease in returns. In the years after the crisis, the returns were back at pre-crisis levels. For approximately the last 8 years, we see a decrease in the returns. Especially in the last 3 years, there were some big losses.

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Figure 4: Swiss Re US wind cat bond performance index returns and the 12-month moving average in percentage points.

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Figure 5: Autocorrelation function for Swiss Re US wind cat bond performance index returns.

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Figure 6: Swiss Re US wind cat bond performance index returns in percentage points, monthly total damage in billions of 2019 US Dollars and monthly temperature anoma-lies in Celcius degrees.

In table 2, we have the 10 biggest monthly damages and the cat bond returns in the same month. The returns in these months are all negative or close to zero except for the return in October 2005. From this table, we again observe that cat bond returns are sensitive to US storm damages.

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Date Damage rcb 1 Aug 2004 21.74 -0.45 2 Sep 2004 50.08 -1.12 3 Sep 2005 163.64 -0.10 4 Oct 2005 39.67 0.90 5 Sep 2008 43.94 -1.85 6 Apr 2011 18.75 -0.30 7 Oct 2012 55.68 0.02 8 Aug 2017 99.29 -1.03 9 Sep 2017 59.45 -5.00 10 Oct 2018 16.29 -0.70

Table 2: 10 biggest monthly damages in billions of 2019 US dollars and the cat bond returns in the corresponding months in percentage points.

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Figure 7: Swiss Re US wind cat bond performance index returns, return on the BBB corporate index and return on the BB corporate index in percentage points.

In tables 3 and 4 we have the 10 biggest monthly losses and the 10 highest monthly returns from the BB corporate bond index respectively. Furthermore, the returns on cat bonds in the corresponding months are in these tables. If we look at the losses, half of the corresponding cat bond returns are also negative. The other returns are not higher than the historical average return. Especially for the big losses on the BB index, we find big losses for the cat bonds as well. If we look at the highest returns on the BB corporate bonds, we obtain 9 positive returns for cat bonds. From them, 6 are above the average return.

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Date BB index ret rcb 1 jun 2002 -8.15 0.52 2 Jul 2002 -4.36 0.47 3 Mar 2005 -2.91 0.59 4 Sep 2008 -7.12 -1.85 5 Oct 2008 -15.13 -2.09 6 Nov 2008 -3.47 0.14 7 May 2010 -2.86 -0.43 8 Aug 2011 -2.89 -0.26 9 Jun 2013 -2.84 0.44 10 Mar 2020 -9.27 -3.13

Table 3: 10 biggest monthly losses from the ICE BofA BB US High Yield Index and the cat bond returns in the corresponding month in percentage points.

Date BB index ret CAT USW

1 dec 2008 6.45 0.33 2 Jan 2009 7.07 0.14 3 Apr 2009 7.59 0.52 4 May 2009 4.91 -0.01 5 Jul 2009 4.64 1.11 6 Sep 2009 4.42 4.46 7 Oct 2011 4.93 0.95 8 Jan 2019 4.30 1.26 9 Apr 2020 4.78 0.89 10 Jul 2020 5.00 1.23

Table 4: 10 highest monthly returns from the ICE BofA BB US High Yield Index and the cat bond returns in the corresponding month in percentage points.

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Figure 8: Cat bond returns, the BB corporate spread, BBB corporate spread and the 3-month US treasury bill rate in percentage points.

Figure 9 shows the cat bond return from price movements together with the BB corporate spread and the monthly damage. Obviously, the returns from price movements are lower than the total returns. However, the shape is almost

identical. Hence, if we compare the returns from price movement with the

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Figure 9: Cat bond returns from price movements in percentage points, the BB corpo-rate spread in percentage points and the monthly US storm damage in billions of 2019 US dollars.

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4.1

Data used for the scenarios

For the financial variables we have 2000 scenarios of the monthly values for each climate pathway. We have them from January to December for the period from 2021 until 2059. Furthermore, we have projections of of frequency and severity of US storm damages for each climate pathway. The scenarios as well as the pro-jections for storm damages are provided by ORTEC Finance. ORTEC Finance is an independent provider of technology and solutions for risk and return man-agement. The scenarios are from ClimateMAPS, which quantifies the systemic impact of climate change on the real economy and financial markets. This tool is developed together with Cambridge Econometrics and combines climate sci-ence with macro economic and financial modeling, to obtain real world scenario sets. In short, the methodology behind the scenarios is as follows. For each climate pathway, the impact of global warming and technological and policy changes are based on robust climate science. These assumptions are then used in the macro econometric model of Cambridge Econometrics. This model is a global macro econometric model that accounts for interactions between econ-omy, energy sector and environment. The output from this model, which are differences in annual growth rates per country, are then used in the ORTEC Finance stochastic financial model. Based on stylized facts and economic rea-soning, this model translates the climate adjusted GDP shocks into scenarios for various financial variables. For examples of other studies that use these sce-narios, see Bongiorno, Claringbold, Eichler, Jones, Kramer, Pryor, and Spencer (2020) and Ma, Caldecott, and Volz (2020) (p. 276-299).

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Figure 11: Projections of frequency of US extreme weather events of a failed transition.

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Figure 13: Projections of frequency of US extreme weather events of a Paris transition.

Figure 14: Projections of total damage from US extreme weather events for a Paris transition.

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5

Results

5.1

Model selection

When selecting the best model, we start by running a model with all our ex-planatory variables included. However, we have two measures to account for

direct climate change effects: T emperatureAnomalytand

M A12T emperatureAnomalyt. We have to select one of the two, since these

variables are obviously correlated. Using both will cause multicollinearity. The

same holds for log(volumet) and log(issuet), the two variables to account for

possible liquidity premiums.

In table 5 we have the results of different time series linear regression models, all with different combinations of the variables mentioned above. The results are obtained by OLS. The first thing we observe is that in none of the six models, there is a significant effect for any of the two variables directly related to climate

change. Both log(volumet) and log(issuet) are negative in all models except

model 4, where log(issuet) is positive. Moreover, log(volumet) is statistically

significant at a 10%-level in model 5 and at a 5%-level in model 1. From this it seems that the decrease in cat bond returns over time is rather due to increasing market volume than through increasing temperature anomalies. This result is a bit in contrast with what Morana and Sbrana (2019) found. They found a negative effect of temperature volatility on the cat bond multiple. That Morana and Sbrana (2019) found a negative affect for temperature volatility, could be explained by the fact that temperature volatility is a different variable than temperature anomaly. Furthermore, more temperature volatility could lead to increasing uncertainty which lowers the multiple. However, since Morana and Sbrana (2019) did not account for (changes in) market volume, the negative effect they found could also be the result of temperature volatility moving in the same direction as market volume. That this is the case for the tempera-ture anomaly is confirmed by table 17 in the appendix. In this table, we have the output from a model with the 12-month moving average and without the

variables log(volumet) and log(issuet). When we do not account for (changes

in) market volume, the negative coefficient of M A12T emperatureAnomalyt

be-comes significant at a 10%-level.

Since there is no significant effect for T emperatureAnomalytor

M A12T emperatureAnomalytin any of the first four models, two models

with-out these variables are included in column 5 and 6. The models with volumet

(model 1,2 and 5) have higher R2and adjusted R2than the models with issuet

(model 3,4 and 6). For the AME, RMSE and AICcthe same is true. Therefore,

we decide to proceed with volumetas a variable to account for liquidity

premi-ums. In other words, we have to choose the best model out of model 1, 2 and 5. When comparing the measures of fit, we see that model 5 is better than the

others in terms of adjusted R2 and AIC

c. Furthermore, the RMSE of model

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Dependent variable: rcb,t (1) (2) (3) (4) (5) (6) BBspreadt−1 0.066∗ 0.059 0.070∗ 0.063 0.064 0.069∗ (0.040) (0.040) (0.041) (0.041) (0.040) (0.041) ∆BBspreadt −0.155 −0.157 −0.152 −0.159 −0.152 −0.151 (0.132) (0.132) (0.133) (0.132) (0.131) (0.132) ∆BBspread2 t −0.179∗∗∗ −0.177∗∗∗ −0.184∗∗∗ −0.180∗∗∗ −0.177∗∗∗ −0.184∗∗∗ (0.053) (0.053) (0.053) (0.053) (0.053) (0.053) U S3Mt− 1 0.061 0.065 0.087∗ 0.080∗ 0.062 0.087∗ (0.047) (0.047) (0.045) (0.045) (0.047) (0.045) Damaget −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) T emperatureAnomalyt 0.131 0.039 (0.203) (0.203) M A12T emperatureAnomalyt −0.306 −0.596 (0.409) (0.381) log(V olumet) −0.244∗∗ −0.133 −0.210∗ (0.120) (0.148) (0.107) log(Issuet) −0.119 0.020 −0.109 (0.115) (0.133) (0.104) February −0.462 −0.453 −0.453 −0.445 −0.457 −0.452 (0.299) (0.299) (0.301) (0.299) (0.298) (0.300) March −0.474 −0.432 −0.443 −0.423 −0.437 −0.432 (0.305) (0.299) (0.307) (0.300) (0.299) (0.301) April −0.432 −0.423 −0.425 −0.415 −0.426 −0.423 (0.300) (0.300) (0.302) (0.300) (0.299) (0.301) May −0.223 −0.240 −0.237 −0.231 −0.243 −0.242 (0.302) (0.301) (0.305) (0.301) (0.300) (0.302) June −0.234 −0.253 −0.255 −0.247 −0.256 −0.261 (0.301) (0.299) (0.303) (0.300) (0.299) (0.301) July 0.141 0.112 0.112 0.117 0.110 0.104 (0.302) (0.298) (0.304) (0.299) (0.298) (0.300) August 0.785∗∗ 0.758∗∗ 0.753∗∗ 0.761∗∗ 0.756∗∗ 0.745∗∗ (0.305) (0.301) (0.307) (0.302) (0.301) (0.303) September 1.263∗∗∗ 1.243∗∗∗ 1.234∗∗∗ 1.244∗∗∗ 1.241∗∗∗ 1.228∗∗∗ (0.311) (0.309) (0.313) (0.310) (0.309) (0.311) October 0.793∗∗ 0.773∗∗ 0.777∗∗ 0.778∗∗ 0.774∗∗ 0.772∗∗ (0.308) (0.306) (0.310) (0.307) (0.306) (0.308) November 0.022 0.012 0.003 0.013 0.012 0.001 (0.305) (0.304) (0.307) (0.305) (0.304) (0.306) December −0.351 −0.364 −0.375 −0.364 −0.364 −0.378 (0.303) (0.302) (0.305) (0.303) (0.302) (0.304) Constant 2.586∗∗ 2.032 1.024 0.890 2.420∗∗ 1.014 (1.174) (1.257) (0.777) (0.775) (1.144) (0.774) Observations 224 224 224 224 224 224 R2 0.379 0.380 0.370 0.377 0.378 0.370 Adjusted R2 0.325 0.325 0.314 0.322 0.326 0.318 MAE 0.74 0.74 0.77 0.75 0.74 0.76 RMSE 1.02 1.03 1.06 1.03 1.02 1.05 AICc 615.67 615.51 618.98 616.37 613.71 616.61 Residual Std. Error 0.906 (df = 205) 0.906 (df = 205) 0.913 (df = 205) 0.907 (df = 205) 0.905 (df = 206) 0.910 (df = 206) F Statistic 6.953∗∗∗(df = 18; 205) 6.966∗∗∗(df = 18; 205) 6.683∗∗∗(df = 18; 205) 6.895∗∗∗(df = 18; 205) 7.358∗∗∗(df = 17; 206) 7.108∗∗∗(df = 17; 206) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

Table 5: Results of models with different combinations of T emperatureAnomalyt,

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When we look at the financial variables in model 5, we see that ∆BBspread2t has a negative and statistically significant effect at a 1%-level on cat bond re-turns. The other three financial variables all have expected signs. The sign of

BBspreadt−1 is positive, since spread levels should partly determine coupon

payments. The sign of ∆BBspreadt−1 is negative, since increase in spreads

should lead to a decrease in market value of bonds and the other way around.

Lastly, the signs of U S3Mt−1 is positive, which is as expected with similar

reasoning as for BBspreadt−1. However, BBspreadt−1, ∆BBspreadt−1 and

U S3Mt−1 all do not have a significant effect on cat bond returns.

To optimize our model, we estimate models with different combinations of the in-significant financial variables in model 5. In table 6 we have the results of model

5 together with all seven models with other combinations of BBspreadt−1,

∆BBspreadt−1and U S3Mt−1. The first thing we observe is that the effects of

BBspreadt−1 and ∆BBspreadt−1 become significant, at least at a 10%-level,

when these variables are included separately. The signs of BBspreadt−1 and

∆BBspreadt−1 remain unchanged in all models. U S3Mt−1 does not become

significant in any of the models. The sign of U S3Mt−1 also remains the same

in all models.

By looking at the measures of fit, we see that the differences are very small.

The adjusted R2is the slightly higher for model 5 compared to the other

mod-els. The AMEs of model 8 and model 11 are the highest. Moreover, the value

of the RMSE of model 8 is the best. Model 9 has the lowest AICc value. Since

model 8 has the best measures of fit in terms of predictive power and its adjusted

R2 and AIC

c are not much worse compared to the other models, we proceed

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Dependent variable: rcb,t (5) (7) (8) (9) (10) (11) (12) (13) BBspreadt−1 0.064 0.081∗∗ 0.043 0.060∗ (0.040) (0.037) (0.036) (0.033) ∆BBspreadt −0.152 −0.150 −0.230∗ −0.213∗ (0.131) (0.132) (0.123) (0.120) ∆BBspread2 t −0.177∗∗∗ −0.216∗∗∗ −0.172∗∗∗ −0.211∗∗∗ −0.138∗∗∗ −0.143∗∗∗ −0.188∗∗∗ −0.188∗∗∗ (0.053) (0.041) (0.053) (0.041) (0.047) (0.046) (0.039) (0.039) U S3Mt− 1 0.062 0.061 0.031 0.016 (0.047) (0.047) (0.043) (0.043) Damaget −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ −0.029∗∗∗ (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) log(V olumet) −0.210∗ −0.206∗ −0.262∗∗∗ −0.258∗∗ −0.256∗∗ −0.279∗∗∗ −0.271∗∗ −0.282∗∗∗ (0.107) (0.107) (0.100) (0.100) (0.104) (0.099) (0.104) (0.099) February −0.457 −0.476 −0.461 −0.481 −0.452 −0.456 −0.484 −0.484 (0.298) (0.298) (0.299) (0.298) (0.299) (0.299) (0.301) (0.300) March −0.437 −0.455 −0.443 −0.461 −0.443 −0.446 −0.479 −0.479 (0.299) (0.298) (0.299) (0.299) (0.300) (0.299) (0.301) (0.300) April −0.426 −0.392 −0.433 −0.399 −0.451 −0.450 −0.403 −0.404 (0.299) (0.298) (0.300) (0.298) (0.300) (0.300) (0.301) (0.300) May −0.243 −0.285 −0.256 −0.297 −0.245 −0.253 −0.319 −0.320 (0.300) (0.298) (0.301) (0.299) (0.302) (0.301) (0.301) (0.300) June −0.256 −0.286 −0.261 −0.290 −0.249 −0.253 −0.298 −0.298 (0.299) (0.298) (0.299) (0.298) (0.300) (0.300) (0.301) (0.300) July 0.110 0.113 0.107 0.110 0.099 0.100 0.099 0.099 (0.298) (0.298) (0.298) (0.298) (0.299) (0.298) (0.300) (0.300) August 0.756∗∗ 0.713∗∗ 0.752∗∗ 0.710∗∗ 0.768∗∗ 0.763∗∗ 0.698∗∗ 0.698∗∗ (0.301) (0.299) (0.302) (0.300) (0.302) (0.302) (0.302) (0.301) September 1.241∗∗∗ 1.209∗∗∗ 1.236∗∗∗ 1.205∗∗∗ 1.254∗∗∗ 1.249∗∗∗ 1.205∗∗∗ 1.204∗∗∗ (0.309) (0.308) (0.310) (0.309) (0.310) (0.310) (0.311) (0.310) October 0.774∗∗ 0.782∗∗ 0.765∗∗ 0.773∗∗ 0.747∗∗ 0.747∗∗ 0.748∗∗ 0.748∗∗ (0.306) (0.306) (0.307) (0.307) (0.307) (0.306) (0.309) (0.308) November 0.012 −0.030 0.011 −0.031 0.031 0.026 −0.035 −0.035 (0.304) (0.302) (0.304) (0.302) (0.305) (0.304) (0.305) (0.304) December −0.364 −0.370 −0.362 −0.368 −0.351 −0.353 −0.357 −0.358 (0.302) (0.302) (0.302) (0.302) (0.303) (0.302) (0.305) (0.304) Constant 2.420∗∗ 2.348∗∗ 3.096∗∗∗ 3.019∗∗∗ 3.142∗∗∗ 3.409∗∗∗ 3.354∗∗∗ 3.488∗∗∗ (1.144) (1.143) (1.022) (1.021) (1.055) (0.988) (1.056) (0.992) Observations 224 224 224 224 224 224 224 224 R2 0.378 0.374 0.373 0.369 0.370 0.368 0.359 0.359 Adjusted R2 0.326 0.325 0.324 0.323 0.321 0.323 0.313 0.316 MAE 0.74 0.76 0.73 0.74 0.75 0.73 0.79 0.76 RMSE 1.02 1.03 0.98 0.99 1.03 0.99 1.06 1.02 AICc 613.71 612.76 613.18 612.20 614.10 612.30 615.50 613.31 Residual Std. Error 0.905 (df = 206) 0.905 (df = 207) 0.906 (df = 207) 0.907 (df = 208) 0.908 (df = 207) 0.907 (df = 208) 0.913 (df = 208) 0.912 (df = 209) F Statistic 7.358∗∗∗(df = 17; 206) 7.723∗∗∗(df = 16; 207) 7.685∗∗∗(df = 16; 207) 8.099∗∗∗(df = 15; 208) 7.600∗∗∗(df = 16; 207) 8.090∗∗∗(df = 15; 208) 7.779∗∗∗(df = 15; 208) 8.358∗∗∗(df = 14; 209) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

Table 6: Results of models with different combinations of BBspreadt−1 ,U S5t−1 and

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When we look at the coefficients of model 8, we see that the months that in the peak period of the hurricane season significantly affect the cat bond re-turns, while other months do not. August and October both have a positive and significant effect at a 5%-level of around 0.75. September has a positive and significant effect at a 1%-level of 1.21. The months around the peak of the hurri-cane season do have have positive signs (July and November). All other months have negative signs. The months outside the peak of the hurricane season do not have a significant effect on cat bond returns. Therefore we construct other vari-ables that accounts for seasonality effects in order to improve our model. Since the coefficients in the months August, September and October are all

signifi-cant at least at a 5%-level and positive, we construct the variable AugSepOctt.

AugSepOctt is a dummy variable that equals 1 if montht equals either

Au-gust, September or October and 0 otherwise. Moreover, we estimate a model with a categorical variable that has the following 4 levels: August, September, October and the other months. In this model the last level is the reference level. In table 7, we have the results for model 8, 14, 15 and 16. Here, model 14 has

the same explanatory variables as model 8, but without the variable montht.

Model 15 has the same explanatory variables as model 8, but with the variable

AugSepOcttinstead of the variable montht. Lastly, model 16 also has the same

explanatory variables as model 8, but with the categorical variable described

above instead of montht. When we compare the adjusted R2s of the models, we

see that it is still the highest for model 8. For model 15 and 16, we see that the dummy variable and the new categorical variable both have a significant effect

and improve the model in terms of AICc. However, by using these variables we

lose predictive power in terms of AME and RMSE. Hence, since model 8 still has the best values for three of the measures of fit, we conclude that the original

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Dependent variable: rcb,t (8) (14) (15) (16) BBspreadt−1 0.043 0.048 0.048 0.044 (0.036) (0.041) (0.036) (0.036) ∆BBspreadt −0.150 −0.076 −0.129 −0.138 (0.132) (0.142) (0.126) (0.127) ∆BBspread2 t −0.172∗∗∗ −0.171∗∗∗ −0.182∗∗∗ −0.176∗∗∗ (0.053) (0.057) (0.051) (0.052) Damaget −0.029∗∗∗ −0.018∗∗∗ −0.028∗∗∗ −0.029∗∗∗ (0.004) (0.005) (0.004) (0.004) log(V olumet) −0.262∗∗∗ −0.229∗∗ −0.251∗∗ −0.255∗∗ (0.100) (0.113) (0.100) (0.100) February −0.461 (0.299) March −0.443 (0.299) April −0.433 (0.300) May −0.256 (0.301) June −0.261 (0.299) July 0.107 (0.298) August 0.752∗∗ 0.987∗∗∗ (0.302) (0.147) September 1.236∗∗∗ 1.476∗∗∗ (0.310) (0.233) October 0.765∗∗ 1.005∗∗∗ (0.307) (0.231) November 0.011 (0.304) December −0.362 (0.302) AugSepOctt 1.145∗∗∗ (0.147) Constant 3.096∗∗∗ 2.768∗∗ 2.733∗∗∗ 2.791∗∗∗ (1.022) (1.132) (1.003) (1.001) Observations 224 224 224 224 R2 0.373 0.151 0.336 0.346 Adjusted R2 0.324 0.131 0.318 0.322 MAE 0.73 0.87 0.81 0.78 RMSE 0.98 1.15 1.04 1.02 AICc 613.18 656.17 603.21 604.09 Residual Std. Error 0.906 (df = 207) 1.027 (df = 218) 0.910 (df = 217) 0.908 (df = 215) F Statistic 7.685∗∗∗(df = 16; 207) 7.743∗∗∗(df = 5; 218) 18.304∗∗∗(df = 6; 217) 14.237∗∗∗(df = 8; 215) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

Table 7: Results of models with different specifications for montht.

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monthly damage in billions of 2019 US dollars. However, other specifications could improve our model. In table 8, we have model 8 together with three mod-els with other specifications for damaget. Since small damages are less likely to

affect cat bond returns than big damages, we expect the effect per US dollar to be higher for bigger damages. Therefore we replaced the monthly damage by the squared monthly damage in model 17. Note that we could not use the damage and the squared damage together, since they are correlated. Further-more, it could be the case that for big damages that trigger cat bonds, even bigger damages do not lead to an even worse return, since the cat bonds will be triggered anyway. To account for this, we introduce a maximum damage X. This means that for a maximum damage X, all damages that are bigger than X will be set equal to X. Hence, for all damages higher than X, the effect on cat bond returns is the same. For X we try all integer values from 1 to 100 billion US dollars and compare the measures of fit. We do this for a model with just monthly damage (model 18) and with squared monthly damage (model 19). For model 18 it turns out that setting X equal to 59 billion results in the

best R2and AICc, while setting X equal to 17 billion results in the lowest AME

(0,722). Setting X equal to 39 billion leads to the lowest RMSE (0.990). For X equal to 59 billion, the AME (0.729) and RMSE (0.991) are not much higher. Hence we decide to set the maximum damage to 59 billion US dollars in model 18.

For model 19, setting X equal to 27 billion is optimal in terms of R2and AIC

c.

Setting X equal to 17 billion leads to the lowest AME (0.679) and setting X equal to 20 billion leads to the best value for the RMSE (0.966). However, again

the optimal maximum damage in terms of R2and AIC

c does not lead to much

worse AME (0.710) and RMSE (0.973). Hence, in model 19, we set X equal to 27 billion US dollars.

If we look at model 17, we see that the squared damage has a negative

sig-nificant effect on cat bond returns. However, the R2s decreased a lot compared

to model 8. Furthermore, the AICc faced a big increase. The other measures

of fit are roughly similar. The new specifications for the damage variable in model 18 and 19 are also both negative and significant. Furthermore, they lead

to a big increase in both R2s and a big decrease in AIC

c. For model 19, the

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. Dependent variable: rcb,t (8) (17) (18) (19) BBspreadt−1 0.043 0.048 0.045 0.041 (0.036) (0.039) (0.034) (0.034) ∆BBspreadt −0.150 −0.162 −0.126 −0.109 (0.132) (0.140) (0.123) (0.123) ∆BBspread2 t −0.172∗∗∗ −0.165∗∗∗ −0.184∗∗∗ −0.184∗∗∗ (0.053) (0.056) (0.049) (0.049) Damaget −0.029∗∗∗ (0.004) Damage2 t −0.0001∗∗∗ (0.00003) Damage59t −0.058∗∗∗ (0.007) Damage272 t −0.004∗∗∗ (0.0005) log(V olumet) −0.262∗∗∗ −0.256∗∗ −0.253∗∗∗ −0.276∗∗∗ (0.100) (0.106) (0.093) (0.093) February −0.461 −0.475 −0.448 −0.478∗ (0.299) (0.317) (0.280) (0.279) March −0.443 −0.486 −0.397 −0.455 (0.299) (0.318) (0.280) (0.280) April −0.433 −0.495 −0.363 −0.384 (0.300) (0.318) (0.281) (0.280) May −0.256 −0.329 −0.181 −0.257 (0.301) (0.319) (0.282) (0.281) June −0.261 −0.268 −0.254 −0.275 (0.299) (0.318) (0.280) (0.280) July 0.107 0.102 0.114 0.107 (0.298) (0.317) (0.279) (0.279) August 0.752∗∗ 0.656∗∗ 0.798∗∗∗ 0.821∗∗∗ (0.302) (0.320) (0.282) (0.282) September 1.236∗∗∗ 0.968∗∗∗ 1.432∗∗∗ 1.405∗∗∗ (0.310) (0.324) (0.292) (0.290) October 0.765∗∗ 0.5990.973∗∗∗ 0.990∗∗∗ (0.307) (0.324) (0.289) (0.289) November 0.011 0.023 −0.004 0.006 (0.304) (0.323) (0.285) (0.285) December −0.362 −0.356 −0.369 −0.358 (0.302) (0.321) (0.283) (0.283) Constant 3.096∗∗∗ 3.002∗∗∗ 3.016∗∗∗ 3.237∗∗∗ (1.022) (1.087) (0.956) (0.956) Observations 224 224 224 224 R2 0.373 0.292 0.450 0.452 Adjusted R2 0.324 0.238 0.408 0.409 Residual Std. Error (df = 207) 0.906 0.962 0.848 0.847 MAE 0.73 0.74 0.74 0.71 RMSE 0.98 0.97 0.99 0.97 AICc 613.18 640.14 583.54 583.02 F Statistic (df = 16; 207) 7.685∗∗∗ 5.346∗∗∗ 10.602∗∗∗ 10.657∗∗∗ Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

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To check whether including a measure for the the reinsurance cycle would have improved our model, we have the results of a model with the same explanatory variables as model 19 and the Guy Carpenter US Property Catastrophe Rate-On-Line Index in table 18 in the appendix. The data for this index is obtained from ARTEMIS (ARTEMIS, 2020b). As mentioned, we can not use the index in our final model, since we do not have projections for it. This is unfortunate, since including this index would have improved all measures of fit. Moreover, the coefficient is significant.

In figure 15, we have the auto-correlation function of the residuals of model 19. As one can see from the figure, there is only a very small significant au-tocorrelation in the residuals for the second lag. However, a Breusch-Godfrey test for serial autocorrelation up to lag 20 results in a p-value of 0.595, meaning that the null hypothesis that there is no autocorrelation in the residuals is not rejected. This means that there is no information left in the residuals in terms

of autocorrelation. Hence, adding autoregressive terms for rcb,t should not be

necessary.

Figure 15: Autocorrelation function of the residuals of model 19.

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variables. We check this by fitting an ARIMAX model to the cat bond re-turns. This is done with the function auto.arima from the package forecast

in R. This function returns the best ARIMAX model according to the AICc

values and determines, based on a ADF test, how many first differences are needed to make the model stationary. For the exogenous variables, we use the same explanatory variables as in model 19. The function auto.arima returns an ARIMA-order of (1, 0, 1), meaning that including one autoregressive (AR)

term and one moving average (MA) term increase the model fit in terms of AICc.

In table 9, we have the results of model 19 and model 20, which is the ARI-MAX(1,0,1) model. Both the AR and MA term are significant in model 20. All

other coefficients are roughly similar. Furthermore, the AICcis a bit lower

com-pared to model 19. However, the AR and MA term lead to an increase in the AME. The RMSE is similar for both models The increase in the AME is large

relative to the decrease in AICc. Therefore, since we , we do not include the AR

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Dependent variable: rcb,t (19) (20) rcb,t−1 0.960∗∗∗ (0.040) t−1 −0.898∗∗∗ (0.061) intercept 3.237∗∗∗ 3.121∗ (0.919) (1.762) BBspreadt−1 0.041 0.004 (0.033) (0.042) ∆BBspreadt −0.109 −0.140 (0.118) (0.116) ∆BBspread2 t −0.184∗∗∗ −0.177∗∗∗ (0.047) (0.046) Damage272 t −0.004∗∗∗ −0.004∗∗∗ (0.0004) (0.0004) log(V olumet) −0.276∗∗∗ −0.252 (0.090) (0.178) February −0.478∗ −0.469∗ (0.268) (0.255) March −0.455∗ −0.446∗ (0.269) (0.256) April −0.384 −0.384 (0.269) (0.257) May −0.257 −0.253 (0.270) (0.258) June −0.275 −0.269 (0.269) (0.257) July 0.107 0.107 (0.268) (0.256) August 0.821∗∗∗ 0.827∗∗∗ (0.271) (0.259) September 1.405∗∗∗ 1.415∗∗∗ (0.279) (0.267) October 0.990∗∗∗ 0.991∗∗∗ (0.278) (0.265) November 0.006 0.016 (0.273) (0.260) December −0.358 −0.352 (0.272) (0.258) Observations 224 224 MAE 0.71 0.79 RMSE 0.97 0.97 AICc 583.02 578.55 Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

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Now that we have selected our final model, the next step we take is checking for multicollinearity. We do this by computing the variance inflator factors (VIF). The VIF for an explanatory variable is the ratio of the variance of the full model and the variance of a model with only that explanatory variable. Following James, Witten, Hastie, and Tibshirani (2013), VIF values should not exceed 5. As one can see in table 10, this is the case. Hence, there is no evidence for multicollinearity.

BBspreadt−1 ∆BBspreadt ∆BBspread2t Damage272t log(V olumet) montht

VIF 1.36 1.85 1.95 1.19 1.03 1.36

Table 10: VIF values model 19.

Lastly, we test for heteroskedasticity. This is done by using a Breusch-Pagan test. We observe a p-value of 0.12, meaning that the null hypothesis of no

heteroskedasticity is not rejected. In other words, there is no evidence for

heteroskedasticity. Therefore the final model is given by the following equa-tion:

rcb,t=3.237 + 0.041BBspreadt−1− 0.109∆BBspreadt− 0.184∆BBspread

2 t

− 0.004Damage272

t− 0.276 ∗ log(V olumet) − 0.4781(Montht= F ebruary)

− 0.4551(Montht= M arch) − 0.3841(Montht= April)

− 0.2571(Montht= M ay) − 0.2751(Montht= J une)

+ 0.1071(Montht= J uly) + 0.8211(Montht= August)

+ 1.4051(Montht= September) + 0.9901(Montht= October)

+ 0.0061(Montht= N ovember) − 0.3581(Montht= December)

, where1(A) is the indicator function that returns 1 if condition A is satisfied

and 0 otherwise.

5.2

Future implications

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The average BB spread per month for the 2000 scenarios is given in figure 16. Due to the COVID-19 crisis, spreads increased in 2020. With the expected end of the pandemic in the course of 2021, spreads are expected to steadily return to normal levels. Therefore, we see that spreads in the first years after 2020 decrease for all climate pathways. Then, the spreads under a Paris transi-tion increase a bit. This can be explained by the transitransi-tion measures that will be adopted by the government, which will lead to price corrections on financial markets. After this increase for the Paris transition, the spreads under a Paris transition and under no global warming move quite similar. Though, the spread under a Paris transition is slightly higher. For a failed transition, we observe two short term peaks. From these, the second one is much more extreme. They can be explained as follows. In the scenario set, it is assumed that at some point in time, the world realizes that the transition has failed. Then, negative market sentiment will cause strong price corrections on the stock market, which leads to by higher spreads. At a certain point, the financial markets calm down. Though, stock prices are lower and spread levels are higher than before, due to increased risk and lower expected economic growth. As figure 16 shows, the peaks only lasts a few years. Therefore, the peaks themselves will not affect our conclusions for the long term.

Figure 16: Average BB spread per month of the 2000 scenarios in percentage points.

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to test if there exists a trend. A Mann-Kendall test is less sensitive to outliers. The test results in a p-value of 0.61. This means that the null hypothesis that there is no trend can not be rejected. Therefore, we use data from 2000 to 2020 to compute current expected damages.

Figure 17: Average damage per storm over the years.

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Figure 18: Projected yearly growth in percentage relative to 2020 for a Paris transition.

Figure 19: Projected yearly growth in percentage relative to 2020 for a failed transition.

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years. Until approximately 2035, the returns in the Paris transition and failed transition pathways are very similar. Then for a short period, the returns under a failed transition increase and are higher than those under a Paris transition. This can be explained by the different spreads in both pathways. However, after 2040, we see that the returns for a failed transition decrease much faster. In 2059, the return is even below 1%, while the return under a Paris transition is still more than 3%.

Nevertheless, we should be careful while interpreting these result. For the dam-age, we took for each month historical averages multiplied with expected growth. By the definition of our final model, big damages that exceed 27 billion 2019 US dollars are set equal to 27 billion 2019 US dollars. However, the expected monthly damage only exceeds 27 billion under the failed transition pathway in September 2058 and September 2059. In practice, damages are not equal to expected damage each year. Most of the time, monthly damages are relatively small or even equal to zero. Big damages occur only occasionally. But these damages are then set equal to 27 billion, which results in a much lower average damage variable. Hence, the effect of the damage variable is overestimated un-der the approach above. Therefore, a scenario analysis with simulated random damages should give more realistic results.

Figure 20: Expected yearly returns in percentage points using expectations of the ex-planatory variables.

5.2.1 Simulating monthly storm damages

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In table 11, we have the parameter λ and the p-values of the Poisson disper-sion test for each month of the year. As one can see, the null hypothesis that the data follows a Poisson distribution is not rejected for any of the months. Hence, we assume that the number of storms for each month follows a Poisson distribution with the parameter λ from the table below.

Janauary February March April May June July August September October November December λ 1.10 1.10 1.24 1.14 1.29 0.81 0.50 0.55 1.10 0.75 0.40 0.90 p-value 0.70 0.46 0.89 0.70 0.89 0.38 0.78 0.40 0.65 0.94 0.59 0.70

Table 11: Poisson rates and the p-values of the corresponding Poisson dispersion tests.

Figure 21 shows the average and standard deviation of storm damages per month. From the figure we see that these numbers are different per period of the year. Ideally, we would have used a different distribution for the damage per storm for each month. However, we do not have enough data to fit a dis-tribution for each month. For example, for the month November, we only have 8 observations for the period from 2000 until 2020. In order to have enough data points to fit distributions, we divided the year in two periods. In figure 21, we see that the average as well as standard deviation is roughly similar for the period from November until July. Then for August, September and October, which form the peak of the hurricane season, both, the average and standard de-viation increase. Therefore, the two periods for which we estimate a probability distribution are the following:

• calm period: January, February, March, April, May, June, July, November and December;

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Figure 21: Average and standard deviation of individual storm damages per month.

Tables 12 and 13 show the parameter estimates for the distributions of the dam-age per storm and the p-values of the goodness-of-fit tests, for the calm period and the hurricane period respectively. The parameter estimates are obtained by maximum likelihood estimation.

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Lognormal Gamma Pareto Parameters µ = −0.6205 α = 0.6608 α = 0.1289 σ = 1.75 β = 0.4888 β = 22e − 04 p-values χ2 0 0.28 0 KS 0.037 0.855 0 CvM 0 0.517 0 AD 0 0.281 0

Table 12: Parameter estimates and p-values for the χ2, Kolmogorov-Smirnov,

Cramer–von Mises and Anderson– Darling tests for the calm period.

Log normal Gamma Pareto

Parameters µ = 0.0887 α = 0.2835 α = 0.168 σ = 2.7594 β = 0.0223 β = 0.0028 p-values χ2 0.52 0.037 0 KS 0.689 0.267 0.001 CvM 0.355 0.03 0.023 AD 0.37 0.028 0.006

Table 13: Parameter estimates and p-values for the χ2, Kolmogorov-Smirnov,

Cramer–von Mises and Anderson– Darling tests for the hurricane period.

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Figure 23: Histogram and fitted density for the damage per storm in the hurricane period.

5.2.2 Future implications using scenarios

For our scenarios of returns, we start with 2000 scenarios of BB spreads and

com-pute from this the BBspreadt−1, ∆BBspreadtand ∆BBspread2t for each

sce-nario. For the damages we do the following. First let i = (J anuary, F ebruary, ...,

December). We start by drawing for each month i a random number Ni from

the Poisson distribution with the parameter λi. Then we draw for each month

Nidamages from the severity distribution of month i and sum them. This gives

the monthly damages. We do this 2000 times for each year, where we

multi-ply for each year the λi’s and the damages by the expected yearly growth of

that year. For the no global warming pathway, we do the same, but without growth. Furthermore, we assume that the market volume stays at the current level. When we have computed the monthly returns scenarios, we can compute the yearly returns in the 2000 scenarios.

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